Displacement Flow of Miscible Fluids with Density andViscosity ContrastbyAli EtratiBSc, University of Tehran, 2011MASc, University of Victoria, 2013a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mechanical Engineering)The University of British Columbia(Vancouver)August 2018© Ali Etrati, 2018The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Displacement Flow of Miscible Fluids with Density and Viscosity Con-trastsubmitted by Ali Etrati in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Mechanical Engineering.Examining Committee:Ian Frigaard, Mechanical Engineering, Department of MathematicsSupervisorKamran Alba, Mechanical Engineering Technology, University of HoustonCo-SupervisorGregory Lawrence, Civil EngineeringUniversity ExaminerColin Macdonald, Department of MathematicsUniversity ExaminerBruce Sutherland, Department of Physics and of Earth and Atmospheric Sciences,University of AlbertaExternal ExaminerAdditional Supervisory Committee Members:Gwynn Elfring, Mechanical EngineeringSupervisory Committee MemberNeil Balmforth, Department of MathematicsSupervisory Committee MemberiiAbstractWe study downward displacement flow of buoyant miscible fluids with viscosityratio in a pipe, using experimental, numerical and mathematical approaches. In-vestigation of this problem is mainly motivated by the primary cementing processin oil and gas well construction. Our focus is on displacements where the degree oftransverse mixing is low-moderate and thus a two-layer, stratified flow is observed.An inertial two-layer model for stratified density-unstable displacement flowsis developed. From experiments it has been observed that these flows develop for asignificant range of parameters. Due to significant inertial effects, existing modelsare not effective for predicting these flows. The novelty of this model is that theinertia terms are retained and the wall and interfacial stresses are modelled. Withnumerical solution of the model, back-flow, displacement efficiency and instabilityonset predictions are made for different viscosity ratios.The experiments are conducted in a long pipe, inclined at an angle whichis varied from vertical to near-horizontal. Viscosity ratio is achieved by addingxanthan gum to the fluids. At each angle, flow rate and viscosity ratio are var-ied at fixed density contrast. Density-unstable flows regimes are mapped in the(Fr,Recos β/Fr)-plane, delineated in terms of interfacial instability, front dynam-ics and front velocity. Amongst the many observations we find that viscosifyingthe less dense fluid tends to significantly destabilize the flow, for density-unstableconfiguration. Different instabilities develop at the interface and in the wall-layers.The results are compared to the inertial two-layer model. In density-stable experi-ments we mostly focus on the effects of viscosity ratio on displacement efficiencyand stability of wall-layer. Unique instabilities appear in the case of shear-thinningdisplacements. Displacement efficiency decreases with increasing viscosity ratio,iiiflow rate and inclination angle.Finally, a number of three-dimensional parallel numerical simulations are com-pleted in the pipe geometry, covering both density-stable and unstable flows. Un-steady Navier-Stokes equations are solved and the Volume of Fluid (VOF) methodis used to capture the interface between the fluids. The results give us great insightinto several features of these flows that were not available from experiments or 2Dsimulations.ivLay SummaryIn this thesis we study displacement of a fluid inside a pipe by another fluid withdifferent properties. Such flows occur in many industrial processes, but our studyis driven mainly by the primary cementing stage of well construction, where thedrilling mud has to be displaced by the cement slurry, downward in the casing andthen upward in the gap between the casing and earth. Successful displacement ofmud by the slurry is crucial to ensure mechanical integrity and sealing of the well.Depending on the fluid viscosities and densities, the imposed flow rate and the pipeinclination, markedly different flows can develop, ranging from fullymixed to stablecounter-current flow. Previous studies have focused on fluids with equal viscosities,which is rarely the case in industrial applications. Here, we employ experimental,analytical and computational approaches to study the effects of viscosity differenceon displacement efficiency.vPrefaceThe four chapters describing the research in this thesis have been published, sub-mitted, or are in preparation for journal publication.A version of Chapter 2 is published in Physics of Fluids [39]. The authorsof this chapter are myself and I. Frigaard. I was the lead investigator, responsiblefor major areas of concept formation, data collection and analysis. I. Frigaard wasthe supervisory author on this project and was involved throughout the project inconcept formation and manuscript composition.A version of Chapter 3 is published in Physics of Fluids [40]. The authorsof this chapter are myself, K. Alba and I. Frigaard. I was the lead investigator,responsible for all major areas of concept formation, data collection and analysis,as well as manuscript composition. I. Frigaard and K. Alba were involved in theearly stages of concept formation and contributed to manuscript edits. I. Frigaardwas the supervisory author on this project and was involved throughout the projectin concept formation and manuscript composition.A version of Chapter 4 is submitted for publication. The authors of this chapterare myself and I. Frigaard. I was the lead investigator, responsible for all majorareas of concept formation, data collection and analysis, as well as manuscriptcomposition. I. Frigaard was involved in the early stages of concept formationand contributed to manuscript edits. I. Frigaard was the supervisory author onthis project and was involved throughout the project in concept formation andmanuscript composition.A version of Chapter 5 is being prepared for publication. The authors ofthis chapter are myself and I. Frigaard. I was the lead investigator, responsiblefor all major areas of concept formation, data collection and analysis, as well asvimanuscript composition. I. Frigaard was involved in the early stages of conceptformation and contributed to manuscript edits. I. Frigaard was the supervisoryauthor on this project and was involved throughout the project in concept formationand manuscript composition.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Primary cementing . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Unstable & mixed flows . . . . . . . . . . . . . . . . . . 101.2.3 Lock-exchange flows . . . . . . . . . . . . . . . . . . . . 111.2.4 Density-unstable displacement flows . . . . . . . . . . . . 131.2.5 Density-stable displacement flows . . . . . . . . . . . . . 171.2.6 Two-layer flows . . . . . . . . . . . . . . . . . . . . . . . 201.3 Conclusions & scope of thesis . . . . . . . . . . . . . . . . . . . 23viii2 A two-layer model for the inertial regime . . . . . . . . . . . . . . . 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Two-layer model . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Dimensionless equations . . . . . . . . . . . . . . . . . . 322.3 Frictional closure models & equilibrium solutions . . . . . . . . . 342.3.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.4 Momentum equilibrium . . . . . . . . . . . . . . . . . . 372.4 Well-posedness and linear stability . . . . . . . . . . . . . . . . . 412.4.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . 432.4.2 Artificial diffusion . . . . . . . . . . . . . . . . . . . . . 462.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.5.1 Lock-exchange flow (Fr→ 0) . . . . . . . . . . . . . . . 512.5.2 Displacement flow (Fr > 0) . . . . . . . . . . . . . . . . 532.6 Summary & discussion . . . . . . . . . . . . . . . . . . . . . . . 673 Two-layer density-unstable displacement flow experiments with vis-cosity ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 713.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3.1 Main features . . . . . . . . . . . . . . . . . . . . . . . . 743.3.2 Stability & regime classification . . . . . . . . . . . . . . 763.3.3 Front dynamics . . . . . . . . . . . . . . . . . . . . . . . 853.3.4 Front velocity measurement . . . . . . . . . . . . . . . . 893.4 Summary & discussion . . . . . . . . . . . . . . . . . . . . . . . 944 Density-stable displacement flow experiments with viscosity ratio . . 974.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . 1004.1.1 Vertical displacements . . . . . . . . . . . . . . . . . . . 1004.1.2 Inclined displacements . . . . . . . . . . . . . . . . . . . 1064.1.3 Notes on stability . . . . . . . . . . . . . . . . . . . . . . 110ix4.2 Summary & discussion . . . . . . . . . . . . . . . . . . . . . . . 1155 Full numerical simulation of displacement flows . . . . . . . . . . . 1175.1 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.2 Density-stable displacement flows . . . . . . . . . . . . . . . . . 1205.2.1 Vertical displacements . . . . . . . . . . . . . . . . . . . 1215.2.2 Inclined displacements . . . . . . . . . . . . . . . . . . . 1305.2.3 Summary of results . . . . . . . . . . . . . . . . . . . . . 1335.3 Density-unstable displacement flows . . . . . . . . . . . . . . . . 1355.3.1 Main features . . . . . . . . . . . . . . . . . . . . . . . . 1365.3.2 Cyclic flow regime . . . . . . . . . . . . . . . . . . . . . 1375.3.3 Front dynamics . . . . . . . . . . . . . . . . . . . . . . . 1465.3.4 Revisiting front velocity . . . . . . . . . . . . . . . . . . 1505.3.5 Revisiting two-layer models . . . . . . . . . . . . . . . . 1555.3.6 Summary of results . . . . . . . . . . . . . . . . . . . . . 1606 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . 1626.1 Key findings in density-unstable flows . . . . . . . . . . . . . . . 1626.2 Key findings in density-stable flows . . . . . . . . . . . . . . . . 1686.3 Implications for industrial applications . . . . . . . . . . . . . . . 1686.4 Limitations of the present study . . . . . . . . . . . . . . . . . . 1716.5 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 173Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176A Research methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 189A.1 Displacement flow-loop . . . . . . . . . . . . . . . . . . . . . . . 189A.1.1 Flow visualization & image processing . . . . . . . . . . 191A.1.2 Fluid preparation . . . . . . . . . . . . . . . . . . . . . . 193A.2 OpenFOAM simulations . . . . . . . . . . . . . . . . . . . . . . 195A.2.1 Transport models . . . . . . . . . . . . . . . . . . . . . . 196A.2.2 Geometry & mesh . . . . . . . . . . . . . . . . . . . . . 196A.2.3 Initial & boundary conditions . . . . . . . . . . . . . . . 197A.2.4 Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 197xA.3 Numerical solution of two-layer model . . . . . . . . . . . . . . . 199A.3.1 Advection Equation . . . . . . . . . . . . . . . . . . . . . 199A.3.2 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . 201A.3.3 Convergence Check . . . . . . . . . . . . . . . . . . . . . 202B Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 204xiList of TablesTable 3.1 Full range of physical and dimensionless parameters used ininclined density-stable experiments. . . . . . . . . . . . . . . . 73Table 4.1 Full range of physical and dimensionless parameters used indensity-stable experiments. . . . . . . . . . . . . . . . . . . . 98Table 5.1 Density-stable cases solved using OpenFOAM. . . . . . . . . . 121Table A.1 Effect of xanthan gum concentration on shear-thinning fluidrheology. Fitted κˆ and n values are based on µˆ = κˆ Ûγn−1. . . . . 194Table B.1 The important dimensionless groups used in this thesis. Sub-scripts H,L denote heavy and light fluids, and subscripts 1,2refer to displacing and displaced fluids, respectively. . . . . . . 204xiiList of FiguresFigure 1.1 A simple schematic of the primary cementing process. a) Afterthe well-bore is drilled the casing is inserted. The casing andthe annulus are initially filledwith the drillingmud. A sequenceof fluids are pumped inside to displace the mud downward inthe casing and upward in the annulus. b) Ideally the mud iscompletely removed and only pure cement remains in the annulus. 2Figure 1.2 A simple schematic of the downward displacement flow prob-lem. Initially the two fluids are separated. . . . . . . . . . . . 5Figure 1.3 Growth of two-dimensional Rayleigh-Taylor instabilities whena heavy fluid (white) rests on top a light fluid (black). Perturba-tions in the interface grow exponentially with time. The resultsare from numerical simulations in OpenFOAM (see Chapter 5). 11Figure 1.4 Summary of flow regimes in near-horizontal iso-viscous dis-placement flows [123]. Domains identified by i are inertial, andthose identified by v are viscous. Symbols denote sustainedback-flow (), temporary back-flow (C), stationary interface(B) and instantaneous (•). The vertical dashed line is Fr = 0.9.Figure is taken from [123]. . . . . . . . . . . . . . . . . . . . 15xiiiFigure 1.5 Summary of flow regimes in inclined iso-viscous displace-ment flows [4]. The solid and dashed lines denote χ = χc andRet cos β = 500− 50Fr respectively. Blue symbols representinstantaneous displacements. The regimes are diffusive (nosuperposed symbol), viscous (superposed circles) and inertial(superposed squares). Figure is taken from [4]. . . . . . . . . 16Figure 1.6 Growth of Kelvin-Helmholtz instabilities when two fluids withdifferent densities move with different directions. These snap-shots are for a gravity driven counterflow of a heavy fluid(white) and a light fluid (black) in a plane channel. The resultsare from numerical simulations in OpenFOAM. . . . . . . . . 21Figure 2.1 Schematic of the two-layer model used. The flow is in theaxial z−direction, with (x,y) in the pipe cross-section and x isperpendicular to gravity. H and L refer to heavy and light fluidlayers, respectively. α is the area fraction of the cross-sectionoccupied by the heavy layer. The geometric parameters aregiven in (2.11)-(2.13). . . . . . . . . . . . . . . . . . . . . . 30Figure 2.2 Comparison of the equilibriumfluxes and velocities for Re/Fr2 cos β=43.58, m = 1 and Fr = 1. The curves represent Model 1 (B),Model 2 (), Model 3 (#) and numerical solution of [83] (C). 38Figure 2.3 Comparison of the equilibriumfluxes and velocities for Re/Fr2 cos β=129.4, m = 1 and Fr = 1. Curves marked as in Fig. 2.2. . . . . 38Figure 2.4 Comparison of the equilibriumfluxes and velocities for Re/Fr2 cos β=129.4, m = 4 and Fr = 1. Curves marked as in Fig. 2.2. . . . . 39Figure 2.5 Comparison of the equilibriumfluxes and velocities for Re/Fr2 cos β=129.4, m = 0.25 and Fr = 1. Curves marked as in Fig. 2.2. . . 39Figure 2.6 Comparison of the equilibrium fluxes (Model 2) for differentvalues of Re/Fr2 cos β andm. The curves represent Re/Fr2 cos β=(34.86,103.53,200) andm = 1 for (,#, ♦), respectively. (O, M)show fluxes for Re/Fr2 cos β = 103.53 with m > 1 and m < 1,respectively, with the solid lines corresponding to m = (10,0.1)and the broken lines corresponding to m = (4,0.25). . . . . . . 41xivFigure 2.7 Map of stability regimes for steady displacement flow withm = 1 and interface height: (a) h¯ = 0.3; (b) h¯ = 0.5; (c) h¯ = 0.7.The blue region represents stable displacements (C2d≥ v2w), thegreen region is well-posed but viscously unstable (0 ≤C2d≤ v2w)and the red region is ill-posed (C2d< 0). . . . . . . . . . . . . 47Figure 2.8 Map of stability regimes for steady displacement flow withm = 1 and interface height: (a) h¯ = 0.1; (b) h¯ = 0.9. Regimesidentified as in Fig. 2.7. . . . . . . . . . . . . . . . . . . . . . 48Figure 2.9 Effect of viscosity ratio on the linear stability map for h¯ = 0.5with (a) m = 0.25, and (b) m = 4. . . . . . . . . . . . . . . . 48Figure 2.10 Growth rate (amplification factor) of instabilities (ωI ) vs. wave-length (λ) for a stable case (a) Ret cos β = 130, Fr = 0.9, aviscous unstable case (b) Ret cos β = 250, Fr = 0.9 and an un-stable (ill-posed) case (c) Ret cos β = 43.6, Fr = 2.0. In allcases h = 0.9 and m = 1 (see Fig. 2.8(b)). The lines denote ωIfrom Inviscid Kelvin-Helmholtz (ikh) analysis (- -), ViscousKelvin-Helmholtz (vkh) analysis (–), vkh with η1 = 10−2 (M)and with η1 = η2 = 10−2 (#). In all cases the imaginary part ofthe unstable frequency has been plotted. Note that asωI→−∞,the wave amplitude goes to zero exp i(−iωI ) = exp(ωI ) → 0. . 50Figure 2.11 Comparison of the amplification factor of the unstable case inFig. 2.10(c) with η1,2 = 10−2 (#), η1,2 = 10−3 () and η1,2 =10−4 (M). The cut-off wavelength becomes smaller with η. . . 50Figure 2.12 Time evolution of a stable lock-exchange flow solved using thetwo-layer model. The domain length is L = 160 and simulationtime is tmax = 80. The simulation parameters are ∆x = 0.1,∆tmax = 10−3 and CFLmax = 0.1 and the flow parameters areFr = 10−4, β = 80◦,m = 1.0 and Ret cos β = 45. (a) Showsevolution of α from an initial condition and (b) is the spatio-temporal diagram. . . . . . . . . . . . . . . . . . . . . . . . 52xvFigure 2.13 (a) Lock-exchange flow with Ret cos β = 45 and m = 0.25 (M),m = 1 (#) and m = 4 (). Increasing the viscosity ratio resultsin a thicker heavy fluid layer and smaller front velocity. Theresults are plotted at t = (16,32,48,64). (b) Comparison of twoexchange flows with Ret cos β = 45 but different inclinations:Ret = 260, β = 80◦ (#) and Ret = 90, β = 60◦ (M). The resultsare plotted at t = (8,24,40,56). . . . . . . . . . . . . . . . . . 52Figure 2.14 Lock-exchange flow with Ret cos β = 50 and m = 1. The frontvelocity w f is equal to the equilibrium velocity of the frontheight wH (αf ). . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 2.15 Front velocity of the heavy layer w f in lock-exchange flow. Thehorizontal and vertical broken lines show the plateau region of[109] where w f = 0.7, and the transition point Ret cos β = 50.The solid line is shock velocity from lubrication model. Thefilled symbols denote stable exchange flows. . . . . . . . . . . 55Figure 2.16 Time evolution of a displacement flow solved using the two-layer model. The displacement parameters are Re = 92,Fr =0.4, β = 85◦,m = 1.0 and Ret cos β = 20. (a) shows evolutionof α from an initial condition at δt = 6 intervals and (b) is thespatio-temporal diagram. . . . . . . . . . . . . . . . . . . . . 56Figure 2.17 An instantaneous displacement flow with a frontal shock. Thedisplacement parameters are Re = 230,Fr = 1.0, β = 85◦,m =1.0 and Ret cos β = 20. . . . . . . . . . . . . . . . . . . . . . 56Figure 2.18 A stable, instantaneous displacement flow with a bump at thefront. The displacement parameters are Re = 320,Fr = 1.4, β =85◦,m = 1.0 and Ret cos β = 20. . . . . . . . . . . . . . . . . 57Figure 2.19 An unstable displacement flow with a bump at the front. Thedisplacement parameters are Re = 230,Fr = 1.0, β = 75◦,m =1.0 and Ret cos β = 60. . . . . . . . . . . . . . . . . . . . . . 57xviFigure 2.20 (a) Effect of increasing Fr at a constant Ret cos β. At a flowratea shock forms at the front and at higher Fr the inertial bumpappears. The bump height αf grows with Fr until it reachesαf = 1. Also, backflow stops at critical Fr at each Ret cos β.The results shown here are for Ret cos β = 40with Fr = 0.1 (#),0.4 (M), 0.8 () and 1.0 (O). (b) Effect of Ret cos β at constantFr . The plotted results are for Fr = 0.1 and Ret cos β = 10 (#),20 (M), 40 () and 60 (O). With increasing Ret cos β, backflowbecomes stronger and at Ret cos β ≈ 50 the flow becomes unsta-ble. All results are plotted at t = 34. The fluids were separatedat x = 10 at t = 0. . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 2.21 Amarginally stable displacement flow with Ret cos β = 50 andFr = 0.6. Small amplitude instabilities appear at long timest = O(100), compared to unstable displacements Ret cos β > 50where instabilities grow very quickly. . . . . . . . . . . . . . 59Figure 2.22 (a)w f vs Fr for constant values of Ret cos β. The symbols (#,M,O,,/,I,.,♦) correspond to Ret cos β= (0,10,20,40,50,60,80,100),respectively. The broken line represents w f = Fr . (b) w fvs Ret cos β for constant values of Fr . The symbols (#,M,O,,/,I,.,♦) correspond toFr = (0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.4),respectively. The filled symbols denote stable displacementsand the superposed circles denote presence of backflow. Thesolid lines are front velocities found from the lubrication model. 60Figure 2.23 Classification of our results in thr (Fr,Ret cos β)-plane. Dis-placements with backflow are colored in red and the ones with-out backflow are colored in blue. The superposed circle denotesunstable displacements. The thick broken line represents back-flow prediction of [4] χ = χc = 116.32 and the thick solid line isbackflow prediction using our two-layer model χ = χc = 112.5.The horizontal line is Ret cos β = 50. . . . . . . . . . . . . . . 61xviiFigure 2.24 A viscous displacement with temporary backflow. Displace-ment parameters are Ret cos β = 20 and Fr = 0.4. Leading(w f ) and trailing (wb f ) front velocities are plotted in (b). Thebackflow velocity becomes negative at long times, meaningthe backflow stops and the trailing front start moving in thedisplacement direction. . . . . . . . . . . . . . . . . . . . . . 62Figure 2.25 Comparison of some unstable cases with varying degree ofinstability. In both top plots Ret cos β = 50 at (a) Fr = 0.6 and(b) Fr = 1.0. In other two plots Fr = 1.0 and (c) Ret cos β = 60and (d) Ret cos β = 100. In (a) instabilities do not appear untilt ≈ 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 2.26 Effect of viscosity ratio on a displacement flow with Fr = 1.0and Ret cos β = 40. The curves represent m = 1 (#), 4 (M),0.5 (), 0.25 (O) and 0.1 (♦). As the heavy fluid becomesmore viscous (m < 1), displacement becomes more stable andefficient. All plots are at t = 36 and the fluids are initiallyseparated at x = 10. . . . . . . . . . . . . . . . . . . . . . . 64Figure 2.27 Plots of w f vs. Ret cos β for (a) Fr = 0.4 and (b) Fr = 1.0. Thesymbols represent m = 0.1 (#), 0.25 (), 0.5 (M), 1 (O), 2 (I)and 4 (♦). The filled symbols denote stable displacements andthe superposed circles denote presence of backflow. . . . . . . 65Figure 2.28 Backflow prediction at different viscosity ratios. The solidline in (a) shows prediction for m = 1. The thick broken anddash-dot lines below it show predictions for m = 2 and m = 4,respectively. The thin lines above it show predictions for m =0.5, m = 0.25 and m = 0.1, respectively. Variation of criticalvalues χc corresponding to these lines, with m are plotted in (b). 65xviiiFigure 2.29 Front velocity and regime classification for displacement flowswith m = 4. The symbols (#,M,O,,/,I,.) in (a) denote Fr =(0.2,0.4,0.6,0.8,1.0,1.2,1.4). The solid lines are front veloc-ities found from the lubrication model. The thick line in (b)represents backflow prediction χ = χc = 63.5 from lubricationmodel. Same as Fig. 2.23, displacements with backflow arecolored in red and the ones without backflow are colored inblue. The superposed circle denotes unstable displacements. . 66Figure 2.30 Front velocity and regime classification for displacement flowswith m = 0.25. The symbols (#,M,O,,/,I) in (a) denoteFr = (0.2,0.4,0.6,0.8,1.0,1.2). The solid lines are front ve-locities found from the lubrication model. The thick line in(b) represents backflow prediction χ = χc = 155.7 from lu-brication model. Same as Fig. 2.23 and 2.29, displacementswith backflow are colored in red and the ones without backfloware colored in blue. The superposed circle denotes unstabledisplacements. . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 2.31 Variation of plateau velocity wpfwith Fr for m = 0.25 (M),m = 1 (#) and m = 4 (). . . . . . . . . . . . . . . . . . . . . 68Figure 3.1 A simple schematic of the experimental set-up. At tˆ = 0 thegate valve is closed, separating the two fluids. . . . . . . . . . 72Figure 3.2 Spatio-temporal diagrams of experiments at β = 75◦ and Fr ≈1.5. The viscosity ratios are (a) m = 1, (b) m = 0.94 withµˆeH = 5.9mPasn and µˆeL = 5.56mPasn, (c) m = 2.062, (d)m = 0.48, (e) m = 5.52 and (f) m = 0.14. . . . . . . . . . . . . 75xixFigure 3.3 Snapshots of different two-layer displacement flow regimes:(a) A stable displacement flow with β = 60◦, Fr = 1.54 andRet cos β = 45.33. Both fluids are shear-thinning (κˆH ,L =9.7mPasn, nH ,L = 0.76). (b) A wavy displacement flow withβ = 60◦, Fr = 0.85, Ret cos β = 84.14 and m = 0.14. Theheavy fluid is shear-thinning (κˆH = 11.8 mPasn, nH = 0.74).(c) A cyclic displacement flow with β = 60◦, Fr = 0.79 andRet cos β= 245. (d)An inertial displacement flowwith β= 75◦,Fr = 3.78, Ret cos β = 57.52 and m = 0.20. The light fluid isshear-thinning (κˆL = 11.3mPasn, nL = 0.72). The field of viewin all snapshots is 1540×20mm2 located 1670mm downstreamof the gate valve. . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 3.4 Snapshots of some unstable displacement flows with wavesat the interface. Displacement parameters are (a) β = 75◦,Fr = 0.78, Ret cos β = 44.5 and m = 0.12, (b) β = 75◦, Fr =0.38, Ret cos β = 39.63 and m = 0.097, (c) β = 75◦, Fr = 0.39,Ret cos β = 78.69 and m = 0.39 and (d) β = 60◦, Fr = 0.79,Ret cos β = 91.44 and m = 7.21. . . . . . . . . . . . . . . . . 78Figure 3.5 (a) Spatio-temporal plot and (b) mean concentration evolutionfor the same experiment of Fig. 3.3c. The horizontal linesin (a) and arrows in (b) denote times t = 99 and 112, whichcorrespond to the last two snapshots in Fig. 3.3c. The peakin C¯ and white spot region in (a) show the temporary mixingof the fluids across the pipe. This event is followed by a fallin C¯ or the dark region in (a), where the front is cutoff andthe streams separate. This is a characterstic of the flows herenamed as intermittent. . . . . . . . . . . . . . . . . . . . . . 79xxFigure 3.6 Interfacial stability map of two-layer displacements for (a) alldisplacements, (b) iso-viscous displacements m = 1, (c) dis-placement with m > 1 and (d) displacement with m < 1. Cir-cle and square symbols denote Newtonian and shear-thinningflows. Triangles in (b) are data points taken from [4]. Thecolors represent stable (l), wavy (l), cyclic (l) and inertial(l). The diffusive/mixed experiments are presented by (l).The superposed circles denote displacements with sustainedback-flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 3.7 Some shear-thinning displacement flows showing instabilitiesgrowing from (a) the wall-layer and (b) behind the tip movingat the center of the pipe. . . . . . . . . . . . . . . . . . . . . 85Figure 3.8 Different fronts observed in the experiments: a) frontal shock,b) unsteady shock, c) slump, d) central and e) mixed. Thesnapshot below (a) is the view of the same front in the top viewmirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 3.9 Map of different fronts for (a) all experiments, (b) m ≈ 1, (c)m > 1 and (d) m < 1. The colors represent the frontal typesobserved in Fig. 3.8: shock (l), unsteady shock (l), slump(l), central (l) and mixed (l). The horizontal and verticalbroken lines are at Ret cos β = 200 and Fr = 1, respectively. . 87Figure 3.10 Snapshots and profile at t = 4.7−11.9. Displacement parame-ters are Fr = 0.38, Ret cos β = 127.04 and m = 1. The field ofview is x = 4−35 and the time difference between the snapshotsis ∆t = 0.654. . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 3.11 Mean concentration profiles for a displacement with Fr = 0.79and Ret cos β = 13.34, κˆL = 0.0103Pasn2 and nL = 0.75 fromimages taken with camera 1 and 2 are plotted in (a) and (b). Themeasured front velocity Vf from each figure is plotted in (c).The broken line denotes Vf measured in the early stages of thedisplacement usingCt = 0.05 in (a), and the solid black and redlines show measured Vf from data in (b) using Ct = 0.05 and0.15, respectively. The final steady front velocity is Vf = 1.177. 90xxiFigure 3.12 Change in front velocity Vˆf with inclination angle β, andimposed velocity Vˆ0 for (a) iso-viscous Newtonian displace-ments with νˆ = 1mm2 s−1 (b) and shear-thinning displacementswith κˆ ≈ 10mPasn and n ≈ 0.7 . The symbols denote differ-ent imposed velocities: Vˆ0 = 10mms−1 (Fr = 0.39± 0.016)(J), Vˆ0 = 20mms−1 (Fr = 0.793±0.044) (s), Vˆ0 = 40mms−1(Fr = 1.55±0.153) (#), Vˆ0 = 70mms−1 (Fr = 2.687±0.171)(I), Vˆ0 = 100mms−1 (Fr = 3.84±0.090) (n), Vˆ0 = 120mms−1(Fr = 4.377±0.054) (t). . . . . . . . . . . . . . . . . . . . 91Figure 3.13 Front velocity for different Fr for (a) m = 1, (b) m > 1 and(c) m < 1. The symbols denote Fr = 0.395± 0.02 (J), Fr =0.792±0.066 (s), Fr = 1.574±0.147, l Fr = 2.665±0.135(I), Fr = 3.818±0.126 (n), Fr = 4.373±0.112 (t). . . . . . 92Figure 3.14 Front velocity values Vf for all inclined experiments with a)m > 1, b) m = 1 and c) m < 1. The colormap scale in all figureshas been limited to values between 1 and 2. Superposed blackand red circles denote steady and unsteady shock fronts. . . . 93Figure 3.15 Comparison of the measured front velocities with: (a) the pre-diction from the lubrication model for Ret cos β < 50; the pre-diction of (3.4) for, b) m = 1, c) m > 1 and d) m < 1. Thetriangles in (b) are data taken from [4]. The dashed lines indi-cated Vˆf /Vˆt experiments = Vˆf /Vˆt prediction. The colormap inall figures shows Ret cos β. . . . . . . . . . . . . . . . . . . 94Figure 4.1 A simple schematic of the experimental flow-loop. . . . . . . 98Figure 4.2 Results fromadisplacement flowwith β= 30◦, Vˆ0 = 95.54mms−1and m = 2.02. The plots on left (a, c, e) are from camera 1 andthe ones on right (b, d, f) are from camera 2. The mean con-centration profiles in (c, d) correspond to the snapshots in (a,b) and are plotted at ∆tˆ = 1 and ∆tˆ = 1.5 intervals. . . . . . . 99xxiiFigure 4.3 Snapshots of iso-viscous vertical displacements with a) Vˆ0 =9.05mms−1, b) Vˆ0 = 20.32mms−1, c) Vˆ0 = 42.93mms−1, d)Vˆ0 = 67.86mms−1, e) Vˆ0 = 100.16mms−1 and f) Vˆ0 = 110.55mms−1.The length of each snapshot is Lˆ = 47.7dˆ and plotted at timesthat interface is at xˆ ≈ 124dˆ. . . . . . . . . . . . . . . . . . . 101Figure 4.4 Snapshots of vertical experimentswith imposed velocity of Vˆ0 ≈42mms−1 with a) µˆ= 1mPas andm = 1, b) µˆe = 5.28mPas andm = 1, c) m = 2.04, d) m = 0.51, e) m = 5.38, and f) m = 0.19.All snapshots are at 90dˆ from the gate-valve and cover Lˆ = 72dˆ.In each plot the time difference between consequent snapshotsis constant and 3−3.25s. . . . . . . . . . . . . . . . . . . . 103Figure 4.5 a) Mean concentration profiles and b) spatio-temporal diagramfor the same experiment as in Fig. 4.3b. The dashed lines in (b)have different slopes and drawn for eye guidance. The velocitieswith which different mean concentrations move are plotted in(c). Thewiggles at C¯ ≈ 0.9 in (a) are due to instabilities growingat the walls and can be seen as dark lines in (b). . . . . . . . 104Figure 4.6 Effect of increasing imposed flow rate on a vertical shear-thinning displacement flowwith κˆH ,L = 10.8mPasn and nH ,L =0.733. The mean velocities are Vˆ0 = 17.22mms−1 (), Vˆ0 =44.2mms−1 (t), Vˆ0 = 73.22mms−1 (s), Vˆ0 = 96.16mms−1(n) and Vˆ0 = 110.53mms−1 (l). . . . . . . . . . . . . . . . . 105Figure 4.7 Front velocity vs mean velocity for vertical displacements withdifferent viscosity ratios. The symbols correspond to: µˆ =1mPas and m = 1 (l), µˆ = 4.13− 7.85mPas and m = 1 (n),m = 1.75−2.51 (s), m = 0.12−0.24 (J), m = 0.43−0.56 (t).Symbols with thick lines are Vˆtip and others are VˆS . . . . . . . 105Figure 4.8 Snapshots of iso-viscous displacements at β = 60◦ with a) Vˆ0 =15.09mms−1, b) Vˆ0 = 37.87mms−1, c) Vˆ0 = 63.85mms−1, d)Vˆ0 = 77.98mms−1, e) Vˆ0 = 93.48mms−1 and f) Vˆ0 = 109.89mms−1.The length of each snapshot is Lˆ = 19dˆ and plotted at timesthat interface is at xˆ ≈ 124dˆ. . . . . . . . . . . . . . . . . . . 107Figure 4.9 Mean concentration profiles for the snapshots of Fig. 4.8. . . . 108xxiiiFigure 4.10 Snapshots of inclined displacements with Vˆ0 ≈ 42mms−1 at a)β = 10◦, b) β = 30◦, c) β = 45◦, d) β = 60◦, e) β = 75◦ and f)β = 10◦. Field of view is 38dˆ. . . . . . . . . . . . . . . . . . 108Figure 4.11 Effect of increasing β on displacement flows with a) Vˆ0 ≈42mms−1 and b) Vˆ0 ≈ 97mms−1. The symbols denote β = 10◦(l), β = 30◦ (s), β = 45◦ (J), β = 60◦ (I), β = 75◦ () andβ = 85◦ (n). . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 4.12 Front velocity vs mean velocity for iso-viscous displacementsat different inclinations. The data correspond to: β = 85◦ (l),β = 75◦ (s), β = 60◦ (J), β = 45◦ (n), β = 30◦ (t), β = 10◦(I), β = 0◦ (). All values correspond to Vˆtip. . . . . . . . . . 110Figure 4.13 Front velocity vs mean velocity for displacements with a) β =75◦ and b) β = 85◦. The symbols correspond to: µˆ = 1mPasand m = 1 (l), µˆ = 4.13− 9mPas and m = 1 (n), m = 1.75−2.51 (s), m = 4.6− 10.1 (I), m = 0.43− 0.56 (t) and m =0.1−0.2 (J). Symbols with thick lines are Vˆtip and others areVˆS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 4.14 Snapshots of a displacement showing instabilities from walllayer. Displacement parameters are β = 30◦, κˆH = 14.7mPasn,nH = 0.68 and Vˆ0 = 39.13mms−1. The field of view is Lˆ = 44dˆand snapshots are plotted at times t = [10−31]. . . . . . . . . 111Figure 4.15 Snapshots of a displacement showing instabilities from bottomlayer. Displacement parameters are β = 10◦ , κˆL,H = 10mPasn,nL,H = 0.74 and Vˆ0 = 19.27mms−1. The field of view is Lˆ =68dˆ and snapshots are plotted at times t = [104−127]. . . . . 112Figure 4.16 Snapshots of a displacement with interfacial instabilities. Dis-placement parameters are β = 85◦ , κˆH = 12.3mPasn, nH = 0.7and Vˆ0 = 100.17mms−1. The field of view is Lˆ = 47dˆ andsnapshots are plotted at times t = [110−143]. . . . . . . . . . 113xxivFigure 4.17 a) Kelvin-Helmholtz instability prediction based on (4.2). Blueand red symbols represent stable and unstable flows respec-tively. The dashed line denotes equality in (4.2). b) Stabilitymap from all inclined experiments. Hollow symbols denotestable displacements. Colors denote wall-layer (blue), bottom-layer (red) or both (green) instabilities. The symbols representinclination: β = 85◦ (#), β = 75◦ (), β = 60◦ (M), β = 45◦ (♦),β = 30◦ (O), β = 10◦ (I). . . . . . . . . . . . . . . . . . . . . 114Figure 5.1 Mesh topology for pipe simulations. The pipe diameter is thesame as the experimental set-up. The cutaway is to showcasethe mesh in the axial direction. . . . . . . . . . . . . . . . . . 118Figure 5.2 The computational grid used in our the pipe cross-section.The number of cell faces in the cross-sections are 1360, 3300and 4800, respectively. The black and white colors show theconcentration field for a density-stable displacement. . . . . . 120Figure 5.3 Snapshots and velocity profiles of case v1 (top) and case v3(bottom) to show the effect of Vˆ0 on growth of the spike at thefront. In each figure the first snapshot is from the central planeand the second is a 3D view of the pipe. The last snapshotshows the streamlines in a moving frame for case v1. . . . . . 122Figure 5.4 Velocity profiles of a) case v1 at xˆ = 0.7m (C), 0.78m (M),0.8m (), 0.82m (∗), 0.84m (B), 0.9m (#), 1.0m (×) and b)case v3 at xˆ = 0.8m (C), 0.94m (M), 0.96m (), 0.98m (∗),1.0m (B), 1.2m (#). . . . . . . . . . . . . . . . . . . . . . . 123Figure 5.5 Results for case v1* at times tˆ = 5s (top) and tˆ = 30s (bottom).At each time the first snapshot is from the central plane and thesecond is 3D view of the pipe. The length and concentrationof the added mixed region is 100mm and 0.3, respectively. . . 123Figure 5.6 Snapshots of central plane for case v2* at times tˆ = [15,17, ...,25]s.124xxvFigure 5.7 a) Mean concentration profiles for case v1 (dashed lines) andv1d (solid lines). b) Comparison of V(C¯) for Case v1 (4),case v1d (#) and experiment (×). The results for case v2d arepresented in c & d. . . . . . . . . . . . . . . . . . . . . . . . 125Figure 5.8 Snapshots of central plane for case v4 at times tˆ = [0.5,1.5, ...,4.5]s.The last snapshot is at tˆ = 11s. . . . . . . . . . . . . . . . . . 126Figure 5.9 a) Comparison of V(C¯) for case v4 (# against experiment (×).b, c, d) V(y), C(y) and m(y) for case v4 at xˆ = 0.2m (C), 0.3m(M), 0.31m (), 0.33m (∗), 0.35m (B), 0.37m (#), 0.47m (×)at time tˆ = 5s. The snapshot at the bottom is at time tˆ = 5s. . . 127Figure 5.10 Snapshots of central plane for case v5 at times tˆ = [2,3, ...,10]s.The last snapshot is at tˆ = 17s. . . . . . . . . . . . . . . . . . 128Figure 5.11 Snapshots of central plane for case v6 at times tˆ = [0.5,1, ...,4.5]s.The last snapshot is at tˆ = 12s. The image at the bottom isC = 0.5 contours, coloured with dimensionless velocity mag-nitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Figure 5.12 a) Plot of V(C¯) for case v5 (#) and case v6 (×). The filledsymbol at C¯ = 0 is Vmax for a shear-thinning fluid with n =0.516. b, c, d) V(y), C(y) and m(y) for case v5 at xˆ = 0.63m(C), 0.65m (M), 0.67m (), 0.69m (∗) and 0.71m (#) at timetˆ = 7.5s. The snapshot at the bottom is at time tˆ = 7.5s. . . . . 129Figure 5.13 Comparison of the results for case i1 against experimental re-sults. The snapshots are at times tˆ = [3,3.5,4]s. At each time,the first snapshot in central plane, the second is the side 3Dview and third one is from experiments. . . . . . . . . . . . . 131Figure 5.14 Snapshots for case i1 at tˆ = 14.5s. The cross-sections from leftto right are at xˆ = [1,1.1,1.2]m, respectively. The red solid linein all images shows C = 0.9. The cross-sections also show thein plane velocity vectors. . . . . . . . . . . . . . . . . . . . 132xxviFigure 5.15 Snapshots and velocity profiles at central plane (top) and cross-sections (bottom) of the pipe for case i2. The snapshotsare at times tˆ = [3,3.5, ...,5.5]s. The cross-sections are atxˆ = [0.4,0.6,0.8,1]m at time tˆ = 5.5s. The velocities are di-mensionless. . . . . . . . . . . . . . . . . . . . . . . . . . . 132Figure 5.16 a) Mean concentration profiles and b) V(C¯) from numericalsimulation (#) of case i2 and experiments (×). The red symbolat C¯ = 0 is Vmax for a shear-thinning fluid with n = 0.7. . . . . 133Figure 5.17 Snapshots of case i3 at times tˆ = [1,3, ...,9]s. The last twosnapshots are at tˆ = [11,13]s. The image at the bottom is fromthe experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 5.18 a) Mean concentration profiles and b) V(C¯) from numericalsimulation (l) of case i3 and experiments (×). The filledsymbol at C¯ = 0 is Vmax for a shear-thinning fluid with n = 0.516.134Figure 5.19 Spatio-temporal diagrams of displacement flows with β = 75◦and Ret cos β = 126.1 and a) Fr = 1.0, b) Fr = 1.564, c) Fr =3.1281 and d) Fr = 3.91. . . . . . . . . . . . . . . . . . . . . 137Figure 5.20 Spatio-temporal diagrams of displacement flows with β = 60◦and Ret cos β = 243.6 and a) Fr = 1.173, b) Fr = 1.7595, c)Fr = 2.346 and d) Fr = 3.1281. . . . . . . . . . . . . . . . . 138Figure 5.21 Top: β = 60◦, Ret cos β = 87, Fr = 0.857 and m = 0.138. Thecenter plane snapshots are plotted at tˆ = 25,27s. Cross-sectionsare at xˆ = [1.0,1.02,1.04]m at tˆ = 27s. Middle: β = 60◦,Ret cos β = 201, Fr = 0.782 and νˆH ,L = 1.2mm2 s−1. The cen-ter plane snapshots are plotted at tˆ = 20,22s. Cross-sectionsare at xˆ = [0.64,0.66,0.68]m at tˆ = 20s. Bottom: β = 60◦,Ret cos β = 87, Fr = 0.857 and m = 7.21. Snapshots are plot-ted at tˆ = 13,14s. The 3D isosurfaces are C = 0.5 viewed fromthe top. The red solid lines represent C = 0.5. . . . . . . . . . 139xxviiFigure 5.22 Spatio-temporal diagrams from a) experiment with Fr = 0.788and Ret cos β = 245.52, b) numerical simulation with Fr =0.782 and Ret cos β = 243.6 (case 1) and c) numerical simu-lation with Fr = 0.788 and Ret cos β = 245.1 (case 2). Thex-axis is dimensionless distance from the gate-valve. . . . . . 140Figure 5.23 Effect of the initial maximum time-step on dispersion of con-centration field. The initial time-steps are a) ∆tˆi = 0.01 and b)∆tˆi = 0.5. The contours denote C = 0.1− 0.9. The snapshotsare at tˆ = [0.5,1.5, ...,4.5]s. The snapshot in (c) show the lastsnapshot of (a). The result in (d) are from a simulation with afiner mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Figure 5.24 Snapshtos of the simulations in Fig. 5.22c. The center planesnapshots (from top to bottom) and the cross-section snapshots(from left to right) are plotted at times ta = 20.75, tb = 24.48 andtc = 28.73, respectively. The cross-sections are at xˆ = 0.625mat tˆ = 40s and the initial interface is at xˆ = 0.2m. The red solidlines represent C = 0.5. The rectangle in the first snapshotshows the probe window used to create Fig. 5.25. . . . . . . . 142Figure 5.25 Temporal evolution of a) 〈u〉x(y,t), b) 〈C〉x(y,t), c) top: ∆u∗,bottom: 〈v2〉x(0,t) (black line) and 〈v2 +w2〉x(0,t)(red line), d)〈v2〉x(y,t), e) top: −∂〈C〉x/∂y |0, bottom: the transverse sepa-ration between the extrema of 〈u〉x , f) instantaneous Reynoldsnumber vs instantaneous gradient Richardson number (5.6). . 144Figure 5.26 Multiple laminar-turbulent cycles in lock-exchangeflowPIV/LIFmeasurements of [126]. Dimensionless quantities are denotedby the ˜ superscript. Note that time is nondimensionalized withτN = 2pi/√2gAtsinβ/d. Also 2z˜∗ ≡ ∆y∗. Three ramp-cliff cy-cles can be seen in (a), each one beginning at t˜ia(i = 1,2,3). Theimage is taken from [126]. . . . . . . . . . . . . . . . . . . . 145xxviiiFigure 5.27 Front shape and streamlines for a displacement with β = 75◦,Ret cos β = 126, Fr = 0.1955 (top) and Fr = 0.391 (bottom).The streamlines are in moving frame with Vˆf and are plottedat center plane. The red line represents C = 0.5. The 3Disosurfaces are C = 0.5 viewed from the top of the pipe. Thefront velocities are Vf = 4.3743 (top) and Vf = 2.6562 (bottom). 147Figure 5.28 Iso-contours of concentrationC = 0.1−0.9 and secondary flowvectors for the displacement flow of Fig. 5.27: Fr = 0.1955(top) and Fr = 0.391 (bottom). The cross sections from leftto right are at xˆ = [1.67,1.68,1.69]m. The solid line showC = 0.1−0.9. The velocities are scaled with Vˆf . . . . . . . . 148Figure 5.29 Front shape, streamlines and contours for a displacement withβ = 75◦, Ret cos β = 126 and Fr = 0.7820. The front velocityis Vf = 2.1. Plots are at times tˆ = [30,30.5,31,31.5]s. . . . . . 149Figure 5.30 Iso-contours of concentrationC = 0.1−0.9 and secondary flowvectors for the displacement flow of Fig. 5.29 at tˆ = 31s (top)and tˆ = 31.5s (bottom). The cross sections from left to right areat xˆ = [1.67,1.68,1.69]m. The solid lines show C = 0.1−0.9.The velocities are scaled with Vˆf . . . . . . . . . . . . . . . . 150Figure 5.31 Front shape, streamlines and secondary flows for a displace-ment with β = 85◦ and Fr = 1.5640 at tˆ = 25s. The frontvelocity is Vf = 1.505. The cross sections from left to right areat xˆ = [1.66,1.68,1.70]m. . . . . . . . . . . . . . . . . . . . 151Figure 5.32 Front shape, streamlines and secondary flows for a displace-ment with β = 10◦, Ret cos β = 89.24, Fr = 1.6422 and m =5.37 at tˆ = 18s. The front velocity is Vf = 1.8927. The crosssections from left to right are at xˆ = [1.60,1.62,1.64]m. . . . . 152Figure 5.33 Comparison of the front velocity values at a) β = 75◦ andRet cos β = 126 and b) β = 60◦ and Ret cos β = 243.6 fromexperiments (n) and numerical simulations (l). The solid anddashed lines are curve-fits from (1.6) and (3.4), respectively.Superposed symbols denotes displacements with back-flow. . 152xxixFigure 5.34 Comparison of V(C¯) for β = 60◦ and a) Fr = 0.79 and b)Fr = 3.91 from numerical simulations (#) and experiments (×). 153Figure 5.35 a) Spatio-temporal and b) mean concentration profiles for thesame simulation as Fig. 5.34b. The front velocities in (c) arecomputed using Ct = 0.01 (#) and Ct = 0.1 (×). The concen-tration profiles are plotted at equal intervals for t = 1.05−59.85. 154Figure 5.36 The same plot as Fig. 5.33b. The added triangle symbols arefront velocities from simulations with using Ct = 0.1. . . . . . 155Figure 5.37 a) Comparison of V(C¯) between 3D simulation (#) and 2LMsolution (red solid line) for a displacement flow with β = 80◦,Ret cos β = 20 and Fr = 0.6. b) Time evolution of V(C¯) from3D simulation for C¯ = 0.1 (solid line), C¯ = 0.3 (dashed line)and C¯ = 0.4 (dashed-dotted line). The right axis correspondsto C¯ = 0.6 (dotted line). . . . . . . . . . . . . . . . . . . . . . 156Figure 5.38 a) Time evolution of secondary flow strength for the displace-ment flow of Fig. 5.37. b) Maxima of 〈v2〉1/2 (×) and 〈w2〉1/2(#). c)Mean concentration profile and secondary flow strengthmultiplied by 10 at t = 25.75. . . . . . . . . . . . . . . . . . . 157Figure 5.39 Left: Mean concentration profiles for 3 displacement flows atβ = 85◦ with a) χ < χc, c) χ = χc and e) χ > χc. Right:Mean concentration profiles (left axis) and flux of light fluidlayer qL (right axis) for the displacement flows on the left.The dashed and solid lines correspond to t = [57.75,63] in (b),t = [19.73,22.26] in (d) and t = [26.77,29.4] in (f), respectively. 159Figure 6.1 Regime classifications fromprevious studies: a) near-horizontal[123] and b) inclined [4], both for iso-viscous displacementflows. c) Summary of density-unstable displacement flow find-ings from this thesis in terms of stability and back-flow. . . . . 166xxxFigure 6.2 Dependency of dimensionless parameters and the resulting flowregimes on changing physical quantities. The physical quan-tities shown are density difference (∆ρ), imposed flow (V0),viscosity (µ), inclination angle from vertical (β) and pipe (cas-ing) diameter (d). The symbols and lines are described inFig. 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Figure A.1 A simple schematic of the experimental set-up. . . . . . . . . 190Figure A.2 An image of the displacement flow-loop set-up in ComplexFluids Lab at UBC. . . . . . . . . . . . . . . . . . . . . . . . 191Figure A.3 a) Average light intensity as a function of ink concentration.The ink-water solution becomes saturated at 800mg/L. b)The top two images are calibration images for the black andclear fluids. The third image an unprocessed snapshot from anexperiment. The last image is processed using the calibrationimages. The red broken line is the depth-averaged concentration.192Figure A.4 Rheometry data for different concentrations of xanthan gum inwater. All solutions have 3% Glycerol. . . . . . . . . . . . . . 194Figure A.5 Mesh topology for simulations in a pipe. Pipe diameter is thesame as the experimental set-up. . . . . . . . . . . . . . . . . 197Figure A.6 The computational grid used in our the pipe cross-section. Thenumber of cell faces in the cross-sections are 1360, 3300 and4800, respectively. . . . . . . . . . . . . . . . . . . . . . . . 198Figure A.7 Effect of mesh size on results for a displacement flow withRet cos β = 50, Fr = 1, β = 60◦ and m = 1. Mesh sizes are∆x = 0.05 (#), ∆x = 0.1 (), ∆x = 0.2 (O) and ∆x = 0.5 (M).In all cases artificial diffusivity is η = 10−3. A large mesh sizesmears any shocks in the solutions and result in smaller frontalshock height and higher front velocity. With a finer mesh,instabilities with shorter wavelengths are resolved, however theartificial diffusivity keeps the problem well-posed. . . . . . . 202xxxiFigure A.8 Effect of artifical diffusivity on the results at t = 18. Thedisplacement parameters are the same as Fig. A.7. In all casesthe mesh size is ∆x = 0.1. Artificial diffusivities are η = 10−3(#), η = 10−4 (O) and η = 10−5 (M). Diffusivity of η = 10−3does not smear the shock but decreases the amplitude of theinstabilities. Reducing η beyond 10−4 does not change theresults since the artifical viscosity becomes smaller than thenumerical diffusivity. . . . . . . . . . . . . . . . . . . . . . . 203xxxiiList of Abbreviations2lm Two-Layer Modeldns Direct Numerical Simulationikh Inviscid Kelvin-Helmholtzkh Kelvin-Helmholtzrt Rayleigh-Taylorvkh Viscous Kelvin-HelmholtzxxxiiiAcknowledgmentsFirst I would like to thankmy dear supervisor Professor Ian Frigaard who helped mea great deal throughout my research. Without his knowledge, support and patienceI would not be able to finish this project. Whenever stuck with an ill-posed problemor a broken pipe in hand, I would hear "don’t worry, be happy".I am also grateful to Dr. Kamran Alba whose motivational speeches and metic-ulous feedbacks helped me from day one. Running the experiments would not bepossible without his constant guidance.At University of British Columbia I had the privilege of working with andlearning from many bright minds. As a teaching assistant, a research assistant, agraduate student, a lab supervisor or a seminar attendant, I enjoyed every moment.Last but not least I would like to thank all my friends, peers and officematesin Vancouver. We went through lots of ups and downs together, and shared manymoments of hope, despair, laughter and silence. Thank you all.I acknowledge the financial support I received from the University of BritishColumbia, Natural Science and Engineering Research Council of Canada andSchlumberger.xxxivI am still“one too many mornings,and a thousand miles behind.”– Bob DylanxxxvChapter 1IntroductionIn this thesiswe study displacement flowof twomiscible fluidswith different densityand viscosities in an inclined pipe. Such flows occur in many industrial processes.Our study is driven by cementing operations performed on oil and gas wells [84],mainly the primary-cementing stage of oil/gas well construction, where the drillingmud has to be displaced by the cement slurry, downward in the casing and thenupward in the annulus between the casing and earth. Successful displacement ofmud by the slurry is crucial to ensure mechanical integrity and sealing of the well.1.1 Primary cementingPrimary cementing is the process of placing cement in the annulus between casingand the formations exposed to the wellbore. The main objectives of primarycementing is to provide hydraulic sealing, or zonal isolation, prevent migration offluids such as water or gas towards surface, mechanically support the casing andprotect it from corrosion from hot formation brines and hydrogen sulfide. Withouta good cementing job and complete isolation in the well-bore, the well may neverreach its full production potential, and costly remedial processes are required. About15% of primary cementing jobs fail, costing the oil and gas industry an estimated450 million USD annually in remedial cementing work [84]. The importance ofa well-designed primary cementing job cannot be overstated. In addition to thehigh short-term costs (remedial processes and reduced extraction), a lack of high-1(a) (b)Zone 1Zone 2Zone 3Zone 4CementDrilling mudWash fluidSpacer fluidCement SlurryFigure 1.1: A simple schematic of the primary cementing process. a) Afterthe well-bore is drilled the casing is inserted. The casing and the annulusare initially filledwith the drillingmud. A sequence of fluids are pumpedinside to displace the mud downward in the casing and upward in theannulus. b) Ideally themud is completely removed and only pure cementremains in the annulus.quality primary cementing has additional long-term costs in plug and abandonmentof wells at the end of their life cycle. Additionally, it can result in catastrophies;The well-documented DeepWater Horizon explosion in 2010 at Gulf of Mexicooccurred during primary cementing.During the primary cementing stage, a large volume of cement slurry is pumpedinside the casing. Thewell and the annulus are filledwith drillingmud. The pumpedcement must displace the drilling mud downward inside the casing, and upward inthe annulus until mud removal is complete and the annulus is filled with cement.A simple schematic of this stage can be seen in Fig. 1.1. The common practice inprimary cementing is to pump fluid stages into the casing. A typical fluid sequenceconsists of drilling mud, wash, spacer, lead slurry, main slurry, tail slurry, spacerand drilling mud. Although rubberized plugs sometimes separate fluid stages,operational constraints often prevent their usage. Moreover, plugs (or wipers) areonly routinely used before the tail slurry and after the lead slurry, and not betweenthe spacer and wash fluids and the drilling mud. As a result, even when plugs are2used several fluid-fluid displacements and interfaces are present during the casingdisplacement. The resulting flow regime (at each interface) depends on the densitydifference, flow rate, inclination angle and rheological characteristics of the fluids.In end-of-life operations such as plug cementing usually no plugs are used. Plugcementing is a crucial process in abandonment of wells, where several cement plugsare places inside the well-bore to provide a zonal-isolation, so that fluids do notleak to the environment.Depending on composition, the density of well cement slurries may rangefrom as low as 720kgm−3 (foamed cements) to as high as 2400kgm−3 (high-density systems) [73], but more common would be in the range 1650−1900kgm−3.Much of the focus in the scientific and technical literature has been on the annulusdisplacement and not the downward casing displacement. The cementing jobpractices are mostly based on early rule-based systems concerned with annulardisplacement, dating back to 1960s; see [18, 23, 78]. The common practice forlaminar displacement flows is to pump sequentially more dense fluids (higherdensity by 10%) into the pipe to achieve stable flow during the upward annulusdisplacement. For instance, weighted preflushes (spacers) are made more densethan the mud by 60− 240kgm−3 using weighting agents such as barite, hematiteand calcium carbonate [65, 101]. Since the cement is generally more dense thanthe mud, this type of density hierarchy is inevitable, resulting in density-unstableconfigurations going down the casing. Density-stable configurations can also arise,e.g., pumping a wash (basically water) is also common.As will be shown throughout this thesis, the fluids quality and configuration atthe end of the casing displacement greatly depends on displacement parameters.Consequently, the fluids sequence assumed to arrive at bottom-hole and enter theannulus might be quite far from the designed conditions, during the cementing job.It should be noted that an efficient displacement might not necessarily result inan efficient cementing job. For instance the quality of the cement depends on thedegree of mixing during displacement. Also, thin layers of the mud remaining atthe walls in the annulus can dehydrate and allow gas migration, a so-called micro-annulus. In more extreme cases (and particularly when the annulus is eccentric) astatic mud channel may form. i.e., blocking a section of the annulus. According to[73], a study of 15,500 wells in the Gulf of Mexico showed that as a well becomes315 years old, it has a 50% probability of being affected by sustained casing pressure.The percentage of wells suffering from this problem was about 35% in the Gulf ofMexico, and similar numbers have been reported for the North Sea. Most of theproblems related to loss of cement integrity can be traced back to improper cementplacement.The density difference between the cement slurry, thewash and spacer fluids andthemud, can significantly affect the displacement process. A variety of additives aremixed with the cement slurry to change its density. Additionally, most oil and gaswells are very long, typically 2−10km, and often are inclined towards horizontal tomaximize production. Thus the angle at which displacement happens changes fromvertical to horizontal which can result in markedly different displacement regimes.Another important factor in the efficiency of the displacement is the rheologicalbehaviour of the fluids. Most drilling muds and cement slurries are shear-thinningand have a yield stress. The rheology of cement slurries depends on their chemicalcomposition, the additives used, and the water-to-cement ratio. Modeling thecomplete cementing process, through experiments or simulations, is not achievabledue to the range of length scales associated with the problem, and uncertaintiessuch as wash-outs, eccentricities, etc. Therefore one should first isolate the variousphenomena involved in the process and study the combined effects in a step-by-stepmanner.1.2 BackgroundAs mentioned above, in actual primary cementing several fluids of different prop-erties are pumped in a sequence, resulting in several fluid interfaces. This results ina very complex problem. In this thesis we focus on displacement flow of two fluidsin a pipe. A simple schematic of the problem is depicted in Fig. 1.2. The pipe hasa diameter of dˆ, length of Lˆ, and is inclined at an angle β. Note that throughoutthis thesis we use the hat (ˆ) superscript to denote dimensional quantities. Theinclination angle can change from vertical (β = 0◦) to near-horizontal (β ≈ 90◦).Initially, the two fluids are separated and are at rest. Then the displacing fluid (fluid1) is fed to the pipe with a mean imposed velocity Vˆ0, to displace the displaced fluid(fluid 2) out of the pipe. Our underlying motivation in studying these flows is to be4Vˆ0xˆyˆgˆdˆLˆDisplacing FluidDisplaced Fluidgˆ cosgˆ sinFigure 1.2: A simple schematic of the downward displacement flow problem.Initially the two fluids are separated.able to make pragmatic predictions of the flow regimes and overall flow behaviour,in parameter ranges of industrial interest.Study of mixing between two fluid stages becomes of critical interest in that thecement slurry is used to seal the well; ineffective displacement and/or mixing/con-tamination severely compromise this ability. This process involves flows that rangefrom laminar through to fully turbulent, with positive and negative fluid densitydifferences, and significant rheological differences.The simplest model system consists of two (miscible) Newtonian fluids alongan inclined pipe, which is characterised by 6 dimensionless parameters1 Reynolds(Re), Atwood (At), modified Froude (Fr) and Péclet (Pe) numbers, viscosity ratio(m) and pipe inclination angle:Re =ˆ¯ρVˆ0dˆˆ¯µ, Fr = Vˆ0√2 ˆ¯ρ|∆ρˆ|gˆdˆ , Pe =Vˆ0dˆDˆm, At =|∆ρˆ|2 ˆ¯ρ, m =µˆ2µˆ1, β, (1.1)where ˆ¯ρ = (ρˆ1 + ρˆ2)/2 is the mean density, ˆ¯µ the mean viscosity, Dˆm moleculardiffusivity, and ∆ρˆ = ρˆ1 − ρˆ2. Subscripts 1 and 2 denote the displacing and dis-1The important dimensionless groups used in this thesis are listed in Table B.1.5placed fluids, respectively. However, considering the flows we study, there are twoassumptions we make that simplify the problem:1. Although the fluids we are concerned with are miscible, molecular diffusionis very small, i.e., Pe 1. Therefore the flow is in the immiscible limit of adisplacement flow; however surface tension does not play a role.2. The density difference between the fluids is small (At 1), therefore the flowis in the Boussinesq limit, meaning the density difference effects on inertiaare negligible, however strong buoyancy effects are still possible via controlof the modified Froude number Fr , through the flow rate.With the assumptions of large Pe and density ratios close to 1 (small At), only 4dimensionless groups remain: Re, Fr , m and β.Viewed in terms of density difference, buoyant displacement flows in ducts canbe categorized into density-stable (ρˆ1 < ρˆ2) and density-unstable (ρˆ1 > ρˆ2). In bothcases we regard the flow as being in the downward direction. In the density-stablecase, the lighter fluid is on top of the heavier fluid and in the unstable case the heavierfluid is on top. Each configuration results in markedly different flow regimes. Inthe iso-viscous, non-buoyant limit (∆ρˆ = 0) the problem is simply advection of theinitial interface in a single-fluid pipe flow. Therefore the interface evolves with thePoiseuille profile, moving with the maximum pipe velocity at the center. In thebuoyant case (ρˆ1 , ρˆ2), the heavier fluid is accelerated in the axial direction relativeto the lighter fluid due to axial buoyancy ∆ρˆgˆ cos β. This force, in the density-stablecase, acts to retard the displacing (light) fluid therefore making the displacementmore efficient, and in density-unstable displacements, causes back-flow of the lightfluid, i.e., a flow back upstream against the imposed velocity. In both cases, thetransverse component of buoyancy ∆ρˆgˆ sin β tends to stratify the fluids.The problem complexity arises from both the large number of parametersinvolved and the richness of the phenomena observed, e.g., ranging from efficient,steady displacements, through stratified viscous flows, convective-diffusive inertialflows to fully diffusive mixed displacements. In the following section we will givean overview of the different fundamental problems which are closely related tothis study. Although these studies may not be directly comparable to the general6displacement flow problem, they help understand various phenomena observed inthe complex flow in hand, e.g., various hydrodynamic stability problems that ariseat different flow regimes. We will also review a number of two-fluid displacementflows with range of parameters different to ours, that cover different limits of theproblem. This helps to (i) differentiate the scope of this work from the existingliterature, (ii) summarize the gaps in knowledge regarding displacement flows, and(iii) better highlight the contributions of this thesis.• As mentioned earlier, we are concerned with flows with high Péclet number,such that Pe 1. As a result, in absence of mixing and instabilities, theinterface remains sharp and we can consider the flow to be analogous toan immiscible flow with no interfacial surface tension. Surface tension andmiscibility at the interface can have significant effects on onset of instabilities,as will be discussed later. Additionally, once the interface becomes unstablethe resulting flow regimes will be very different.• There is a rich history of work on gravity driven flows. Such flows havebeen studied in open and confined geometries. Examples include studieson gravity currents which are of significant importance in geophysics andlock-exchange flows in pipes. Naturally, in the limit of no imposed flowVˆ0 → 0 the density-unstable displacement flow asymptotically behaves asa lock-exchange flow. Gravity currents are of less relevance compared tothe range of parameters we study, but will be discussed since often similarapproaches and models are used to model both flows.• Depending on the flow parameters, several stability problems arise in dis-placement flows. Since the resulting flow regimes with a set of displacementparameters are not known a priori, a thorough stability analysis of these flowsis not possible. Even for much simpler problems, such as steady multi-layerflows, stability analysis studies are often limited to linear and onset prediction,two-dimensional channels, non-buoyant or iso-viscous flows. Nevertheless,the most relevant of these works will be reviewed.• Density-stable and unstable displacements result in very different flow be-haviors. This requires separate studies and different analyses for each flow.7Whereas the density-unstable flow is closely related to gravity-driven flows,the density-stable flow is similar in some ways to core-annular flows. There-fore relevant studies to each problem will discussed separately.• The theses of Dr. Taghavi [116] and Dr. Alba [1] are the most relevant to thiswork and will be reviewed in more depth.1.2.1 OverviewFluid-fluid displacement flows have been studied by numerous researchers in ge-ometries such as pipes, channels, capillary tubes, and Hele-Shaw cells using mis-cible or immiscible fluids, as well as extensively in porous media for oil recoveryapplications. One can categorize these flows based on whether the two fluids aremiscible or immiscible. Without significant instabilities, high Péclet number misci-ble displacements can be considered as immiscible with zero surface tension, i.e.,this is equivalent mathematically to the large capillary number limit of an immisci-ble flow. Note that by definition (Pe = Vˆd dˆ/Dˆm), high Péclet number correspondsto flows with very small molecular diffusivity. If Pe Lˆ/dˆ the flow reaches thelimit of Taylor dispersion [6, 128]. With the typical range of Pe ∼ 104−107 in in-dustrial applications, for a pipe diameter of 10cm the Taylor dispersion limit wouldrequire Lˆ ∼ 1−1000km, which is not common. Therefore although the flows hereare considered high Pe, in terms of time and length scales they are far from Taylordispersion regimes.A lot of the early studies on displacement flows are focused on low Reynolds,viscous flows in capillary tubes and Hele-Shaw cells since they are analogous toflows in porous media. One of the early immiscible displacement studies was theseminal work of Taylor [129]. He measured the amount of a viscous fluid left at thewall of a capillary tube when displaced by air. He found that at smallCa = Vˆt µˆ2/σˆ,the value of this fraction (m) changes as√Ca, and then reaches an asymptotic limitof 0.56. Cox [24] later found an asymptotic value of 0.6 for larger Ca. Zukoski[141] experimentally studied the influence of viscosity, surface tension and tubeinclination angle on bubble velocity. He found that the velocity increases to amaximum value as the tube is inclined away from vertical position to θ = 45◦, andthen decreases at higher inclinations.8High-Péclet-number miscible displacements were first studied in detail in [21,86, 92, 137]. The experiments of Petitjeans and Maxworthy [86] along withcomputational work of Chen andMeiburg [21] focused on displacement of a viscousfluid in capillary tubes. They measured the amount of a viscous fluid left on thewall, when displaced by a less viscous fluid, as a function of Pe and viscosityratio. More recently this problem was studied through axisymmetric and three-dimensional Stokes simulations by Vanaparthy and Meiburg [136].Yang and Yortsos [137] developed asymptotic solutions for viscous, miscible,iso-dense displacements in capillary tubes and Hele-Shaw cells with long aspectratios ( = Hˆ/Lˆ). Considering → 0 with finite Pe, the leading order equationsresult in transverse flow equilibrium approximation, where convective transversemixing is retained in the advection-diffusion equation of concentration, howeverpressure changes only in the axial direction. They found a flux function F(C) forconcentration, and showed that the concentration profiles are either completely self-similar, or consist of shocks. Using the same analytical approach and Hele-Shawexperiments, Lajeunesse et al. [69] studied downward density-stable displacement.They found that beyond a critical viscosity ratio and critical velocity, shocks formin the concentration profiles and 3D instabilities develop in the flow. Later in[70] they found the bounds between 3 different flow types in (U,m)-plane forboth Hele-Shaw cells and capillary tubes, where m is viscosity ratio and U =8µˆ2Vˆ0/Rˆ2∆ρˆgˆ is normalized velocity. In flow domain 1, for small m and U >U12(m) the concentration profiles are self-spreading and the velocity of the leadingedge is equal to Vmax , i.e., the maximum Poiseuille pipe velocity. In domain2 where U < U12(m) and U < U23(m) an internal contact shock exists travellingwith velocity VS < Vmax . U12(m) and U23(m) denote the boundaries between thedomains. Domain 3, with large m and U > U23(m), consists of flows with frontalshockwithVS >Vmax . In experiments, flows of domain 1 and 2were well predicted.In domain 3 however the tip velocities from experiments were smaller than VS fromthe model. Stability of these flows was later investigated in [71].Density-driven stability of miscible flows in Hele-Shaw cells were studied byFernandez et al. [41, 42]. More recently Meiburg and co-workers have continuedthese studies computationally and analytically, considering flow stability e.g. [48,49, 85], but focusing primarily at less inertial ranges than we study.91.2.2 Unstable & mixed flowsStability of single or two-fluid flows have long been studied dating back to worksof Helmholtz, Kelvin, Rayleigh and Reynolds in the 19th century (see [37] for acomprehensive review of hydrodynamic stability). From experiments of OsborneReynolds we know at Re ' 2000 the pipe flow of a Newtonian fluid becomesunstable to small disturbances. At sufficiently large Re the flow becomes fullyturbulent. In two-fluid displacement flow, with density and viscosity difference,the transition criteria for turbulent flow is not known. Note that while in a simplepipe or channel flow the disturbances grow from the walls, here turbulence canbe induced by other sources such as buoyancy. For turbulent flows, estimates ofmixing from Taylor dispersion type analyses e.g. [80, 117], are often adequate forthe required accuracy of prediction. Thus, the main focus of research to date hasbeen on laminar flows, as also here. We will mainly focus on two major sources ofinstability: density and viscosity variation across the interface of the fluids.In density-unstable displacements the density configuration is mechanicallyunstable. Neglecting the effects of imposed flow and viscosity, and assuming thepipe inclination is close to vertical, one expects the fluids to mix. Stability of suchproblem, in the simple case of a more dense fluid resting on top a less dense fluid,was first studied by Rayleigh [95] and Taylor [127]. It has been shown that thisconfiguration is unstable in the inviscid limit, and gives rise to Rayleigh-Taylor(Rayleigh-Taylor (rt)) instabilities [37, 110].Growth and different stages of RT instabilities in miscible fluids were studiedexperimentally and numerically in [22, 28, 76, 77]. In vertical ducts, developementof such instabilities leads to rapid mixing of the fluids across the pipe. This resultsin a diffusive flow with self-similar cross-section averaged concentration profiles.Such flows have been studied experimentally in vertical [29, 32, 33] and tilted[75, 105, 107, 108, 139, 140] ducts. In this regime, the evolution of the meanconcentration profile can be described by a diffusion equation∂C∂ tˆ= DˆM∂2C∂ xˆ2, (1.2)where DˆM is a macroscopic diffusion coefficient. The experimental values of10Figure 1.3: Growth of two-dimensional Rayleigh-Taylor instabilities when aheavy fluid (white) rests on top a light fluid (black). Perturbations in theinterface grow exponentially with time. The results are from numericalsimulations in OpenFOAM (see Chapter 5).DˆM can be obtained plotting C(xˆ, tˆ) as a function of the similarity variable xˆ/√tˆand fitting an error function. A similar collapse was seen in displacement flowexperiments of Alba et al. [4], with the similarity variable of (xˆ − Vˆ0 tˆ)/√tˆ. Thesestudies will be discussed in more detail shortly.The density-unstable configuration does not necessarily result in a turbulentmixed flow. Density difference, viscosity and inclination angle, particularly, andthe imposed velocity in case of displacement flows, play a significant role. Theresulting flow can be anything from fully diffusive (discussed above) to laminarcounter-current flow. In §1.2.3 and 1.2.4, the different flow regimes and transitioncriteria from one to another will be reviewed.1.2.3 Lock-exchange flowsIn absence of an imposed flow, the heavy-light configuration (ρˆ1 > ρˆ2) in a ductdevelops an exchange flow. Although not directly relevant to primary cementing,the lock-exchange flow is of interest here since it is essentially similar to density-unstable displacement flow as Fr→ 0.Exchange flow of iso-viscous miscible fluids in a long vertical pipe was studiedexperimentally by Debacq et al. [32]. The length and diameter of the pipe wasdˆ = 20mm and Lˆ = 4m, respectively. The range of density differences studied was11At = 2× 10−5 to 10−1. They found that at high Atwood numbers (At ≥ 4× 10−3),the flow becomes fully diffusive and the concentration profiles C(xˆ, tˆ) become self-similar. The effective macroscopic diffusivity was found to be 105 times higherthan molecular diffusivity. At lower Atwood numbers (1.5×10−4 ≤ At ≤ 2×10−3)the flow still remains diffusive but with higher fluctuations in the collapsed profiles.The mixing at these Atwoord numbers are due to counterflow of the two fluids thatresult in instabilities. At still lower density differences (At < Atm = 1.5× 10−4)the flow becomes non-diffusive, meaning the fluids do not mix efficiently and theconcentration profiles do not collapse.Later, they studied the effects of pipe diameter (dˆ = 2−44mm) and viscosity (νˆ =1−16×10−6 m2 s−1) in [33]. They found that depending on the Reynolds number,the flow can be turbulent-diffusive, convective-diffusive or stable counterflow. Inthe turbulent-diffusive regime, the flow is turbulent and is characterized by rapid andeffective mixing of the fluids across the pipe. In the convective-diffusive regime, themixing is less efficient and the flow structures become visible. In the stable regimethe mixing is further reduced and the fluid finger moves at a constant velocity atthe center of the pipe. The transition from stable to diffusive regimes occurs byincreasing At and dˆ or decreasing νˆ. It was also found that the local density contrastδρˆ and not the global ∆ρˆ drives the front motion. The local contrast itself dependson the mixing of the fluid, therefore as the fluids mix, δρˆ reduces and the fluidvelocities reduce.Seon et al. [105–109] followed a similar experimental approach and studiedbuoyant mixing, front dynamics and macroscopic diffusion in tilted pipes. Beforediscussing the results, let us first introduce two characteristic velocities used inlock-exchange and displacement flows:Vˆt =√Atgˆdˆ, Vˆν =Atgˆdˆ2νˆ. (1.3)Vˆt is an inertial velocity scale that is determined by balancing buoyancy and pressureforces, and Vˆν is a velocity found from a balance between buoyancy and viscousforces. Note that Vˆt is similar to the velocity of inertial Taylor bubbles [30]. Now a12Reynolds number can be defined based on Vˆt for gravity driven flows:Ret =ˆ¯ρVˆt dˆµˆ. (1.4)Note that with the definition of Fr in 1.1, we can writeFr =Vˆ0Vˆt, Ret =ReFr. (1.5)In [109] three different regimeswere observed in tilted pipes: For Ret cos β / 50there is a constant counterflow of the two fluid layers, with no transverse mixing,and the normalized front velocity Vf = Vˆf /Vˆt increases linearly with Ret cos β. ForRet cos β ' 50, the front velocity scales only with Vˆt , so that Vf reaches a plateau.In near-vertical pipes, the fluids mix and the flow becomes diffusive, similar tothose in [32, 33]. In this regime Vf drops from the constant value of regime 2.In [105], using Laser Induced Fluorescence (LIF) measurements, it was confirmedthe front velocity is related to the local density contrast Vf ∝ δC0.5f , where Cf isthe concentration at the front (Cf = δρˆ/∆ρˆ). This explains the drop in Vf for thediffusive flows. In near-horizontal lock-exchange flow experiments of [107], itwas also found that when Ret cos β / 50 the front velocity scales with the viscousscale Vˆf ' 0.0145Vˆν cos β, and for Ret cos β ' 50 with the inertial velocity scaleVˆf = 0.7Vˆt .1.2.4 Density-unstable displacement flowsIn this section we will review the previous works on miscible density-unstabledisplacement flows, mostly dedicated to Newtonian iso-viscous fluids in near-horizontal [119, 120, 123] and inclined pipes [4]. These studies can be seen asan extension of lock-exchange flows discussed above, and similar characteristicvelocities and parameters will be used to present the results. With the added effectof imposed velocity the symmetry of the problem breaks and the flow behaviourwill naturally deviate from exchange flows at high Froude numbers. These studieswill be reviewed in terms of flow regimes, front velocity variations, and back-flow,which are all of importance in context of primary cementing.13Near-horizontal ductsIn a series of studies, Taghavi et al. studied density-unstable displacement flowof miscible fluids in near-horizontal ducts, using analytical [118], numerical andexperimental [119–121, 123] approaches. In [119] the effect of adding a meanimposed flow to the exchange flow of miscible fluids was studied. The experimentalprocedure was similar to the exchange flow studies mentioned earlier. They foundthat at small values of Vˆ0, the dynamics of the flow is similar to the exchangeflow. As Vˆ0 is increased, the flow becomes more stable and the front velocityVf = Vˆf /Vˆ0 ≈ 1.3. At higher velocities the flow becomes turbulent and Vf ≈ 1. In[121] they studied effects of moderate viscosity ratios in near-horizontal ducts. Ina series of experiments in a pipe inclined at β = 85◦ they found that in generalviscosifying the displacing fluid improves the displacement efficiency and reducesthe front velocities. The effects of viscosity ratio on back-flow and stability howeverwere not studied.Later in [123] they classified near-horizontal β ∈ (83◦,85◦,87◦) displacementflows as (i) viscous, where no instabilities were observed at the interface of thefluids and (ii) inertial, where instabilities and local mixing were observed at theinterface. A map of their results in (Fr,Recos β/Fr)-plane is shown in Fig. 1.4.When Fr / 0.9, viscous and inertial flows were observed at Ret cos β / 50 andRet cos β ' 50, respectively, similar to the lock-exchange flows of [107]. For Fr /0.9, the viscous flows were further categorized as (v1) exchange flow dominated for2(Recos β)/Fr2 > χc = 116.32 and (v2) temporary back-flow for 2(Recos β)/Fr2 <χc = 116.32. The dimensionless parameter χ = 2(Recos β)/Fr2 represents thebalance of axial buoyancy and viscous stresses. The critical value χc is found usinga lubrication/thin-filmmodel (see [117, 120]), such that for χ > χc back-flowwouldbe found, meaning the trailing front (light fluid) moves in the opposite direction ofthe displacement. For Fr ' 0.9 the flow is (i2) inertial with temporary back-flowfor χ > χc and (v3) imposed flow dominated otherwise.For each regime they approximated the displacement front velocity, using ablend of analytical models and experimental fits to find dimensionless closureexpressions. for χ < χc a lubrication model was used to predict Vf (see [118] fordetail). In the inertial regime the following expression was fitted to their measured147.2. Displacement in pipes0 1 2 3 4 5 6020406080100120Vˆ0Vˆt≡ FrVˆνcosβVˆt≡RecosβFrβ = 83oβ = 85oβ = 87ov1v2v3i1i2χ = χcFigure 7.10: Classification of our results for the full range of experimentsin the first and second regimes (Re < 2300) in Table 7.1: sustained backflow (•, §), stationary interface (.), temporary back flow (J, /) and instan-taneous displacement (•). Data point with filled symbols are viscous andwith hollow symbols are inertial. The horizontal bold line shows the firstorder approximation to the inertial-viscous transition (Ret cosØ = 50, from[135]). The dotted line and its continuation (the heavy line) represent theprediction of the lubrication model for the stationary interface, ¬ = ¬c. Thevertical dashed-line is Vˆ0/Vˆt = 0.9. The thin broken lines are only illustra-tive and show an estimate for the turbulent shear flow transition, implyingto the third fully mixed regime. These are based on Re = 2300. Regionsmarked with vj (j=1,2,3) and ij (j=1,2) are explained in the main text.where they limit the velocity of the trailing front moving upstream. In-ertial eÆects are also significant local to the leading displacement front,where they usually appear in the form of an inertial bump. However, inthe bulk of the flow energy is dissipated by viscosity. The front velocitycan be well predicted by (7.13).(d) Viscous temporary back flow regime: These flows are found in aregime bounded by (7.17) and Fr = Vˆ0/Vˆt . 0.9, marked by by v2 inFig. 7.10. As with regime i2 this regime is transitionary showing a pro-gressive change from exchange-dominated to imposed flow-dominated160Figure 1.4: Summary of flow regimes in near-horizontal iso-viscous displace-ment flows [123]. Domains identified by i are inertial, and those identi-fied by v are viscous. Symbols denote sustained back-flow (), tempo-rary back-flow (C), stationary erface (B) an instantaneous (•). Thevertical dashed line is Fr = 0.9. Figure is taken from [123].Vˆf valu sVˆfVˆt= 0.7+0.595Fr +0.362Fr2, (1.6)so that at Fr → 0 the plateau value of 0.7 from inertial lock-exchange flows (see[109]) would be recovered.Inclined ductsUsing the same experimental approach, Alba et al. [4] extended the range ofRet cos β by conducting experiments in highly inclined pipes β ∈ (0◦−70◦). Theyclassified their flow regimes in (Fr , Ret cos β)-plane as:(i) Fully-diffusive: For Ret cos β > 500−50Fr the fluids are completely mixed157.2. Displacement in pipes0 1 2 3 4 5 6020406080100120Vˆ0Vˆt≡ FrVˆνcosβVˆt≡RecosβFrβ = 83oβ = 85oβ = 87ov1v2v3i1i2χ = χcFigure 7.10: Classification of our results for the full range of experimentsin the first and second regimes (Re < 2300) in Table 7.1: sustained backflow (•, §), stationary interface (.), temporary back flow (J, /) and instan-taneous displacement (•). Data point with filled symbols are viscous andwith hollow symbols are inertial. The horizontal bold line shows the firstorder approximation to the inertial-viscous transition (Ret cosØ = 50, from[135]). The dotted line and its continuation (the heavy line) represent theprediction of the lubrication model for the stationary interface, ¬ = ¬c. Thevertical dashed-line is Vˆ0/Vˆt = 0.9. The thin broken lines are only illustra-tive and show an estimate for the turbulent shear flow transition, implyingto the third fully mixed regime. These are based on Re = 2300. Regionsmarked with vj (j=1,2,3) and ij (j=1,2) are explained in the main text.where they limit the velocity of the trailing front moving upstream. In-ertial eÆects are also significant local to the leading displacement front,where they usually appear in the form of an inertial bump. However, inthe bulk of the flow energy is dissipated by viscosity. The front velocitycan be well predicted by (7.13).(d) Viscous temporary back flow regime: These flows are found in aregime bounded by (7.17) and Fr = Vˆ0/Vˆt . 0.9, marked by by v2 inFig. 7.10. As with regime i2 this regime is transitionary showing a pro-gressive change from exchange-dominated to imposed flow-dominated1604.3. Regime classification and leading order approximations0 2 4 6 80200400600800FrRecosβ/FrFigure 4.17: Classification of our results for the full range of experiments,presented in the (Fr,Re cos β/Fr)-plane: (i) instantaneous displacementflows are colored in blue and non-instantaneous flows in red; (ii) fully dif-fusive flows have no superposed symbol; (iii) non-diffusive flows are markedas viscous (superposed circles) or inertial (superposed squares). The heavyline represents the p ediction f viscous b ck flows, from the lubricationmodel in [130], (χ = χc = 116 32). The thick broken li e representsRe cos β/Fr = −50Fr + 500. The point of intersection of the two linesis Fr ≈ 4.62 and Re cos β/Fr ≈ 270.81Figure 1.5: Summary of flow regimes in inclined iso-viscous displacementflows [4]. The solid and dashed lines denote χ = χc and Ret cos β =500− 50Fr respectively. Blue symbols represent instantaneous dis-placements. The regimes are diffusive (no superposed symbol), viscous(superposed circles) and inertial (superposed squares). Figure is takenfrom [4].across the pipe, with mean concentration diffusing/dispersing relative to themean flow. Similar to lock-exchange flows the mean concentration profilesare self-similar with the similarity variable being (xˆ− Vˆ0tˆ)/√tˆ.(ii) Viscous: For χ < χc = 116.32 and Ret cos β < 500−50Fr instantaneous two-layer flow develops with no significant instabilities observed at the interfaceof the layers. A lubrication model can be used to predict Vf .(iii) Intermittent (inertial): For χ > χc = 116.32 and Ret cos β < 500− 50Fr ,the displacement flows are not fully diffusive nor fully viscous and theyexhibit a range of behaviours: instability, partial mixing, forward flows orcounter-current.The map of different flow regimes from [4] can be seen in Fig. 1.5. In the dif-fusive regime it was observed that the measured macroscopic diffusion coefficientswere an order of magnitude larger than those predicted from Taylor dispersion.16They approximated DˆM values withDˆM = DˆM ,Ex + dˆVˆ0(c0 + c1(β)/√Fr), (1.7)where DˆM ,Ex = 5×103(Vˆt dˆ)(1+3.6tan β)2(Vˆt/Vˆν)3/2 is the approximation for lock-exchange flows used in [107], and c0 = 0.6618 and c1(β)= 0.9054−1.838tan βwerefitted from displacement flow experiments.For front velocity, the lubrication model of [123] was used in the viscousregime. In the inertial regime, for χ > χc and Ret cos β < 500− 50Fr a curve-fitwas developed for front velocity asVˆf /Vˆt = Fr −0.002337(Ret cos β+50Fr −500)(1−0.98Fr +1.03Fr2), (1.8)to give Vˆf = Vˆ0 when Ret cos β = 500−50Fr .More recently Amiri et al. [5] studied density-unstable displacement flow ofmiscible fluids with small density differences in vertical pipes. They observedthat given Ret < 79+Fr +2Fr2, stable displacements with a finger moving at thecenter of the pipe can develop. Displacement flow of buoyant immiscible fluids wasstudied by Hasnain et al. analytically [54] and experimentally [55]. They found thatwith immiscible fluids completely different patterns than those of miscible flowswould develop, and the displacement efficiency could be significantly improvedaccording to the wetting properties of the displacing liquid.1.2.5 Density-stable displacement flowsDensity-stable displacements lead to less exotic and more predictable flow patternsthan density-unstable flows. Alba et al. [2] studied iso-viscous density-stabledisplacement flows experimentally. They found that in general density-stable dis-placements are very efficient compared to density-unstable flows. Flow stability andthe effects of viscosity contrast, however, have not been studied. Viscosity ratiosand shear-thinning effects add complexity to the flow problem and dimensionlessparameter space, but also move the work closer to industrial applications, whereviscosity ratio is used to control the displacement process.Although stability of these flows have not been studied, a number of two-17fluid studies have addressed stability in relevant flow orientations. Hickox [56]studied long-wave stability of axisymmetric flow of two fluids separated by acylindrical interface (radius R1) flowing within a vertical pipe (radius R2). Forunidirectional flows, he found that for viscosity ratiom > 1 asymmetric disturbancesare always unstable and axisymmetric disturbances are unstable when R1/R2→ 1,i.e., thin film of viscous fluid at the wall is unstable. Density difference doesnot affect stability at R1/R2 → 1. When R1/R2 → 0, asymmetric disturbancesare unstable and axisymmetric disturbances can become unstable with densitydifference (unstable spikes). Joseph and Renardy [64] studied non-buoyant flow oftwo fluids with different viscosities and found that when the core is less viscous theflow is unstable. Hu and Joseph [60] studied the same problem at short wavelengthlimit and found that viscous wall-layers are always unstable. In more recent works,Selvam and Meiburg [103] studied stability of miscible core-annular flows withviscosity contrast and the effects of interface thickness on stability. They foundthat having a finite interface thickness has a uniformly stabilizing effect, henceinstabilities only appear for thin interfaces. In comparison to immiscible studies([56]) the flow is not unstable for all small viscosity differences, and a criticalviscosity ratio exists. They also found that miscible core-annular flows are unstableat vanishing Reynolds number (Re) and high Péclet number (Pe).Gabard and Hulin [45] studied the influence of viscosity and shear-thinning oniso-density miscible displacement of a more viscous fluid in a vertical tube. Theyreported a large decrease of the residual film thickness observed when displacinga shear-thinning fluid. Rashidnia [94] found that a spike forms during the dis-placement of a liquid by a more viscous miscible liquid in a vertical tube. Whenbuoyancy is weak compared to viscous forces, the main finger is axisymmetric andstable. Balasubramaniam et al. [7] studied experimentally interface instabilities ofmiscible fluids with viscosity ratios and small Re. They reported that downwarddisplacement with a heavier, more viscous fluid leads to interfaces instabilities andgrowth of an asymmetric, sinuous spike. During upward displacements (density-stable), they observed an axisymmetric finger at the leading front, with a spike insome cases. This is similar to numerical results of Chen and Meiburg [21] at lowReynolds numbers, where they found that a finger of the more viscous, displacingfluid forms at the front. Scoffoni et al. studied experimentally the downward misci-18ble displacement in a vertical pipe for Re ' 1, [102]. They reported that at certainviscosity ratios and flow rates, the interface between the two fluids can destabilize.At higher velocities, they observed asymmetric “corkscrew” type instabilities, sim-ilar to those observed in core-annular flows of immiscible fluids. When the effectof gravity is dominant the interface is flat.Stability of non-buoyant multi-layer flow of miscible fluids with viscosity con-trast has also been investigated by several authors (see [36, 38, 46, 93, 99, 103, 104,125]). These studies include two-layer Poiseuille flow or core-annular flow with amixed layer at the interface of the fluids. A detailed review of such flows is givenby Govindarajan and Sahu [47]. Selvam et al. [103] studied stability of misciblecore-annular flows with viscosity contrast in a pipe, and compared the results tothose of immiscible flows. They found that with high Péclet number, the interfacethickness has a stabilizing effect at all Reynolds numbers. As interface thickness δis increased, the shorter wavelengths are stabilized and the maximum growth rateshifts towards longer wavelengths. Compared to immiscible flows, they showedthat the flow is not unstable for all viscosity ratios. Instead the the flow is stablefor all Reynolds numbers below a critical viscosity ratio. As the viscosity ratio isincreased the flow becomes unstable at smaller Reynolds numbers.D’Olce et al. [36] studied experimentally core-annular flow of neutrally buoyantmiscible fluids with the less viscous fluid at the core. They showed two distinctaxisymmetric patterns develop in these flows: pearls for small cores and Reynoldsnumbers and mushrooms otherwise. Later, Selvam et al. [104] studied convec-tive/absolute instabilities in these flows via linear stability analysis and numericalsimulations. They found that above a critical value of viscosity ratio, miscible core-annular flows are absolutely unstable. A thicker interface decreases the absoluteinstability. At large viscosity ratios, the onset of pearl & mushroom instabilitiesappears to be similar to an inverse bamboo wave, as observed earlier by [45].Talon and Meiburg [125] studied plane Poiseuille flow of miscible fluids withviscosity contrast in the Stokes flow regime. For two-layer flows they identifiedfour types of instability: two interfacial modes with large growth rates and two bulkmodes that grow more slowly. For two-layer flows, they showed that the stabilitydepends strongly on the position of the interface. Sahu and Govindarajan [99]studied stability of two-layer flows in channel using linear stability analysis and19three dimensional simulations in moderate Reynolds numbers. They showed thatthe overlap of the layer of viscosity stratification (interface with finite thickness) andthe critical layer of dominant disturbances provides the mechanism for instability.The thickness of the interface affects the stability of the flow. They also showedthat the stability is sensitive to the location of the interface.1.2.6 Two-layer flowsAs discussed above, when the fluids are mixed across the duct simple diffusionor dispersion models are adequate to describe the flow. These flows develop athigh Ret cos β values where the flow becomes turbulent because of significant axialbuoyancy, or at high Rewhere the flow becomes turbulent. Given the high viscosityof fluids often used in cementing processes, or high inclination angles, a stratifiedtwo-layer flow develops, with the heavy fluid slumping underneath the light fluid.Most of the existing two-layer models for displacement flows are limited to2D channels, in the form of lubrication/thin-film approximations (δ = dˆ/Lˆ → 0)for viscous flows. Taghavi et al. [118] developed a model for heavy-light andlight-heavy displacement of generalized Newtonian fluids in a 2D channel. Seonet al. [106] developed a simple model for viscous lock-exchange flows assumingquasi-parallel flow. They found that at short times front propagation is controlledby inertia and at long times by viscous effects, where the front velocity decreases toa steady valuesV∞f . The same approach was extended to pipe displacement flows in[120]. It was found that when χ = χc the light fluid layer becomes stationary at thetrailing edge. This critical values was found to be χc ≈ 116.32 for pipes and 69.94for plane channels. The significance of χc is that for χ > χc sustained back-flowoccurs, which was discussed in 1.2.4. Moyers-Gonzalez et al. [83] proposed asemi-analytical closure approximation for viscous flow of two Herschel-Bulkleyfluids in pipes. The method consisted of exact area-averaged momentum balanceand approximation of interfacial and wall shear stresses.Stability of parallel two-layer flowsDensity and/or viscosity difference results in different velocities in the heavy andlight fluid layers. This gives rise to several interfacial stability problems, the20Figure 1.6: Growth of Kelvin-Helmholtz instabilities when two fluids withdifferent densities move with different directions. These snapshots arefor a gravity driven counterflow of a heavy fluid (white) and a light fluid(black) in a plane channel. The results are from numerical simulationsin OpenFOAM.most classical being that of two fluid layers moving with different velocities, i.e.,Kelvin-Helmholtz (Kelvin-Helmholtz (kh)) instability [37]. This problem was firstremarked by Helmholtz (1871), then solved by Kelvin (1890) and described interms of vorticity dynamics by Batchelor (1967). For inviscid flows of two fluidswith different densities it can be shown that the interface is always unstable to shortwave-length disturbances if VH , VL , with V being the velocity in each layer. Anexample of KH instabilities can be seen in Fig. 1.6. One the most famous studies onthese instabilities are the experiments of Thorpe [130], who measured the growthrate and threshold of instability. Stability of the flow in stratified environmentsdepends on both density and velocity fields. A detailed review and analysis ofinternal waves and interfacial waves in multi-layer and stratified flows is written bySutherland [114].In two-layer gravity driven and displacement flows KH type instabilities havebeen observed. The classical KH instability analysis is for inviscid flow of two fluidswith discontinuous density and velocity at the interface. However, in physicalviscous flows, viscosity stabilizes short wave-length disturbances. As soon asinstabilities grow the fluids mix resulting in a mixed layer at the interface anddecrease in density gradient, which has a stabilizing effect.Multi-layer flow stability has a long history dating back to the classical study of21Yih [138]. However, much of this literature concerns onset and an initial base statewith no mixing, i.e. immiscible with zero surface tension. In the context of misciblemulti-fluid flows, linear stability studies can proceed by assuming a quasi-steadyparallel base state, e.g. [38, 46, 93]. In addition, flows studied are often simplified toplane channel flowswhere the base flow is easily found, e.g. [97, 98]. Thesemethodsundoubtedly give insights and are adaptable to non-Newtonian configurations.The stability of 2D channel displacement flow was studied using the weightedresidual approach in [3], which is effectively an inertial perturbation of the lubri-cation model. While the results found are consistent with more complex analyses(e.g. Orr-Sommerfeld) at long wavelengths, flow complexity prohibits simple ex-tension to pipe flows. In the multi-phase flow context, Picchi et al. [88–91] haverecently studied stratified Newtonian/Shear-thinning (Carreau fluid) combinations,developing two-fluid model closures and studying long-wave instabilities.Gravity currentsWe discussed gravity-driven flows in form of lock-exchange flows at high incli-nations and in confined geometries. There is also a large literature concerninggravity currents, i.e., high Reynolds number gravity driven flows in horizontal ornear-horizontal environments which are of significant importance in the oceano-graphic, meteorological and geophysical contexts. Such flows are mostly studiedin unconfined geometries.Analytical modelling of gravity currents dates back to the seminal work ofBenjamin [12]. Later on this model was improved and compared against exper-iments by Shin et al. [111]. Dynamics of the current were studied by Simpsonand Britter [112] over a horizontal surface and by Britter and Linden [19] down anincline. Huppert [61] studied propagation of two-dimensional and axisymmetricviscous gravity current over a horizontal surface. A lubrication type model wasused and the results were compared against experiments. A similarity solutionfor shape and rate of propagation of the current was found. Such flows have alsobeen studied in sloping channels [13], linearly stratified ambient fluids [81], innon-Boussinesq regimes [14, 67, 96] and for viscous currents [115]. Analytically,a vorticity based model was developed by Borden and Meiburg [16] and extended22by Khodkar et al. [66] for unsteady flows. Shallow-water type formulations havealso been used to model gravity currents in unconfined and confined geometries(see [58, 79, 134, 135]). Recently high-resolution numerical simulations of gravitycurrents have been performed by Meiburg and co-workers (see [14, 50–53]) andCantero et al. [20]. See [82] for an in-depth review of computational approachesfor gravity and turbidity currents. These studies provide very useful informationon dynamics of the flow, e.g., the behaviour close to the front, and help benchmarkand improve simplified models for gravity currents. Unfortunately, no such effortshave been made to simulate displacement flows.We should note the key differences between the scope of the flows we are con-cerned with and those found in gravity currents. Gravity current flows are studiedmostly in high-Reynolds number inviscid limit, meaning Ret values are much largerthan the range of interest in context of industrial displacement flows. The large Retassumption justifies neglect of the friction terms. The second important differ-ence is that gravity current flow studies are limited to horizontal inclinations, thuscos β ≈ 0. In industrial displacement flows however, we are interested in inclinedpipes. Therefore the characterizing parameter here is Ret cos β. As mentionedearlier, at high Ret cos β values the two fluids become completely mixed acrossthe pipe and the flow is essentially fully-diffusive. Furthermore, gravity currentsare mostly studied in unconfined domains, rectangular ducts or with partial-depthrelease. All these lead to different flow dynamics compared to displacement flowsin pipes, although the insights gained are useful.1.3 Conclusions & scope of thesisA number of problems closely related to displacement flows were described above.The number of dimensionless parameters associated with displacement flows posesa challenge for a simple solution for industrial applications, and at the same timepresents several interesting fluidmechanics related problems. Despite all the studiesreviewed, a number of question remain unanswered in context of displacement flows.At sufficiently high flow rates, Taylor dispersion can be used to adequatly de-scribe the flow. However, achieving fully turbulent flows might not be possibledue to high viscosity of typical fluids used in oil/gas well production. Therefore23the focus so far has been on laminar flows. It was shown that given high enoughRet cos β, density-unstable flows become fully-diffusive and the flow can be de-scribed by self-similar mean concentration profiles. Just as with turbulent Taylordispersion, these fully diffusive flows allow one to predict approximately howmuchmixing occurs between successive fluids. Therefore the main focus of this thesiswill remain on stratified or non-diffusive flow regimes. Here, we will summarizethe gaps in knowledge regarding displacement flows to highlight the objectives andcontributions of this work:• One limitation of the experimental work to date is the iso-viscous assumption.In industrial processes of interest, the fluids rarely have the same viscosity.The effects of viscosity contrast on displacement efficiency, stability, back-flow and other phenomena in displacement flows have not been studied yet.Here, we investigate the effects of viscosity contrast in both density-stableand unstable displacement flows.• The stability of density-unstable displacement flows has not been studiedin any detail. The viscous-inertial transition along χ = χc is observed tocoincide with back-flows (see [117, 120]), but inertial flows are observed toarise from instabilities (e.g. roll wave onset) which have not been predictedto date. Secondly, the existence of stable viscous back-flows at lower Frshows that χ > χc is not a sufficient condition for instability. This problemis non-trivial in the displacement flow context. Using a two-layer modeldeveloped later in the thesis and the experimental data, we categorize thetwo-layer displacement flows in terms of stability and make predictions forthe onset of instabilities.• Unlike gravity currents, the front dynamics of displacement flows have notbeen studied. First, front velocity measurements and therefore reported dis-placement efficiencies might be sensitive to the flow conditions near the front.Second, it is known that near the front strong secondary flows exist, whichinvalidates the parallel flow assumption of 1D models. In both experimentsand numerical simulation we look at the front dynamics to (i) get an idea ofthe applicability of lubrication models for front velocity predictions, and (ii)24study sensitivity of front velocity measurements in different flow regimes.• The existing simplified models for displacement flows are limited to lubri-cation approximations where inertia is neglected. This limits their validityto small Re. A new inertial two-layer model is developed for two fluidswith different viscosities. Retaining the inertia terms and modelling viscousstresses allows for instability prediction.• While growth of instabilities and asymmetric spikes due to viscosity ratio isreported by other authors, stability and viscosity ratio effects in density-stabledisplacement flows have not been investigated.• Finally, numerical simulations of miscible displacement flows are limitedto 2D channels or capillary tubes. The flow regimes in plane channelsare significantly different to those observed from experiments in pipes. Wecomplete a series of 3Dnumerical simulations in pipeswith long aspect ratios.The results are compared against the experiments and the two-layermodel. Toour knowledge, there are no other 3D displacement flow simulations availablein the regimes covered by our experiments.Themajor contribution of this thesis is to address the above-mentioned questionsthrough analytical, experimental and numerical tools, to help better understand andpredict the fluid-fluid displacement flow problem. The structure of this thesis is asfollows:Chapter 2 We develop a two-layer inertial model for stratified density-unstabledisplacement flows. The complete model for the displacement flow consistsof mass and momentum equations for each fluid, resulting in a set of fournon-linear equations. By integrating over each layer and eliminating thepressure gradient we reduce the system to two equations for the area andmean velocity of the heavy fluid layer. The wall and interfacial stressesappear as source terms in the reduced system. The final system of equationsis solved numerically using a robust, shock-capturing scheme. The equationsare stabilized to remove non-physical instabilities. A linear stability analysisis able to predict the onset of instabilities at the interface and together with25numerical solution, is used to study displacement effectiveness over differentparametric regimes. Backflow and instability onset predictions are made fordifferent viscosity ratios.Chapter 3 We present the experimental results for density-unstable displacementflows in inclined pipes. We consider viscosity ratios in the range 1/10 to10. Our focus is on displacements where the degree of transverse mixingis low-moderate and thus a two-layer, stratified flow is observed. A widerange of parameters is covered in order to observe the resulting flow regimesand to understand the effect of the viscosity contrast. The inclination of thepipe (β) is varied from near horizontal β = 85◦ to near vertical β = 10◦. Ateach angle, flow rate and viscosity ratio are varied at fixed density contrast.Flow regimes are mapped in the (Fr,Recos β/Fr)-plane, delineated in termsof interfacial instability, front dynamics and front velocity. The results arecompared against the two-layer model introduced in Chapter 2.Chapter 4 Density-stable displacement flows with viscosity ratio are discussedin Chapter 4. Here, we present the experimental results, with the sameset-up and fluids as in Chapter 3. The effect of adding a viscosity contrastbetween the fluids on the displacement efficiency, and stability of density-stable displacements is studied.Chapter 5 We present three-dimensional numerical simulations of the displace-ment flow problem. We outline the numerical model developed in Open-FOAM. The results for a number density-stable and unstable cases are bench-marked against experiments. Several unanswered questions from previouschapters are addressed and the different phenomena observed in the experi-ments, such as front dynamics are investigated.Chapter 6 We summarize the work conducted, list the main contributions of thethesis, draw conclusions, and present recommendations for future work.26Chapter 2A two-layer model for the inertialregimeWe start the thesis with a chapter1 that is specifically focused at modelling theinertial/intermittent range of flows. These flows are characterized by large valuesof Re/Fr cos β, but with Re not high enough for the flow to become turbulent.Note that by inertial displacement flows, we mean stratified flows where transversebuoyancy separates the fluids into two layers, but is not strong enough to stabilizethe flow. As a result interfacial instabilities are always present and transversemixing occurs locally. Currently there are no predictive models for these flowregimes and none that might be used to predict the transition from viscous toinertial (or potentially inertial to diffusive displacement flows). As our interest isin modelling flows where both inertia and buoyancy are important, we develop atwo-layer model for these flows by keeping the inertial terms in the momentumequations and accounting for viscous effects only via wall and interfacial stressclosure expressions. This allows us to make regime predictions and study instabilityonset from a basal inertia dominated flow, rather than a viscous dominated flow.This also allows us to study rheological effects via the stress closure models,although here we restrict to Newtonian fluids of differing viscosity.1A version of this chapter has been published as: A. Etrati and I.A. Frigaard. A two-layer modelfor buoyant inertial displacement flows in inclined pipes. Phys. Fluids, 30(2) (2018): 022107.272.1 IntroductionThe two-layer model that we derive simplifies to a one-dimensional two-equationmodel, which is similar to two-layer shallow-water equations. The main noveltyarises from the displacement flow context, as the style of model derivation andapproach, together with the numerical implementation, is similar to the two-fluidmodel that has been used extensively to study laminar and turbulent, gas/liquidflows in pipelines [10, 25, 59, 62, 63, 89, 100, 124]. A thorough review of thesemodels, their linear and nonlinear stability analysis and numerical solutions hasbeen given in [31]. Similar approaches have also been utilized in modelling ofgravity currents [26, 27], for roll-wave prediction in granular flows [34, 35], for thestability of core-annular flows [8, 9] and for stratified flows [10].Although we borrow from the existing literature in our study, it is importantto outline the difference between the two flows. In pipeline flows each phaseis injected into the pipe separately, i.e., the flux of each phase is controlled. Indisplacement flows however, the fluxes are not known a priori. In a densityunstable displacement the pipe is initially filled with the light fluid and the heavyfluid is pumped in downwards at a fixed flux. The individual layer fluxes qˆH and qˆLare therefore found from the solution of the two-equation system at each positionand time. Other key differences with gas/liquid flows are that: (a) here the densitydifferences are very small thus the flow is in Boussinesq regime; (b) the layervelocities have the same order of magnitude; (c) back-flow (of the light layer) mayor may not occur. In comparison, in gas/liquid systems the gas phase velocity ismuch higher than the liquid velocity, and usually in the turbulent regime. Lastly,we are interested in displacement of miscible fluids, thus surface tension does notplay a role. Therefore the resulting flow regimes in miscible displacement flowsare very different to those observed in pipeline flows.Nevertheless, we study the same two-equation system at its core. As pointedout by many authors using two-fluid models, the resulting model is ill-posed undercertain conditions. We will later show how constraining this condition is within theparameters range we are interested to study. This problem arises due to simplifi-cation of the physical model by removing the diffusive terms from the momentumequations, which allows rapid growth of unphysical shortwave instabilities in the28solution [11, 63, 113]. To overcome this, authors have suggested using numericaldiffusivity [63] or to include surface tension terms [10, 43] to introduce a cut-offwavelength. Numerical diffusivity is grid dependant and requires using a rathercoarse grid. Surface tension introduces difficult-to-implement third-order terms,which are unphysical in the context of miscible flows. Another method suggestedin [59] is to add artificial diffusion terms to both momentum and mass equations,which can make the system unconditionally stable. This method has been suc-cessfully used by others [15, 43, 44], allowing the use of two-fluid models beyondthe stability limit dictated by ill-posedness of the model. Here we adopt a similarapproach, as discussed in more detail later.Finally, the characteristics method (e.g. as utilized in [58, 79, 134]) requires thesystem to be hyperbolic, which is restrictive in studying the range of parameterswe are interested in. The numerical method used here is able to resolve our modelnumerically in non-hyperbolic parameter ranges.An outline of this chapter is as follows. Section 2.2 following, derives the strat-ified two-layer model of displacement flow in dimensionless form. Well posednessand stability are addressed in §2.4. We study linear stability of the base flows andthen explore stabilization of the model to remove non-physical instabilities in arobust way. Our results section starts (§2.5.1) with predictions from exchange flows(Fr = 0), where our inertial model successfully predicts the experimental stabilitythreshold Recos β/Fr ≈ 50. For displacement flows (§2.5.2) we start by explor-ing general features, in which we show that the broadly similar characteristics arefound as in past experimental studies. The model is then used to classify regimesvia the behaviour of the displacement front. This is applied to both iso-viscousflows and those with significant viscosity ratios. The paper ends with a brief dis-cussion. Frictional closures and computational details are listed in supplementaryappendices.2.2 Two-layer modelOur interest is in stratified displacements that evolve as shown schematically inFig. 2.1, with the heavy layer moving forward at the bottom of the pipe. The pipeis inclined at angle β from vertical. The pipe is initially filled with the light fluid.29SLSHSiHLWˆ0Vˆt⌘ FrFigure 2.1: Schematic of the two-layer model used. The flow is in the axialz−direction, with (x,y) in the pipe cross-section and x is perpendicularto gravity. H and L refer to heavy and light fluid layers, respectively. αis the area fraction of the cross-section occupied by the heavy layer. Thegeometric parameters are given in (2.11)-(2.13).The heavy fluid is then pumped into the pipe with the mean imposed velocity Wˆ0.Transverse buoyancy (∆ρˆgˆ sin β) will tend to stratify the fluid whereas the differencein layer velocities and the viscosity contrast at the interface might encourage mixingof the fluids. The axial component of buoyancy (∆ρˆgˆ cos β) accelerates the heavylayer so that there is generally a velocity difference between the layers. The interfacebetween the layers elongates (in the axial direction) and might becomes unstabledue to mechanismsmentioned earlier. The efficiency of the displacement is dictatedby how much faster the heavy fluid advances compared to the mean flow. The flowis described by the following dimensional system of equations. Note that the hatsuperscript denotes dimensional parameters.ρˆk[∂uˆ∂ tˆ+ uˆ · ∇ˆuˆ]= −∇ˆpˆ+ ρˆk gˆ+ ∇ˆ · τˆk in fluid k, (2.1)∇ˆ · uˆ = 0. (2.2)Here k = (H,L) denotes the heavy or light fluids. The interface is assumed to beflat in the transverse plane, with height yˆ = hˆ(zˆ, xˆ, tˆ), and evolves according to the30kinematic equationDDtˆ[hˆ(zˆ, xˆ, tˆ)− yˆ] = 0. (2.3)To simplify the system we integrate the momentum equations (2.1) over each layerto get two area-averaged z-momentum equationsρˆH[∂∂ tˆAˆH wˆH +∂∂ zˆAˆH wˆ2H]= AˆH[−∂ pˆ0∂ zˆ+ ρˆH gˆ cos β]− [τˆH SˆH + τˆiH Sˆi] , (2.4)ρˆL[∂∂ tˆAˆLwˆL +∂∂ zˆAˆLwˆ2L]= AˆL[−∂ pˆ0∂ zˆ+ ρˆL gˆ cos β+∆ρˆgˆ sin β∂ hˆ∂ zˆ]− [τˆL SˆL − τˆiL Sˆi] , (2.5)and the mass conservation equation∂∂ tˆAˆH +∂∂ zˆ[AˆH wˆH ] = 0. (2.6)Here Aˆk and Sˆk(hˆ) are the fluid layer areas and wetted perimeters respectively, wˆkis the mean axial velocity and τˆk the wall shear stress in each layer. The interfacialshear stress at the heavy and light sides of the interface is τˆi,k and Sˆi(hˆ) is theinterface length. We have eliminated the static pressure by integrating across thepipe and pˆ0 is the pressure at the bottom of the pipe. In simplifying the equationsto get (2.4) and (2.5), we have made the following assumptions:1. Streamwise spatial variations occur over lengths larger than the pipe diameter.2. Streamwise viscous stress gradients are less relevant than shear stresses at thewalls and the interface. The latter are integrated to give averaged stresses.3. The momentum correction factor is unity, that is∫ α0 wˆ2 dAˆ = wˆ2H AˆH and∫ 1−α0 wˆ2 dAˆ = wˆ2L AˆL .The first assumption is the long-wave approximation, needed to achieve the sim-plified 1D model. It also lets us assume pressure equilibrium between the layers.The friction terms (τˆ) need closures which will be discussed shortly. To close the31system we also need to write the constant flow rate conditionAˆH wˆH + AˆLwˆL =pidˆ24Wˆ0. (2.7)2.2.1 Dimensionless equationsSince our model is focused at the study of inertial regimes, as defined in [4], theappropriate scale for the velocity comes from balance of buoyancy and inertiaVˆt =√12∆ρˆˆ¯ρgˆdˆ =√Atgˆdˆ, (2.8)where At = (ρˆH − ρˆL)/(ρˆH + ρˆL) is a densimetric Atwood number. Therefore thevariables are scaled aszˆ = dˆz, wˆk = Vˆtwk, tˆ =dˆVˆtt, pˆ0 = ˆ¯ρVˆ2t p0, (2.9)and the geometric parameters asAˆH =pidˆ24α, AˆL =pidˆ24(1−α), Sˆk = pidˆ4 Sk . (2.10)The scaled geometric functions α and Sk are defined in terms of dimensionlessinterface height h = hˆ/dˆ, as follows:α(h) = 1pi[cos−1(1−2h)−2(1−2h)√h(1− h)]. (2.11)SH (h) = 4picos−1(1−2h) = 4− SL(h), (2.12)Si(h) = 8pi√h(1− h), (2.13)αh(h) ≡ dαdh = Si(h). (2.14)The shear stresses will be scaled asτˆ =ˆ¯µVˆtdˆτ = ˆ¯ρVˆ2tFrReτ, (2.15)32where Re = ˆ¯ρWˆ0dˆ/ ˆ¯µ is the Reynolds number based on the imposed velocity Wˆ0,average density ˆ¯ρ = (ρˆH + ρˆL)/2, geometric mean viscosity ˆ¯µ =√µˆH µˆL , and thepipe diameter dˆ. Closure expressions for the wall and interfacial shear stresseswill be detailed in the next section. Here Fr = Wˆ0/Vˆt is the Froude number whichmeasures the relative strength of the imposed flow to that induced by buoyancy.From definitions of Re and Fr , we can see that the ratio Re/Fr does not depend onWˆ0 and is a Reynolds number based on the velocity scale VˆtRet ≡ ReFr =ˆ¯ρWˆ0dˆˆ¯µVˆtWˆ0=ˆ¯ρVˆt dˆˆ¯µ. (2.16)We will use Ret and Re/Fr interchangeably throughout the rest of this paper.The experiments that have been performed in [4, 122] are characterized by small|At |, but significant ranges of Fr . Thus, using these dimensionless parameters, andassuming At ∼ 0, (2.4)–(2.7) become:∂∂tαwH +∂∂zαw2H = α[−∂p0∂z+ cos β]− FrRe(τHSH + τi,HSi), (2.17)∂∂t(1−α)wL + ∂∂z(1−α)w2L = (1−α)[−∂p0∂z− cos β+2∂h∂zsin β]−FrRe(τLSL − τi,LSi), (2.18)∂α∂t+∂∂z(αwH ) = 0, (2.19)αwH + (1−α)wL = Fr . (2.20)Note that in (2.17) and (2.18), p0 is the pressure at the bottom of the pipe, includingthe hydrostatic pressure of the mean flow ( ˆ¯ρgˆdˆ cos β). We now have a four-by-foursystem of equations for wH , wL , p0 and α. We further simplify the model, reducingit to two equations. By summing (2.17) and (2.18) we get∂∂z[αw2H + (1−α)w2L] = −∂p0∂z+ (2α−1)cos β+2(1−α)sin β∂h∂z(2.21)− FrRe[τHSH + τLSL + (τi,H − τi,L)Si],This expression can be integrated to compute p0 from the eventual solution. Here33however, we use this expression to eliminate ∂p0/∂z. Secondly, wL can be elimi-nated using (2.20), so that the axial momentum balance in the heavy layer becomes∂wH∂t+∂∂z[w2H (12−α)− (Fr −αwH )2(1−α)]+2sin β1−ααh∂α∂z= 2(1−α) [cos β−T],(2.22)where T contains the friction termsT = 12FrRe{1α(τHSH + τi,HSi)− 11−α (τLSL − τi,LSi)}. (2.23)Combining with the mass conservation equation, we have the following system forV = (wH,α)T :∂∂tV+ ∂∂zF(V) = S(V), (2.24)with F(V) and S(V) being the flux and source vectorsF(V) = ©«w2H (12 −α)−(Fr −αwH )2(1−α) +2g(α)sin βαwHª®®¬, (2.25)S(V) =(2(1−α) [cos β−T]0). (2.26)The geometric function g(α) is the following integralg(α) =∫ α01− α˜αh(α˜) dα˜ = h(α)−∫ h0α dh, (2.27)which varies monotonically from g(0) = 0 to g(1) = 0.5.2.3 Frictional closure models & equilibrium solutionsWe examine here the closure models for the wall and interfacial shear stresses,needed to complete the two-layer model. Here we seek models that are goodapproximations to the actual stresses but are easy to implement numerically. Weintroduce 3 different models and compare them using the steady-state equilibriumcondition (cos β = T ), for which the friction terms are balanced by axial buoyancy.34The closure models are based on using average velocities, hydraulic diameters andappropriate friction factors. The wall shear stresses are expressed as:τˆk = fkρˆk |wˆk |wk2; fk =16Rek; Rek =ρˆk |wˆk |Dˆkµˆk, (2.28)where µk and Dˆk are the viscosity and the hydraulic diameter for each fluid layer,thus Rek is the local Reynolds number for each layer. The dimensionless wall shearstress for each layer will beτH =8√mwHDH; τL = 8√mwLDL, (2.29)where m = µˆL/µˆH is the viscosity ratio. Only laminar flows are considered here.The expressions for the hydraulic diameters DH and DL as well as the interfacialshear stress will be discussed for each model described below.2.3.1 Model 1We adopt (2.29) directly for the wall shear stresses. We write a single interfacialshear stress τi and choose the friction factor depending on which fluid is draggingthe other oneτˆi =fHρˆH (wˆH − wˆL)|wˆH − wˆL |2if wH > wL,fLρˆL(wˆH − wˆL)|wˆH − wˆL |2if wH < wL .(2.30)The hydraulic diameters used for fk for flows with no back-flow (wH,wL > 0), aredefined using the following expressions (see [17, 132]):DH =4αSH + Si; DL =4(1−α)SLif wH > wL, (2.31)DH =4αSH; DL =4(1−α)SL + Siif wH < wL, (2.32)DH =4αSH; DL =4(1−α)SLif wH ∼ wL . (2.33)35if back-flow occurs (wH > 0,wL < 0) then:DH =4αSH + Si; DL =4(1−α)SL + Si. (2.34)With the above model τi is dominated by the faster fluid.2.3.2 Model 2Ullmann et al. [133] suggested applying correction factors Fk to the wall shearstress expressions (2.28), such that in the limiting cases of single fluid flows theyyield the correct values. The correction factors are written asτH =2√mSH + SiαwH ·FH ; FH = g11− 11+ XwLwHg12, (2.35)τL = 2√mSL + Si1−α wL ·FL; FL = g22−X1+ XwHwLg21, (2.36)whereg11 =SHSH + Si; g12 =4pi+2SHSH + SL, (2.37)g22 =SLSL + Si; g21 =4pi+2SLSH + SL, (2.38)are geometric functions andX =1m1−αα. (2.39)The functions g11 and g22 change the hydraulic diameters of the layers in differentlimiting cases. If m 1 or α→ 1 and wH wL , FH → 1 and FL→ g22. Thus thehydraulic diameter of the light fluid changes from (SL + Si)/(1−α) to SL/(1−α).Similarly, when m 1 or α→ 0 and wH wL , FL → 1 and FH → g11. Thefunctions g12 and g21 are defined to give correct values of shear stress in the caseof a single fluid laminar pipe flow where At = 0 and m = 1.The correction factor for the interfacial shear stress isFiH =11+ X; FiL =11+ 1X. (2.40)36and the integrated τi can be written asτi =1√m2(SH + Si)α(wH −wL)FiH if wH > wL,√m2(SL + Si)1−α (wH −wL)FiL if wH < wL .(2.41)The first case corresponds to flows where the heavy fluid dominates the stress andis dragged by the light fluid. The second case is where the light fluid dominates theinterfacial shear stress (small α or m).Note that there is a single interfacial shear stress τiH = τiL = τi when usingmodels 1 or 2, and the friction term T will beT = 12FrRe{1ατHSH +1α(1−α)τiSi −11−ατLSL}. (2.42)2.3.3 Model 3The third model is described in [83] and does not fit directly into the friction factorframework, although also based on area-averaged momentum balances. The modelis focused at estimating the viscous and interfacial stresses for a two-layer Herschel-Bulkley flow. The model proceeds by calculating exactly the analogous two-layerplane channel flow and then geometrically mapping to the pipe. Correction factorsare then introduced to ensure the accuracy of the model for single phase flows; see[83] for full details.2.3.4 Momentum equilibriumWe now compare the closure models through solution of the steady momentumequation in which axial buoyancy forces are balanced by friction forces (i.e. T =cos β). This equilibrium equation is:2ReFrcos β =1ατHSH +1α(1−α)τiSi −11−ατLSL . (2.43)This relationship has a number of different applications. First note that there is noz-dependency in (??). Thus, equilibrium solutions, say (wH,wL,α), are also steady370 0.2 0.4 0.6 0.8 100.20.40.60.810 0.2 0.4 0.6 0.8 100.20.40.60.811.21.41.6a bFigure 2.2: Comparison of the equilibrium fluxes and velocities forRe/Fr2 cos β = 43.58, m = 1 and Fr = 1. The curves represent Model 1(B), Model 2 (), Model 3 (#) and numerical solution of [83] (C).0 0.2 0.4 0.6 0.8 100.20.40.60.811.21.40 0.2 0.4 0.6 0.8 100.511.522.53a bFigure 2.3: Comparison of the equilibrium fluxes and velocities forRe/Fr2 cos β = 129.4, m = 1 and Fr = 1. Curves marked as in Fig. 2.2.z-independent solutions of (2.24), i.e. stratified flows. These base flows are alsorelevant for the study of gas/liquid flow in pipelines, where wH , wL and h are setindependently. In this case solving the equilibrium equation requires changing twoparameters and solving iteratively to find the third one. The equilibrium solutionsare of less relevance for gravity currents, where the flows are not steady.Our interest here is in displacement flows, where the sum of the volumetric380 0.2 0.4 0.6 0.8 100.511.50 0.2 0.4 0.6 0.8 100.511.522.533.5a bFigure 2.4: Comparison of the equilibrium fluxes and velocities forRe/Fr2 cos β = 129.4, m = 4 and Fr = 1. Curves marked as in Fig. 2.2.0 0.2 0.4 0.6 0.8 100.20.40.60.811.21.40 0.2 0.4 0.6 0.8 100.511.522.5a bFigure 2.5: Comparison of the equilibrium fluxes and velocities forRe/Fr2 cos β = 129.4, m = 0.25 and Fr = 1. Curves marked as inFig. 2.2.39fluxes is fixed. The geometric parameters α, SH , SL and Si are all functionsof h only. Thus, wL and the velocity difference ∆w can be found for a given hand wH . This equilibrium solution has direct relevance to thin film/lubricationmodels of displacement, in which it is assumed that the momentum equations haveevolved rapidly to this steady solution and the displacement is predicted by solvingthe kinematic equation (2.19) in time for the evolution of h. In these models thekey quantity determining the displacement is the flux qH through the lower fluidlayer, which we normalize with Fr . These models are used predictively in thelaminar/viscous range of displacement flows. Apart form using these equilibriumsolutions to compare frictional pressure models, we will study the linear stabilityof these solutions later in §2.4.1.A range of results are shown in Figs. 2.2-2.5. Model 2 shows excellent agree-ment with the numerical solution over the whole range of α and other parameters.Consequently, this is the closure model we adopt throughout the paper.Note that when the flux is scaled with Fr the results vary only with m andwith χ = 2 ReFr2cos β, which is the appropriate dimensionless number when axialbuoyancy is balanced by axial viscous forces, e.g. as is used in lubrication/thin filmmodels of displacement. Figure 2.6 shows the effects of changing χ and m on theflux.In the preceding Figs. 2.2-2.5, we also see that the flux qH is often non-monotoneas a function of α. In the displacement context, the interface often evolves into ashock, the position and speed of which can be calculated directly from qH and itsderivative using the equal areas rule. The result of doing this is the prediction ofthe displacement front velocity w f , i.e. that would result in a lubrication/thin-filmdisplacement. The front velocity so computed increases with m, implying a lessefficient displacement.400 0.2 0.4 0.6 0.8 100.20.40.60.811.21.41.61.8Figure 2.6: Comparison of the equilibrium fluxes (Model 2) for differentvalues of Re/Fr2 cos β and m. The curves represent Re/Fr2 cos β =(34.86,103.53,200) and m = 1 for (, #, ♦), respectively. (O, M) showfluxes for Re/Fr2 cos β = 103.53 with m > 1 and m < 1, respectively,with the solid lines corresponding to m = (10,0.1) and the broken linescorresponding to m = (4,0.25).2.4 Well-posedness and linear stabilityThe system (2.24) is strictly hyperbolic when the Jacobian A of the flux F has tworeal eigenvalues. The Jacobian of F is the matrixA =∂F∂V =((1−2α)wH +2αwL 2 1−ααh sin β−(wH −wL)2α wH)(2.44)and its eigenvalues areλ± = (1−α)wH +αwL ±√α(1−α)(2αhsin β−[wH −wL]2). (2.45)41The characteristics are real only when the term in the square root is positive. Thatis ifwH −wL ≤(2sin βαh) 12. (2.46)When the criterion (2.46) is not met, the system is no longer hyperbolic, theeigenvalues become complex and instabilities start to grow in the solution. Writingthe inequality (2.46) using dimensional parameters we getwˆH − wˆL ≤[pi2(ρˆH − ρˆLρˆH + ρˆL)gˆdˆ2Sˆisin β] 12, (2.47)which is the inviscid Kelvin-Helmholtz (Inviscid Kelvin-Helmholtz (ikh)) criterionfor fluids with different densities. This can also be derived by carrying out a linearstability analysis on (2.24), while ignoring the friction terms, or considering thefriction terms but in the short wavelength limit. According to (2.46), in the absenceof a density contrast the flow is always unstable unless the velocities are equal.With density contrast the transverse component of buoyancy (sin β) can stabilizethe flow. The term αh = Si in the denominator also implies that in the limits ofα→ 0,1, i.e., a thin (or thick) heavy layer, the eigenvalues remain real.This condition on hyperbolicity of the system is rather strict, specially in incli-nations away from horizontal where sin β decreases, or in inertial flows. In inertialflows the buoyancy is balanced primarily by inertia, resulting in relatively highvelocity difference. This ill-posedness of the model has been addressed by manyauthors in gas/liquid flows e.g. see [31]. The well-posedness condition (2.46) ispartly a consequence of the simplified two-layer model, i.e., it is a direct result ofassuming two distinct velocities at the interface, and does not necessarily predictthe onset of instabilities in the actual displacement flow. In the actual flow theshear layer thickness at the interface becomes immediately finite due to viscosity.Nonetheless, if the two-layer model is well-posed, then interfacial instabilities canbe resolved by adequate numerical accuracy. Thus, in gas/liquid pipeline flowsmany authors have focused only on the well-posed range of parameters (interfaceheight and layer velocities). When the model becomes ill-posed, short-wave stabi-lizing mechanisms, such as artificial viscosity or surface tension, must be utilized42to make numerical solution of the model possible. Therefore reliable predictionof instabilities in the actual displacement, requires great amount of care to avoidresolving instabilities that are unphysical (the result of simplified modelling) ornumerical. We will focus on the stability of the model and difficulties in numericalsolution of if in the next sections.2.4.1 Stability analysisWenow look at the stability of stratified displacement flows and the two-layermodel.It should be mentioned that our focus is on interfacial instabilities of the laminarflow and not the shear-mode instabilities that might eventually cause turbulent flowin each fluid layer. Stabilizing or destabilizing mechanisms at the interface aredefined in terms of the fluid velocities, interface height, density contrast, viscositycontrast and the inclination of the pipe. A linear stability analysis helps us to see thestabilizing or destabilizing effects of the frictional terms and also guides us towardseffective methods for treating the ill-posedness of the model.We first find a steady-state, base solution V¯ from the equilibrium equation of(??), i.e. as studied in §2.3.4. In place of α¯ we consider h¯ as the steady height, asthis simplifies the analysis below. The base solution is linearly perturbed:wH = wH +w′H ; h = h+ h′, (2.48)where |h′ |, |w′H | 1, meaning the perturbations are small compared to all otherquantities. Linearization of the momentum equation about (wH,h) results in∂w′H∂t+ (1−2α)wH∂w′H∂z−2(1−α)wL∂w′L∂z+[2(1−α)sin β−αh(w2H −w2L)] ∂h′∂z= −2(1−α)T ′. (2.49)Note that we include wL and w′L for convenience, although due to (2.20) they arenot independent variables. Here T ′ is the perturbed friction term:T ′ = ∂T∂hh′+∂T∂wHw′H . (2.50)with the derivatives of T evaluated at (wH,h). The velocity perturbations w′H and43w′L can be eliminated from (2.49) by first linearizing the continuity equations ineach layer∂h′∂t+wH∂h′∂z+ααh∂w′H∂z= 0, (2.51)∂h′∂t+wL∂h′∂z− 1−ααh∂w′L∂z= 0. (2.52)Differentiating (2.49) with respect to z and multiplying by −α/αh we get∂2h′∂t2+2 [wH −α∆w] ∂2h′∂t∂z+[w2H −α(w2H −w2L)−2(1−α)ααhsin β]∂2h′∂z2− (1−α)ααh∂T ′∂z= 0, (2.53)where the velocity perturbations have been eliminated from T ′. We now use amonochromaticwave to represent the perturbed interface height h′= h0 exp i(kz−ωt)in (2.53), with k and ω being the wavenumber and angular frequency. After divid-ing (2.53) by h0 exp i(kz−ωt) and some rearranging we get the following dispersionequationω2−2(ak − bi)ω+ (ck −2di)k = 0, (2.54)wherea = (1−α)wH +αwL; c = (1−α)w2H +αw2L −2(1−α)ααhsin β (2.55)b = (1−α) ∂T∂wH; d = −(1−α)ααh∂T∂h+ (1−α)wH ∂T∂wH.The derivatives can be either calculated numerically or obtained explicitly (forconstant friction factors). Physically, a is the reference velocity, Cd =√a2− c is thedynamic wave velocity, Cν = d/b is the wave velocity at the onset of instabilities,and vw =Cν−a is the continuity wave velocity (see [25, 62]). Writingω =ωR+iωI ,the steady state solution is unstable wheneverωI > 0, leading to exponential growthof the small perturbations, with ωI being the amplification factor. In the inviscid44limit where b = d = 0, the roots of the dispersion equation areω± = k(a±Cd). (2.56)Therefore, ωI is non-zero whenever C2d = a2 − c < 0, which is the same as (2.46).In this case, the root leading to growth of instabilities is ω = k(a+Cd) and theamplification factor is ωI = k√c− a2. It is obvious that as k→∞ the amplificationfactor becomes infinite, i.e. short wavelength instabilities (λ = 2pi/k → 0) growexponentially. The wave velocity C = ω/k isC = a±Cd (2.57)= (1−α)wH +αwL ±√α(1−α)(2αhsin β−[wH −wL]2). (2.58)which is equal to the characteristics given by (2.45) and is independent of wave-length. As long as the term in the square root is positive, the amplification factorωIis zero. Thus the criteria for well-posedness of the problem and ikh are the same.When the friction terms are kept (Viscous Kelvin-Helmholtz (vkh)) the roots areω± = ak − bi±√(a2− c)k2− b2 +2(d− ab)ki. (2.59)Different wavelengths travel at different speeds, leading to the dispersion of insta-bilities. Neutral stability is achieved when ωI = 0 and the maximum growth ratecan be found from ∂ωI/∂k = 0. Going back to (2.54) and writing ω = ωR we getω2R −2akωR + ck2 = 0, (2.60)bωR − dk = 0. (2.61)From the second equation ωR = kCν. Then the criterion for instability for theviscous case can be rewritten as0 < (Cv − a)2 +α(1−α)([wH −wL]2− 2αhsin β). (2.62)45orC2d < v2w, (2.63)The second bracket in (2.62) is the ikh criterion and the first bracket is the addi-tional effect of the shear stress terms, or the difference between the wave velocityobtained from vkh and ikh, at the inception of instability. The criterion (2.63) saysinstabilities grow when continuity waves (moving at speed vw) overtake dynamicwaves (moving at a±Cd). With the friction terms included, the problem can nowbe well-posed (C2d≥ 0) but viscous unstable according to (2.62). In this rangethe unsteady hyperbolic system can be solved numerically and instabilities can beresolved given sufficient numerical accuracy and a stable scheme.We vary Fr , Re, β and m over a wide range to find the neutral viscous andinviscid stability boundaries for different values of h. The results can be seen inFigs. 2.7 & 2.8 for m = 1. We observe that the well posed region decreases ash¯ increases, whereas the linearly stable regime actually increases in size. In theshallow layer limit h→ 0 the stable and the viscous unstable regions grow and theproblem becomes ill-posed only at high Fr . As h→ 1 it seems like transition fromstable to viscous unstable and to ill-posed depends primarily on Re.In Figure 2.9 we fix h¯ = 0.5 and study the effects of m (m = 0.25 and m = 4).Compared to the iso-viscous case (Fig. 2.7 (b)), decreasing (increasing) m has thesame qualitative effect as decreasing (increasing) h¯). The variations with m seemto affect primarily the regime or well posedness, rather than that of linear stability.2.4.2 Artificial diffusionThe two-layer model is based on a long wavelength assumption, enabling us toassume pressure equilibrium between the layers. Thus, short wavelength instabili-ties can not be reliably predicted by the model and should be filtered out from theresults. It has been shown that adding a diffusion term to the two-layer model cankeep it well-posed [59]. This changes the system to:∂∂tV+A ∂∂zV−E ∂2∂z2V = S(V), (2.64)460 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rRe/Frcosβ0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F r0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rb ca0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rRe/Frcosβ0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F r0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rb ca. . . .Re/Frcosβ0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F r0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rb caFigure 2.7: Map of stability regimes for steady displacement flow with m = 1and interface height: (a) h¯ = 0.3; (b) h¯ = 0.5; (c) h¯ = 0.7. The blueregion represents stable displacements (C2d≥ v2w), the green region iswell-posed but viscously unstable (0 ≤ C2d≤ v2w) and the red region isill-posed (C2d< 0).470 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rRe/Frcosβ0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F ra bFigure 2.8: Map of stability regimes for steady displacement flow with m = 1and interface height: (a) h¯ = 0.1; (b) h¯ = 0.9. Regimes identified as inFig. 2.7.0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F rRe/Frcosβ0 0.5 1 1.5 2 2.5 3 3.5 4050100150200250F ra bFigure 2.9: Effect of viscosity ratio on the linear stability map for h¯ = 0.5 with(a) m = 0.25, and (b) m = 4.where the diffusivity matrix E can be designed such that the system becomesconditionally or unconditionally well-posed. For now, we add two (arbitrary)viscosities η1 and η2 to the momentum and kinematic equations, respectively.These viscosities can represent both numerical and artificial viscosities, used toinvestigate the effect of diffusion on the stability of the model.We set the diffusivity matrix as E11 = η1, E22 = η2 and E12 = E21 = 0. Therefore,two third-order derivatives ∂3h′/∂t∂2z and ∂3h′/∂3z will appear in the linearized48momentum equation. These terms add complex, k-dependent terms to the dis-persion equation. Considering only the inviscid case for now, to see how the ikhconstraint changes, we find:ω2− [2ak − i(η1 +η2)k2]ω+ [c− i(η1 +η2)wH k −2iη2α∆wk −η1η2k2] k2 = 0,(2.65)where ∆w = wH −wL . Solving for the roots of this equation gives us:ω = ak − i2(η1 +η2)k2±√(a2− c)k2− 14(η2−η1)2k4 + ik3(η2−η1)α∆w. (2.66)Including friction and diffusion terms we getω = ak − bi− i2(η1 +η2)k2±√(a2− c)k2− b2 + b(η2−η1)k2− 14 (η2−η1)2k4 +2(d− ab)ik + ik3(η2−η1)α∆w.(2.67)Figure 2.10 shows how the amplification factor ωI changes with wavelengthλ in a stable, a viscously unstable and an ill-posed case. In the stable case, ωIfrom ikh analysis is zero for all wavelengths and for vkh analysis it is negative.In the viscous unstable case, ωI from the ikh analysis remains zero but ωI fromvkh analysis is positive (but finite) for small wavelengths. In the ill-posed (inviscidunstable) case ωI →∞ for both ikh and vkh analysis.The effect of adding a diffusive term to the system can be seen in Figs. 2.10and 2.11. Adding artificial diffusion to the momentum equation only results ina cut-off wavelength, but ωI reaches a maximum before dropping to −∞. Whenequal values of artificial diffusivity are added to both momentum and kinematicequations the amplification factor remains much smaller before reaching cut-offwavelength. As expected, using a smaller diffusivity results in a smaller cut-offwavelength. Generally, we want to filter out instabilities with wavelengths smallerthan the pipe diameter. Meaning we wish to have a cut-off frequency correspondingto wavelength of λc ≈ 1.4910 -2 10 -1 10 0 10 1 10 2-0.200.20.40.60.8110 -2 10 -1 10 0 10 1 10 2-0.200.20.40.60.8110 -2 10 -1 10 0 10 1 10 2-0.200.20.40.60.81(c)(b)(a)ω IFigure 2.10: Growth rate (amplification factor) of instabilities (ωI ) vs. wave-length (λ) for a stable case (a) Ret cos β = 130, Fr = 0.9, a viscousunstable case (b) Ret cos β = 250, Fr = 0.9 and an unstable (ill-posed)case (c) Ret cos β = 43.6, Fr = 2.0. In all cases h = 0.9 and m = 1 (seeFig. 2.8(b)). The lines denote ωI from ikh analysis (- -), vkh analysis(–), vkhwith η1 = 10−2 (M) andwith η1 = η2 = 10−2 (#). In all cases theimaginary part of the unstable frequency has been plotted. Note that asωI→−∞, the wave amplitude goes to zero exp i(−iωI )= exp(ωI )→ 0.10-3 10-2 10-1 100 101 10200.20.40.60.81Figure 2.11: Comparison of the amplification factor of the unstable case inFig. 2.10(c) with η1,2 = 10−2 (#), η1,2 = 10−3 () and η1,2 = 10−4 (M).The cut-off wavelength becomes smaller with η.502.5 ResultsWe present the results from numerical solution of the developed two-layer modelin this section. For each case the values of Re, Fr , β and m are set. The totallength of the domain is 70 ≤ L ≤ 160 and a mesh size of ∆x = 0.1 is used in allcases. The time-step ∆t is set such that CFLmax ≤ 0.1, so that typically ∆t ≤ 10−3.Artificial viscosity is also kept constant at η1 = η2 = 10−3 for consistency. For theinitial condition, it is assumed the fluids are separated over a length of Li = 10, sothat α is changed from 1 to 0 over Li. The total solution time tmax is varied for eachcase such that steady solutions are achieved. By steady solution we mean constantpropagation speeds. From the initial distribution of α, we solve the equilibriumequation to initialize wH . At the inlet, (wH,α)= (Fr,1) is fixed and at the outlet freeboundary condition is used. However the numerical solution is stopped before thelight/heavy front reaches the inlet/outlet. The numerical method used is detailed in§A.3. Convergence of the results were checked by changing mesh size and artificialdiffusivity values (see §A.3.3).2.5.1 Lock-exchange flow (Fr→ 0)In the limiting case of Fr→ 0 (no imposed flow) the problem in hand becomes thatof the lock-exchange flow. Although lock-exchange flows are not the focus of thisstudy and are studied extensively by other researchers, we briefly talk about themas a starting point and to compare two-layer model with the existing studies. In ournumerical solutions we have used a very small Fr = 10−4 rather than Fr = 0.An example viscous exchange flow is shown in Fig. 2.12. Starting from theinitial condition, two fronts (heavy and light) propagate in the opposite directions.The interface is stretched constantly with no instabilities appearing. Figure 2.13(a)shows the interface evolution of Fig. 2.12, and the effect of viscosity ratio m. Whenthe heavy fluid is more viscous (m < 1), the height of the heavy front becomes largerand it moves slower (smaller w f ) compared to the iso-viscous case. For mass tobe conserved, the light layer thus moves backwards at a higher speed (larger wb f ).Increasing m above 1 has the opposite effect, with the light fluid becoming thickerand moving at a lower speed.Figure 2.13(b) compares the results of two cases with similar numerical param-510 50 100 1500816243240485664720 50 100 15001020304050607000.20.40.60.81a bFigure 2.12: Time evolution of a stable lock-exchange flow solved using thetwo-layer model. The domain length is L = 160 and simulation time istmax = 80. The simulation parameters are ∆x = 0.1, ∆tmax = 10−3 andCFLmax = 0.1 and the flow parameters are Fr = 10−4, β = 80◦,m = 1.0and Ret cos β = 45. (a) Shows evolution of α from an initial conditionand (b) is the spatio-temporal diagram.20 40 60 80 100 120 14000.20.40.60.8120 40 60 80 100 120 14000.20.40.60.81a bFigure 2.13: (a) Lock-exchange flow with Ret cos β = 45 and m = 0.25 (M),m = 1 (#) and m = 4 (). Increasing the viscosity ratio results in athicker heavy fluid layer and smaller front velocity. The results areplotted at t = (16,32,48,64). (b) Comparison of two exchange flowswith Ret cos β = 45 but different inclinations: Ret = 260, β = 80◦ (#)and Ret = 90, β = 60◦ (M). The results are plotted at t = (8,24,40,56).52eters and Ret cos β = 45, however, at different inclinations. What is interesting isthat at the lower inclination a sharp shock develops at the front, in contrast to thesmooth front at the higher inclination. At a slightly higher Ret cos β the shape ofthe interface changes. Results for Ret cos β = 50 are plotted in Fig. 2.14(a). In theearly stages, a bump appears at the heavy front and grows to a steady height. Sincethe height and velocity of the front is steady, we can compare the front velocityfound from two-layer solution w f to the equilibrium velocity wH (α = αf ), whereαf is the steady area fraction of the bump and wH is the corresponding equilibriumvelocity (see Section 2.3.4). Figure 2.14 shows that the front velocity of the bumpis indeed equal to the equilibrium velocity.Note that Ret cos β ≈ 50 coincides with the transition point from viscous toinertial exchange flows found by Seon et al. [109]. They found that up to Ret cos β ≈50, the exchange flow is viscous and w f changes linearly with Ret cos β. ForRet cos β ≥ 50, the flow becomes inertial regime where w f plateaus at value of 0.7.Figure 2.15 shows results obtained by our model for Fr = 0 and Ret cos β ≤ 200.Although a clear transition in regime occurs at Ret cos β = 50, w f does not plateaucompletely, but rather changes slowlywith Ret . This disagreement can be explainedby the fact that the two-layer assumptions break at the front where secondary flowsare significant and the streamlines are not parallel.Note that the non-monotonicity of wH indicated in Fig. 2.14 is not necessarilyindicative of any non-uniqueness. In our equilibrium computations we specify thetotal flow rate and fraction of heavier fluid, which leads to computation of velocitiesin each layer and the pressure gradients. In different multi-phase flow applicationsit is common to specify individual flow rates of the two layers and calculate thelayer thicknesses and pressure drops, which can result in a non-unique solution; seee.g. [72, 87, 131]. Here no such non-uniqueness occurs.2.5.2 Displacement flow (Fr > 0)General behaviourNow we focus on displacement flows, i.e., Fr > 0. Displacement parameters areFr , Ret cos β, β and m. For now we focus on iso-viscous displacements where530 50 100 15000.20.40.60.810 0.25 0.5 0.75 100.10.20.30.40.50.60.70.8a bFigure 2.14: Lock-exchange flow with Ret cos β = 50 and m = 1. The frontvelocity w f is equal to the equilibrium velocity of the front heightwH (αf ).m = 1 and later discuss viscosity ratio effects. Additionally, by representing theresults in (Fr,Ret cos β) plane, we only work with two parameters and assume alleffects due to β are captured by using Ret cos β.Figure 2.16 shows the results for a stable displacement flow with Fr = 0.4and Ret cos β = 20. Similar to exchange flows, there are two fronts present inthis displacement, the leading front moving forward in the displacement direction(the heavy fluid) and the trailing front, moving backward here (the light fluid).The imposed flow in displacements flows breaks the symmetry of the exchangeflows, resulting in different velocities of leading and trailing fronts. When thetrailing front evolves with velocity of opposite direction to the mean flow, we termthe flow a backflow. The interface between the two fluids is constantly beingstretched throughout the displacement. From Fig. 2.16b we can see the differentslopes of the fronts in the initial stages of the displacement compared to latertimes. Both the leading and trailing fronts achieve a constant velocity eventually.Increasing Fr decreases backflow velocity and beyond a critical value preventsbackflow completely. This can be seen in Fig. 2.16-2.18 and Fig. 2.20(a). Thecritical Fr depends on both Ret cos β and m. An instantaneous displacement flow,where no backflow occurs, can be seen in Fig. 2.17. Another difference betweenthe displacement flows of Fig. 2.16 and 2.17 is formation of a sharp shock at the540 50 100 15000.20.40.60.81Figure 2.15: Front velocity of the heavy layer w f in lock-exchange flow. Thehorizontal and vertical broken lines show the plateau region of [109]where w f = 0.7, and the transition point Ret cos β = 50. The solid lineis shock velocity from lubrication model. The filled symbols denotestable exchange flows.leading front, evident from the color jump in Fig. 2.17 compared to the gradualchange in color in Fig. 2.16. At yet higher Fr a bump appears at the front. Thisbump develops in the early stages of displacement and reaches a steady heightand velocity, similar to the one discussed in the lock-exchange flows. The heightincreases with Fr and eventually reaches αf = 1.When Ret cos β increases, transverse buoyancy becomes weaker and less effi-cient in keeping the fluid layers separated and stable. Interfacial instabilities canbe seen in Fig. 2.19, between the light front and behind the bump. From linearstability analysis in Sec. 2.4.1, we saw that the stable region is mostly confinedto small Fr and Ret cos β values. In the full unsteady, non-linear numerical solu-tions also we observed that a large number of displacement flows become unstablevery quickly. The amplitude, wave length and speed of instabilities vary with550 20 40 600612182430364248546066720 20 40 6001020304050607000.20.40.60.81a bFigure 2.16: Time evolution of a displacement flow solved using the two-layer model. The displacement parameters are Re = 92,Fr = 0.4, β =85◦,m = 1.0 and Ret cos β = 20. (a) shows evolution of α from an initialcondition at δt = 6 intervals and (b) is the spatio-temporal diagram.0 20 40 600481216202428320 20 40 6005101520253000.20.40.60.81baFigure 2.17: An instantaneous displacement flow with a frontal shock. Thedisplacement parameters are Re = 230,Fr = 1.0, β = 85◦,m = 1.0 andRet cos β = 20.560 20 40 6002468101214161820222426280 20 40 60051015202500.20.40.60.81a bFigure 2.18: A stable, instantaneous displacement flow with a bump at thefront. The displacement parameters are Re= 320,Fr = 1.4, β= 85◦,m=1.0 and Ret cos β = 20.0 20 40 6004812162024283236400 20 40 600510152025303500.20.40.60.81a bFigure 2.19: An unstable displacement flow with a bump at the front. Thedisplacement parameters are Re = 230,Fr = 1.0, β = 75◦,m = 1.0 andRet cos β = 60.displacement parameters.Figure 2.20(a) compares interfaces of cases at Ret cos β = 40, with increasingFr , all at equal t. With increasing Fr , backflow becomes weaker, the leadingfront forms a shock and an inertial bump grows which increases in height. Alldisplacements remain stable at Ret cos β = 40. In Fig. 2.20(b), Fr is kept constantand Ret cos β is increased. With increasing Ret cos β, backflow becomes stronger,57-10 0 10 20 30 40 50 60 7000.10.20.30.40.50.60.70.80.91-10 0 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.91a bFigure 2.20: (a) Effect of increasing Fr at a constant Ret cos β. At a flowratea shock forms at the front and at higher Fr the inertial bump appears.The bump height αf grows with Fr until it reaches αf = 1. Also,backflow stops at critical Fr at each Ret cos β. The results shownhere are for Ret cos β = 40 with Fr = 0.1 (#), 0.4 (M), 0.8 () and 1.0(O). (b) Effect of Ret cos β at constant Fr . The plotted results arefor Fr = 0.1 and Ret cos β = 10 (#), 20 (M), 40 () and 60 (O). Withincreasing Ret cos β, backflow becomes stronger and at Ret cos β ≈ 50the flow becomes unstable. All results are plotted at t = 34. The fluidswere separated at x = 10 at t = 0.a shock is formed at the leading front and eventually the displacement becomesunstable.Front velocity & regime classificationOur main objectives with studying displacement flows are to i) see how frontvelocity and thus displacement efficiency changes with Fr and Ret cos β, ii) predicttransition to the inertial, unstable regime and iii) predict occurrence of backflow.Front velocities w f are plotted in Fig. 2.22(a) and 2.22(b), against Fr and Ret cos β,respectively. The first thing to notice is that w f always increases with Fr , i.e., wˆ fincreases with Wˆ0. Looking at Fig. 2.22(b) we see that w f plateaus at some valueof Ret cos β for each Fr . The plots in Fig. 2.22(b) in fact show similar behavior toFig. 2.15 for lock-exchange flows. The onset of the plateau region seems to shifttowards smaller Ret cos β when Fr is increased. We found that the plateau velocity580 20 40 60 8000.20.40.60.81Figure 2.21: A marginally stable displacement flow with Ret cos β = 50 andFr = 0.6. Small amplitude instabilities appear at long times t =O(100),compared to unstable displacements Ret cos β > 50 where instabilitiesgrow very quickly.wpfchanges linearly with Fr , meaning in the inertial regime w f = f (Fr).Figure 2.22(b) also shows that all displacements with Ret cos β > 50 are un-stable, and that backflow is prevented at higher Fr . We can see these better inFig. 2.23. At Ret cos β = 50, the flows are marginally stable, i.e., instabilities havesmall amplitudes and appear at a much longer time-scale compared to unstablecases. These flows can be compared to actual displacement flows where interfacialinstabilities appear in form of waves and are not strong enough to cause mixing.We now classify our results in the (Fr,Ret cos β)-plane. Figure 2.23 summarisesthe data from all of our inclined, iso-viscous numerical solutions. As a reminder, thequantity Recos β/Fr = Ret cos β is a Reynolds number based on the inertial velocityscale and is independent of imposed velocity Wˆ0. The effect of imposed velocityis captured in Fr = Wˆ0/Vˆt . In Fig. 2.23 we can see that all unstable displacementsoccur at Ret cos β > 50. Displacements at Ret cos β = 50 are marginally stable (seeFig. 2.21), and are stable up to Fr ≈ 0.5. The thick broken line in this figure590 0.2 0.4 0.6 0.8 1 1.2 1.400.20.40.60.811.21.41.60 20 40 60 80 10000.20.40.60.811.21.41.6a bFigure 2.22: (a) w f vs Fr for constant values of Ret cos β. The symbols (#,M,O,,/,I,.,♦) correspond to Ret cos β= (0,10,20,40,50,60,80,100), re-spectively. The broken line represents w f = Fr . (b) w f vs Ret cos βfor constant values of Fr . The symbols (#,M,O,,/,I,.,♦) correspondto Fr = (0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.4), respectively. The filled sym-bols denote stable displacements and the superposed circles denotepresence of backflow. The solid lines are front velocities found fromthe lubrication model.shows backflow prediction of [120], which corresponds to Ret cos β / 58Fr . Thiscriterion was developed using thin-film/lubrication model of the viscous regimedisplacements. As discussed in §2.3.4, in such models a single parameter, χ =2Recos β/Fr2 governs the dynamics at long times. The critical value χc = 116.32was found as the limit above which there is always a backflow, as used in [4]. Weused the same approach, but using our two-layer model and Model 2 for frictionterms (see Sec. 2.3.4) and found critical value of χc = 112.5 when m = 1. This isshown in Fig. 2.23 as the thick solid line.In [123], Taghavi et al. studied near horizontal displacement flows experimen-tally. Although the focus of our two-layer model is not necessarily limited tonear-horizontal displacements, we will compare our results to those of [123] sincethe range of the parameters they studied is comparable to ours. In their study, theyreported that when Fr ≤ 0.9, displacements are viscous given Ret cos β ≤ 50 andwhen Fr ≥ 0.9, viscous flows were observed when χ / χc. First, we could argue600 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6020406080100Figure 2.23: Classification of our results in thr (Fr,Ret cos β)-plane. Dis-placements with backflow are colored in red and the ones withoutbackflow are colored in blue. The superposed circle denotes unstabledisplacements. The thick broken line represents backflow predictionof [4] χ = χc = 116.32 and the thick solid line is backflow predictionusing our two-layer model χ = χc = 112.5. The horizontal line isRet cos β = 50.that a more intuitive transition point compared to Fr = 0.9 would be intersection ofRet cos β = 50 and χ = χc, which corresponds to Fr = 0.86 using their model, andFr = 0.89 using ours. Secondly, from our results we see that beyond Fr = 0.89,displacements remain unstable for Ret cos β ≥ 50. However, clearly from Fig. 2.23the lubrication backflow prediction breaks at this point. Note that compared to theirresults we have not distinguished between temporary backflow and instantaneousdisplacements. In our classification, we have marked all displacements wherebackflow is zero, or stops at some time after an initial backflow, as displacementswithout backflow (blue circles in Fig. 2.23). For viscous displacements with tem-porary backflow we increased the length and time of numerical simulations so that610 50 100 1500501001502000 50 100 150 2000.640.650.660.670.680.690.70.710.720.73-0.0200.020.040.060.080.1a bFigure 2.24: Aviscous displacement with temporary backflow. Displacementparameters are Ret cos β = 20 and Fr = 0.4. Leading (w f ) and trailing(wb f ) front velocities are plotted in (b). The backflow velocity becomesnegative at long times, meaning the backflow stops and the trailing frontstart moving in the displacement direction.the front and backflow velocities reach steady values. As it can be seen in Fig. 2.24,at very long times the backflow velocity becomes zero and eventually negative,meaning the trailing front initially moves backwards but moves in the displacementdirection at longer times.In our regime classification, to follow that of [4], we have named all displace-ments with instabilities as inertial. In [4], no comments were made on the strengthof instabilities in inertial flows, e.g., whether they are finite amplitude waves at theinterface or result in partial mixing of the fluids. Similarly, in our numerical solu-tions we did not distinguish between different unstable cases, except for Ret cos βwhere instabilities grow at long times. However, it is noteworthy to comment on theobserved instabilities. In Fig. 2.25, the results of four different unstable displace-ments are plotted. It is evident that the wave length and amplitude of instabilitiesare different, specially comparing Fig. 2.25(d) to the first three. Although the two-layer model cannot account for mixing due to its inherent limitations, we can arguethat the large amplitude instabilities similar to the ones in 2.25(d) correspond toactual displacements where mixing occurs at the interface, resulting in a three layerdisplacement. The small amplitude instabilities of the first three figures, most likely620 10 20 30 40 50 60 700481216202428323640440 10 20 30 40 50 60 700 4 8 1216202428320 10 20 30 40 50 60 700 4 8 1216202428320 10 20 30 40 50 60 700 4 8 121620242832a bc dFigure 2.25: Comparison of some unstable cases with varying degree of in-stability. In both top plots Ret cos β = 50 at (a) Fr = 0.6 and (b)Fr = 1.0. In other two plots Fr = 1.0 and (c) Ret cos β = 60 and (d)Ret cos β = 100. In (a) instabilities do not appear until t ≈ 32.appear as dispersive waves at the interface, with trivial mixing. However we shouldsay that exact prediction of these instabilities requires a more thorough analysis ofthe cut-off wavelength due to artificial diffusion introduced in the numerical model(see Sec. 2.4.2).Viscosity ratioNow we investigate viscosity ratio effects on displacement efficiency and stabil-ity. Generally, viscosifying the heavy fluid (m < 1) results in more efficient andstable displacement, whereas viscosifying the light fluid destabilizes the flow anddecreases efficiency. An example displacement with varying viscosity ratio canbe seen in Fig. 2.26. Decreasing the viscosity ratio progressively increases the630 10 20 30 40 50 60 7000.20.40.60.81Figure 2.26: Effect of viscosity ratio on a displacement flow with Fr = 1.0and Ret cos β = 40. The curves represent m = 1 (#), 4 (M), 0.5 (),0.25 (O) and 0.1 (♦). As the heavy fluid becomes more viscous (m < 1),displacement becomes more stable and efficient. All plots are at t = 36and the fluids are initially separated at x = 10.displacement efficiency by decreasing the front velocity, suppressing instabilitiesand decreasing the interface length.Front velocities for Fr = 0.4 and 1 are plotted in Fig. 2.27(a) and 2.27(b),respectively, for different values of m. When m 1, front velocities decreasesignificantly and seem to plateau at higher Ret cos β. We used back-flow predictionfrom lubrication model (χc) for the viscous regime with m = 1. However, wecan also find χc for different viscosity ratios. These are plotted in Fig. 2.28(a)as lines with different slopes. The classification in this figure is for iso-viscousdisplacements and the thick solid line is χc(m = 1). With increasing m, χc andhence the slope of this line decreases. The two lines below the iso-viscous lineare for m = 2 and m = 4. The three lines above it are for m = 0.5, m = 0.25 andm = 0.1 respectively. In Fig. 2.28(b), we can see variation of χc with m. Aswith the iso-viscous case, we expect these values to correctly predict backflow inthe viscous range. To verify this and find critical Ret cos β where displacements640 20 40 60 800.40.50.60.70.80.911.10 20 40 60 8011.11.21.31.41.5a bFigure 2.27: Plots of w f vs. Ret cos β for (a) Fr = 0.4 and (b) Fr = 1.0. Thesymbols represent m = 0.1 (#), 0.25 (), 0.5 (M), 1 (O), 2 (I) and 4(♦). The filled symbols denote stable displacements and the superposedcircles denote presence of backflow.0 0.5 1 1.50204060801000 1 2 3 4406080100120140160180a bFigure 2.28: Backflow prediction at different viscosity ratios. The solid linein (a) shows prediction for m = 1. The thick broken and dash-dot linesbelow it show predictions for m = 2 and m = 4, respectively. The thinlines above it show predictions for m = 0.5, m = 0.25 and m = 0.1,respectively. Variation of critical values χc corresponding to theselines, with m are plotted in (b).6510 20 30 40 500.20.40.60.811.21.41.61.80 0.5 1 1.50102030405060a bFigure 2.29: Front velocity and regime classification for displacement flowswith m = 4. The symbols (#,M,O,,/,I,.) in (a) denote Fr =(0.2,0.4,0.6,0.8,1.0,1.2,1.4). The solid lines are front velocities foundfrom the lubrication model. The thick line in (b) represents backflowprediction χ = χc = 63.5 from lubrication model. Same as Fig. 2.23,displacements with backflow are colored in red and the ones withoutbackflow are colored in blue. The superposed circle denotes unstabledisplacements.become unstable, we completed a number of simulations usingm = 4 andm = 0.25.Front velocity variation and regime classification for m = 4 and m = 0.25 canbe seen in Fig. 2.29 and 2.30, respectively. When stable, it appears that the frontvelocity predictions are in line with the results predicted by the lubrication model.At m = 4, the front velocities are higher than those of m = 1 (and lower m), theyplateau at lower Ret cos β for each Fr . The back-flow prediction also works wellup to Ret cos β ≈ 40. The significant difference to iso-viscous displacements is theonset of instabilities. The iso-viscous cases became unstable at Ret cos β ≈ 50 forall Fr . For m = 4 however, the critical Ret cos β where the flow becomes unstabledrops with Fr . This may be partly due to the higher front velocities compared toiso-viscous displacements.Results for m = 0.25 can be seen in Fig. 2.30. Compared to iso-viscous dis-placements, front velocities are smaller and plateau at higher Ret cos β. Againstable flows are well represented by the lubrication model predictions. The back-6620 40 60 800.20.40.60.811.21.40 0.5 10102030405060708090a bFigure 2.30: Front velocity and regime classification for displacement flowswith m = 0.25. The symbols (#,M,O,,/,I) in (a) denote Fr =(0.2,0.4,0.6,0.8,1.0,1.2). The solid lines are front velocities foundfrom the lubrication model. The thick line in (b) represents backflowprediction χ = χc = 155.7 from lubrication model. Same as Fig. 2.23and 2.29, displacements with backflow are colored in red and the oneswithout backflow are colored in blue. The superposed circle denotesunstable displacements.flow prediction now works well up to Ret cos β ≈ 80 and instabilities appear athigher Ret cos β. Finally, the plateau velocities for m = 0.25, m = 1 and m = 4 arecompared in Fig. 2.31. A linear curve-fit for each plot givesm = 0.25 : wpf≈ 0.53Fr +0.68, (2.68)m = 1 : wpf≈ 0.55Fr +0.73, (2.69)m = 4 : wpf≈ 0.69Fr +0.71. (2.70)2.6 Summary & discussionWe have proposed a simple model for inertial density-unstable displacement flows,where a two-layer flow develops. In our model we retain the inertial and buoy-ancy terms of the momentum equations and use laminar closures for friction terms.670 0.2 0.4 0.6 0.8 1 1.2 1.40.70.80.911.11.21.31.41.51.6Figure 2.31: Variation of plateau velocity wpfwith Fr form = 0.25 (M), m = 1(#) and m = 4 ().The unsteady, non-linear equations were solved numerically for a large number ofparameter regimes, to provide a numerical experiment to compare with analysisand existing experimental results. The underlying model becomes linearly unsta-ble for sufficiently large Ret cos β. However, the resulting short wavelength ikhinstabilities, which are inherent with assuming discontinuous layer velocities, arenon-physical and suppressed by using artificial diffusion. This allowed us to solvenominally unstable displacements at larger Ret cos β, as needed to explore relevantparameter regimes.The successful aspects of the model are as follows. First, the stabilized modelpredicts qualitative features of observed wave-like instabilities, which grow andsaturate nonlinearly. The formation of an inertial bump/tip at the propagating frontis also observed experimentally, e.g. [119]. Secondly, for smaller values of Frthe model predicts loss of stability above a critical Ret cos β, which is found to beapproximately 50 for m = 1. This value is as found experimentally. The modelalso predicts variations in this critical transition as m is varied: large m reduces68stability, smaller m increases stability. These predicted regime transitions may betested experimentally. In the stable regime the prediction of front velocities by thelubrication model is as expected. However, the plateau-like front velocities attainedin unstable regimes were not predicted and again provide testable information forexperiments.Less positively, the variation in back-flow prediction χc(m) does not appearto provide a good prediction of the destabilization of the viscous flow for Fr '1, (even for m = 1). Part of the reason is that the developed two-layer modelmay still be improved physically (e.g. the inertial terms are approximated with nocorrection factors, the form of artificial diffusion might be changed, etc) and suchimprovements will affect the flows. The other aspect is that this backflow predictionis simply based on a viscous lubrication flow, whereas the two layer model leadsto a stability calculation. The results will be compared against experiments in thenext chapter.69Chapter 3Two-layer density-unstabledisplacement flow experimentswith viscosity ratioIn this chapter1, we study characteristics of miscible density-unstable displacementflows in inclined pipes. By density-unstable we mean the heavier fluid displaces theless dense fluid downward in the pipe. As discussed in §1.2.4, depending on the flowparameters, different flow regimes develop in density-unstable displacements. Themain focus of this chapter is to see the effect of viscosity ratio on the overall regimeclassification of [4] and revisit the inertial two-layer displacement flows that are notwell studied. Particularly the transition from stable to unstable regimes and onsetof instabilities have been paid attention to. We present the experimental results andcompare them to the predictions of the two-layer model (Two-Layer Model (2lm))developed in Chapter 2, as well as previous iso-viscous studies ([4, 123]).3.1 Problem setupIn density-unstable displacement flows there is a constant influx of the heavy fluidin an inclined pipe initially filled with a less dense fluid. The resulting flow regime1A version of this chapter has been published as: A. Etrati, K. Alba and I.A. Frigaard. Two-layerdisplacement flow of miscible fluids with viscosity ratio: Experiments. Phys. Fluids, 30(5) (2018):052103.70depends on the density difference, flow rate, inclination angle and rheologicalcharacteristics of the fluids. For these flows and in the high Péclet number andBoussinesq limit, the problem is characterised by 4 dimensionless parameters:Re =ˆ¯ρVˆ0dˆˆ¯µ, Fr =Vˆ0√Atgˆdˆ, β, m =µˆLµˆH, (3.1)where Vˆ0 is the imposed mean velocity, ˆ¯ρ = (ρˆH + ρˆL)/2 is the mean density,ˆ¯µ = (µˆH µˆL)1/2 is the mean viscosity, dˆ is the pipe diameter, gˆ the gravitationalacceleration, β the pipe inclination from vertical and m the viscosity ratio. Sub-scripts H and L denoting heavy and light fluids. The Boussinesq flow assumptionis valid when the density difference is small, which is the case in flows studied here.Strong buoyancy effects are still possible via control of the modified densimetricFroude number Fr , through the flow rate. The Froude number defined in (3.1)is the ratio of the mean imposed velocity Vˆ0 to a buoyant-inertial velocity scaleVˆt =√Atgˆdˆ, which is the inertial velocity scale arising from balancing buoyancystresses with inertial stress; see [109]. The ratio of buoyancy to viscous forces inthe axial direction can be written as a Reynolds number based on Vˆt times cos βReFrcos β =ˆ¯ρVˆ0dˆˆ¯µVˆtVˆ0cos β = Ret cos β. (3.2)To simplify the analysis, we work with Fr , Ret cos β andm only. The strength of theimposed flow relative to the characteristic buoyant velocity scale is controlled viaFr only. The buoyant Reynolds Ret cos β is independent of imposed velocity andhere is controled via β and ˆ¯µ. Note that in our experiments, the density differenceAt and pipe diameter are fixed, the variable parameters are the flow rate Vˆ0, theinclination angle (β) and the viscosity of the fluids (κˆH ,L , nH ,L). The full range ofthe parameters of our experiments are given in Table 3.1.3.2 Experimental procedureOur experiments are carried out in a transparent acrylic pipe with an internaldiameter of dˆ = 19.05mm and length of Lˆ = 4m. The pipe is made of two separate71Vˆ0xˆyˆGate valvegˆ⇢ˆL⇢ˆHC = 0C = 1Figure 3.1: A simple schematic of the experimental set-up. At tˆ = 0 the gatevalve is closed, separating the two fluids.pieces, which are put inside fish-tanks and are carefully connected together to avoiddisturbance of the fluid flow. The acrylic fish-tanks are filled with a glycerol-watersolution to correct for the light refraction due to the acrylic walls. Initially, the pipeis filled with the displaced fluid and a gate valve separates it from a shorter pipe,which is initially filled with the displacing fluid. The pipes are fitted in a frame thatprovides vertical and horizontal supports for the fish-tanks at several points alongtheir length and can be tilted to any angle between horizontal (β = 90◦) and vertical(β = 0◦). Additionally, there are internal supports inside each fish-tank to fix theacrylic pipe and prevent it from bending. All supports are designed to allow forvisual access to the flow from both front and top.The displacing fluid is fed to the pipe from a pressurized tank (∼ 10psi) whichallows feeding the pipe at all angles without the disturbances caused by pumpingthe fluids directly to the pipe. The tank pressure is held constant and the flowrate for each experiment is controlled by setting a needle-valve at the outlet. Tovisualize the flow, black ink is added to the displaced fluid. The pipe is backlit usingLight-Emitting Diode (LED) stripes and light diffusers are used tomake the lightinguniform. Two high-speed, black and white cameras (one for each fish-tank) are usedto take images of the flow throughout the experiments. First-surface mirrors areused to provide a top view of the pipe as well as a side view. We use a rotameterand a needle valve to measure and control the mean flow rate in each experiment.Image processing is conducted using custom scripts written in MATLAB, where72Parameter Rangeβ (◦) 10,30,45,60,75,85Vˆ0 (mms−1) 10−110[κˆ (mPasn),n] [1,1],[3,0.85],[10,0.7]At 3.5×10−3Fr 0.38−4.76Re 11−2153Re/Fr cos β 3.34−480m 0.1−11Table 3.1: Full range of physical and dimensionless parameters used in in-clined density-stable experiments.the light intensity at each pixel is translated to a depth-averaged concentration valueC, between 0 and 1, using the well-known Beer-Lambert law. The experimentalset-up and measurement techniques are described in more detail in §A.1.The density difference (At) is fixed in all of our experiments and the variableparameters are the flow rate (Vˆ0), the inclination angle (β) and the viscosity of thefluids (κˆH ,L , nH ,L). The full range of the parameters of our experiments are givenin Table 3.1.We have used xanthan gum to achieve a viscosity contrast in our fluids. Therheological behaviour of xanthan gum solutions is measured with a Malver BohlinGemini HR Nano rheometer. A smooth cone-and-plate geometry with 40mmdiameter, 4◦ cone angle and 30 µm gap at the cone tip, is used for all rheometryand the temperature is controlled by a Peltier system. A shear rate ramp is applied,varying over the range of 0.001− 100s−1, with 100 data points and 400s sweeptime. The xanthan gum solutions are modeled as power law fluids, with the effectiveviscosity written as µˆ = κˆγn−1, where κˆ is the consistency and the power-law indexis n < 1. We use the strain rate range 1−100s−1 to fit the fluid consistency, κˆ, andpower-law index, n, from a log-log plot of the effective viscosity versus strain rate.This range of shear rate covers that found in our experiments.In our experiments we consistently prepare and use two xanthan gum solutionswith concentrations 100 and 300mgL−1. The shear-thinning effects in the lowconcentration solutions are minimal (κˆ ≈ 3mPasn and n ≈ 0.85) and the higher73concentration is also only weakly shear-thinning (κˆ ≈ 10mPasn and n ≈ 0.7). Othersolutions such as Glycerol were tested to obtain constant, moderate viscosity ratiosof 2−4. However it was found that keeping the density difference small (∆ρˆ/ρˆ <0.01) while achieving significant viscosity ratio was problematic. For Newtonianfluids m = µˆL/µˆH is defined as the viscosity ratio. For shear-thinning fluids we usea characteristic effective viscosity defined asµˆe,k = κˆk(3nk +14nk)nk (8Vˆ0dˆ)nk−1, (3.3)for each fluid and use m = µˆe,L/µˆe,H as the viscosity ratio. The mean effectiveviscosity used for Reynolds numbers (3.1 and 3.2) is defined as ˆ¯µ = (µˆe,L µˆe,H )1/2.With these definitions and the range of flow rates given in Table 3.1, in Newtoniandisplacements the effective viscosity of the more viscous fluid is typically µˆe ≈2.7mPas at Fr = 0.4 and µˆe ≈ 1.9mPas at Fr = 4.3. For the more shear-thinningexperiments, the effective viscosity is about µˆe ≈ 8.5mPas at the lowest Fr andµˆe ≈ 4.2mPas at the highest.3.3 ResultsFirst, we discuss the main features of density-unstable displacement flows withviscosity ratio in §3.3.1. Then we distinguish between stable and different unstabletwo-layer displacement flows in §3.3.2. In §3.3.3, we discuss various frontal regiondynamics observed in the experiments. Measurements of the front velocity arepresented in §3.3.4.3.3.1 Main featuresBefore discussing in detail the stability and efficiency of displacement flows, webriefly describe the main observations in presence of viscosity ratio. Figure 3.2shows spatio-temporal diagrams for a series of experiments at β = 75◦ and Fr ≈ 1.5,but different viscosity ratios. The x-axis shows dimensionless distance from thegate valve (x = xˆ/dˆ) and the y-axis is dimensionless time (t = tˆVˆ0/dˆ). The col-ormap represents mean concentration C¯(x,t). The displacement flow of Fig. 3.2ais iso-viscous Newtonian, whereas the fluids in the flow of Fig. 3.2b are shear-7410 20 30 40 50 6015 30 45 60 75 90 10512000.20.40.60.8110 20 30 40 50 601020304050607080 00.20.40.60.8110 20 30 40 50 6010203040506070809000.20.40.60.8110 20 30 40 50 6010 30 50 70 90 11013015000.20.40.60.8110 20 30 40 50 601020304050607080 00.20.40.60.8110 20 30 40 50 601020304050607000.20.40.60.81(a) (b)(c) (d)(e) (f)Figure 3.2: Spatio-temporal diagrams of experiments at β = 75◦ and Fr ≈ 1.5.The viscosity ratios are (a) m = 1, (b) m = 0.94 with µˆeH = 5.9mPasnand µˆeL = 5.56mPasn, (c) m = 2.062, (d) m = 0.48, (e) m = 5.52 and (f)m = 0.14.75thinning with the same xanthan gum concentrations (m = 0.94). Comparing thetwo experiments, it is evident that viscosifying both fluids (at the given flow param-eters), suppresses instabilities and prevents back-flow, resulting in a more efficientdisplacement.The spatio-temporal diagram of Fig. 3.2a shows growth of strong instabilitiesand mixing of the fluids at t ' 90. back-flow and instabilities are visible in Fig. 3.2c& e, where the displaced fluid is more viscous (m > 1). However the instabilities arenot as strong as the iso-viscous case (Fig. 3.2a). Instead, they are interfacial waveswith finite amplitude. When the displacing fluid is viscosified, the flow becomesmore stable (Fig. 3.2d & f). The most efficient case is the displacement of Fig.3.2f, where m = 0.14.It should be noted that when viscosifying one or both fluids, the bulk viscosityof the flow ˆ¯µ= (µˆH µˆL)1/2 increases and consequently Ret cos β decreases, meaningaxial viscous forces become more dominant compared to axial buoyancy, thereforemixing becomes suppressed. However, other instabilities may arise at the interfacedue to the viscosity contrast. To isolate the effect ofm on stability and displacementefficiency, both Fr and Ret cos β must be fixed. With 2lm it was predicted thatincreasing m at constant Fr and Ret cos β decreases displacement efficiency andmakes the flow more unstable. We explore this and other predictions of 2lm in thefollowing sections.3.3.2 Stability & regime classificationIn this section, we intend to describe the transition criterion between stable andunstable flows. Our focus for now, will be mostly on the onset of instabilities,the transition from stable to unstable displacements and finally classification ofdifferent stratified flow regimes. By stratified flows we mean displacement flowswhere the fluids form two (or three) distinct layers along the pipe, in contrast tofully-diffusive flows where the fluids completely mixed across the pipe. The fluidlayers can have completely stable interfaces, or have instabilities in form of wavesor local mixing, all of which were classified as non-diffusive in [4].Interfacial instability in two-layer displacement flows can develop due to ve-locity difference between the fluid layers or a jump in viscosity. Note that in the761 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 (b)(a)(c) (d)Figure 3.3: Snapshots of different two-layer displacement flow regimes: (a) Astable displacement flow with β = 60◦, Fr = 1.54 and Ret cos β = 45.33.Both fluids are shear-thinning (κˆH ,L = 9.7mPasn, nH ,L = 0.76). (b)A wavy displacement flow with β = 60◦, Fr = 0.85, Ret cos β = 84.14and m = 0.14. The heavy fluid is shear-thinning (κˆH = 11.8 mPasn,nH = 0.74). (c) A cyclic displacement flow with β = 60◦, Fr = 0.79and Ret cos β = 245. (d) An inertial displacement flow with β = 75◦,Fr = 3.78, Ret cos β = 57.52 and m = 0.20. The light fluid is shear-thinning (κˆL = 11.3mPasn, nL = 0.72). The field of view in all snapshotsis 1540×20mm2 located 1670mm downstream of the gate valve.77(a)(b)(c)(d)Figure 3.4: Snapshots of some unstable displacement flows with waves atthe interface. Displacement parameters are (a) β = 75◦, Fr = 0.78,Ret cos β = 44.5 andm = 0.12, (b) β = 75◦, Fr = 0.38, Ret cos β = 39.63and m = 0.097, (c) β = 75◦, Fr = 0.39, Ret cos β = 78.69 and m = 0.39and (d) β = 60◦, Fr = 0.79, Ret cos β = 91.44 and m = 7.21.flow regimes that we are interested in, other mechanisms such as double-diffusion,surface tension or turbulence are not relevant. Without a viscosity contrast, thedestabilizing mechanism is the difference in mean velocity of the fluid layers thatcauses Kelvin-Helmholtz (kh) type of instabilities, whereas transverse buoyancytends to stabilize the flow. As noted in [4, 119], the imposed flow rate can alsostabilize the flow in some range by preventing back-flow and thus suppressing khinstabilities. In the presence of a viscosity contrast, stability of the flow is directlyaffected by a viscosity jump at the interface, and also indirectly due to changesin the mean velocity and thickness of the fluid layers. The indirect changes wereconfirmed with 2lm where it was shown that displacements with viscosity ratioof greater than 1, i.e., less viscous fluid displacing the more viscous fluid, areprogressively unstable.We have identified fourmajor type of stratified displacements in our experimentsregarding interfacial stability (see Fig. 3.3): (a) stable, (b) wavy, (c) cyclic and (d)inertial. Snapshots of some displacements in these flow regimes are shown in Fig.7890 100 110 120 130 140 15055 65 75 85 95 105115125 00.10.20.30.40.50.60.70.80.9190 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.91(a) (b)Figure 3.5: (a) Spatio-temporal plot and (b) mean concentration evolution forthe same experiment of Fig. 3.3c. The horizontal lines in (a) and arrowsin (b) denote times t = 99 and 112, which correspond to the last twosnapshots in Fig. 3.3c. The peak in C¯ and white spot region in (a)show the temporary mixing of the fluids across the pipe. This event isfollowed by a fall in C¯ or the dark region in (a), where the front is cutoffand the streams separate. This is a characterstic of the flows here namedas intermittent.3.3a-d. The transition from one regime to another is not always very clear since theclassification is done visually and is limited by the time-scale of the experiments.In some case the instabilities grow at very long times and at high flow rates the longtime behaviour of the flow cannot be studied since the flow is limited to the lengthof the pipe.In stable displacements (Fig. 3.3a) the interface remains sharp with no visibleinstabilities. In these flows, transverse buoyancy is strong enough to stabilize theinterface. However, even here instabilities might grow at a very long time, e.g.if the flow parameters are very close to the onset criterion, as was observed in2lm. If the interface becomes very long such that Lˆ/dˆ 1, thin-film/lubrication ortwo-layer models can be applied to predict back-flow, front velocity and the shapeof the interface.In the wavy unstable regime (Fig. 3.3b), waves with different amplitudes,wavelengths and speeds appear at the interface. Some example displacementsin this regime can also be seen in Fig. 3.4. The instabilities change from single-79wavelength waves moving at the same speed (Fig. 3.4a) to dispersive waves movingin both directions (Fig. 3.4b & c). From top view of the pipe, we can see thatthe waves in this regime remain left-right symmetrical (Fig. 3.4d). As the flowbecomes more unstables some waves break and form a three-layer structure behindthe front. However the instabilities are not strong enough to cause complete mixingacross the pipe and cut the heavy layer.As we enter the cyclic regime the transverse mixing becomes stronger andthree-dimensional (Fig. 3.3d). In some case the two layers completely mix and thefront is cut off from the displacing stream and a secondary front appears behind it.Upstream of this mixed region the flow remains stratified and retains its two-layerformation. This cyclic growth of instabilities andmixing eventsmakes front velocitymeasurements very challenging. The spatio-temporal and mean concentration C¯evolution of the experiment of Fig. 3.3c is shown in Fig. 3.5. The mixing eventfollowed by the cutting of the heavy layer, are seen as a bright and dark regionsin the spatio-temporal diagram and the spike and dip in the mean concentration att = 99 and t = 112, respectively.The fourth regime, termed inertial here, is observed mostly at higher flow rates(Fig. 3.3d). The imposed flow seems to stabilize the flow to some extent bypreventing back-flow. Therefore the strong mixing that arises due to strong counterflow of the layers is prevented. However local mixing and dispersion effects arepresent at the interface and at the front.A stability map of all displacement flow experiments is presented in the(Fr,Ret cos β)-plane in Fig. 3.6a. A few of our inclined experiments were fullydiffusive and are colored black in the map. The broken line in Fig. 3.6a is the pre-dicted boundary between non-diffusive and diffusive flow regimes for iso-viscousdisplacements from [4]. Due to small number of experiments in the diffusive regimewhich were also limited to iso-viscous displacements, we cannot make predictionsfor this criterion for different viscosity ratios. The rest of the points are two-layerdisplacements and are classified with different colors.Previously the transition from stable to unstable regime was predicted using theback-flow criterion calculated from a viscous lubrication model. For iso-viscousNewtonian displacements, back-flow occurs when χ > χc = 112.5 (see [? ]). Thesolid line in Fig. 3.6a corresponds to χ = χc, meaning that two-layer models80Figure 3.6: Interfacial stability map of two-layer displacements for (a) alldisplacements, (b) iso-viscous displacements m = 1, (c) displacementwith m > 1 and (d) displacement with m < 1. Circle and square symbolsdenote Newtonian and shear-thinning flows. Triangles in (b) are datapoints taken from [4]. The colors represent stable (l), wavy (l), cyclic(l) and inertial (l). The diffusive/mixed experiments are presentedby (l). The superposed circles denote displacements with sustainedback-flow. In (a) the solid line is back-flow prediction from 2lm and thebroken line is Ret cos β = 500−50Fr . In (b) the solid line is back-flowprediction for m = 1. In (c) the solid and dashed lines show back-flowpredictions for m = 2 and 4, respectively and in (d) the solid, dashed,and dot-dashed lines are back-flow predictions for m = 0.5, 0.25 and0.1, respectively. The thick broken lines in (b-d) are instability onsetpredictions from 2lm for m = 1, m = 4 and m = 0.25.81predict back-flow above this line. As we can see from the stability map, similarto the prediction from 2lm, onset of instabilities does not coincide with back-flowprediction. Instead it appears that wavy instabilities occur below this line, foundonly above a critical Ret cos β, at large enough Fr . At a higher Ret cos β the fluidlayers start mixing, resulting in cyclic and inertial regimes. Transition betweencyclic and inertial regimes is Fr dependent.The superposed circles in Fig. 3.6 denote displacements with sustained back-flow, meaning the light layer moves near the top of the pipe in the opposite directionof the displacement. This, however, does not mean that in flows without a back-flowthe light layer velocity is positive everywhere along the pipe. In other words, therecan be counter-current flow at a point along the interface. However in flows with noback-flow, the velocity of the light layer becomes non-negative as C¯→ 1. Althoughdisplacements with back-flow seem to be limited to Fr < 1 in Fig. 3.6, cyclic flowsthat are characterised by strong counter-current flow that lead to sudden mixing ofthe layers are observed up to Fr ≈ 2.6.In Fig. 3.6b-d respectively, separate regime maps are presented for iso-viscousdisplacements (m ≈ 1), unstable viscosity ratios (m > 1) where µˆH < µˆL , and stableviscosity ratios (m < 1) where µˆH > µˆL . First we compare our results to thoseof the existing iso-viscous displacements [4, 123] and 2lm. To this end we alsohave re-examined the iso-viscous experiments from [4] and classified in terms ofthe observed flow types here. These are included in Fig. 3.6b (triangles) withthe newer data. A first observation is that the classifications of the 2 data setsare the same, demonstrating experimental consistency, although the flow loop wascompletely rebuilt between the experimental studies. Also we do not observe anysignificant difference between the Newtonian m = 1 studies and those with mildlynon-Newtonian rheology (correctly m ≈ 1).The solid and broken lines in Fig. 3.6b are back-flow and instability onset predic-tions from 2lm, which are χ = χc(m = 1) = 112.5 and Ret cos β = 50, respectively.All flows with sustained back-flow are confined to χ > χc and Fr < 1. Note thatthe lines χ = 112.5 and Ret cos β = 50 cross at Fr = 0.89. In 2lm it was observedthat the back-flow prediction based on χc works up to Ret cos β ≈ 50. BeyondRet cos β = 50 back-flow is progressively suppressed as Fr is increased. Here toowe see that sustained back-flow is limited to Fr < 1, but temporary back-flowmight82occur at higher Fr . These results agree with those of [123] for near-horizontaldisplacements: at Fr > 1 only temporary back-flow occurs. In [4] it was reportedthat instantaneous displacements were observed up to Fr ≈ 2, i.e., displacementswithout sustained or temporary back-flow. Combining all these studies, we canconclude the following regarding back-flow of iso-viscous displacements. WhenRet cos β < 50 back-flow exists if χ > χc. When Ret cos β > 50, flows with sus-tained back-flow are limited to Fr < 1, however temporary back-flow might occurup to Fr ≈ 2, meaning the light fluid layer moves backwards from the initial gatevalve but will slow down and stop eventually.In terms of stability, when χ < χc and Ret cos β < 50, only stable displacementsare observed. This is in agreement with the linearly stable regime predicted with2lm: onset of linear instabilities for iso-viscous displacements occurs at Ret cos β ≈50. In the region where χ < χc and Ret cos β > 50, we observe both stable andwavy interfaces. The wavy displacements however correspond to displacementswith small amplitude waves that grow at long times. At larger Fr we also havea few inertial displacements. Our interpretation of this region is that the visualclassification is not perfect, due to the convective nature of unstable flows and thefixed length of experimental tube, i.e. for a sufficiently long tube flows classifiedas stable might develop observable waves and eventually become inertial at largeamplitudes.Above Ret cos β ≈ 150, we see cyclic displacements at Fr / 2.6 and inertial forhigher Fr . Our interpretation is that beyond Ret cos β ≈ 150 transverse buoyancy isnot strong enough to prevent the layers from mixing. Strong counter-current flow atFr / 2.6 causes sudden growth of instabilities and mixing of the layers. Howeverbuoyancy is not strong enough to result in a fully mixed flows, therefore the fluidsstratify again. As Fr is increased. counter-current flow is reduced and the imposedflow stabilizes the displacement but local mixing still occurs at the interface.We now look at the displacements with viscosity ratio. It was shown in 2lmthat for Newtonian fluids with m > 1 the interface becomes unstable. The onsetprediction is shown in Fig. 3.6c for m = 4. From our experiments we can also seethat stable displacements only appear at small Ret cos β and Fr . The transition fromwavy to inertial regime also looks Fr-dependent. From back-flow prediction weknow that χc(m) decreases withm, meaningwhenm > 1, at each Fr back-flow starts83at a smaller Ret cos β. back-flow predictions for m = 2 and m = 4 are plotted in Fig.3.6c. As for iso-viscous displacements it seems that at small Ret cos β back-flowoccurs when χ > χc, and at higher Ret cos β only for Fr / 1. However the transitionpoint is not clear since both χc and instability onset prediction are for Newtonianfluids with constants viscosities. In the experiments, the fluids are slightly shear-thinning, and the effective viscosities in our dimensionless parameters were definedonly using the mean velocity. Additionally, the instability onset prediction of 2lmonly predicts instabilities due to velocity difference between the layers and cannotmake predictions for instabilities due to jump in the shear rate at the interface.When m < 1 (Fig. 3.6d) the stability map looks more similar to iso-viscouscases. In 2lm it was predicted that these flows become unstable at a constantRet cos β, higher than 50 for iso-viscous cases. The prediction for m = 0.25 isplotted in Fig. 3.6d, corresponding to Ret cos β = 60. At all viscosity ratios itseems like transition from intermittent to inertial is at a critical Fr ∈ (2,3), givenRet cos β ' 150. The transition from stable to wavy regime can be predicted from2lm. In stable flows, back-flow occurs when χ > χc(m). When the flow becomesunstable back-flow is limited to cases with Fr / 1. At Ret cos β > 60, as Fr isincreased beyond χ = χc the amplitude of the waves become smaller. However, incontrast to Fig. 3.6b, no stable cases are seen at χ < χc and Ret cos β > 60.Although our focus has been on interfacial stability of displacement flows, itis noteworthy to mention other types of instabilities observed in our experiments,mainly instabilities growing from wall-layers. Some experiments with these insta-bilities can be seen in Fig. 3.7a. The instabilities grow from a thin layer of lightfluid left behind by the heavy fluid. Although the configuration is density-unstable,the onset of instabilities here seem to be due to viscosity contrast, since they weremost prominent and visible in the shear-thinning experiments. In addition, thereis no symmetry in the growth of instabilities, meaning they do not appear to growin the central plane of the pipe, at the lowest point. Therefore the draining flowin these thin layers, in the azimuthal direction, might play a role. Another majorinstability is observed when the displacing front moves along the center of the pipeas seen in Fig. 3.7b (discussed in the next section).841 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 (a) (b)Figure 3.7: Some shear-thinning displacement flows showing instabilitiesgrowing from (a) the wall-layer and (b) behind the tip moving at thecenter of the pipe.3.3.3 Front dynamicsIn previous studies on iso-viscous displacement flows, front velocities have beenmeasured using a threshold value on C¯ and tracking its location with time (see Fig.3.11). However, no comments were made on the shape of the (heavy) front andthe flow structure near this region. Figure 3.8 shows the different type of frontsobserved in our experiments, which show very different behaviours at the front. Wehave made a map of the different front types in Fig. 3.9a for all experiments, andin Fig. 3.9b-d we have separated the data for different viscosity ratios.In Fig. 3.8a, a sharp frontal shock forms during the displacement. This canbe compared to the roll-up of the heavy head in gravity currents and lock-exchangeflows [53], the difference being the pipe geometry. In some unstable cases khinstabilities are shed from this front and destabilize the interface behind it or forma mixed layer at the interface. In the mirror view of this front we can see rollingup of the sides of the tip, suggesting presence of strong secondary flows. Theunsteady fronts shown in Fig. 3.8b show similar behaviour (roll-ups in the mirrorview), however in tip moves from the center of the pipe to the bottom wall and upagain throughout the displacement. By looking at Fig. 3.9 we can see that these85(b)(a)(c)(d)(e)Figure 3.8: Different fronts observed in the experiments: a) frontal shock, b)unsteady shock, c) slump, d) central and e) mixed. The snapshot below(a) is the view of the same front in the top view mirror.fronts are confined to small Froude numbers (Fr / 1) andmoderate Ret cos β values(20 / Ret cos β / 200). Beyond Ret cos β ≈ 200 the front is mixed (Fig. 3.8e) forall Fr . Below Ret cos β ≈ 200, given Fr ' 1 the heavy front slumps beneath thelight fluid and becomes flat and elongated (Fig. 3.8c).With Direct Numerical Simulation (dns) simulations of lock-exchange flowswith no-slip boundary condition, Hartel et al. [53] observed that in a frame movingwith the tip speed, the flow of the light fluid approaching the front bifurcates ata certain height, so that some of the light fluid is overrun by the gravity current.Thus, a stagnation point exists on the “nose" of the heavy current, below andbehind the foremost part of the front, creating a density-unstable region at theheavy front. Although the simulations of [53] are for lock-exchange flows in arectangular channel, some of their findings might be used to explain the frontdynamics of displacement flows. In almost all of our experiments (as far as imagequality and accuracy allows) we see that the foremost part of the front is raisedabove the bottom wall. This suggests that here too there is a stagnation point at thefront, and some of the light fluid remains below the heavy displacing fluid. At longtimes and far behind the front, the light layer gets washed away slowly and becomes86Figure 3.9: Map of different fronts for (a) all experiments, (b)m ≈ 1, (c)m > 1and (d) m < 1. The colors represent the frontal types observed in Fig.3.8: shock (l), unsteady shock (l), slump (l), central (l) and mixed(l). The horizontal and vertical broken lines are at Ret cos β = 200 andFr = 1, respectively.thinner. Right behind the stagnation point at the front, strong secondary flow arisesin the azimuthal direction, resulting in squeezing of the light fluid upwards andforming the rolled-up shape of the front apparent in the mirror image (see Fig. 3.4dand 3.8a).A key difference between displacement flows and lock-exchange flows is theeffect of the imposed flow. In lock-exchange flows and in a moving frame, theincoming light fluid flow is uniform. With the added imposed flow however, thevelocity field downstream of the moving front has a Poiseuille profile. In a moving871 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 15 20 25 30 3500.10.20.30.40.50.60.70.80.91(b)(a)Figure 3.10: Snapshots and profile at t = 4.7−11.9. Displacement parametersare Fr = 0.38, Ret cos β = 127.04 and m = 1. The field of view isx = 4−35 and the time difference between the snapshots is ∆t = 0.654.frame this means the incoming velocity is smaller at the pipe center compared tolock-exchange flows, and changes sign (becoming negative) when Vˆmax = 2Vˆ0 > Vˆfor 2Fr > Vˆf /Vˆt . This could move the stagnation point upwards so that at some Frit moves to the top of the tip, resulting in the slump type front. Note that even inthe slump type front the tip is lifted and some of the light fluid is overrun by theheavy fluid (see Fig. 3.8c). However, it is not possible to see the exact shape ofslump type fronts either our side images of the pipe or the mirror images. If thefront moves very close to the bottom wall it simply becomes difficult to visualize,since accuracy of the images are worst at top and bottom portions of the pipe. Ifthe front moves along the center, it results only in a small concentration, since fromthe side we view the depth-averaged concentration field, again making it difficult tovisualize and track the front.By looking at the unsteady front (Fig. 3.8b), it appears that the unsteadybehaviour is due to formation of vortices and growth of instabilities in the lightlayer underneath the tip. This can be seen in Fig. 3.8b as a kh type instability inthe light layer at the front. The vortex then pulls the heavy tip downward causingvortices to appear at the top and pulling the tip upward. Looking at the front maps,it appears that these unsteady fronts are at the transition points from shock typefronts to slump type fronts, suggesting a critical Fr where the stagnation point88constantly moves from bottom of the tip to the top and back. Another noteworthypoint is the transition from one front type to another during different stages of adisplacement. Figure 3.10 shows early stages of a displacement, where the frontmoves from slump, to unsteady and then to a steady shock type.Above Ret cos β ≈ 200 the front becomes unstable and mixed. Looking atthe stability maps (Fig. 3.6), this corresponds to cyclic and inertial regimes.The constant mixing and cutting of the front makes front velocity measurementschallenging in these regimes, and requires some temporal averaging. Interestingly,all of the experiments with central front, where the flow remained stratified behindthe front, but the front moved at the center of the pipe, were observed at β = 10◦and with a shear-thinning displacing fluid.3.3.4 Front velocity measurementWe now present the measured front velocities from our experiments to see theeffects of viscosity ratio on the displacement efficiency. The front velocity Vˆf wasmeasured using a threshold value Ct on the concentration profile evolution. Figure3.11 shows the concentration profile evolution of a displacement flow, and themeasured front velocity using Ct = 0.05. As mentioned in the previous discussionof front dynamics, the Vˆf measurement is most reliable with steady frontal shocks(Fig. 3.8a), and most challenging when the front is mixed (Fig. 3.8e). Withfrontal shocks the measured Vˆf is not sensitive to the chosen threshold value Ct .In Fig. 3.11c we have plotted Vf measured using Ct = 0.05 (solid black line) andCt = 0.15 (solid red line). In the slump type, choosing a large Ct can result insmaller Vˆf however. The challenge in the unsteady and mixed fronts is changes inthe concenration profiles at C¯ → 0 and thus oscillations in measured Vˆ(C¯ → 0).Nevertheless, in all our measurements we to used Ct = 0.05−0.1 for consistency.Figure 3.12 shows variation of dimensionless front velocity Vf ≡ Vˆf /Vˆ0 withinclination angle β and the imposed velocity, for displacementswithm≈ 1. Inwater-water displacements (Fig. 3.12a), Vf increases with β and reaches a maximum atβ ≈ 60◦ for Fr = 0.78. As Fr is increased, themaxima are shifted to higher β andVfdecreases at every β, expect at β = 10◦. Note that at constant Fr , as β is increasedRet cos β decreases, so we move from top to bottom in (Fr,Ret cos β)-plane. Figure8910 20 30 40 50 60 7000.10.20.30.40.50.60.70.80.9190 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.9140 60 80 100 120 14011.11.21.31.41.51.61.71.81.92(a) (b) (c)Figure 3.11: Mean concentration profiles for a displacement with Fr = 0.79and Ret cos β = 13.34, κˆL = 0.0103Pasn2 and nL = 0.75 from imagestakenwith camera 1 and 2 are plotted in (a) and (b). Themeasured frontvelocity Vf from each figure is plotted in (c). The broken line denotesVf measured in the early stages of the displacement using Ct = 0.05 in(a), and the solid black and red lines showmeasuredVf from data in (b)using Ct = 0.05 and 0.15, respectively. The final steady front velocityis Vf = 1.177.3.12 shows variation of Vf for shear-thinning displacements of fluids with the sameeffective viscosities. Here, Vf decreases with β at all Fr . In these shear-thinningdisplacements, the effective viscosity changes from µˆe ≈ 8.4mPas at Fr ≈ 0.39 toµˆe ≈ 4.2mPas at Fr ≈ 4.3 and they are all in the stable and wavy regimes and thefronts are shock or slump type. At lower inclinations (β ≤ 45◦), Vf decreases withFr . At β = 60◦ the front velocities are equal and at higher β the order is reversed,i.e., Vf increases with Fr .In Fig. 3.13 we have plotted the difference between front and mean velocities∆Vˆ = Vˆf − Vˆ0 scaled by Vˆt at different Fr to see the variation with Ret cos β.The plots are separated for different viscosity configurations. In the iso-viscousdisplacements (Fig. 3.13a), we see that ∆Vˆ increases with Ret cos β at all Fr , up toRet cos β ≈ 150 and then decreases. At lower Fr , the scaled velocity differences arevery similar. As Fr increases the difference becomes larger. The plot form > 1 (Fig.3.13b) shows a markedly different behaviour. Although the range of Ret cos β isdifferent to the iso-viscous cases, it can be seen that the velocity difference plateausat low Ret cos β and slightly decreases at high Ret cos β and low Fr .The plots for m < 1 (Fig. 3.13c) are very scattered but show similar behaviour900 20 40 60 800.511.522.533.50 20 40 60 800.511.522.533.5(a) (b)Figure 3.12: Change in front velocity Vˆf with inclination angle β, and im-posed velocity Vˆ0 for (a) iso-viscous Newtonian displacements withνˆ = 1mm2 s−1 (b) and shear-thinning displacements with κˆ ≈ 10mPasnand n ≈ 0.7 . The symbols denote different imposed velocities:Vˆ0 = 10mms−1 (Fr = 0.39±0.016) (J), Vˆ0 = 20mms−1 (Fr = 0.793±0.044) (s), Vˆ0 = 40mms−1 (Fr = 1.55± 0.153) (#), Vˆ0 = 70mms−1(Fr = 2.687± 0.171) (I), Vˆ0 = 100mms−1 (Fr = 3.84± 0.090) (n),Vˆ0 = 120mms−1 (Fr = 4.377±0.054) (t).to m = 1 at low Ret cos β. It should be noted that the scatter in all plots here isdue to a number of reasons. Firstly, the imposed velocities and thus Fr are notexactly the same. The standard deviations for each Fr are indicated in the figurecaption. Second, for m > 1 and m < 1 the viscosity ratios are not constant. Finally,we should mention the inherent error in finding a steady Vˆf for unsteady or mixedfronts. Another factor is that the scaling with Vˆt , while reasonable at low Fr is lessrelevant at larger Fr where Vˆ0 is the relevant scale. However, scaling with Vˆ0 orsome combination e.g. max{Vˆt,Vˆ0} also does not produce a significantly better datacollapse.Maps of dimensionless front velocity valuesVf = Vˆf /Vˆ0 are plotted for differentviscosity ratios in Fig. 3.14. The colormap scale has been limited to maximumvalue of 2 to better see the variations. It is evident that displacement efficiencyis significantly reduced for m > 1, with most of the data points showing Vf ≥ 1.5.When m < 1 more points with Vf < 1.3 can be seen compared to iso-viscous910 100 200 300 400 50000.511.522.533.50 50 100 150 200 250 300 35000.511.522.533.50 100 200 300 40000.511.522.533.5(a)(b) (c)Figure 3.13: Front velocity for different Fr for (a) m = 1, (b) m > 1 and (c)m < 1. The symbols denote Fr = 0.395±0.02 (J), Fr = 0.792±0.066(s), Fr = 1.574±0.147,l Fr = 2.665±0.135 (I), Fr = 3.818±0.126(n), Fr = 4.373±0.112 (t).displacements. A similar trend in all cases is very high values of Vf at small Fr ,which is expected since at low Fr the characteristic velocity scale is Vˆt . At Fr > 1displacements are efficient at low Ret cos β. As Ret cos β is increased Vf increases(efficiency decreases). At high enough Ret cos β the front becomes mixed whichreduces Vf .Finally, it is interesting to compare the front velocities against existing predic-tions and closures. For stable two-layer displacement flows, the lubrication model920 1 2 3 4 50501001502002503003504004505000 1 2 3 4 50501001502002503003504004505000 1 2 3 4 505010015020025030035040045050011.11.21.31.41.51.61.71.81.92(a) (b) (c)Figure 3.14: Front velocity values Vf for all inclined experiments with a)m > 1, b) m = 1 and c) m < 1. The colormap scale in all figures hasbeen limited to values between 1 and 2. Superposed black and redcircles denote steady and unsteady shock fronts.can be used to find a shock velocity Vf (χ,m). In the previous chapter the frontvelocities from the 2lm were compared against those from the lubrication modeland it was shown that they agree well for small Fr and for Ret cos < 50. At highervalues of Ret cos β the front velocities from the 2lm plateau. The front velocitiesfrom experiments are compared to lubrication model predictions for Ret cos β < 50and all m in Fig. 3.15a. The values of Fr , χ and m from each experiment wereused in the lubrication model to find the front velocity. We find that the predictionworks well for Fr ≤ 3. At higher Fr the lubrication model under-predicts.In earlier experimental works [4, 123], the focus was on producing a simpli-fied prediction of the front velocity so as to be able to estimate the displacementefficiency. In these inertial/unstable regimes we were unable to derive a simpli-fied predictive model and instead relied upon fitting experimentally measured frontvelocities to a closure expression motivated by dimensional analysis. In [4] thiswas:Vˆf /Vˆt = Fr −0.002337(Ret cos β+50Fr −500)(1−0.98Fr +1.03Fr2), (3.4)which was a modification of an earlier expression in [123] that covered a narrowerrange of Fr & Ret cos β. Our data is compared against (3.4) in Fig. 3.15b for m = 1and appears to fit this expression just as well as that in [4]. The prediction for m > 1930 2 4 60123456501001502002503003504004500 2 4 60123456710152025303540450 2 4 60123456501001502002503000 2 4 6012345650100150200250300(a) (b)(d)(c)Figure 3.15: Comparison of the measured front velocities with: (a) theprediction from the lubrication model for Ret cos β < 50; the pre-diction of (3.4) for, b) m = 1, c) m > 1 and d) m < 1. The tri-angles in (b) are data taken from [4]. The dashed lines indicatedVˆf /Vˆt experiments = Vˆf /Vˆt prediction. The colormap in all figuresshows Ret cos β.and m < 1 are only slightly worse; see Figs. 3.15c & d. This suggests that at leastthe leading order effects are well represented by (3.4), which is consistent with thenotion that these regimes are inertial, rather than viscous dominated.3.4 Summary & discussionIn this chapter we presented an experimental investigation of displacement flow ofmiscible viscous and shear-thinning fluids, where the displacing fluid is heavier94than the displaced fluid. Our focus has been mostly on two-layer flow regimes andthe transition from stable to unstable flows. We also studied the effect of viscosityratio on flow stability, back-flows and efficiency of displacement flow. This dataand associated analysis constitute the main novel contribution of the paper.Sustained back-flow, i.e. where the lighter displaced fluid moves back upstreamagainst the imposed flow, occurs consistently when χ > χc and Fr ≤ 1. At higherFr , given χ > χc, temporary back-flows might occur up to Fr ≈ 2. Along theinterface, counter-current flows can be found up to Fr ≈ 2.6. This characterizationdoes not appear to be much affected by m, except that χc changes.In terms of flow regimes, we have classified flows into 5 different regimes:a mixed/diffusive regime (as in [4]) and 4 regimes relating to different two-layerregimes. We have produced maps of these regimes in the (Fr,Ret cos β)-plane. Interms of predicting instability, we have seen that χ > χc is not the correct criterion.While this may appear slightly contradictory to earlier work [4, 123], we notethat these studies did not define or classify stability/instability. Instead they werefocused at prediction of front velocity (hence displacement efficiency), and ourfront velocity predictions here are consistent with the earlier work.The flow becomes unstable at a critical Rec(Fr,m). For m ≤ 1, this criticalvalue seems to be a function of m only, or very weak function of Fr , that is Rec(m)for m ≤ 1. Viscosifying the light fluid (m > 1) has more significant effect onthe stability of the displacement flow, compared to m < 1. When m > 1, the stableregion is limited to a very small region of parameter space and Rec(Fr,m) decreaseswith Fr . From experiments we see that at the interface, small amplitude waveswith short or long wavelengths can exist between χ = χc and Ret cos β = Rec. Inthe iso-viscous flows, the flow seems to be stable in these regions, meaning theinterface is smooth or has waves with finite amplitudes that do not grow noticeablyin time. When m , 1, both the velocity difference between the fluid layers andthe viscosity contrast at the interface contribute to growth of instabilities. AboveRet cos β ≈ 150 the stabilizing effect of transverse buoyancy weakens and strongmixing can occur between the fluids. The strong counter-current flow betweenthe fluid layers causes sudden mixing of the layer, however transverse buoyancy,although weak, stratifies the layers again. Up to Fr ≈ 2.6 the observed flows areintermittent/cyclic, characterized by sudden mixing of the layers and cutting off95of the front. When Fr ' 2.6 the imposed flow becomes strong enough to reducecounter-current flow, the displaced light fluid layer becomes thin and local mixingoccurs at the interface.Displacement front dynamics were investigated in detail. Different behavioursat the front were identified and mapped out for different viscosity ratios. AboveRet cos β ≈ 200, the front becomes mixed. At lower Ret cos β, a frontal shockforms, with roll-ups vortex-like structures along the sides, near the pipe walls.When Fr > 1, the front becomes flat and elongated. In all cases, thin layers of thedisplaced fluid go beneath the displacing front. In flows with a shock type front,strong secondary flow develops in these thin layers and in the azimuthal direction,driving the vortex-like structures, which can be observed in our mirror images. Inthe shear-thinning experiments we see that instabilities grow from these thin layers.Finally, we see that displacement efficiency is significantly affected by the vis-cosity ratio. Viscosifying the displacing fluid increases efficiency and viscosifyingthe displaced fluid reduces the efficiency. This trend conforms to our intuition.This change in efficiency is more evident at higher Fr , where buoyancy effects arereduced.96Chapter 4Density-stable displacement flowexperiments with viscosity ratioIn the previous chapters we studied density-unstable displacement flows. In thischapter we focus on the density-stable configuration, meaning the less dense fluiddisplaces the heavier fluid downward in the pipe. The experimental procedure isthe same as density-unstable experiments described in Chapter 3. More than 260experiments are completed in a pipe covering a broad range of inclination angles,flow rates and viscosity configurations. Viscosity contrast between the fluids isobtained by adding xanthan gum to water, while Glycerol is used to achieve densitydifference. The main objective of this chapter is to study the effects of viscositycontrast on flow patterns and efficiency of density-stable displacement flows.In buoyant displacement flows, the displacement efficiency is governed by thebalance of different forces and timescales. These balances are estimated throughdimensionless combinations of a number of dimensional flow parameters. Themainparameters that we control experimentally are the the density difference, imposeddisplacement velocity, viscosities and inclination angle. Similar to density-unstableflows described in the previous chapter, the relevant dimensionless groups for thisproblem are the Atwood number, At, the Reynolds number, Re, the Froude number,Fr , and the viscosity ratio m. Note that the viscosity ratio is now defined asm = µˆe,H/µˆe,L . The effective viscosity for each fluid k =H,L is defined in Eqn. 3.3.We only focus on small At, the significance of which is that density differences97Vˆ0xˆyˆGate valvegˆ⇢ˆL ⇢ˆHC = 0C = 1Figure 4.1: A simple schematic of the experimental flow-loop.Parameter Rangeβ (◦) 0,10,30,45,60,75,85Vˆ0 (mms−1) 10−110[κˆ (mPasn),n] [1,1],[3,0.85],[10,0.7]At 0.0035Fr 0.38−4.76Re 11−2153m 0.1−11Table 4.1: Full range of physical and dimensionless parameters used indensity-stable experiments.affect the flow significantly only through the buoyancy force, captured by the Froudenumber: Fr = Vˆ0/√Atgˆdˆ, but not through the acceleration of the individual fluids(Boussinesq approximation). The buoyancy parameter χ defined above is the ratioof buoyancy to viscous forces. The mean effective viscosity used for Reynoldsnumbers is defined as ˆ¯µ = (µˆe,L µˆe,H )1/2. With these definitions and with the rangeof flow rates given in Table 4.1, the effective kinematic viscosity of themore viscousfluid is typically µˆe/ρˆ≈ 2.7−8.5mm2 s−1 at Fr = 0.4 and µˆe/ρˆ≈ 1.9−4.2mm2 s−1at Fr = 4.3.Figure 4.2 shows examples of processed images from both cameras, as well asevolution of the mean concentration C¯(xˆ, tˆ), for an iso-viscous displacement flow98 90 100 110 120 130 140 1508010012014016000.20.40.60.8110 20 30 40 50 60 702040608010000.20.40.60.8110 20 30 40 50 60 7000.10.20.30.40.50.60.70.80.911 0.90.80.70.60.50.40.30.20.10 1 0.90.80.70.60.50.40.30.20.10 (a) (b)(c) (d)(e) (f)90 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.91(d)Figure 4.2: Results from a displacement flowwith β = 30◦, Vˆ0 = 95.54mms−1and m = 2.02. The plots on left (a, c, e) are from camera 1 and the oneson right (b, d, f) are from camera 2. The mean concentration profiles in(c, d) correspond to the snapshots in (a, b) and are plotted at ∆tˆ = 1 and∆tˆ = 1.5 intervals.99at β = 30◦. The mean concentration profiles plotted in Figs. 4.2c-d correspond tothe snapshots in Figs. 4.2a-b, respectively, and represent cross-sectional averagedvalues C¯(xˆ, tˆ), whereas the snapshots show depth-averaged concentrationsC(xˆ, yˆ, tˆ).The spatio-temporal plots in Figs. 4.2e-f are another representation of C¯(xˆ, tˆ). Notethat throughout this chapter, the pipe diameter dˆ, mean velocity Vˆ0 and dˆ/Vˆ0 areused to make lengths, velocities and time dimensionless.4.1 Experimental resultsIn general, density-stable displacements are more efficient compared to iso-dense ordensity-unstable displacements and are characterised by a typically sharp interfaceseparating the fluids (see Fig. 4.2). There is a transient period in the initial stages ofdisplacement after the gate valve is opened. This can be seen in the first snapshotsin Fig. 4.2a. This is due to different reasons. (i) Right after the gate valve is openedthere is a pressure difference at the interface due to the applied back-pressure behindthe gate valve. (ii) There is typically a hydrostatic pressure imbalance since theinterface is initially perpendicular to the pipe. (iii) The flow starts from zero-velocity and attains its full flow rate only after some (start-up) time. Therefore,initially the fluids slightly mix when the gate valve is opened but soon stabilize dueto buoyancy. The final shape and angle of the interface depends on the displacementparameters and will be discussed shortly. It is not obvious if a truly steady interfaceshape or wall-layer thickness is achievable.4.1.1 Vertical displacementsFigures 4.3a-f show snapshots of iso-viscous vertical displacements for increasingmean velocity. The snapshots represent depth-averaged concentration values. Atthe lowest mean velocity (Fig. 4.3a), the heavy fluid is being displaced by a columnof the light fluid (left to right in the snapshots and vertically downwards in theactual experiments). Buoyancy acts only in the axial (vertical) direction. Thedisplacement is very efficient with only a small amount of the heavy fluid leftbehind at the pipe wall. There is a mixed, dilute region between the the light andheavy fluids. This is due to the initial mixing when the gate valve is opened. Wewill refer to the front of the pure light fluid and the front of the mixed region as first100(a)(b)(c)(d)(e)(f)Figure 4.3: Snapshots of iso-viscous vertical displacements with a) Vˆ0 =9.05mms−1, b) Vˆ0 = 20.32mms−1, c) Vˆ0 = 42.93mms−1, d) Vˆ0 =67.86mms−1, e) Vˆ0 = 100.16mms−1 and f) Vˆ0 = 110.55mms−1. Thelength of each snapshot is Lˆ = 47.7dˆ and plotted at times that interfaceis at xˆ ≈ 124dˆ.and second fronts, respectively. From Fig. 4.3a-c it appears that the shape of thefirst and second fronts are very similar.As the mean velocity increases, a progressively pointed spike forms at thefront: compare Fig. 4.3c with the flat front in Fig. 4.3a. The spike formationhas been observed in both experiments and simulations in tubes and channels; seee.g. [2, 7, 94]. Since the velocity profile is parabolic downstream and upstreamof the moving front, but flat at the interface, there must be secondary flows bothbehind and in front of the interface. At low velocities buoyancy is strong enoughcompared to viscous forces to keep the interface flat. However, as χ decreases,the light fluid starts to move faster at the center of the pipe due to the parabolicvelocity profile, i.e. the secondary flow is pulling the interface forward. At highermean velocities the initial mixed region is dispersed downstream. Note too that asthe fluids are miscible, any mixed fluid across the interfacial layer is progressivelydispersed by the secondary flows.Finally, we observe behind the front there are residual wall layers, which appearto be thicker as the displacement velocity increases and the interface becomes less101flat. In all snapshots of Fig. 4.3 instabilities can be seen growing from the residualwall layers. Stability will be discussed in Sec. 4.1.3.Figure 4.4 shows vertical displacement flows with the same mean velocity ofVˆ0 ≈ 42mms−1, but different viscosity ratios. The iso-viscous case in Fig. 4.4a isthe same experiment as Fig. 4.3c, where a spike starts to grow. In Fig. 4.4b bothfluids are shear-thinning, iso-viscous with an effective viscosity of µˆe = 5.28mPas.Compared to the water-water displacement, the shear-thinning displacement showsthicker residual wall-layers and strong asymmetrical instabilities at the finger inter-face. The spike at the front also grows and becomes unstable.Figures 4.4c-d have modest viscosity ratios of m = 2.04 and m = 0.51, respec-tively. Figure. 4.4c looks similar to the iso-viscous case. However, in Fig. 4.4dwhere the light fluid is more viscous, the spike looks less dispersed. The effects ofmore extreme viscosity ratios can be seen in Fig. 4.4e-f, withm = 5.38 andm = 0.19.The results are not surprising: displacing the more viscous fluid (Fig. 4.4e) resultsin fingering and thus thick and unstable wall layers emerge and a long and unstablespike at the front. Viscosifying the displacing fluid makes the displacement stableand efficient and prevents formation of a spike (Fig. 4.4).We now seek to find the effect of both imposed velocity and viscosity ratioon the displacement efficiency of vertical displacement, via analysis of the frontvelocities. In displacement flows it is common to find the fastest front velocity,that we refer to here as the tip velocity Vtip, since its inverse gives a measure ofdisplacement efficiency. In density-stable displacementsVtip is expected to be closeto 1.0, specially at lower flow rates where the interface is sharp and flat and thedisplacement is efficient. For example, in [2] a thresholding method was used tofind the front velocity Vf for a C¯ value between 0.1 and 0.2: this was found to bevery close to 1.0. However, when spikes form the tip velocity may be larger.We first explain how front velocities are obtained. The tip velocity is found byusing a small threshold value of C¯ = Ct and tracking the front. As an example, welook at the mean concentration profiles (Fig. 4.5a) and the spatio-temporal diagram(Fig. 4.5b) of the experiment of Fig. 4.4a. We can see that the two fronts moveat different speeds. The dashed lines in Fig. 4.5b are drawn as an eye guide tosee the different slopes of the two fronts. In this particular experiment, there is afrontal shock (the red dashed line), where the displacing fluid is separated from102 1 0.90.80.70.60.50.40.30.20.10 1 0.90.80.70.60.50.40.30.20.10 1 0.90.80.70.60.50.40.30.20.10 1 0.90.80.70.60.50.40.30.20.10 1 0.90.80.70.60.50.40.30.20.10 1 0.90.80.70.60.50.40.30.20.10 (a) (b)(c) (d)(e) (f)Figure 4.4: Snapshots of vertical experiments with imposed velocity of Vˆ0 ≈42mms−1 with a) µˆ = 1mPas and m = 1, b) µˆe = 5.28mPas and m = 1,c) m = 2.04, d) m = 0.51, e) m = 5.38, and f) m = 0.19. All snapshotsare at 90dˆ from the gate-valve and cover Lˆ = 72dˆ. In each plot the timedifference between consequent snapshots is constant and 3−3.25s.1030 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.50.60.70.80.9190 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.91(a)90 100 110 120 130 140 1508090100110120130140150 00.20.40.60.81(c)(b)0 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.50.60.70.80.9190 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.91(a)90 100 110 120 130 140 1508090100110120130140150 00.20.40.60.81(c)(b)Figure 4.5: a) Mean concentration profiles and b) spatio-temporal diagramfor the same experiment as in Fig. 4.3b. The dashed lines in (b) havedifferent slopes and drawn for eye guidance. The velocities with whichdifferent mean concentrations move are plotted in (c). The wiggles atC¯ ≈ 0.9 in (a) are due to instabilities growing at the walls and can beseen as dark lines in (b).the mixed region and the concentration value drops from C¯ ≈ 0.85 to C¯mix ≈ 0.3.The concentration profiles suggest that the mixed region is growing in length andthat C¯mix drops in time. Therefore, instead of using one threshold value we plotthe velocity of different concentration values. The results are plotted in Fig. 4.5c,where V(C¯) = Vˆ(C¯)/Vˆ0 is the velocity with which each value of C¯ is propagating.The velocity of the frontal shock here is VS = 1.10 and the leading front is movingahead with Vtip = 1.18.In Fig. 4.6a, V(C¯) is plotted for a range of shear-thinning vertical displacements1040.5 0.75 1 1.25 1.5 1.75 200.10.20.30.40.50.60.70.80.91(a)90 100 110 120 130 140 150 16000.20.40.60.81(b)Figure 4.6: Effect of increasing imposed flow rate on a vertical shear-thinningdisplacement flow with κˆH ,L = 10.8mPasn and nH ,L = 0.733. Themean velocities are Vˆ0 = 17.22mms−1 (), Vˆ0 = 44.2mms−1 (t), Vˆ0 =73.22mms−1 (s), Vˆ0 = 96.16mms−1 (n) and Vˆ0 = 110.53mms−1 (l).0 20 40 60 80 100 120020406080Figure 4.7: Front velocity vs mean velocity for vertical displacements withdifferent viscosity ratios. The symbols correspond to: µˆ = 1mPas andm = 1 (l), µˆ = 4.13− 7.85mPas and m = 1 (n), m = 1.75− 2.51 (s),m = 0.12−0.24 (J), m = 0.43−0.56 (t). Symbols with thick lines areVˆtip and others are VˆS .105at different mean velocities (same rheology in both fluids). The correspondingconcentration profiles are plotted in Fig. 4.6b. All cases have spikes at the frontwhich is evident from increase in V(C¯) as C¯ → 0. However, the shock velocityVS exists only at small mean velocities. In the iso-dense, limit (χ ≈ 0) the tipof the front moves with the maximum pipe velocity, Vtip → (1+ 3n)/(1+ 3) forshear-thinning fluids, and the flow is fully dispersive. With the shear-thinning fluidused (nH = 0.733), the maximum pipe velocity ahead of the front is Vmax = 1.846.Thus, at high mean velocities (small χ), we expect Vtip → 1.846, but only if theexperimental resolution allows us to threshold at a very small C¯t value. This canbe checked from numerical simulations in Chapter 5.Figures 4.5 & 4.6 and the above discussion illustrate some of the difficultywith front velocity measurement. Thus below, to compare the front velocities ofdifferent displacements, we will report both VS and Vtip as front velocities whenthey exist and are unique (Vtip > VS). However, when VS does not exists we takeonly Vtip as the front velocity. The values for our vertical displacements are plottedin Fig. 4.7. In all cases the front velocities increase with Vˆ0. Tip velocities are muchhigher than shock velocities when m ≥ 1. In these flows taking a high thresholdvalues can significantly under predict the front velocities. On the other hand,from the perspective of displacement efficiency, using Vtip as the front velocitymay significantly under predict the efficiency where the narrow dispersive spike sfollowed by a much wider displacing shock.4.1.2 Inclined displacementsWe now focus on inclined displacements (β ≥ 10◦). As an example, Figs 4.8a-f show snapshots of iso-viscous experiments at β = 60◦ with increasing meanvelocities. An interesting observation is that although the initial contamination ofthe heavy fluid occurs for all experiments, when the pipe is inclined the mixedregion disappears after some a short transient and the front becomes sharp again.This might be a type of Boycott effect related to the effects of transverse buoyancy.Another difference compared to vertical displacements is the shape of the spike.As mean velocity is increased the flow becomes more stratified and the front movesfaster at the top of the pipe. At the highest velocities (Fig. 4.8e-f) it appears that a106(f)(d)(c)(b)(a)(e)(a)(b)(c)(d)(e)(f)Figure 4.8: Snapshots of iso-viscous displacements at β = 60◦ with a)Vˆ0 = 15.09mms−1, b) Vˆ0 = 37.87mms−1, c) Vˆ0 = 63.85mms−1, d)Vˆ0 = 77.98mms−1, e) Vˆ0 = 93.48mms−1 and f) Vˆ0 = 109.89mms−1. Thelength of each snapshot is Lˆ = 19dˆ and plotted at times that interface isat xˆ ≈ 124dˆ.thin spike grows andmoves closer to the center of the pipe. The correspondingmeanconcentration profiles are plotted in Fig.4.9 and better show how, with increasingVˆ0, the interface becomes progressively stretched and a thicker layer of the heavyfluid remains at the bottom of the pipe. The bottom residual layer can becomeunstable and get washed away by the displacing layer as seen in Fig. 4.8f.Figures 4.10a-f illustrate the effect of changing the inclination angle at a fixedflow rate. In the near-vertical case (Fig. 4.10a) the front is dispersive, similar tovertical displacements. At intermediate inclinations 30◦ ≤ β ≤ 60◦, the dispersivefront disappears and the interface is slightly tilted. At high inclinations β= [75◦,85◦](Fig. 4.10e & f) the interface becomes stretched and the light fluid moves at thetop of the pipe. This effect can also be seen from the corresponding concentrationprofiles in Fig. 4.11. The effect is pronounced at higher Vˆ0 (see Fig 4.11), as thehighly inclined displacements show longer interface lengths. In general, for iso-viscous displacements the interface becomes stretched β and 1/χ, as discussed in[2].Overall, density-stable displacement flows seem to be characterised by three10790 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.91Figure 4.9: Mean concentration profiles for the snapshots of Fig. 4.8.(a)(b)(c)(d)(e)(f)(a)(b)(c)(d)(e)(f)Figure 4.10: Snapshots of inclined displacements with Vˆ0 ≈ 42mms−1 at a)β = 10◦, b) β = 30◦, c) β = 45◦, d) β = 60◦, e) β = 75◦ and f) β = 10◦.Field of view is 38dˆ.10890 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.9190 100 110 120 130 140 15000.10.20.30.40.50.60.70.80.91(a) (b)Figure 4.11: Effect of increasing β on displacement flows with a) Vˆ0 ≈42mms−1 and b) Vˆ0 ≈ 97mms−1. The symbols denote β = 10◦ (l),β = 30◦ (s), β = 45◦ (J), β = 60◦ (I), β = 75◦ () and β = 85◦ (n).main features. (i) Generally a thin layer of the heavy fluid remains behind at thepipe wall, unless m 1; (wall-layer). (ii) Often a thicker layer of the heavy fluidremains at the bottom of the pipe (bottom layer). This occurs in inclined pipes andcan be partly due to azimuthal drainage and partly to behaviour of the front. (iii) Thefrontal region exhibits different characteristic behaviours. At small imposed flowrates the displacement front remains perpendicular to gravity, with a very thin layerof the heavy fluid remaining behind, and the effect of inclination angle is minimal.As the flow rate increases the relative importance of axial buoyancy diminishes: thefront shape is affected by the parabolic velocity profile and the effect of inclinationangle on the thickness of the bottom layer becomes more significant.Front velocities of iso-viscous experiments are compared in Fig. 4.12. Allvalues correspond to the velocity of the tip (with threshold Ct ∈ [0.05,0.15] used).At all β, the front velocity increases with Vˆ0. The values for β ∈ [10◦,60◦] arevery similar, whereas at higher inclinations, at each Vˆ0, Vˆf increases with β. Theexception is high front velocity values for vertical displacements.The effect of viscosity ratio on displacement efficiency is most significant inthe near-horizontal displacement. The front velocities for the two highly inclinedcases are plotted in Fig. 4.13. At all flow rates Vˆf decreases with m. The lowestm, where the displacing fluid is much more viscous than the displaced fluid, thefront velocities remain close to Vˆ0 and only increase at high flow rates, where the1090 20 40 60 80 100 12001020304050Figure 4.12: Front velocity vs mean velocity for iso-viscous displacements atdifferent inclinations. The data correspond to: β = 85◦ (l), β = 75◦(s), β = 60◦ (J), β = 45◦ (n), β = 30◦ (t), β = 10◦ (I), β = 0◦ ().All values correspond to Vˆtip.effective viscosity drops. The shear-thinning displacements with m ≈ 1 show betterdisplacement efficiency than water-water displacements.4.1.3 Notes on stabilityIgnoring the frontal region, two main type of instabilities can appear in density-stable displacement flows: 1) instabilities growing from the layers of the heavyfluid remaining at the pipe walls (thin-layer instabilities) and 2) inertial interfacialinstabilities that appear at higher Reynolds numbers. Here we should note keydifferences between displacement flows studied here and two-fluid systems reviewedearlier that make direct comparison challenging. Whereas in core-annular flows thebase state is steady flow, here the wall-layer thickness grows from zero at the initialinterface location to a finite length near the front. As the displacement progressesthe flow can be assumed steady at the leading order. Close to the front however1100 20 40 60 80 100 1200102030405060(a)0 20 40 60 80 100 120010203040506070(b)Figure 4.13: Front velocity vsmean velocity for displacementswith a) β = 75◦and b) β = 85◦. The symbols correspond to: µˆ= 1mPas andm = 1 (l),µˆ = 4.13−9mPas and m = 1 (n), m = 1.75−2.51 (s), m = 4.6−10.1(I), m = 0.43− 0.56 (t) and m = 0.1− 0.2 (J). Symbols with thicklines are Vˆtip and others are VˆS .Figure 4.14: Snapshots of a displacement showing instabilities from walllayer. Displacement parameters are β = 30◦, κˆH = 14.7mPasn,nH = 0.68 and Vˆ0 = 39.13mms−1. The field of view is Lˆ = 44dˆ andsnapshots are plotted at times t = [10−31].the assumption is not valid. Furthermore, here both density and viscosity contrastsexist. Flow with the less viscous fluid at the wall are not relevant since whenthe displacing fluid is more viscous the wall-layer get washed away. In summary,wall-layer instabilities in vertical displacements are most relevant to core-annularflows with density and viscosity contrast, in the limit of thin layer of more viscousfluid at the wall.111Figure 4.15: Snapshots of a displacement showing instabilities from bottomlayer. Displacement parameters are β = 10◦ , κˆL,H = 10mPasn, nL,H =0.74 and Vˆ0 = 19.27mms−1. The field of view is Lˆ = 68dˆ and snapshotsare plotted at times t = [104−127].The stability of vertical displacements was briefly discussed in §4.1.1. Instabil-ities in the wall-layer were observed unless form 1, in which case the heavy fluidis completely displaced and no wall-layer exists (see Fig. 4.3). All instabilities areasymmetric in our experiments. Because of the disturbances at the early distancesit is not possible to see onset of instabilities. Instead, in flows with no or smallviscosity ratios the instabilities appear as streaks of the heavy fluid being advectedwith the flow. At high viscosity ratios, and when both fluids are shear-thinning,the growth rates are higher. It appears that all instabilities are convective and notabsolute, meaning they move only in the direction of the flow. Another interestingobservation is that the thin spike at the front becomes unstable when the displacedfluid is shear-thinning.The stability problem is very different when the pipe is inclined. Now buoyancyis destabilizing wall-layers and stabilizing in the bottom layer. The stabilizing effectreduces as the pipe is inclined towards vertical. The thickness of the bottom layerdepends on inclination, viscosity ratio and imposed velocity. Two examples ofthin-layer, low Reynolds number instabilities can be seen in Figures 4.14 and4.15. In the first case, instabilities appear as streaks growing from the wall-layer as the displacing finger moves forward. These instabilities were observedmostly when the displaced fluid (wall-layers) were more viscous, suggesting the112Figure 4.16: Snapshots of a displacement with interfacial instabilities. Dis-placement parameters are β = 85◦ , κˆH = 12.3mPasn, nH = 0.7 andVˆ0 = 100.17mms−1. The field of view is Lˆ = 47dˆ and snapshots areplotted at times t = [110−143].onset of instabilities is viscosity driven and not due to buoyancy. In the secondcase (Fig. 4.15), instabilities can be seen growing from the bottom-layer. Theseinstabilities were observed at near-vertical inclinations (β = 10◦) when the displacedfluid was shear-thinning. In the displacement shown in the Fig. 4.15, both fluids areshear-thinning. The onset of instabilities happens at long times where the front is atxˆ f > 80dˆ. Once the instabilities grow they seem to evolve with the velocity profile.The unique shape observed right behind the front (the last snapshot in Fig. 4.15), canbe due to the secondary flow behind the front, where the axial velocity deceleratesat the pipe center and accelerates closer to the pipe wall.Finally, interfacial Kelvin-Helmholtz type instabilities can grow due to inertiaand viscosity contrast. The cause of this instability is the velocity difference betweenthe two fluid layers and the onset criterion is similar to two-layer density-unstabledisplacement flows. Ignoring the viscous effects, the Inviscid Kelvin-Helmholtzcriterion for two fluids in a pipe was derived in Chapter 2 and for density-stableflows isuˆL − uˆH >[pi2(ρˆH − ρˆLρˆH + ρˆL)gˆdˆ2Sˆisin β]1/2, (4.1)where uL,H is average velocity in each layer and Sˆi is the interface length. The abovecriterion suggests KH instabilities grow when the velocity difference is high. Since1130.1 0.25 0.5 1 2 4 10 2010110 210 31 2 3 4 5 6 71234567(a) (b)Figure 4.17: a) Kelvin-Helmholtz instability prediction based on (4.2). Blueand red symbols represent stable and unstable flows respectively. Thedashed line denotes equality in (4.2). b) Stability map from all inclinedexperiments. Hollow symbols denote stable displacements. Colorsdenotewall-layer (blue), bottom-layer (red) or both (green) instabilities.The symbols represent inclination: β = 85◦ (#), β = 75◦ (), β = 60◦(M), β = 45◦ (♦), β = 30◦ (O), β = 10◦ (I).no back-flow of the heavy layer occurs in density-stable displacements, this can becorrelated with high front velocities instead. From front velocity measurements weknow that front velocity increases with Vˆ0, β and m. The right-hand side of (4.1)suggests the instabilities can be triggered when At and/or sin β decreases, or whenSˆi increases. The first two are intuitive, since ∆ρˆgˆ sin β is transverse buoyancy andis stabilizing. The third condition suggests the interface is most unstable wheneach fluid takes half of the cross-section, and becomes stable when the bottom-layer becomes thin. This further proves that the thin-layer instabilities are viscositydriven and not KH type. Based on (4.1) and front velocity measurements we cancome up with a simple stability criterion assuming i) the velocity in the heavy layeris small and the instabilities are driven by the velocity of the displacing fluid, ii)assume uˆL = Vˆf , iii) Si = Sˆi/dˆ = 0 when Vf = Vˆf /Vˆ0 = 1 and Si = 1 when Vf = 2.The last assumption is based on the fact that when the displacement is efficient theheavy layer gets washed away completely, and in the worst case (for stability) theinterface length is maximum (Si = 1) when the displacing fluid is moving with Vˆ0.Based on these we can simplify (4.1) for density-stable displacements to114VˆfVˆt>[pi2sin βVf −1]1/2. (4.2)For each experiment we can compute left and right-hand sides of (4.2) andcheck for stability. The results are plotted in Fig. 4.17a for all experiments withβ ∈ [30◦,85◦]. Based on (4.2), flows with unstable interface must lie above thedashed line. The red symbols, denoting unstable experiments, all lie above this linesuggesting the prediction is valid. The red data point below this line corresponds toan experiment with a very high viscosity ratio (m = 32). First thing to note is thatthe unstable points in Fig. 4.17-a represent both KH and bottom-layer instabilities.Although, (4.2) is greatly simplified we see that all unstable flows lie above thedashed line. One limitation of this prediction is that the direct effects of viscositycontrast at the interface are not taken into account. The indirect effect of viscosityratio, via changing the front velocity, however, are accounted for.A stability map of all inclined experiments is presented in Fig. 4.17b. Generally,we can see that most unstable cases have m > 1, the exception being displacementsat β = 10◦ where wall-layer instabilities appear even with m ≤ 1.4.2 Summary & discussionIn this chapter we have presented an experimental investigation of displacementflow of miscible viscous and shear-thinning fluids, where the displacing fluid is lessdense than the displaced fluid and the flow is downwards along the inclined pipe,i.e. density-stable downward displacements. In contrast to [2], where bulk featuresof iso-viscous density-stable displacement flows were studied, here we have mostlyfocused on effects of viscosity ratio on the efficiency of displacements (throughthe front velocity), and on various instabilities observed when we have a viscositycontrast.For vertical flows the bulk displacements are typically very efficient, but in allcases secondary flows (relative to the front) result in a dispersive spike that advancesahead of the main front. This feature increases with displacement flow rate andmay destabilize at higher flow rates. The other main feature is a residual layer ofheavy fluid left behind at the wall. If the displacing fluid is more viscous there is115little/no residual layer, but form > 1 the wall layer persists (growing thicker withm)and this wall layer is generally unstable. Asymmetric streaks (surface waves) areobserved moving in the direction of flow. The instabilities are amplified for higherviscosity ratios.For inclined displacements we find that initial mixing effects are reduced, thefluids stratify approximately perpendicular to gravity (more so at slower flow rates)and the interface is generally quite sharp. The interface spike advances ahead of themain front and orients downwards towards the center of the pipe, but is not presentat intermediate inclinations.Residual layers in inclined displacements are either a thin wall layer or a thickerbottom layer, both of which destabilize. The wall layer instabilities are fairly similarto those for vertical flows, appearing a streaks. The bottom layer instabilities takethe form of regularly spaced roll waves. For both instabilities a crude instabilitycriterion based on the KH-mechanism appears to have reasonable predictive ability.116Chapter 5Full numerical simulation ofdisplacement flowsIn the two previous chapters we looked into displacement flows in pipe through anexperimental approach. In Chapter 2, we developed a simple model for density-unstable displacement flows in the inertial regime. In this chapter, we present theresults from a number of numerical simulations of displacement flows in a pipe,within the parameter range of the experiments. This allows direct comparison withexperiments to benchmark the numerical model.We first introduce the numerical model in the following section. The resultsare presented for density-stable (§5.2) and density-unstable (§5.3) displacementflows, with slightly different objectives. The density-stable simulations are mostlydesigned for comparison against the experiments of Chapter 4 to benchmark thenumerical model and revisit the observed phenomena in the experiments such asinstabilities and frontal spikes. In the density-unstable simulations, however, afew cases are directly compared against the experiments, and the rest are morephenomenological to address the uncertainties discussed in Chapters 2 and 3.117Lˆdˆ = 19.05mmFigure 5.1: Mesh topology for pipe simulations. The pipe diameter is thesame as the experimental set-up. The cutaway is to showcase the meshin the axial direction.5.1 Numerical modelThe numerical simulations are implemented and solved using OpenFOAM1. Thegoverning equations are the full Navier-Stokes equations, and the two incompress-ible liquids are modeled using Volume of Fluids (VOF) method, first proposed byHirt and Nichols [57]. In this method the volume of a fluid in a cell is computedas CVcell, with C being the phase fraction in the cell and Vcell is the volume of thecell. The momentum and continuity equations are∂ρU∂t+∇ · ρUU = ∇p+∇ ·2µS+ ρg, (5.1)∇ ·U = 0, (5.2)1http://www.openfoam.com118where S = [(∇U)+ (∇U)T ]/2 is the mean rate of strain tensors and ρ and µ are themixture density and viscosity found fromρ = Cρ1 + (1−C)ρ2, (5.3)µ = Cµ1 + (1−C)µ2, (5.4)withC ∈ [0,1] being the fluid 1 phase fraction. The transport equation for the phasefraction C is∂C∂t+∇ ·CU+∇ · [C(1−C)Ur ] = 0. (5.5)Note that we have set the diffusion term zero. The molecular diffusion coefficientwas also set to zero, giving infinite Schmidt number (Sc). The diffusion in thesolution is only numerical diffusions controlled by the mesh size. The additionalterm ∇ · [C(1−C)Ur ] in (5.5) is a compression term, with Ur being the relativevelocity between the two-phases, also called as compression velocity. It can beadded to the classical transport equation to compress the interface and result in asharper interface between the two phases. The C(1−C) inclusions ensures thisterm is only active near the interface.For the rheology of the fluids, constant viscosity and power-law models wereused for Newtonian and shear-thinning fluids respectively. For Newtonian fluidswe set constant values for dynamic viscosity µˆH ,L of the fluids. For shear-thinningfluids, the viscosity can be modelled as power-law µˆ = κˆγn−1.Figure 5.2 shows the mesh used in a cross-section of the pipe. In the coarsestmesh (mesh 1), the block in the center has 20×20 cells, and the four blocks have20× 12 cells, which get gradually finer at the wall, resulting in 1360 cells at eachcross-sectional slice. The axial mesh size was chosen so that long aspect ratios areavoided. We used mesh 1 in some cases where a very long domain was used, andfiner meshes with shorter domains.The boundary conditions used are as follows. For velocity, no-slip conditionswere set at the pipe wall. At the inlet fully-developed Poiseuille profile was used andat the outlet zero-gradient condition. For phase fraction α constant value of 1 wasused at the inlet and zero-gradient everywhere else. For the initial condition, we setα = 1 at a length Li from the inlet, so that the initial interface between the two fluids119(a) (b) (c)Figure 5.2: The computational grid used in our the pipe cross-section. Thenumber of cell faces in the cross-sections are 1360, 3300 and 4800,respectively. The black and white colors show the concentration fieldfor a density-stable displacement.would be at x = Li. Since no back-flow occurs in density-stable displacements thislength was a few diameters from the inlet, typically Li = 5d.Explicit Euler method was used for time marching. The time-step ∆t is dictatedby the CFL number. After some iterations, we found the maximum allowable CFLof 0.1 for both momentum and phase fraction solvers works best. Higher CFLvalues (up to 0.8) were used without numerical difficulty, however to avoid highdegree of numerical diffusion we kept the CFL number below 0.1 for all casespresented here.The simulations were run in parallel on a custom-built 24-core workstationin UBC. Running the code in parallel significantly reduces the wall-clock time,and the gain in efficiency increases with the number of cells. The whole domainwas decomposed into 24 sub-domains, each assigned to a single processor. Thenumerical model is described in more detail in §A.2.5.2 Density-stable displacement flowsWe now present and discuss the results of the numerical simulations listed inTable 5.1. Most of the cases have the same displacement parameters as in thephysical experiments, in order to do direct qualitative and quantitative comparisons.The goal is to benchmark the numerical model, and look at the flow field, frontdynamics and other unknowns/uncertainties in the experiments. Similar to the120Case β (◦) Vˆ0 (mms−1) [κˆH (mPasnH),nH ] Lˆ (mm) No. of cellsv1 0 20.2 [1, 1] 1300 1.768×106v1* 0 20.2 [1, 1] 1300 1.768×106v2* 0 40.2 [1, 1] 1300 1.768×106v3 0 100.2 [1, 1] 1300 1.768×106v4 0 42 [10.8, 0.73] 1300 5.28×106v5 0 42 [81.0, 0.516] 1300 1.768×106v6 0 100.2 [81.0, 0.516] 2500 2.72×106i1 75 70 [15, 0.674] 2500 2.72×106i2 85 100.17 [12.3,0.7] 2500 2.72×106i3 60 100.9 [81.0, 0.516] 2500 2.72×106Table 5.1: Density-stable cases solved using OpenFOAM.experiments we will break the results into vertical and inclined displacements.Cases v1-v6 are all vertical displacements and i1-i4 are inclined. The densities usedin all simulations are ρˆL = 998grcm−3 and ρˆH = 1005grcm−3, so that At = 0.0035.The velocities and viscosities are listed in Table 5.1.5.2.1 Vertical displacementsWe start the comparison with a number of vertical displacements. Cases v1and v3 are both iso-viscous with mean velocities of Vˆ0 = 20.2mms−1 and Vˆ0 =100.2mms−1, respectively. Snapshots of the displacement and velocity profilesclose to the front are shown in Fig. 5.3. These two cases have the same displace-ment parameters as the experiments of Fig. 4.3b & e. Both central plane and 3Dview of the pipe are presented since the 3D view is close to what is seen in theexperimental images. The snapshots of case v1 show that even at low velocities athin spike forms at the front at the center of the pipe. The low concentration of thespike makes it undetectable in the experiments, and the 3D view, after averagingacross the pipe - and only barely visible in the central plane view. As the velocityis increased the thickness and concentration of the spike increases, as seen in snap-shots of case v3. The enlarged snapshots in Fig. 5.3 is the same snapshot as thetop, with streamlines plotted in a moving frame reference. They show clearly theexistence of secondary flows close to the front and circulation zones just behind the121Figure 5.3: Snapshots and velocity profiles of case v1 (top) and case v3 (bot-tom) to show the effect of Vˆ0 on growth of the spike at the front. In eachfigure the first snapshot is from the central plane and the second is a 3Dview of the pipe. The last snapshot shows the streamlines in a movingframe for case v1.front.The velocity profiles in the central plane for cases v1 and v3 are plotted inFig. 5.4a & b, respectively, at the same instances as in Fig. 5.3. With low imposedvelocity, the velocity profile is close to Poiseuille profile behind the front. Theprofiles at xˆ = 700mm and 780mm show that at about one diameter behind theinterface the flow field is steady. The maximum velocity is smaller than maximumpipe velocity due to higher velocities in the wall-layers. Close to the interface thevelocity profile of the displacing fluid becomes flat, with overshoots in the velocityof the displaced fluid. As we go further along the spike, the velocity is higher in thedisplaced layer and drags the light fluid forward at the center of the pipe. Eventuallythe velocity profile reaches that of fully-developed pipe flow in the heavy layer.To recreate the initial condition of the experiments, where the fluids mix afteropening of the gate-valve, in cases v1* and v2* we have added a mixed region withC = 0.3 for xˆ ∈ [0.05,0.06]mm separating the fluids at tˆ = 0. Case v1* has thesame mean velocity of case v1. The snapshots right after the displacement starts1220 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.800.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.800.10.20.30.40.50.60.70.80.91(a) (b)Figure 5.4: Velocity profiles of a) case v1 at xˆ = 0.7m (C), 0.78m (M), 0.8m(), 0.82m (∗), 0.84m (B), 0.9m (#), 1.0m (×) and b) case v3 atxˆ = 0.8m (C), 0.94m (M), 0.96m (), 0.98m (∗), 1.0m (B), 1.2m (#).Figure 5.5: Results for case v1* at times tˆ = 5s (top) and tˆ = 30s (bottom). Ateach time the first snapshot is from the central plane and the second is3D view of the pipe. The length and concentration of the added mixedregion is 100mm and 0.3, respectively.123Figure 5.6: Snapshots of central plane for case v2* at times tˆ = [15,17, ...,25]s.and at a later time can be seen in Fig. 5.5. The evolution of the mixed front at latertimes can be seen in Fig. 5.6. The results show that a spike appears and grows atboth fronts. The second front can be seen as a density-stable displacement withdensity difference being 30% of the actual displacement. The first front and itsspike, however, are affected by the mixed region. The spike starts to deform in themixed region and several recirculation zones appearThe mean concentration profiles and V(C¯) of cases v1 and v1* are plotted inFig. 5.7a& b. Themean concentration profiles of case v1* show a similar behaviourto the experiments (Fig. 4.5a) with C¯ of the mixed region dropping in time. TheV(C¯) plots of both cases are compared against the experiment in Fig. 5.7b. Theshock velocity VS for C¯ > 0.3 is the same for all three cases and the mixed frontresults in a higher Vtip.Case v2* has the same displacement flow parameters as the experiment ofFig. 4.5, with a mixed region added to the initial condition. The mean concentrationprofiles are plotted in Fig. 5.7c. The V(C¯) values plotted in Fig. 5.7d are comparedagainst the experimental values and show very good agreement. Here the C¯ of themixed region drops faster and results in a larger increase in Vtip.Unlike the experiments, no instabilities appears in the iso-viscous numericalsimulations. We note that the instabilities in the iso-viscous experiments werebarely visible and grew at long times. Since there is no viscosity contrast and the1240 0.2 0.4 0.6 0.8 1 1.2 1.400.10.20.30.40.50.60.70.80.910 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.91(c) (d)0 10 20 30 40 5000.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.50.60.70.80.91(a) (b)Figure 5.7: a) Mean concentration profiles for case v1 (dashed lines) and v1d(solid lines). b) Comparison of V(C¯) for Case v1 (4), case v1d (#) andexperiment (×). The results for case v2d are presented in c & d.wall-layers are thin, the onset of instabilities are probably due to small disturbancesin the flow which grow because of the density difference.We now look at vertical displacements with viscosity differences to see ifthe wall-layers become unstable. In cases v4-v6 the displaced fluids are shear-thinning. The displacement parameters of case v4 are the same as the experimentof Fig. 4.4e. We have used a finer mesh in this simulation to better resolve the wall-layers and reduce numerical diffusion at the interface. The snapshots at the earlytimes in Fig. 5.8 show growth of roll-ups at the interface. At longer times howeverthese waves become stable and long-wave instabilities remain at the interface. Incomparison, the snapshots of the experiments show significant instabilities at boththe interface and the front. Comparison of V(C¯) in Fig. 5.9a show that the shockvelocities are the same but there is a thicker spike in the experiment. From the125Figure 5.8: Snapshots of central plane for case v4 at times tˆ =[0.5,1.5, ...,4.5]s. The last snapshot is at tˆ = 11s.stability of miscible interfaces discussed earlier, we know that the growth rateand wave-length of instabilities depend on viscosity ratio, Reynolds number andinterface thickness. To see how thick (diffused) the interface is, the values ofconcentration, velocity magnitude and viscosity ratio are plotted at the center-linein Figures 5.9b-c. All figures are plotted at tˆ = 5s, at several point upstream anddownstream the front. At xˆ = 0.2m where the interface is stable and the flow issteady, the interface thickness is δ ≈ 0.05, withC = 0.5 being at the height y = 0.065(see Fig. 5.9c), which is the pointwhere the the velocity gradient changes (Fig. 5.9b).At all other points the interface thickness is higher, due to the instabilities near thefront, and the change in the velocity gradient is more gradual. The effectiveviscosity ratio based on (3.3) for this case is me = 5.35. The plots in Fig. 5.9d showthat at the pipe wall, the viscosity ratios are slightly below this value. What is ofmore importance regarding the stability of the interface, is the viscosity ratio atthe interface. If we take the m(y) values at height where C(y) = 0.5, the viscosityratios are ≈ 3. This lower viscosity ratio at the interface might result in lowergrowth rates for instabilities. However, we should note the significant disturbancesintroduced at the beginning of the experiments, and therefore the difference in the1261 3 5 7 9 11 13 15 1700.050.10.150.20.250.30.350.40.450.50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.050.10.150.20.250.30.350.40.450.50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.200.050.10.150.20.250.30.350.40.450.50 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.800.10.20.30.40.50.60.70.80.91(a) (b)(c) (d)Figure 5.9: a) Comparison of V(C¯) for case v4 (# against experiment (×). b,c, d) V(y), C(y) and m(y) for case v4 at xˆ = 0.2m (C), 0.3m (M), 0.31m(), 0.33m (∗), 0.35m (B), 0.37m (#), 0.47m (×) at time tˆ = 5s. Thesnapshot at the bottom is at time tˆ = 5s.initial conditions. Ignoring the numerical diffusion and the effects of interfacethickness, if the growth rates for the given flow parameters are small, it will takea long time for them to grow in the simulations. In the experiments, however, theflow is already disturbed when the displacement starts.We have also completed two vertical simulations with a highly shear-thinningheavy fluid with κˆH = 81mPasn and nH = 0.516 and mean velocities of Vˆ0 =42mms−1 for case v5 and Vˆ0 = 100.2mms−1 for case v6. The goal is to see ifinstabilities grow faster with a higher viscosity ratio compared to case v4. Case127Figure 5.10: Snapshots of central plane for case v5 at times tˆ = [2,3, ...,10]s.The last snapshot is at tˆ = 17s.Figure 5.11: Snapshots of central plane for case v6 at times tˆ = [0.5,1, ...,4.5]s.The last snapshot is at tˆ = 12s. The image at the bottom is C = 0.5contours, coloured with dimensionless velocity magnitude.1280 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.50.60.70.80.915 10 15 20 25 30 35 4000.10.20.30.40.50.60.70.80.910 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.910 0.5 1 1.5 2 2.5 3 3.5 400.10.20.30.40.50.60.70.80.91(a) (b)(c) (d)Figure 5.12: a) Plot of V(C¯) for case v5 (#) and case v6 (×). The filledsymbol at C¯ = 0 is Vmax for a shear-thinning fluid with n = 0.516. b,c, d) V(y), C(y) and m(y) for case v5 at xˆ = 0.63m (C), 0.65m (M),0.67m (), 0.69m (∗) and 0.71m (#) at time tˆ = 7.5s. The snapshotat the bottom is at time tˆ = 7.5s.v5 has the same mean velocity as case v4, therefore a smaller Reynolds number.The snapshots of case v5 in Fig. 5.10 show growth of waves at the interface inthe early stages of displacement. At tˆ ≈ 6s, the symmetry in the flow breaks andthe displacing finger takes a helical shape. The snapshot at tˆ = 17s shows thatinstabilities start to grow far behind the front where the wall-layer is very thin. Notethat this case has a coarser mesh compare to case v4, therefore the interface is morediffuse in comparison. The form of instabilities at the front are similar to those seen129in the experiments.The results for case v6 (Fig. 5.11) show quick development of waves at theinterface and strong instabilities that result in eventual mixing of the fluids at thefront. The last two images in Fig. 5.11 are central plane concentration and C = 0.5contours coloured by dimensionless velocity, plotted at tˆ = 12s. The contours showasymmetric and helical form of the interface as the finger moves. In both casesv5 and v6 the tip remains sharp and a spike does not form. The plots of V(C¯) inFig. 5.12 show that in both cases the Vtip > Vmax = 1.6807. Figures 5.12b-d arefor case v5 at time tˆ = 7.5s. The interface thickness is δ ∼ 0.15, which is 3 timeslarger than in case v4. Still, the viscosity ratio at the interface is very high. Forthis case, the effective viscosity ratio based on (3.3) is me = 23.51. The m(y) plotsif Fig. 5.12d show that the viscosity ratio at the wall is m(0) = 20 ∼ 25, and ≈ 20when C = 0.5.5.2.2 Inclined displacementsTurning now to inclined displacements, Fig. 5.13 shows central plane, 3D andexperimental images at 2 different times for i3. The upper images show the spike-like penetrating layer of light fluid, only asymmetric. Interestingly, as with densityunstable displacements the tip orients away from the wall and towards the pipecentre (see the discussion in [40]), i.e. downwards here. This is less visible in the3D and experimental images. We also see similar long wave undulations in bothsimulation and experiment, behind the front.The same simulation at a slightly later time is shown in Fig. 5.14, behind thefront. The top image shows an interesting stratification of the light fluid layer in thecentral plane. The red contour showsC = 0.9 and we see a non-monotone variationwith depth. The three cross-sections below provide an explanation. We see aresidual layer of heavy fluid on the walls. The fluid within this layer drains slowlyazimuthally, but at the highest point the azimuthal velocity is zero (symmetry).However, buoyancy forces here pull the residual fluid downwards in a plume, whichis visible in the cross-sections along with the secondary flow that it generates. Thusthe central plane has this interesting vertical concentration stratification, althoughthe flow is otherwise cleanly stratified. Structures such as these are not easily visible1300.1 0.14 0.18 0.22 0.26 0.3 0.34 0.38 0.42 0.46 0.5 0.54 0.58 0.62 0.660.1 0.14 0.18 0.22 0.26 0.3 0.34 0.38 0.42 0.46 0.5 0.54 0.58 0.62 0.660.1 0.14 0.18 0.22 0.26 0.3 0.34 0.38 0.42 0.46 0.5 0.54 0.58 0.62 0.66Figure 5.13: Comparison of the results for case i1 against experimental re-sults. The snapshots are at times tˆ = [3,3.5,4]s. At each time, the firstsnapshot in central plane, the second is the side 3D view and third oneis from experiments.in the experiment.Case i2 has the same displacement parameters as the experiment of Fig. 4.16with interfacial instabilities. The heavy fluid is shear-thinning and the inclinationangle is β = 85◦. The snapshots and velocities can be seen in Fig. 5.15. Thecentral plane snapshots show the emergence of instabilities at the interface and inthe wall-layer. The front becomes very thin in time and the tip moves closer to thecenter of the pipe, as in i3. The velocity profiles show that far behind the frontalregion the velocity is large in the light layer and small in the thick heavy layer. Thecross-sectional slices show that the symmetry breaks due to instabilities and strongsecondary flows exist near the front. The mean concentration profiles and V(C¯)are plotted in Fig. 5.16. The heavy layer is almost static as C¯ → 1. The resultsagree with the experiments and the V(C¯) plot confirms that Vtip is very close to themaximum pipe velocity of the heavy fluid.We completed one set of experiments at β = 60◦with 1000mgL−1 concentrationof xanthan, giving κˆH = 81mPasn and nH = 0.516. The same parameters were usedin case i3. The central plane and 3D snapshots are presented in Fig. 5.17. Stronginstabilities grow at both the front and the interface between the fluids. The frontpropagates in a helical fashion becoming thinner in time but remaining sharp near131Figure 5.14: Snapshots for case i1 at tˆ = 14.5s. The cross-sections from leftto right are at xˆ = [1,1.1,1.2]m, respectively. The red solid line inall images shows C = 0.9. The cross-sections also show the in planevelocity vectors.Figure 5.15: Snapshots and velocity profiles at central plane (top) and cross-sections (bottom) of the pipe for case i2. The snapshots are at timestˆ = [3,3.5, ...,5.5]s. The cross-sections are at xˆ = [0.4,0.6,0.8,1]m attime tˆ = 5.5s. The velocities are dimensionless.1320 20 40 60 80 100 12000.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.50.60.70.80.91(a) (b)Figure 5.16: a) Mean concentration profiles and b) V(C¯) from numericalsimulation (#) of case i2 and experiments (×). The red symbol atC¯ = 0 is Vmax for a shear-thinning fluid with n = 0.7.the tip. This was also observed in the experiments, for both cases v5 and v6. Theseinstabilities and a thick residual layer (barely moving) can be seen in the meanconcentration profiles (Fig. 5.18a). The front velocities agree between experimentand simulation over the mid-range of C¯ (Fig. 5.18b). The simulations give a fastertip velocity; indeed the tip velocity is higher than the maximum pipe velocity of theheavy fluid, similar to case v6.5.2.3 Summary of resultsWith high-resolution simulations performed using OpenFOAM we were able torepeat some of the experiments numerically. In general the agreement betweensimulations and experiments was good, being able to reproduce the main flowfeatures and giving insight into features of the flow not visible/measured in theexperiments.For vertical flows the dispersive spikes and secondary flows are faithfully re-produced in the simulations, as are front velocity variations with C¯. The residuallayering and effects of viscosity ratio are also reproduced, but the instabilities of thewall layer are not fully captured by the simulations. This may be a consequence ofmesh resolution near the wall (although we have refined the mesh to explore this).It may also be that the computations always produce a diffuse layer across the inter-1330.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Figure 5.17: Snapshots of case i3 at times tˆ = [1,3, ...,9]s. The last twosnapshots are at tˆ = [11,13]s. The image at the bottom is from theexperiment.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.800.10.20.30.40.50.60.70.80.910 20 40 60 80 100 12000.10.20.30.40.50.60.70.80.91(a) (b)Figure 5.18: a) Mean concentration profiles and b) V(C¯) from numericalsimulation (l) of case i3 and experiments (×). The filled symbol atC¯ = 0 is Vmax for a shear-thinning fluid with n = 0.516.134face due to numerical diffusion (almost certainly larger than molecular diffusion),which may stabilize what is essentially an interfacial instability; see e.g. [38] andothers. Lastly, it could be that destabilization of the experimental flow initiates withO(1) asymmetric disturbances (e.g. due to opening of the gate valve).In inclined flows the center plane snapshots show that the spike at front indeedmoves close to the center of the pipe. The cross-sectional snapshots confirm thatthin layers of the heavy fluid remain at the pipe wall, on top of the displacingfinger. This creates an unstable density configuration which can destabilize. Thenumerical simulations successfully predict the residual layers, but do not generallyresolve the thin wall-layer instabilities.Overall, the numerical simulations show agreement with the experiments andprovide complementary information on the dynamics of the displacement flows.This agreement provides confidence in using the same computational model fordensity-unstable displacement flows.5.3 Density-unstable displacement flowsIn this section we present the results from numerical simulations of density-unstabledisplacement flows. Similar to the previous chapters we focus on stratified flows.Compared to the density-stable simulations, generallywehave to use longer domainsdue to back-flow at low Froude numbers and development of two-layer flows.Specially for Fr < 1 and χ > χc, where sustained back-flow occurs, the initialinterface must be placed at several diameters from the inlet. Additionally, severalphenomena of interest, e.g. interfacial instabilities and cyclic mixing of the layers,develop at longer times. Therefore, in most cases we have used a domain length ofLˆ = 2000−2500mm.There are a number of questions from density-unstable experiments we seek toanswer, e.g., the dynamics of the front, sensitivity of front velocity measurementsto the threshold value Ct , whether or not wall-layers exist below the heavy fluidlayer. Similar to the density-stable simulations we complete a number cases in eachof the classified regimes in Chapter 3. The goal is to capture the main features ofthe flows observed in the experiments, and address the above-mentioned questions.Finally, we assess some of the assumptions made in development of the two-layer135model, and lubrication models in general.5.3.1 Main featuresFigure 5.19 shows the spatio-temporal diagrams of iso-viscous displacements withfixed Ret cos β = 126.1 and increasing Fr . This covers the same range of parameterscovered in our iso-viscous experiments at β = 75◦ with Ret cos β = 127. We notethat the spatio-temporal diagrams from numerical simulations are typically plottedusing data at ∆tˆ = 0.5s and ∆xˆ = 5− 10mm intervals, resulting in some spatio-temporal filtering. In comparison, the experimental spatio-temporal diagrams areplotted using ∆tˆ = 0.25 timesteps and pixel length of δ xˆ = 1 ∼ 1.1mm. The resultsof Fig. 5.19 show qualitative agreement with the experiments in terms of back-flowand flow regimes (see Fig. 3.6b). At low Fr (Fig. 5.19a) sustained back-flow occursand interfacial instabilities appear in form of travelling waves. As Fr is increasedbeyond Fr = 1, back-flow stops and the flow becomes progressively stable. AtFr = 3.1281 (Fig. 5.19c) no instabilities can be seen. As Fr is further increasedthe flow becomes inertial and instabilities grow at the interface, resulting in localmixing.The displacements of Fig. 5.20 are at lower inclination angle β = 60◦, withRet cos β = 243.6. In the experiments, we categorized these flows as cyclic atFr / 2.5 and inertial at higher Fr . The characteristics observed in the experimentsare again captured. At Fr / 2.5 (Figures 5.20a & b), significant mixing eventsoccur, resulting in cutting off of the heavy layer. These can be seen as dark regionsin the spatio-temporal diagrams. Instabilities move in both forward and backwarddirection due to counter-current flow of the fluid layers. At higher Fr (Figures 5.20c& d), counter-current flow is suppressed, instabilities are convected forward andmixing becomes local.Three cases with interfacial waves can be seen in Fig. 5.21. All three dis-placements have strong counter-current flow between the fluid layers. In the firstdisplacement (snapshots at the top), the heavy fluid is more viscous (m < 1). Thesecond displacement is iso-viscous at higher Ret cos β, and the third displacementshown has the same Fr and Ret cos β values as the first case, with the viscosityof the fluids switched (m > 1). When the heavy fluid is more viscous the fluids1360 20 40 60 80 100 12010203040506000.20.40.60.81(c)0 20 40 60 80 100 1201020304050607000.20.40.60.81(d)0 20 40 60 80 100 120102030405000.20.40.60.810 20 40 60 80 100510152025303540 00.20.40.60.81(a) (b)Figure 5.19: Spatio-temporal diagrams of displacement flows with β = 75◦and Ret cos β = 126.1 and a) Fr = 1.0, b) Fr = 1.564, c) Fr = 3.1281and d) Fr = 3.91.do not mix and the interface remains much sharper, compared to the other twocases. This can be also seen in the experiment of Fig. 3.4c. When m ≥ 1, thewaves break and a mixed layer forms at the interface of the fluids. This was alsoseen in the experiments (see 3.4d). By looking at the snapshots at the center planeand cross-sections, we can see that when m < 1 the heavy fluid protrudes into thelight layer, and when m ≥ the light fluid protrudes into the heavy layer and kh typebillows form that result in local mixing of the fluids.5.3.2 Cyclic flow regimeNow we look at the displacement flows we categorized as cyclic in Chapter 3 inmore detail. The spatio-temporal diagram of the experiment in Fig. 3.5 is plotted1370 20 40 60 80 100 12010203040506000.20.40.60.810 20 40 60 80 100 12010203040506000.20.40.60.810 20 40 60 80 1001020304000.20.40.60.810 20 40 60 80 100 120102030405000.20.40.60.81(c) (d)(a) (b)Figure 5.20: Spatio-temporal diagrams of displacement flows with β = 60◦and Ret cos β = 243.6 and a) Fr = 1.173, b) Fr = 1.7595, c) Fr = 2.346and d) Fr = 3.1281.in Fig. 5.22a. Figures. 5.22b & c show the results from two numerical simulationswith similar flow parameters to the experiment. The axes of Fig. 5.22b & c areadjusted for easier comparison against the experiment, so that in all cases the initialinterface (gate valve) is located at x = 0. We first point out the differences betweenthe two numerical simulation. For convenience wewill call them case 1 (Fig. 5.22b)and case 2 (Fig. 5.22c).Case 2 has a finer mesh compared to case 1 (3.65× 106 cells compared to2.21× 106), and flow parameters closer to the experiment. However, the mixingevent occurs at roughly the same time (t ≈ 35) in the experiment and case 1, whilein case 2 it occurs at t ≈ 25. The unsteady nature of the flow suggests sensitivity tothe initial conditions. Thus, we changed the simulation parameters, e.g., mesh size,time-step, flux limiters, and looked at the early time evolution of the concentration138Figure 5.21: Top: β = 60◦, Ret cos β = 87, Fr = 0.857 and m = 0.138. Thecenter plane snapshots are plotted at tˆ = 25,27s. Cross-sections are atxˆ = [1.0,1.02,1.04]m at tˆ = 27s. Middle: β = 60◦, Ret cos β = 201,Fr = 0.782 and νˆH ,L = 1.2mm2 s−1. The center plane snapshots areplotted at tˆ = 20,22s. Cross-sections are at xˆ = [0.64,0.66,0.68]m attˆ = 20s. Bottom: β = 60◦, Ret cos β = 87, Fr = 0.857 and m = 7.21.Snapshots are plotted at tˆ = 13,14s. The 3D isosurfaces are C = 0.5viewed from the top. The red solid lines represent C = 0.5.13910 20 30 40 50 60 701015202530354000.20.40.60.8110 20 30 40 50 60 701015202530354000.20.40.60.8110 20 30 40 50 60 701015202530354000.20.40.60.81(a) (b)(c)Figure 5.22: Spatio-temporal diagrams from a) experiment with Fr = 0.788and Ret cos β = 245.52, b) numerical simulation with Fr = 0.782 andRet cos β = 243.6 (case 1) and c) numerical simulation with Fr = 0.788and Ret cos β = 245.1 (case 2). The x-axis is dimensionless distancefrom the gate-valve.field.Figures 5.23a& b illustrate iso-contours of concentration (C = 0.1−0.9) at earlytimes for cases 2 and 1, respectively. The first snapshots shown are at tˆ = 0.5s inboth cases. In the numerical simulations the time-step is dictated by CFL number,which was set to 0.1 for all simulations, and a maximum allowable time-step ∆tˆmax .Since the simulations start from a zero-flow condition, ∆tˆmax is used in the initialtime-step, so that ∆tˆi = ∆tˆmax , and then ∆tˆ is calculated according to the CFLcondition. The initial time-steps used in case 1 and 2 were ∆tˆi = 0.5 and in case 2∆tˆi = 0.01, respectively. The large time-step in case 1 results in distortion of theinitial concentration field, which is incidentally close to initial disturbance in the140(a) (b)(c)(d)Figure 5.23: Effect of the initial maximum time-step on dispersion of con-centration field. The initial time-steps are a) ∆tˆi = 0.01 and b)∆tˆi = 0.5. The contours denote C = 0.1− 0.9. The snapshots areat tˆ = [0.5,1.5, ...,4.5]s. The snapshot in (c) show the last snapshot of(a). The result in (d) are from a simulation with a finer mesh.experiments when the gate valve is opened. Because of the dispersive and unstablenature of the flow, the concentration field evolves differently in each case. Thelong-time behaviour of the flows, however, are the same, e.g., in case 1 the fronteventually becomes unsteady. We also completed a numerical simulation with amuch finer mesh and smaller time-step in a short pipe to ensure the results areconvergent. The results at time tˆ = 4.5s is shown in Fig. 5.23d, and Fig. 5.23cdepicts the last snapshot of case 2. The results are in good agreement and theminimal differences are due to the different numerical diffusivities, resulting fromdifferent mesh sizes. Other simulation parameters were changed too, however thechanges in the results were minimal.Laminar-turbulent cyclesIn this section we will focus on the laminar-turbulent cycles that appear in cyclicflows. As discussed in Chapter 3, these flows are characterized by recurring mixing141Figure 5.24: Snapshtos of the simulations in Fig. 5.22c. The center planesnapshots (from top to bottom) and the cross-section snapshots (fromleft to right) are plotted at times ta = 20.75, tb = 24.48 and tc = 28.73,respectively. The cross-sections are at xˆ = 0.625m at tˆ = 40s and theinitial interface is at xˆ = 0.2m. The red solid lines represent C = 0.5.The rectangle in the first snapshot shows the probe window used tocreate Fig. 5.25.events, i.e., sudden bursts of strong turbulence that result in significant transversemixing of the fluid layers across the pipe. After these turbulent periods, the flowrelaminarizes and a stable stratified flow follows. From experiments, these flowswere identified for 200 / Ret cos β / 500 − 50Fr and Fr / 2.6. Similar flowbehaviour in lock-exchange flows were studied experimentally by Tanino et al.[126] using high-resolution PIV and LIF measurements on a section of the pipe.We follow the same analysis here on a numerical simulation with Ret cos β = 245.1and Fr = 0.788 (case 2 discussed in the previous section).The snapshots of Fig. 5.24 show the transition from a stratified shear flow toa turbulent flow. The rectangle drawn on the first snapshot represents a 50mm-long section of the domain used as the measurement window to compute theaveraged quantities, similar to the PIV/LIF section in [126]. Let us first introduce142the quantities used in [126] and in our analysis. We denote the quantities thatwere averaged over the streamwise span of the measurement region by 〈〉x . Then〈v2〉1/2x (y,t) is used as a measure of the strength of turbulence. The quantity ∆u∗ =〈u〉x(y∗H (t),t)− 〈u〉x(y∗L(t),t) is the difference between the maximum and minimumaxial velocities (in the heavy and light fluid layers) and y∗H ,L corresponds to theirtransverse coordinates, respectively. Transverse density gradient is characterizedby the concentration gradient at y = 0 and is denoted by ∂〈C〉/∂y |0. The gradientsdenoted by ∂/∂y |0 are calculated by finding the slope of the line fitted betweeny∗H/2 < y < y∗L/2. Finally, turbulence is characterized by 〈v2〉1/2x (0,t).Temporal evolution of 〈u〉, 〈C〉 and 〈v2〉1/2x are shown in Figures. 5.25a, b & d,respectively. Note that due to the domain length and hence the limited time of theflow solved in the simulation only one cycle occurs in our results. The turbulentburst (mixing event) can clearly be seen as a drop in 〈u〉 across the pipe (Fig. 5.25a),and increase in 〈v2〉1/2x (Fig. 5.25d) at t = 25−30. The mixing of the fluids resultsin a drop in the concentration gradient (Fig. 5.25b).Figures. 5.25c shows the ramp-cliff pattern of mean streamwise velocity ob-served in [126]. At t = 5.32 the front enters the measurement window and ∆u∗ in-creases gradually to its maximum value at t = ta = 20.75 (first snapshot in Fig. 5.24).At this point the kh billows enter the window and start to break, resulting in mixingof the fluids. This is can be seen as a sudden increase in 〈v2〉1/2x and a sud-den drop in −∂〈C〉x/∂y |0. As mixing continues 〈v2〉1/2x reaches its maximum att = tb = 24.48 and then decreases as mixing continues. The ramp phase startsagain at t = tc = 28.73 as the second front enters and ∆u∗ starts to increase again.At t = 30, ∆y∗ reaches its maximum value, concentration gradient increases and∆u∗ = 3.8 ≈ 2(Vf −Vb f )In the stratified shear flows that develop in cyclic regimes, stability is governedby the gradient Richardson number Ri and the Reynolds number. The gradientRichardson number represents the competition between (the destabilizing) shearand (stabilizing) stratification. The Reynolds numbers is simply the competitionbetween inertia and viscosity. Following [126], we define the instantaneous gradient1430 1000 2000 3000 400010 -210 -15 10 15 20 25 30 35 40-0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.10.20.30.40.50.60 10 20 30 4001230 10 20 30 400.40.60.80 10 20 30 4005100 10 20 30 4000.515 10 15 20 25 30 35 40-0.5-0.4-0.3-0.2-0.100.10.20.30.40.500.20.40.60.815 10 15 20 25 30 35 40-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5-3-2-1012345(c)(a) (b)(d)(e) (f)tbta tctbtatcFigure 5.25: Temporal evolution of a) 〈u〉x(y,t), b) 〈C〉x(y,t), c) top: ∆u∗, bot-tom: 〈v2〉x(0,t) (black line) and 〈v2 +w2〉x(0,t)(red line), d) 〈v2〉x(y,t),e) top: −∂〈C〉x/∂y |0, bottom: the transverse separation between theextrema of 〈u〉x , f) instantaneous Reynolds number vs instantaneousgradient Richardson number (5.6).144051015∆U[cm s−1]t˜1a t˜2a t˜2d t˜3dt˜1c t˜2c t˜3ct˜1b t˜2b t˜3bt˜3a t˜4a(a)432100.51〈w2〉1/2x(z = 0)[cm s−1](b)0.60.70.82z˜∗(c)00.51− ∂〈ρ˜〉x∂z˜∣∣∣0(d)5 10 15 20 25 30 35 40 4500.020.04Rit˜(e)Figure 5.26: Multiple laminar-turbulent cycles in lock-exchange flowPIV/LIFmeasurements of [126]. Dimensionless quantities are denoted bythe ˜ superscript. Note that time is nondimensionalized with τN =2pi/√2gAtsinβ/d. Also 2z˜∗ ≡ ∆y∗. Three ramp-cliff cycles can beseen in (a), each one beginning at t˜ia(i = 1,2,3). The image is takenfrom [126].145Richardson number asRi(t) = −g sin β〈ρˆ〉x |0∂〈ρˆ〉x∂ yˆ0(∂〈uˆ〉x∂ yˆ0)−2, (5.6)and the instantaneous Reynolds number as Re∗(t) = ∆u∗Re. Temporal evolution ofRe∗ and Ri are plotted in Fig. 5.25f. The color markers denote Re and Ri valuesat ta, tb and tc. At ta where ∆u∗ is maximum, Re∗ = 3503 and Ri = 0.0073. Atthe turbulent stage where 〈v2〉1/2x is maximum, Re = 2400 and Ri = 0.024. Att = tc where the flow becomes laminar again, Re = 1090 and Ri = 0.045. Thesevalues are in the range reported by [126]. We have to note that although in thedisplacement considered here the Froude number is small, still the imposed flowaffects the results compared to the lock-exchange flow. First, in the lock-exchangeflow constant counter-current flow exists everywhere behind the fronts. Here,counter-current flow is highest close to the leading (heavy) front and decreases atthe trailing (light) front. Second, because of higher velocity in the heavy layer themixing zones get advected forward. Nevertheless, the results here for one cycleshow great agreement with the results of [126], which are plotted in Fig. 5.26.Using a much longer computational domain and a wide range of parameters wouldallow for resolving multiple cycles.5.3.3 Front dynamicsIn §3.3.3 we discussed the different frontal behaviours observed in the density-unstable displacement flows. The fronts were mapped in (Fr,Ret cos β)-plane. Wenow look at the front shapes and plot the streamlines in reference frame movingwith the tip. To do this, we compute u−Vf ex and plot the streamlines in the centerplane of the pipe.Figure 5.27 illustrates the front shape and streamlines for two low-Froude-number displacements with shock-type fronts. Note that both cases are inertial(Ret cos β = 126) with Froude numbers of Fr = 0.1955 and 0.391. The frontshapes look similar to gravity current heads. The 3D isosurfaces show the sideroll-up structures observed in the mirror views in the experiments. The streamlinessuggest stagnation points are slightly below the foremost point of the front. Note146Figure 5.27: Front shape and streamlines for a displacement with β = 75◦,Ret cos β = 126, Fr = 0.1955 (top) and Fr = 0.391 (bottom). Thestreamlines are in moving frame with Vˆf and are plotted at centerplane. The red line representsC = 0.5. The 3D isosurfaces areC = 0.5viewed from the top of the pipe. The front velocities are Vf = 4.3743(top) and Vf = 2.6562 (bottom).that the concentration contours are for C = 0.5. At the lower Froude number, thestreamlines become parallel behind the front and the head is slightly raised from thepipe wall, suggesting a small amount of the displaced fluid is being left over belowthe heavy fluid. The side roll-ups, as discussed in the experiments, are possiblydue to secondary flows which squeeze the displaced fluid out from the wall layers.The length of the side roll-up is ∼ 2.5dˆ in this case. In comparison, at the higherFroude number the front is raised more and the roll-ups are longer.Development of secondary flow at the front can be seen in Fig. 5.28. Ahead ofthe tip a strong upward motion exists in the light fluid. Behind the tip azimuthalmotion develops that brings the light fluid over the heavy fluid and forms the sideroll-ups shown earlier. Another thing to note is that the velocity magnitudes are∼ 0.5Vˆf , which makes the 1D flow assumption close to the front invalid.147Figure 5.28: Iso-contours of concentration C = 0.1−0.9 and secondary flowvectors for the displacement flow of Fig. 5.27: Fr = 0.1955 (top)and Fr = 0.391 (bottom). The cross sections from left to right areat xˆ = [1.67,1.68,1.69]m. The solid line show C = 0.1− 0.9. Thevelocities are scaled with Vˆf .At a slightly higher Froude number the front becomes unsteady. An exampleis shown in Fig. 5.29 for a displacement with same parameters as Fig. 5.27 andFr = 0.782. As the front is further raised above the pipe wall a significant amountof fluid goes under the head. Note that the front velocity Vf = 2.1 is very close tothe maximum pipe velocity. The 3D isosurfaces show that the symmetry breaksand the flow becomes three-dimensional slightly behind the tip. The cross sectionsnapshots in Fig. 5.30 show that the tip is moving slightly below the pipe center. Inaddition to strong azimuthal flow with magnitude of ∼ 0.5Vˆf , instabilities grow inthe wall layer in an off-center plane. These can be seen in the bottom snapshots inFig. 5.30.Figure 5.31 shows a slump type front at a near-horizontal displacement. Thesecondary flow magnitude drops below 0.1Vˆf and even close to tip the interfaceremains flat. The heavy layer becomes thin closer to tip, forming a tongue shape.148Figure 5.29: Front shape, streamlines and contours for a displacement withβ = 75◦, Ret cos β = 126 and Fr = 0.7820. The front velocity is Vf =2.1. Plots are at times tˆ = [30,30.5,31,31.5]s.149Figure 5.30: Iso-contours of concentration C = 0.1−0.9 and secondary flowvectors for the displacement flow of Fig. 5.29 at tˆ = 31s (top) andtˆ = 31.5s (bottom). The cross sections from left to right are at xˆ =[1.67,1.68,1.69]m. The solid lines show C = 0.1−0.9. The velocitiesare scaled with Vˆf .This again invalidates the assumption made in lubrication and two-layer modelsthat the heavy layer area fraction varies only with the interface height.Finally, the central type front is shown in Fig. 5.32 for a near-vertical displace-ment. Ahead of the tip, the light fluid motion is radial and symmetric. Behind thetip clockwise motion develops that forms the helical shape of the interface. Farbehind the front (not shown here) the flow becomes two-layer (see Figures 3.7b and3.8d).5.3.4 Revisiting front velocityIn §5.2 we looked at sensitivity of front velocity measurement on the thresholdvalue Ct . We will do the same here for a series of displacement flows. In §3.3.4 itwas discussed on how different flow behaviours at the front can make front velocitymeasurement challenging. Several attempts at finding curve-fits for Vˆf in different150Figure 5.31: Front shape, streamlines and secondary flows for a displace-ment with β = 85◦ and Fr = 1.5640 at tˆ = 25s. The front ve-locity is Vf = 1.505. The cross sections from left to right are atxˆ = [1.66,1.68,1.70]m.regimes were described first in Chapter 1 and then in Chapter 3. We will compareVˆf found from simulations against experiments and existing curve-fits from [123]& [4].A series of numerical simulations were completed at β = 60◦ and 75◦, withAt = 0.0035, which correspond to Ret cos β = 126 and 243.6, respectively. Thefront velocities were calculated using the same approach described in §5.2. The tipvelocities are plotted in Fig. 5.33 and compared against the experiments and curve-fits. As a reminder, the curve-fit of (1.6) is an extension of lock-exchange flowsfor inertial near-horizontal displacements at small Froude numbers (exchange flowdominated inertial regime in [123]), and the curve-fit of (3.4) is from iso-viscousexperiments of [4], in the range χ > χc and Ret cos β < 500− 50Fr . The twocurve-fits are plotted in Figures 5.33a & b by solid and dashed lines, respectively.Curve-fit of (1.6) is such that it yields front velocity of Vˆf /Vˆt = 0.7 at Fr = 0,making it relevant in the range 50 / Ret cos β / 200, where the flow is two-layer,151Figure 5.32: Front shape, streamlines and secondary flows for a displacementwith β = 10◦, Ret cos β = 89.24, Fr = 1.6422 and m = 5.37 at tˆ = 18s.The front velocity is Vf = 1.8927. The cross sections from left to rightare at xˆ = [1.60,1.62,1.64]m.0 0.5 1 1.5 2 2.5 3 3.50123456c0 0.5 1 1.5 2 2.5 3 3.5 4 4.5012345678c(a) (b)Figure 5.33: Comparison of the front velocity values at a) β = 75◦ andRet cos β = 126 and b) β = 60◦ and Ret cos β = 243.6 from experi-ments (n) and numerical simulations (l). The solid and dashed linesare curve-fits from (1.6) and (3.4), respectively. Superposed symbolsdenotes displacements with back-flow.152-0.5 0 0.5 1 1.5 2 2.500.10.20.30.40.50.60.70.80.910 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.50.60.70.80.91(a) (b)Figure 5.34: Comparison of V(C¯) for β = 60◦ and a) Fr = 0.79 and b) Fr =3.91 from numerical simulations (#) and experiments (×).inertial and the front is not mixed. Its limitation is that it does include any Ret cos βdependency. The fit from (3.4) gives Vˆf /Vˆt = Fr when Ret cos β = 500− 50Fr ,however the exchange flow value of Vf Fr = 0.7 is not achieved as Fr→ 0The results in Fig. 5.33a for Ret cos β = 126 show that Vˆf /Vˆt values from bothsimulations and experiments show agreement with both curve-fits. At Fr ' 1.5,prediction of (3.4) over-predicts front velocities. The results for Ret cos β = 243.6plotted in Fig. 5.33b are more interesting, as predictions from (1.6) and (3.4)deviate as Fr is increased. The computational values are higher than experimentalones at Fr > 1.5. For the experimental data we have used threshold value ofCt = 0.1, similar to [4]. Therefore the agreement with (3.4) is not surprising. In thesimulations, however, we have used a smaller threshold value ofCt = 0.01. It seemslike Vˆf values are sensitive to Ct , specially at higher Froude numbers. To checkthis, we have plotted the experimental and computational values of V(C¯) for twocases where the values agree (Fr = 0.79) and deviate (Fr = 3.91) in Figures 5.34a& b, respectively. The results for Fr = 0.79 show that a shock velocity exists forC¯ < 0.4, making Vˆf insensitive to Ct . On the other hand, the results for Fr = 3.91show strong dependency on Ct for C¯ < 0.2. This is not surprising since the effectsof Fr on front dynamics and formation of long spikes were observed in the previoussections.Spatio-temporal evolution of concentration and temporal evolution of Vf for1530 20 40 60 80 100 12000.10.20.30.40.50.60.70.80.910 20 40 60 80 100 12010203040506000.20.40.60.8110 20 30 40 50 6011.11.21.31.41.51.61.71.81.92(a) (b)(c)Figure 5.35: a) Spatio-temporal and b) mean concentration profiles for thesame simulation as Fig. 5.34b. The front velocities in (c) are computedusing Ct = 0.01 (#) and Ct = 0.1 (×). The concentration profiles areplotted at equal intervals for t = 1.05−59.85.the displacement of Fig. 5.33b are illustrated in Fig. 5.35. At t ≈ 40 instabilitiesgrow near the front and that results in partial detachment of the tip. As a result,depending on the threshold value two distinct fronts can be identified. This can beseen as two different slopes in the spatio-temporal diagram for t ' 45. The meanconcentration profiles also show that the first front is decreasing in C¯ and growingin length, such that its C¯ value drops below 0.1 in the last few profiles. Temporalevolution of Vf for C¯ = 0.01 and C¯ = 0.1 is plotted in Fig. 5.35c. Up to t = 42 bothfront velocities are very close in value. However, the value of Vf for C¯ = 0.1 dropsto 1.33, whereas for C¯ = 0.1 it remains at ≈ 1.92.We recalculated the front velocities of Fig. 5.33b using Ct = 0.1 and plotted theresults in Fig. 5.36. The new values (solid triangles) show great agreement with1540 0.5 1 1.5 2 2.5 3 3.5 4 4.5012345678cFigure 5.36: The same plot as Fig. 5.33b. The added triangle symbols arefront velocities from simulations with using Ct = 0.1.the experimental values and curve-fit of (3.4). The results again confirm that thefront velocity value are sensitive to the threshold value Ct , expect for small Froudenumbers where a shock velocity exists.5.3.5 Revisiting two-layer modelsIn this section we will compare the results from full numerical simulations to thetwo-layer model developed in Chapter 2 for a few stable two-layer flow. The resultsin the previous sections suggest the flow is highly three-dimensional close to thefront. Here we assess the validity of the assumptions made for the two-layer model,and simplified 1D models of displacement flows in general.Figure 5.37 compares the V(C¯) values calculated from the numerical solutionof the two-layer model and a full numerical simulation. The displacement flow isin the viscous regime with χ = 67.1 < χc. The two-layer model was solved untilt = 60 and the full simulation until t = 26. The results in Fig. 5.37a shows thatthe simulations have higher V for C¯ < 0.4 and smaller V for C¯ > 0.4. Lookingat temporal evolution of V for different C¯ values show that the results have notreached steady values yet. The velocities are decreasing with time for C¯ < 0.4 andincreasing for C¯ > 0.4. Interestingly V(0.4) is constant, suggesting that with higher1550 5 10 15 20 251.41.51.61.71.81.920.70.80.911.11.20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.800.10.20.30.40.50.60.70.80.91(a) (b)Figure 5.37: a) Comparison ofV(C¯) between 3D simulation (#) and 2LM so-lution (red solid line) for a displacement flow with β = 80◦, Ret cos β =20 and Fr = 0.6. b) Time evolution of V(C¯) from 3D simulation forC¯ = 0.1 (solid line), C¯ = 0.3 (dashed line) and C¯ = 0.4 (dashed-dottedline). The right axis corresponds to C¯ = 0.6 (dotted line).simulation time the results will be much closer to those of the two-layer model. Inviscous flows the characteristic length and timescales are δ−1 where δ = cot β/χ;See [120]. Therefore the front velocity reaches a constant value at long times aspointed out in §2.5.2 (see Fig. 2.24). For the flow parameters of Fig. 5.37 thiscorresponds to t ≈ 380, which is much larger than the times used in these numericalsolutions. The two-layer model front velocity (Vf = 1.41) however is already closeto the lubrication model prediction (Vf = 1.39).1D flow assumptionLet us use 〈〉 to denote cross-sectional average quantities, such that 〈C〉(x,t) =∫AC(x,y,z,t)dA ≡ C¯(x,t), where A is dimensionless area (∫AdA = 1). Then wedefined the following quantitiesuH = 〈uC〉, qH = uHC¯, (5.7)uL = 〈u(1−C)〉, qL = uL(1− C¯), (5.8)So that qH ,L is flux of the heavy and light fluids and uH ,L is mean velocity ofthe fluids. Note that by definition qH + qL = 1. We now define the strength ofsecondary flow as 〈v2 +w2〉1/2.1560 5 10 15 20 2500.050.10.150.20.250.30 10 20 30 40 50 6000.050.10.150.20.250.3(a) (b)0 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.91(c)Figure 5.38: a) Time evolution of secondary flow strength for the displace-ment flow of Fig. 5.37. b) Maxima of 〈v2〉1/2 (×) and 〈w2〉1/2 (#). c)Mean concentration profile and secondary flow strength multiplied by10 at t = 25.75.In Fig. 5.38a we have plotted 〈v2 +w2〉1/2 for the displacement flow of Fig. 5.37.Even for this viscous flow secondary flow exists along the entire length of theinterface. Two peaks can be seen in the profiles, one near the trailing front anda stronger one close to the leading front, with both decreasing in time. Betweenthese two peaks, i.e., along the interface, 〈v2 +w2〉1/2 has an almost constant valueof ∼ 0.01 which slowly decreases in time. One expects this value to reach zeroat t = O(δ−1), where the slope of the interface becomes very small ∂h/∂x → 0.The maxima of 〈v2〉1/2 and 〈w2〉1/2 are plotted in Fig. 5.38b separately. Bothseem to reach an asymptote at long times. Another thing to note is that 〈w2〉1/2Mis slightly larger than 〈v2〉1/2M . To better see the secondary flow strength over theinterface length, the mean concentration profile and 〈v2 +w2〉1/2 are plotted at the157end of the simulation (t = 25.75). Note that 〈v2 +w2〉1/2 is multiplied by 10 inFig. 5.38c. The secondary flow increases a few diameters ahead of the front, andincreases to 〈v2 +w2〉1/2 ≈ 0.075 at the front. Close to the trailing front it increasesto 〈v2 +w2〉1/2 ≈ 0.02. In both regions, secondary flow is reaches its peak anddecreases to the near constant value of 〈v2 +w2〉1/2 ≈ 0.01 at the interface over alength of l ≈ 10.Stationary layers & back-flowWe now test the criterion for sustained back-flow in viscous flows, that is back-flowoccurs if χ > χc. Take a two-layer displacement flow with sustained back-flow: theflux of the light fluid layer qL is negative at the trailing front. Ahead of the leadingfront qL = 1. Therefore at some point along the interface we must have qL = 0,meaning the velocity of the interface (and not the velocity at the interface) is zero.By increasing the imposed flow Fr the sustained back-flow is prevented, and weget instantaneous displacements, therefore at the trailing front qL > 0. At a criticalcondition we will have qL = 0 at the trailing front. Taghavi et al. [120] found thatthis critical condition coincides with a critical χ = χc, which comes from a balancebetween buoyancy, inertia and viscous forces. In a lubrication/thin-film model theinterface evolves according to the kinematic equation∂α∂t+∂qH∂x= 0, (5.9)where α is the area fraction of the heavy fluid. Note that in the two-layer modelthe kinematic equation was solved along with the momentum equation and α andqH = αuH were found with the solution of the two equations. In lubrication models,it is assumed that axial buoyancy is balanced by viscous forces. In [120] a closurewas found for the the flux qH (h,hx). Assuming t 1, the slope hx becomes verysmall such that qH = qH (h,0). When the light fluid layer is stationary, qL = 0 andqH = 1−qL = 1. Noting that the velocity of the interface is Vi = ∂qH/∂α, when theinterface is stationary (Vi = 0), we would have ∂qH/∂h = 0. Therefore the criticalχc can be found by finding the intersection between qH = 1 and ∂qH/∂α = 0. In[120] this critical values was found to be χc = 116.32 for pipe displacements. Inthe two-layer model we used the solution of the equilibrium equation to find steady1580 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.910 20 40 60 80 100 12000.10.20.30.40.50.60.70.80.910 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.80.910 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.70.80.91-0.200.20.40.60.810 20 40 60 80 100 12000.10.20.30.40.50.60.70.80.9100.20.40.60.810 10 20 30 40 50 6000.10.20.30.40.50.60.70.80.9100.20.40.60.81(c)(a) (b)(d)(e) (f)Figure 5.39: Left: Mean concentration profiles for 3 displacement flows atβ = 85◦ with a) χ < χc, c) χ = χc and e) χ > χc. Right: Meanconcentration profiles (left axis) and flux of light fluid layer qL (rightaxis) for the displacement flows on the left. The dashed and solidlines correspond to t = [57.75,63] in (b), t = [19.73,22.26] in (d) andt = [26.77,29.4] in (f), respectively.159uH and α, and then followed the same procedure to find χc = 112.50.We now look at three numerical simulations with χ values below, equal andabove the critical χc, and check the flux qL at the trailing front. The resultsare plotted in Fig. 5.39. The first displacement (Fig. 5.39a-b) corresponds toan instantaneous displacement with χ = 108.6 < χc. The flux of the light fluidlayer qL has a positive value at the trailing front, meaning uL > 0 as C¯ = α→ 1.Therefore the light fluid is being washed away from the top wall of the pipe. In thesecond case χ = χc = 112.5 (Fig. 5.39c-d) and qL ≈ 0 as C¯ = α→ 1. Finally, withχ = 217.2 > χc (Fig. 5.39e-f) the flux is negative qL < 0 as C¯ = α→ 1, meaningthe trailing front is moving back against the flow (sustained back-flow). For thiscase the interface velocity is zero at x ≈ 22 where ∂qL/∂x = 0.5.3.6 Summary of resultsWe presented the results from full numerical simulations of density-unstable dis-placement flows. Several features of these flows that were observed in the ex-periments were successfully predicted with the numerical model. We focusedon two-layer displacement flows with varying degree of instabilities. A numberof simulations were completed at different flow regimes identified in Chapter 3.The features of the flow were investigated by looking at spatio-temporal diagrams,mean concentration profiles, 2D and 3D iso-contours of concentration, velocitystreamlines and vectors.First, with a series of simulations at β = 75◦ the stabilizing effect of the imposedflow was observed. At Ret cos β = 126 interfacial instabilities appear at low Fr . Byincreasing the imposed velocity, at an intermediate range of Fr the flow becomesstable. With further increase of Fr the flow becomes inertial and the fluids locallymix at the interface. In the wavy regime, finite amplitude interfacial waves grow atthe interface. When the heavy fluid is more viscous (m < 1) mixing is suppressedand the heavy fluid protrudes into the lighter and less viscous fluid. When m ≥ 1the light fluid protrudes into the heavy fluid. The waves break and form a mixedlayer at the interface.At higher Ret cos β cyclic and inertial flow develop. The laminar-turbulentcycles in the cyclic regime were studied in detail using the approach used in lock-160exchange flows [126]. It was found that right before onset of instabilities the velocitydifference between the fluids layers reaches a maximum, where Re∗ ≈ 3500. Thisresults in a burst of turbulence, where the velocity difference starts to decrease andturbulence strength increases. At this point Re∗ ≈ 2400. Then a relaminarizationperiod followswhere the fluids restratify and Reynolds number drops to Re∗ ≈ 1000.Although in the simulation one cycle was resolved, the ramp-cliff pattern of ∆u∗suggests similar cycles will appear at longer times.We looked at the flow field for different front types observed in the experiments.At low Fr the front shape is similar to gravity heads. A stagnation point existsjust below the foremost point of the tip. The light fluid going below the head issqueezed out in the azimuthal direction. This secondary motion shapes the sideroll-ups at the front. Higher Fr / 1 marks the transition from shock to slump typefronts, where the front moves in an unsteady motion. The light fluid remainingbelow the tip becomes unstable and vortices appear in the streamlines. At higherFr the front becomes stretched and slightly raised from the bottom of the pipe. Inall front types significant secondary flow develops ahead and along the tip, whichbreak the 1D flow assumption.Similar to density-stable simulations, the results here show excellent agreementwith the experiments. Several directions can be taken in the future with fullnumerical simulations, including improvements to predictive simplified models,and predicting the transition to diffusive and turbulent flows.161Chapter 6Conclusions and future workIn this thesis we studied displacement flow of two miscible fluids with densityand viscosity contrasts in an inclined pipe, using experimental, computational andmathematical approaches. In terms of density difference, both density-stable anddensity-unstable displacement were considered. In this chapter we will summarizethe key findings in both flows and the contributions from this thesis. We thenconclude by noting the limitations of the present study and future directions thatcould be undertaken to improve our understanding of displacement flows.6.1 Key findings in density-unstable flowsAt sufficiently large Ret cos β buoyancy is strong enough to cause rapid mixing ofthe fluids across the pipe. As discussed in Chapter 1, these flows can be adequatelydescribed using Taylor dispersion type models. Therefore this thesis was focusedon two-layer stratified flows, i.e., flows with weak or local transverse mixing.In Chapter 2 we developed an inertial two-layer model to describe these flows.In contrast to lubrication/thin-film models where inertia is neglected, buoyancyand inertia were retained, and the wall and interfacial stresses were modelled assource terms. By omitting the pressure gradient, the problem was reduced to twoequations: momentum and mass conservation of the heavy fluid layer. Artificialdiffusion was used to filter out unphysical instabilities in the solution.First, by solving the equilibrium equation (balance of buoyancy and friction162terms), accuracy of the stress closure was checked by comparing against exactstresses for different interface heights and viscosity ratios. Critical values of χ,above which back-flow occurs, were also found for different viscosity ratios. Bynumerical solution of the model, back-flow and instability onset predictions weremade for two-layer displacements with viscosity ratio.For iso-viscous (m ≈ 1) and viscosity stable (m < 1) displacement flows it waspredicted that the flow becomes unstable above a relatively constant Ret cos β.For iso-viscous flows this critical values was found to be Ret cos β = 50 whichcoincides with viscous/inertial transition for exchange flows. The most notableeffect of viscosity ratio is observed when the less dense fluid is more viscous(m > 1). In these flows, stable interfaces are confined to a small region of small Frand Ret cos β. Regarding back-flow, it was found that above Fr ≈ 1 the lubricationprediction, i.e., sustained back-flow for χ > χc, does not hold.In Chapter 3 we presented the experimental results for two-layer density-unstable displacement flows. Moderate viscosity ratios were achieved by makingdilute xanthan gum solutions. The two-layer displacement flows were mapped in(Fr,Ret cos β)-plane according to interfacial stability and front behaviours for dif-ferent viscosity configurations. In terms of stability, four different flow regimeswere identified:• Stable: In these flows no instability was observed at the interface of thefluid layers. In iso-viscous displacements, stable flows were mostly confinedto Ret cos β < Rec and χ < χc. Interestingly, above Rec = 50 and belowχ = χc both stable and unstable flows were observed. One explanation canbe that instabilities are weak and grow at timescales much longer than ourexperiments. For m < 1 stable flows were only observed for Ret cos β <Rec and χ < χc. Unlike iso-viscous displacements no stable flows wereobserved above Rec ≈ 60. Similar to the prediction of the two-layer model,displacement flows with m > 1 were only stable for small Fr and Ret cos β.• Wavy: Above χc and Rec the interface of the fluid layers becomes unstable.Instabilities appear as finite amplitude waves at the interface. The stabilityof the interface is governed by the competition between stabilizing (trans-verse buoyancy and viscosity) and destabilizing (Reynolds number and shear)163mechanisms. In this flow regime transverse buoyancy is strong enough toprevent significant growth of instabilities up to Ret cos β = 150 ∼ 200.• Cyclic: Above Ret cos β = 150 ∼ 200 and for Fr / 3 the flow is characterizedby recurring laminar-turbulent cycles. Strong counter-current flow betweenthe fluid layers andweak transverse buoyancy destabilizes the flow and causesmixing of the fluids. The mixing is strong enough to cut off the heavy fluidlayer completely and form a secondary front. After these mixing events theflow becomes laminar with a mixed layer with density gradient separatingpure heavy and light fluid layers.• Inertial: With increasing Fr , counter-current flow is suppressed and the flowbecomes more stable compared to the cyclic regime. Due to the high velocityof the heavy layer and inertia of the flow, the interface becomes unstableand the fluids locally mix. However, mixing is not strong enough to cut offthe heavy fluid layer completely. For displacements with m ≤ 1, at smallRet cos β these flows develop only for large Froude numbers (high imposedvelocities), and at Ret cos β > 150 ∼ 200 for Fr ' 3. For m > 1, however,inertial flows develop at Fr ' 2.5 even for small Ret cos β. This is most likelydue to high front velocities in these flows compared to displacements withm ≤ 1.For all viscosity ratios, sustained back-flow (constant flow of the light fluidagainst the flow direction) only occurs for χ > χc and Fr / 1. Above Fr ≈ 1temporary back-flow or strong counter-current flow between the fluid layers canexist. This again confirms the finding of the two-layer model that the lubricationprediction (χc) is only valid for small Fr . In terms of stability, it should be notedthat in the two-layer model only average velocities are considered in the fluid layers.Therefore one of the destabilizing effects of viscosity contrast, which is the jumpin the velocity gradient at the interface, cannot be predicted.It is interesting to view the above features in the (Fr,Ret cos β)-plane and in thecontext of earlier PhD theses on these flows. Figure 6.1 shows the results from [123]and [4], together with our results. First we see an evolution in terms of parameterspace studied. Secondly, because here we have focused on deeper understanding ofthe flows, we report in more depth the different flow types; see Fig. 6.1c.164Displacement efficiency is significantly affected by the viscosity ratio. Ingeneral in two-layer flows viscosifying the displaced fluid results in larger frontvelocities (worse efficiency), and viscosifying the displacing fluid results in smallerfront velocities (better efficiency). However, predicting the effect is not so simplesince by viscosifying one of the fluids Ret cos β changes too. In the (Fr,Ret cos β)-plane displacement flows withm > 1 have higher front velocities compared tom = 1over the whole range of parameters. The inverse is true for m < 1.In §5.3 several features of density-unstable displacement flows were studiedusing three-dimensional OpenFOAM simulations. A detailed analysis of a dis-placement flow in the cyclic regime revealed that a ramp-cliff pattern in the velocitydifferent between the fluid layers is closely related to laminar-turbulent cycles thatdevelop in these flows. In the laminar periods (ramp phase) the velocity differenceand hence the Reynolds number increases until the interface becomes unstable andthe turbulent period (cliff phase) begins. In this period density gradient and velocitydifference drop. Turbulence strength increases to amaximum value before droppingdown. At the end of the turbulent phase Reynolds number drops to Re∗ ≈ 1000 andthe next ramp phase begins.The different front behaviours that were identified in the experiments werestudied using numerical simulations. This was done by plotting the streamlinesin a reference frame moving with the tip. Five main front structures develop intwo-layer displacement flows:• Shock: At small Fr the front shape is similar to gravity-current heads. Plot-ting the streamlines reveals that a stagnation point exists between the foremostpoint of the tip and the bottom wall. Small amounts of the light fluid areover-run by the heavy fluid. A strong secondary flow develops which bringsthe light fluid up in the azimuthal direction. This motion causes side roll-upof the moving front, a feature exclusive to the pipe geometry. Secondary flowvelocities can be more than 50% of the front velocity magnitude.• Unsteady: By increasing Fr the stagnation point rises and more of the lightfluid goes below the heavy layer. Vortices develop below and above the frontand the tip moves from the center of the pipe to the bottomwall in an unsteadymotion. This marks a transition from steady shock to slump type fronts at165Diffusive/MixedSB TB CCLocal mixingInterfacial instabilitiesStableTurbulentc(m)Rec(m)7.2. Displacement in pipes0 1 2 3 4 5 6020406080100120Vˆ0Vˆt≡ FrVˆνcosβVˆt≡RecosβFrβ = 83oβ = 85oβ = 87ov1v2v3i1i2χ = χcFigure 7.10: Classification of our results for the full range of experimentsin the first and second regimes (Re < 2300) in Table 7.1: sustained backflow (•, §), stationary interface (.), temporary back flow (J, /) and instan-taneous displacement (•). Data point with filled symbols are viscous andwith hollow symbols are inertial. The horizontal bold line shows the firstorder approximation to the inertial-viscous transition (Ret cosØ = 50, from[135]). The dotted line and its continuation (the heavy line) represent theprediction of the lubrication model for the stationary interface, ¬ = ¬c. Thevertical dashed-line is Vˆ0/Vˆt = 0.9. The thin broken lines are only illustra-tive and show an estimate for the turbulent shear flow transition, implyingto the third fully mixed regime. These are based on Re = 2300. Regionsmarked with vj (j=1,2,3) and ij (j=1,2) are explained in the main text.where they limit the velocity of the trailing front moving upstream. In-ertial eÆects are also significant local to the leading displacement front,where they usually appear in the form of an inertial bump. However, inthe bulk of the flow energy is dissipated by viscosity. The front velocitycan be well predicted by (7.13).(d) Viscous temporary back flow regime: These flows are found in aregime bounded by (7.17) and Fr = Vˆ0/Vˆt . 0.9, marked by by v2 inFig. 7.10. As with regime i2 this regime is transitionary showing a pro-gressive change from exchange-dominated to imposed flow-dominated1607.2. Displacement in pipes0 1 2 3 4 5 6020406080100120Vˆ0Vˆt≡ FrVˆνcosβVˆt≡RecosβFrβ = 83oβ = 85oβ = 87ov1v2v3i1i2χ = χcFigure 7.10: Classification of our results for the full range of experimentsin the first and second regimes (Re < 2300) in Table 7.1: sustained backflow (•, §), stationary interface (.), temporary back flow (J, /) and instan-taneous displacement (•). Data point with filled symbols are viscous andwith hollow symbols are inertial. The horizontal bold line shows the firstorder approximation to the inertial-viscous transition (Ret cosØ = 50, from[135]). The dotted line and its continuation (the heavy line) represent theprediction of the lubrication model for the stationary interface, ¬ = ¬c. Thevertical dashed-line is Vˆ0/Vˆt = 0.9. The thin broken lines are only illustra-tive and show an estimate for the turbulent shear flow transition, implyingto the third fully mixed regime. These are based on Re = 2300. Regionsmarked with vj (j=1,2,3) and ij (j=1,2) are explained in the main text.where they limit the velocity of the trailing front moving upstream. In-ertial eÆects are also significant local to the leading displacement front,where they usually appear in the form of an inertial bump. However, inthe bulk of the flow energy is dissipated by viscosity. The front velocitycan be well predicted by (7.13).(d) Viscous temporary back flow regime: These flows are found in aregime bounded by (7.17) and Fr = Vˆ0/Vˆt . 0.9, marked by by v2 inFig. 7.10. As with regime i2 this regime is transitionary showing a pro-gressive change from exchange-dominated to imposed flow-dominated1604.3. Regime classification and leading order approximations0 2 4 6 80200400600800FrRecosβ/FrFigure 4.17: Classification of our results for the full range of experiments,presented in the (Fr,Re cos β/Fr)-plane: (i) instantaneous displacementflows are colored in blue and non-instantaneous flows in red; (ii) fully dif-fusive flows have no superposed symbol; (iii) non-diffusive flows are markedas viscous (superposed circles) or inertial (superposed squares). The heavyline represents the p ediction f viscous b ck flows, from the lubricationmodel in [130], (χ = χc = 116 32). The thick broken li e representsRe cos β/Fr = −50Fr + 500. The point of intersection of the two linesis Fr ≈ 4.62 and Re os β/Fr ≈ 270.81(a) (b)(c)Re tcosRe tcosFrRe tcosxFr FrFigure 6.1: Regime classifications from previous studies: a) n ar-horizontal[123] and b) inclined [4], both for iso-viscous displacement flows.c) Summary of density-unstable displacement flow findings from thisthesis in terms of stability and back-flow. The colors represent stable(l), wavy (l), cyclic (l) and inertial (l). Abbreviations denote sus-tained back-flow (SB), temporary back-flow (TB) and counter-currentflow (CC). The transition to diffusive/mixed regime is for iso-viscousflows (Ret cos β = 500− 50Fr). Variation of χc with m can be foundfrom Fig. 2.28. For iso-viscous flows Rec(m) ≈ 50. For m > 1 the stableregime is confined to small Fr and Ret .166Fr ∼ 1.• Slump: At Fr > 1 the front slumps towards the bottom of the pipe andbecomes stretched. The height and width of the front gradually decreasetowards the tip. A dispersive spike, similar to those observed in density-stable displacements, forms at moves closer to the center of the pipe. Anexplanation for this could be that the weaker strength of the incoming flow(in a moving frame) that cannot retard the moving front and form a gravity-current type head. Note that for Fr > 1 the front velocity values are typicallyVˆ0 < Vˆf < 2Vˆ0 whereas high front velocities Vˆf > 2Vˆ0 occur at Fr < 1.• Mixed: At Ret cos β ' 200 the front is affected by mixing and instabilitiesthat develop in the flow. In numerical simulations, since the flow is not ini-tially disturbed, the front develops similar to aforementioned types dependingon Froude number. At longer times, however, the instability of the flow even-tually affects the front region and causes mixing or complete cutting of thefront from the heavy layer. Because of the mixing the local density contrastat the front drops, and Vˆf /Vˆ0 decreases to values closer to 1.• Central: In viscous near-vertical displacements the front moves at the centerof the pipewith Vˆf ≈ 2Vˆ0. Behind the bullet-like tip, helical motion of the flowmakes the moving finger unstable. Far behind this region the flow becomestwo-layer. This suggests that at least near the front region the inclinationangle has some effects not captured by simply using Ret cos β.The discussion above shows that the front shapes and front velocity valuesare closely related: high Vˆf /Vˆ0 values coincide with shock type fronts and shockvelocities in C¯. This makes front velocity measurements insensitive to the thresholdvalueCt . Conversely, at high Fr , where 1 < Vˆf /Vˆ0 < 2 long spikes form at the frontwhich makes the measured values of Vˆf dependent on Ct , i.e., using a smaller Ctresults in a higher measured Vˆf . In unstable flows periodic cutting of the heavylayer and emergence of secondary fronts makes precise Vˆf measurement morechallenging.1676.2 Key findings in density-stable flowsIn Chapter 4 we presented the results from density-stable displacement flow exper-iments with the same range of flow parameters as density-unstable flows. Density-stable configuration results in markedly different flow regimes compared to density-unstable flows. It was found that when the displaced fluid is more viscous displace-ment efficiency decreases and instabilities grow from residual wall layers.In vertical displacements the wall layers become unstable unless when m 1,i.e., when the displacing fluid is significantly more viscous. Front velocity increasesby increasing Vˆ0 and m. Interestingly when both fluids are shear-thinning the frontvelocities are larger than those of water-water displacements, as long and unstablespikes form at the front. By inclining the pipe, the symmetry in the flow breaksand in addition to thin wall layers, a thicker layer of the heavy fluid remains at thebottom section of the pipe. The thickness of this layer increases with β and Vˆ0and m. At all flow rates the front velocity increases with inclination angle givenβ > 0◦, but the increase is modest for intermediate inclinations, say for β / 60◦,and higher for near-horizontal inclinations β ' 75◦. At these inclinations, efficientdisplacements can be achieved by viscosifying the displacing fluid, even at highvelocities.6.3 Implications for industrial applicationsThe results of this thesis can be used in any application where a fluid-fluid displace-ment occurs. Since this study wasmainly motivated by primary and plug cementingof oil and gas wells, we will point out the key implications of our findings in contextof these processes:• The results of this work and previous studies on displacement flows show howthe flow regimes, stability, mixing and displacement efficiency, all stronglydepend on the dimensionless parameters: Reynolds number, Froude numberand viscosity ratio. As an example, in both primary and plug cementingoperations, simply assuming amixed flowwhen pumping a heavy fluid on topa light fluid is wrong. The highly viscous fluids often used in these processesresult in low/moderate Reynolds numbers in many flows, and significantviscosity ratios. Our results suggest that the resulting flow can be two-layer168⇢IncreaseIncrease V0IncreaseµdFigure 6.2: Dependency of dimensionless parameters and the resulting flowregimes on changing physical quantities. The physical quantities shownare density difference (∆ρ), imposed flow (V0), viscosity (µ), inclinationangle from vertical (β) and pipe (casing) diameter (d). The symbols andlines are described in Fig. 6.1.and laminar even at inclinations as low as β = 10◦. Even in strictly verticalpipes, high viscosity can cause the heavy fluid channel through the center ofthe pipe withminimal mixing. The effect of changing the physical parametersin the density-unstable displacement flow is summarized in the flow regimemap in Fig. 6.2. Note that if the inclination angle of the well changes fromvertical to horizontal, β increases (Ret cos β decreases) and the flow regimemight change. However, many wells are cemented in sections with more orless constant β.• From density-stable experiments and simulations we showed that the dis-placement efficiency can strongly depend on viscosity ratio. Assuming inprimary cementing the density-stable interface only exists between the washfluid and the drillingmud, the viscosity ratio ism 1, which can significantlyreduce the displacement efficiency.If the main purpose of the wash fluid is to dilute and thin the drilling mud,the flow rate must be high enough to achieve strong turbulence. If the flow169is laminar the wash fluid will move at the center of the pipe in verticalinclinations. As we go further down the casing and the inclination angleincreases, the wash fluid will move at the top of the pipe, over the drillingmud. If m 1 the interface becomes unstable and some of the drilling mudget washed away from the bottom section of the pipe but the flow is notunstable enough to make up for the decrease in displacement efficiency athigh m.• An important aspect of displacement flows is back-flow of the displaced fluidagainst the flow direction. With both two-layer model and experiments weshowed that sustained back-flow depends on viscosity ratio through χc(m)for Fr / 1. At high Ret cos β sustained back-flow exists for Fr / 1 only.Sustained back-flow coincides with high front velocities, and therefore poordisplacement, i.e., a higher volume of the displacing fluid is required toremove the displaced fluid. Let us assume a rubberized plug is used forthe cement slurry. The fluid sequence ahead of the plug now consists of adensity-unstable flow between the spacer andwash fluids, and a density-stabledisplacement between the wash fluid and the drilling mud. Even if sustainedback-flow (of the wash fluid) does not occur, the trailing front moves slowerthan the mean velocity. Since the plug is moving with mean velocity, itwill eventually catch up with the trailing front. Note that the volume of thespacer fluid is fixed instead of having a constant influx. In time, more ofthe wash fluid accumulates in front of the plug and the heavier spacer fluidmoves forward. Eventually the spacer/wash sequence might get completelyreversed.If a plug is not used, the cement slurry can slump beneath the spacer fluid, andthe spacer fluid beneath the wash fluid or the drilling mud. If these two-layerflows develop, again the leading front at each interface moves faster than thetrailing front of the next interface. As a result, at the end of the flow downthe casing, the flow might consist of 3 or even 4 fluid streams. Even if thesestreams are stably stratified inside the casing, at bottom hole cementing fluidspass through a check valve and other complex geometries in passing into theannulus. Undoubtedly the fluid streams will be mixed at this point.170• In general, increasing the viscosity of the heavier fluidmight be desired for theannulus displacement. However, the overall Reynolds number decreases, andas a result the flow might become stratified. Particularly if the flow regimechanges from diffusive to cyclic the displacement efficiency significantlydecreases. If the flow is in the cyclic regime, it might be better to viscosifythe flow to move to the wavy or stable regime. Then having a viscosity ratioof m < 1 increases the displacement efficiency.• Above all, our results show the complexity of the flow within the casing. Ofparticular relevance is the comment just above, regarding the fate of multi-fluid streams that reach the bottom hole, i.e., theywill be combined andmixedas they pass from inside the casing to annulus. The main industrial point ofunderstanding the casing flow complexity is to be able to estimate the lengthof the casing that contains multiple fluids, either stratified or mixed. This canbe done through flow regime maps, such as Fig. 6.1c. Industrially however,we have to navigate these flow maps by making simple adjustments in fluidcomposition between fluid pairs to adjust rheology, density etc.. Figure 6.2indicates how to navigate these regimes, i.e., the effects of changing densitydifference, viscosity and flow rate. We feel that the industrial implicationsof the work in this thesis will be realized fully when: (a) the flow concepts(regimes) are better understood as part of the common language of cementingdesigns, and (b) when design of the fluids and flow is made to optimizelocation on maps such as Fig. 6.2.6.4 Limitations of the present studyAlthough we have addressed many questions regarding displacement flows withviscosity ratio, there are certainly limitations in the methodology and scope of ourresults.• One of the experimental limitations is the significant disturbance of the flowwhen the gate valve is opened. As we saw in the results, this introduces someuncertainties and transience in the flow. Therefore care should be taken whencomparing the results with theoretical and computational solutions.171• Another limitation regarding the experimental set-up is time and lengthscaleof the experiments. Although the main section of the pipe where the mea-surements were performed was Lˆ ∼ 150dˆ, many features of the flows developor reach steady state at very long times. The initial transient period afteropening the gate valve also adds to this time. For instance, with an imposedvelocity of Vˆ0 = 100mms−1, the front exits the pipe in t = 15 ∼ 30s. Thislimits the maximum flow rates and thus the range of parameters that canbe used. Additionally, some of the instabilities at the interface grow at longtimes, e.g., the regime classification of two-layer density-unstable flows basedon stability can be affected by time scale of the experiments.• To find the transition to the diffusive regime with viscosity ratio, greaterdensity differences must be used. Viscosifying one or both fluids reducesRet cos β. To negate this and achieve high Reynolds numbers Vˆt ∝√∆ρˆmust be increased. This in turn decreases Fr ∝ Vˆ−1t and requires higherimposed velocity Vˆ0. If the fluids are shear-thinning the effective viscositydecreases with Vˆ0. As an example, using the same set-up and fluids we used,at β = 10◦ obtaining a displacement flow with Ret cos β ≈ 500 and m ≈ 4would require At ≈ 0.016. To get an effective viscosity of µˆe ≈ 4 using axanthan gum solution with κˆ = 10mPasn and n = 0.7, the imposed velocitymust be Vˆ0 = 64mms−1 which gives Fr = 1.1. To get Fr = 3 the imposedvelocity must be 174mms−1 which is too fast for the current set-up. Thisjust illustrates the challenges in finding the diffusive/non-diffusive transitionwhen the fluids are more viscous.• A precise criterion for back-flow is not known beyond the viscous regime.In the experiments we found that sustained back-flow occurs for Fr / 1.In both simulations and two-layer model solutions we observed that aboveRet cos β ≈ 50, the back-flow stops for Fr values less than those predicted byχc. For instance, with OpenFOAM simulations with Ret cos β = 126− 243we found that back-flow stops at Fr ≈ 1.1. However to due the limited valuesof Fr covered in the experiments in the Fr = 1−2 range the exact criterioncannot be described.172• Regarding the OpenFOAM simulations, we have to say that although goodagreement with experiments have been observed, high numerical diffusionand dispersion exist in the solutions. This results in thicker, more diffuseinterface between the fluids which may be the reason why the model isunable to predict some of the instabilities observed in the experiments, i.e.,particularly those that results from thin wall layers.• In developing the two-layer model several simplifying assumptions weremade. One of these is the unity shape factors for fluid layers. The sensitivityof the results to this assumption were not checked.6.5 Future directionsBased on the contributions and limitations of this work, the following directionscan be taken• In completing this thesis we performed over 500 experiments to cover a broadrange of parameters for both density-stable and unstable displacements. Thisprevented us from using higher quality and more precise measurement tech-niques such as Particle Image Velocimetry (PIV) and Laser Induced Fluo-rescence (LIF). With our regime classification and different frontal and theinterfacial phenomena observed, PIV/LIF measurements can provide usefulinformation about detail of the flow, and would allow a better comparisonagainst three-dimensional simulations.• Although we used weakly shear-thinning fluids to achieve viscosity ratio,the shear-thinning effects were not studied. An interesting subject wouldbe to repeat some of the experiments using Newtonian fluids with the sameeffective viscosities as xanthan gum solutions, and investigate the shear-thinning effects. Another area of interest is to use highly shear-thinningfluids. However, both studies may require a different scale of flow loop asthe range of shear rates is tied to the annular gap scale and we have practicallimitations on flow rates.• Different stability problems arise in density-stable flows and many of thesewould be interesting to investigate further. In vertical inclinations the stability173problem becomes similar to core-annular flows. Away from vertical and atlow Reynolds numbers two stability problems exist. Stability of the wall-layer around the displacing finger, which is similar to core-annular flows withbuoyancy. The onset of the unique instabilities growing from the bottomlayer can be studied using multi-layer flow stability analysis with buoyancy.• The two-layer model developed here can be extended in different ways. Onepossible direction is to rewrite the model for two concentric layers, i.e., aninertial core-annular flow. This would allow to use the model for verti-cal density-stable displacement flows, and vertical, viscous density-unstabledisplacement flows.• Using the three-dimensional numerical simulations the two-layer model canbe improved in several ways, including corrections for shape factors andimproving the wall and interfacial friction closures. In numerical results,it was observed that always a thin layer of the displaced fluid remains atthe wall. The wall layer thickness increases if the displaced fluid is moreviscous. This can increase the effective friction factor for the heavy layer.For the interfacial stress term, the interface slope effects can be added to theclosure.• The computational model developed in OpenFOAM shows promising resultsin terms of resolving the main features of the displacement flows. Over a 100three-dimensional simulations were completed to first benchmark and thengenerate the results of Chapter 5.First, the model can be easily modified for immiscible flows. Using interfacecompression in the volume of fluids method can make the interface muchsharper. The results can be compared to existing experimental data.Secondly, a more parametric study can be completed to i) find the exactcriterion for onset of sustained back-flow, ii) find the transition from non-diffusive to diffusive flows with and without a viscosity ratio.Thirdly, using the same approach we utilized to study laminar-turbulentcycles, one can compute the turbulence strength and frequency to betterdefine the cycle and inertial regimes, and the transition from one to another.174Finally, the developed model can be used to solve the displacement flowproblem in a plane channel. With the two-dimensional version of the modelwe were able to solve flows in a 20× 1000mm channel, with 100× 5000cells in 2-3 hours using parallel computation. This provides a significantimprovement in accuracy over the existing data in the literature, regardingplane channel displacement flows.175Bibliography[1] K. Alba. Displacement flow of complex fluids in an inclined duct. PhDthesis, 2013. → pages 8, 190, 193[2] K. Alba, S. Taghavi, and I. Frigaard. Miscible density-stable displacementflows in inclined tube. Phys. Fluids, 24(12):123102, 2012. → pages 17,101, 102, 107, 115[3] K. Alba, S. Taghavi, and I. Frigaard. A weighted residual method fortwo-layer non-Newtonian channel flows: steady-state results and theirstability. J. Fluid Mech., 731:509–544, 2013. → pages 22[4] K. Alba, S. Taghavi, and I. Frigaard. Miscible density-unstabledisplacement flows in inclined tube. Phys. Fluids, 25(6):067101, 2013. →pages xiv, xvii, xxi, xxii, xxx, 11, 13, 15, 16, 32, 33, 60, 61, 62, 70, 76, 78,80, 81, 82, 83, 93, 94, 95, 151, 153, 164, 166[5] A. Amiri, F. Larachi, and S. M. Taghavi. Buoyant miscible displacementflows in vertical pipe. Phys. Fluids, 28(10):102105, 2016. → pages 17[6] R. Aris. On the dispersion of a solute in a fluid flowing through a tube.Proceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences, 235(1200):67–77, 1956. ISSN 0080-4630. → pages8[7] R. Balasubramaniam, N. Rashidnia, T. Maxworthy, and J. Kuang.Instability of miscible interfaces in a cylindrical tube. Phys. Fluids, 17(5),2005. → pages 18, 101[8] D. Barnea and Y. Taitel. Stability of annular flow. Int. Commun. Heat MassTransf., 12(5):611–621, 1985. → pages 28[9] D. Barnea and Y. Taitel. Transient-formulation modes and stability ofsteady-state annular flow. Chem. Eng. Sci., 44(2):325–332, 1989. → pages28176[10] D. Barnea and Y. Taitel. Kelvin-Helmholtz stability criteria for stratifiedflow: viscous versus non-viscous (inviscid) approaches. Int. J. Multiph.Flow, 19(4):639–649, 1993. → pages 28, 29[11] D. Barnea and Y. Taitel. Non-linear interfacial instability of separated flow.Chem. Eng. Sci., 49(14):2341 – 2349, 1994. ISSN 0009-2509. → pages 29[12] T. Benjamin. Gravity currents and related phenomena. J. Fluid Mech., 31:209–248, 1968. → pages 22[13] V. Birman, B. Battandier, E. Meiburg, and P. Linden. Lock-exchange flowsin sloping channels. J. Fluid Mech., 577:53–77, 2007. → pages 22[14] V. K. Birman, J. E. Martin, and E. Meiburg. The non-Boussinesqlock-exchange problem. part 2. high-resolution simulations. J. Fluid Mech.,537:125–144, 2005. → pages 22, 23[15] A. Bonzanini, D. Picchi, and P. Poesio. Simplified 1D incompressibletwo-fluid model with artificial diffusion for slug flow capturing inhorizontal and nearly horizontal pipes. Energies, 10(9), 2017. → pages 29[16] Z. Borden and E. Meiburg. Circulation-based models for Boussinesqinternal bores. J. Fluid Mech., 726, 7 2013. ISSN 1469-7645. → pages 22[17] N. Brauner. Liquid-liquid two-phase flow systems. In Modelling andExperimentation in Two-Phase Flow, pages 221–279. Springer, 2003. →pages 35[18] J. W. Brice Jr, B. Holmes, et al. Engineered casing cementing programsusing turbulent flow techniques. Journal of Petroleum Technology, 16(05):503–508, 1964. → pages 3[19] R. E. Britter and P. F. Linden. The motion of the front of a gravity currenttravelling down an incline. J. Fluid Mech., 99(3):531–543, 1980. → pages22[20] M. I. Cantero, J. R. Lee, S. Balachandar, and M. H. Garcia. On the frontvelocity of gravity currents. J. Fluid Mech., 586:1–39, 2007. → pages 23[21] C. Chen and E. Meiburg. Miscible displacements in capillary tubes. part 2.numerical simulations. J. Fluid Mech., 326:57–90, 1996. → pages 9, 18[22] A. W. Cook and P. E. Dimotakis. Transition stages of Rayleigh–Taylorinstability between miscible fluids. J. Fluid Mech., 443:69–99, 2001. →pages 10177[23] M. Couturler, D. Guillot, H. Hendriks, and F. Callet. Design rules andassociated spacer properties for optimal mud removal in eccentric annuli.1990. doi:10.2118/21594-MS. → pages 3[24] B. Cox. On driving a viscous fluid out of a tube. J. Fluid Mech., 14(01):81–96, 1962. → pages 8[25] C. Crowley, G. Wallis, and J. Barry. Validation of a one-dimensional wavemodel for the stratified-to-slug flow regime transition, with consequencesfor wave growth and slug frequency. Int. J. Multiph. Flow, 18(2):249 – 271,1992. ISSN 0301-9322. → pages 28, 44[26] S. D’Alessio, T. Moodie, J. Pascal, and G. Swaters. Gravity currentsproduced by sudden release of a fixed volume of heavy fluid. Stud. Appl.Math., 96(4):359–385, 1996. → pages 28[27] S. D’Alessio, J. Pascal, and T. Moodie. Thermally enhanced gravity drivenflows. J. Comp. Appl. Math., 170(1):1–25, 2004. → pages 28[28] S. Dalziel, P. Linden, and D. Youngs. Self-similarity and internal structureof turbulence induced by Rayleigh–Taylor instability. J. Fluid Mech., 399:1–48, 1999. → pages 10[29] S. B. Dalziel, M. D. Patterson, C. P. Caulfield, and I. A. Coomaraswamy.Mixing efficiency in high-aspect-ratio Rayleigh–Taylor experiments.Phys. Fluids, 20(6):065106, 2008. → pages 10[30] R. M. Davies and G. Taylor. The mechanics of large bubbles rising throughextended liquids and through liquids in tubes. Proceedings of the RoyalSociety of London A: Mathematical, Physical and Engineering Sciences,200(1062):375–390, 1950. ISSN 0080-4630. → pages 12[31] M. de Bertodano, W. Fullmer, A. Clausse, and V. Ransom. Two-FluidModel Stability, Simulation and Chaos. Springer International Publishing,2017. ISBN 978-3-319-44968-5. → pages 28, 42[32] M. Debacq, V. Fanguet, J. Hulin, D. Salin, and B. Perrin. Self similarconcentration profiles in buoyant mixing of miscible fluids in a verticaltube. Phys. Fluids, 13:3097–3100, 2001. → pages 10, 11, 13[33] M. Debacq, J. Hulin, D. Salin, B. Perrin, and E. Hinch. Buoyant mixing ofmiscible fluids of varying viscosities in vertical tube. Phys. Fluids, 15:3846–3855, 2003. → pages 10, 12, 13178[34] C. Di Cristo, M. Iervolino, A. Vacca, and B. Zanuttigh. Influence of relativeroughness and Reynolds number on the roll-waves spatial evolution. J.Hydr. Eng., 136(1):24–33, 2009. → pages 28[35] C. Di Cristo, M. Iervolino, A. Vacca, and B. Zanuttigh. Roll-wavesprediction in dense granular flows. J. Hydr., 377(1):50–58, 2009. → pages28[36] M. d’Olce, J. Martin, N. Rakotomalala, D. Salin, and L. Talon. Pearl andmushroom instability patterns in two miscible fluids’ core annular flows.Phys. Fluids, 20(2):024104, 2008. → pages 19[37] P. G. Drazin and W. H. Reid. Hydrodynamic stability. Cambridgeuniversity press, 2004. → pages 10, 21[38] P. Ern, F. Charru, and P. Luchini. Stability analysis of a shear flow withstrongly stratified viscosity. J. Fluid Mech., 496:295–312, 2003. → pages19, 22, 135[39] A. Etrati and I. Frigaard. A two-layer model for buoyant inertialdisplacement flows in inclined pipes. Phys. Fluids, 30(2):022107, 2018. →pages vi[40] A. Etrati, K. Alba, and I. Frigaard. Two-layer displacement flow of misciblefluids with viscosity ratio: Experiments. Phys. Fluids, 30(5):052103, 2018.→ pages vi, 130[41] J. Fernandez, P. Kurowski, L. Limat, and P. Petitjeans. Wavelengthselection of fingering instability inside hele–shaw cells. Phys. Fluids, 13(11):3120–3125, 2001. → pages 9[42] J. Fernandez, P. Kurowski, P. Petitjeans, and E. Meiburg. Density-drivenunstable flows of miscible fluids in a Hele-Shaw cell. J. Fluid Mech., 451:239–260, 2002. → pages 9[43] W. Fullmer, S. Lee, and M. D. Bertodano. An artificial viscosity for theill-posed one-dimensional incompressible two-fluid model. Nucl. Tech.,185(3):296–308, 2014. → pages 29[44] W. Fullmer, V. Ransom, and M. de Bertodano. Linear and nonlinearanalysis of an unstable, but well-posed, one-dimensional two-fluid modelfor two-phase flow based on the inviscid Kelvin-Helmholtz instability.Nucl. Eng. Des., 268(Supplement C):173 – 184, 2014. ISSN 0029-5493.→ pages 29179[45] C. Gabard and J.-P. Hulin. Miscible displacement of non-Newtonian fluidsin a vertical tube. The European Physical Journal E: Soft Matter andBiological Physics, 11(3):231–241, 2003. → pages 18, 19[46] R. Govindarajan. Effect of miscibility on the linear instability of two-fluidchannel flow. Int. J. Multiph. Flow, 30:1177–1192, 2004. → pages 19, 22[47] R. Govindarajan and K. C. Sahu. Instabilities in viscosity-stratified flow.Annual Rev. Fluid Mech., 46(1):331–353, 2014. → pages 19[48] N. Goyal and E. Meiburg. Miscible displacements in Hele-Shaw cells:two-dimensional base states and their linear stability. J. Fluid Mech., 558:329355, 2006. → pages 9[49] N. Goyal, H. Pichler, and E. Meiburg. Variable density, miscibledisplacements in a vertical Hele-Shaw cell: linear stability. J. Fluid Mech.,584:357–372, 2007. → pages 9[50] Y. Hallez and J. Magnaudet. Effects of channel geometry onbuoyancy-driven mixing. Phys. Fluids, 20:053306, 2008. → pages 23[51] Y. Hallez and J. Magnaudet. A numerical investigation of horizontalviscous gravity currents. J. Fluid Mech., 630:7191, 2009. → pages[52] Y. Hallez and J. Magnaudet. Turbulence-induced secondary motion in abuoyancy-driven flow in a circular pipe. Phys. Fluids, 21:081704, 2009. →pages[53] C. Hartel, E. Meiburg, and F. Necker. Analysis and direct numericalsimulation of the flow at a gravity-current head. part 1. flow topology andfront speed for slip and no-slip boundaries. J. Fluid Mech., 418:189–212,2000. → pages 23, 85, 86[54] A. Hasnain and K. Alba. Buoyant displacement flow of immiscible fluids ininclined ducts: A theoretical approach. Phys. Fluids, 29(5):052102, 2017.→ pages 17[55] A. Hasnain, E. Segura, and K. Alba. Buoyant displacement flow ofimmiscible fluids in inclined pipes. J. Fluid Mech., 824:661687, 2017. →pages 17[56] C. Hickox. Instability due to viscosity and density stratification inaxisymmetric pipe flow. Phys. Fluids, 14(2):251–262, 1971. → pages 18180[57] C. Hirt and B. Nichols. Volume of fluid (VOF) method for the dynamics offree boundaries. J. Comp. Phys., 39(1):201 – 225, 1981. → pages 118, 195[58] A. J. Hogg, M. M. Nasr-Azadani, M. Ungarish, and E. Meiburg. Sustainedgravity currents in a channel. J. Fluid Mech., 798:853–888, 2016. → pages23, 29[59] H. Holmås, T. Sira, M. Nordsveen, H. Langtangen, and R. Schulkes.Analysis of a 1D incompressible two-fluid model including artificialdiffusion. IMA J. Appl. Math., 73(4):651–667, 2008. → pages 28, 29, 46,199[60] H. Hu and D. Joseph. Lubricated pipelining: stability of core-annular flow.part 2. J. Fluid Mech., 205:359–396, 1989. → pages 18[61] H. Huppert. The propagation of two-dimensional and axisymmetricviscous gravity currents over a rigid horizontal surface. J. Fluid Mech.,121:43–58, 1982. → pages 22[62] H. S. Isbin. One-dimensional two-phase flow, Graham B. Wallis,McGraw-Hill, New York (1969). AIChE Journal, 16(6):896–1105, 1970.ISSN 1547-5905. → pages 28, 44[63] R. Issa and M. Kempf. Simulation of slug flow in horizontal and nearlyhorizontal pipes with the two-fluid model. Int. J. Multiph. Flow, 29(1):69–95, 2003. → pages 28, 29[64] D. D. Joseph, M. Renardy, and Y. Renardy. Instability of the flow of twoimmiscible liquids with different viscosities in a pipe. J. Fluid Mech., 141:309–317, 1984. → pages 18[65] P. R. Khalilova, B. E. Koons, D. W. Lawrence, and A. Elhancha.Newtonian fluid in cementing operations in deepwater wells: Friend or foe?2013. → pages 3[66] M. Khodkar, M. Nasr-Azadani, and E. Meiburg. Partial-depth lock-releaseflows. Phys. Rev. Fluids, 2(6), 2017. → pages 23[67] N. A. Konopliv, S. G. Llewellyn Smith, J. N. McElwaine, and E. Meiburg.Modelling gravity currents without an energy closure. J. Fluid Mech., 789:806–829, 2016. → pages 22[68] A. Kurganov and E. Tadmor. New high-resolution central schemes fornonlinear conservation laws and convection–diffusion equations. J. Comp.Phys., 160(1):241–282, 2000. → pages 200181[69] E. Lajeunesse, J. Martin, N. Rakotomalala, and D. Salin. 3d instability ofmiscible displacements in a Hele-Shaw cell. Phys. Rev. Lett., 79:5254–5257, 1997. → pages 9[70] E. Lajeunesse, J. Martin, N. Rakotomalala, D. Salin, and Y. Yortsos.Miscible displacement in a Hele Shaw cell at high rates. J. Fluid Mech.,398:299–319, 1999. → pages 9[71] E. Lajeunesse, J. Martin, N. Rakotomalala, and D. Salin. The threshold ofthe instability in miscible displacements in a Hele-Shaw cell at high rates.Phys. Rev. Lett., 13:799–801, 2001. → pages 9[72] M. Landman. Non-unique holdup and pressure drop in two-phase stratifiedinclined pipe flow. Int. J. Multiph. Flow, 17(3):377 – 394, 1991. ISSN0301-9322. → pages 53[73] A. Lavrov and M. Torsæter. Physics and mechanics of primary wellcementing. Springer, 2016. → pages 3[74] R. LeVeque. Finite Volume Methods for Hyperbolic Problems. CambridgeTexts in Applied Mathematics. Cambridge University Press, 2002. → pages199, 200[75] T. Y. Lin, C. P. Caulfield, and A. W. Woods. Buoyancy-induced turbulentmixing in a narrow tilted tank. J. Fluid Mech., 773:267–297, 2015. →pages 10[76] P. F. Linden and J. M. Redondo. Molecular mixing in Rayleigh–Taylorinstability. part i: Global mixing. Phys. Fluids A: Fluid Dynamics, 3(5):1269–1277, 1991. → pages 10[77] P. F. Linden, J. M. Redondo, and D. L. Youngs. Molecular mixing inRayleigh–Taylor instability. J. Fluid Mech., 265:97–124, 1994. → pages 10[78] C. F. Lockyear, D. F. Ryan, M. M. Gunningham, et al. Cement channeling:how to predict and prevent. SPE Drilling Engineering, 5(03):201–208,1990. → pages 3[79] S. Longo, M. Ungarish, V. D. Federico, L. Chiapponi, and F. Addona.Gravity currents produced by constant and time varying inflow in a circularcross-section channel: Experiments and theory. Advances in WaterResources, 90(Supplement C):10 – 23, 2016. ISSN 0309-1708. → pages23, 29182[80] A. Maleki and I. Frigaard. Axial dispersion in weakly turbulent flows ofyield stress fluids. J. Non-Newt. Fluid Mech., 235:1–19, 2016. → pages 10[81] T. Maxworthy, J. Leilich, J. E. Simpson, and E. H. Meiburg. Thepropagation of a gravity current into a linearly stratified fluid. J. FluidMech., 453:371–394, 2002. → pages 22[82] E. Meiburg, S. Radhakrishnan, and M. Nasr-azadani. Modeling gravity andturbidity currents: Computational approaches and challenges, 2015. →pages 23[83] M. Moyers-Gonzalez, K. Alba, S. Taghavi, and I. Frigaard. Asemi-analytical closure approximation for pipe flows of twoHerschel–Bulkley fluids with a stratified interface. J. Non-Newt. FluidMech., 193:49–67, 2013. → pages xiv, 20, 37, 38[84] E. Nelson and D. Guillot. Well Cementing. Developments in petroleumscience. Schlumberger, 2006. ISBN 9780978853006. → pages 1[85] R. Oliveira and E. Meiburg. Miscible displacements in Hele-Shaw cells:three-dimensional Navier-Stokes simulations. J. Fluid Mech., 687:431–460, 2011. → pages 9[86] P. Petitjeans and T. Maxworthy. Miscible displacements in capillary tubes.part 1. experiments. J. Fluid Mech., 326:37–56, 1996. → pages 9[87] D. Picchi and P. Poesio. Stability of multiple solutions in inclinedgas/shear-thinning fluid stratified pipe flow. Int. J. Multiph. Flow, 84:176 –187, 2016. ISSN 0301-9322. → pages 53[88] D. Picchi and P. Poesio. A unified model to predict flow pattern transitionsin horizontal and slightly inclined two-phase gas/shear-thinning fluid pipeflows. Int. J. Multiph. Flow, 84:279–291, 2016. → pages 22[89] D. Picchi, Y. Manerba, S. Correra, M. Margarone, and P. Poesio.Gas/shear-thinning liquid flows through pipes: Modeling and experiments.Int. J. Multiph. Flow, 73:217–226, 2015. → pages 28[90] D. Picchi, I. Barmak, A. Ullmann, and N. Brauner. Stability of stratifiedtwo-phase channel flows of Newtonian/non-Newtonian shear-thinningfluids. Preprint submitted: arXiv:1706.09209, 2017. → pages[91] D. Picchi, P. Poesio, A. Ullmann, and N. Brauner. Characteristics ofstratified flows of Newtonian/non-Newtonian shear-thinning fluids. Int. J.Multiph. Flow, 97:109–133, 2017. → pages 22183[92] N. Rakotomalala, D. Salin, and P. Watzky. Miscible displacement betweentwo parallel plates: BGK lattice gas simulations. J. Fluid Mech., 338:277–297, 1997. → pages 9[93] T. Ranganathan and R. Govindarajan. Stabilization and destabilization ofchannel flow by location of viscosity-stratified fluid layer. Phys. Fluids, 13:1–3, 2001. → pages 19, 22[94] N. Rashidnia, R. Balasubramaniam, and R. Schroer. The formation ofspikes in the displacement of miscible fluids. Annals of the New YorkAcademy of Sciences, 1027(1):311–316, 2004. → pages 18, 101[95] L. Rayleigh. Investigation of the character of the equilibrium of anincompressible heavy fluid of variable density. Proceedings of the LondonMathematical Society, s1-14(1):170–177, 1882. → pages 10[96] R. Rotunno, J. B. Klemp, G. H. Bryan, and D. J. Muraki. Models ofnon-Boussinesq lock-exchange flow. J. Fluid Mech., 675:1–26, 2011. →pages 22[97] K. Sahu, H. Ding, P. Valluri, and O. Matar. Pressure-driven miscibletwo-fluid channel flow with density gradients. Phys. Fluids, 21:043603,2009. → pages 22[98] K. Sahu, H. Ding, P. Valluri, and O. Matar. Linear stability analysis andnumerical simulation of miscible two-layer channel flow. Phys. Fluids, 21:042104, 2009. → pages 22[99] K. C. Sahu and R. Govindarajan. Linear stability analysis and directnumerical simulation of two-layer channel flow. J. Fluid Mech., 798:889909, 2016. → pages 19[100] Y. Salhi, E. Si-Ahmed, J. Legrand, and G. Degrez. Stability analysis ofinclined stratified two-phase gas–liquid flow. Nucl. Eng. Des., 240(5):1083–1096, 2010. → pages 28[101] C. W. Sauer. Mud displacement during cementing state of the art. 1987. →pages 3[102] J. Scoffoni, E. Lajeunesse, and G. Homsy. Interface instabilities duringdisplacements of two miscible fluids in a vertical pipe. Phys. Fluids., 13(3):553–556, 2001. → pages 19184[103] B. Selvam, S. Merk, R. Govindarajan, and E. Meiburg. Stability of misciblecore–annular flows with viscosity stratification. J. Fluid Mech., 592:23–49,2007. → pages 18, 19[104] B. Selvam, L. Talon, L. Lesshafft, and E. Meiburg. Convective/absoluteinstability in miscible core-annular flow. part 2. numerical simulations andnonlinear global modes. J. Fluid Mech., 618:323–348, 2009. → pages 19[105] T. Séon, J.-P. Hulin, D. Salin, B. Perrin, and E. Hinch. Laser-inducedfluorescence measurements of buoyancy driven mixing in tilted tubes.Phys. Fluids, 18(4):041701, 2006. → pages 10, 12, 13[106] T. Seon, J. Znaien, D. Salin, J.-P. Hulin, E. Hinch, and B. Perrin. Transientbuoyancy-driven front dynamics in nearly horizontal tubes. Phys. Fluids,19:123603, 2007. → pages 20[107] T. Seon, J. Znaien, D. Salin, J.-P. Hulin, E. Hinch, and B. Perrin. Frontdynamics and macroscopic diffusion in buoyant mixing in a tilted tube.Phys. Fluids, 19:125105, 2007. → pages 10, 13, 14, 17[108] T. Seon, J.-P. Hulin, D. Salin, B. Perrin, and E. Hinch. Buoyant mixing ofmiscible fluids in tilted tubes. Phys. Fluids, 16:103106, 2004. → pages 10[109] T. Seon, J.-P. Hulin, D. Salin, B. Perrin, and E. Hinch. Buoyancy drivenmiscible front dynamics in tilted tubes. Phys. Fluids, 17:031702, 2005. →pages xvi, 12, 13, 15, 53, 55, 71[110] D. Sharp. An overview of Rayleigh–Taylor instability. Physica D:Nonlinear Phenomena, 12(1):3 – 18, 1984. ISSN 0167-2789. → pages 10[111] J. O. Shin, S. B. Dalziel, and P. F. Linden. Gravity currents produced bylock exchange. J. Fluid Mech., 521:1–34, 2004. → pages 22[112] J. E. Simpson and R. E. Britter. The dynamics of the head of a gravitycurrent advancing over a horizontal surface. J. Fluid Mech., 94(3):477–495, 1979. → pages 22[113] J. Stuhmiller. The influence of interfacial pressure forces on the characterof two-phase flow model equations. Int. J. Multiph. Flow, 3(6):551 – 560,1977. ISSN 0301-9322. → pages 29[114] B. R. Sutherland. Internal Gravity Waves. Cambridge University Press,2010. → pages 21185[115] B. R. Sutherland, K. Cote, Y. S. D. Hong, L. Steverango, and C. Surma.Non-self-similar viscous gravity currents. Phys. Rev. Fluids, 3:034101,Mar 2018. → pages 22[116] S. Taghavi. From displacement to mixing in a slightly inclined duct. PhDthesis, The University of British Columbia, 2011. → pages 8, 193[117] S. Taghavi and I. Frigaard. Estimation of mixing volumes in buoyantmiscible displacement flows along near-horizontal pipes.Can. J. Chem. Eng., 91:399–411, 2013. → pages 10, 14, 24[118] S. Taghavi, T. Seon, D. Martinez, and I. Frigaard. Buoyancy-dominateddisplacement flows in near-horizontal channels: the viscous limit. J. FluidMech., 639:1–35, 2009. → pages 14, 20[119] S. Taghavi, T. Seon, D. Martinez, and I. Frigaard. Influence of an imposedflow on the stability of a gravity current in a near horizontal duct.Phys. Fluids, 22:031702, 2010. → pages 13, 14, 68, 78[120] S. Taghavi, T. Seon, K. Wielage-Burchard, D. Martinez, and I. Frigaard.Stationary residual layers in buoyant Newtonian displacement flows.Phys. Fluids, 23:044105, 2011. → pages 13, 14, 20, 24, 60, 156, 158[121] S. Taghavi, K. Alba, and I. Frigaard. Buoyant miscible displacement flowsat moderate viscosity ratios and low Atwood numbers in near-horizontalducts. Chem. Eng. Sc., 69:404–418, 2012. → pages 14[122] S. Taghavi, K. Alba, M. Moyers-Gonzalez, and I. Frigaard. Incompletefluid-fluid displacement of yield stress fluids in near-horizontal pipes:experiments and theory. J. non-Newt. Fluid Mech., 167-168:59–74, 2012.→ pages 33[123] S. Taghavi, K. Alba, T. Seon, K. Wielage-Burchard, D. Martinez, andI. Frigaard. Miscible displacement flows in near-horizontal ducts at lowAtwood number. J. Fluid Mech., 696:175–214, 2012. → pages xiii, xxx,13, 14, 15, 17, 60, 70, 82, 83, 93, 95, 151, 164, 166[124] Y. Taitel and A. Dukler. A model for predicting flow regime transitions inhorizontal and near horizontal gas-liquid flow. AIChE Journal, 22(1):47–55, 1976. ISSN 1547-5905. → pages 28[125] L. Talon and E. Meiburg. Plane Poiseuille flow of miscible layers withdifferent viscosities: instabilities in the stokes flow regime. J. Fluid Mech.,686:484–506, 2011. → pages 19186[126] Y. Tanino, F. Moisy, and J.-P. Hulin. Laminar-turbulent cycles in inclinedlock-exchange flows. Phys. Rev. E, 85:066308, Jun 2012. → pages xxviii,142, 143, 145, 146, 161[127] G. Taylor. The instability of liquid surfaces when accelerated in a directionperpendicular to their planes. i. Proceedings of the Royal Society of LondonA: Mathematical, Physical and Engineering Sciences, 201(1065):192–196,1950. → pages 10[128] G. Taylor. The dispersion of matter in turbulent flow through a pipe.Proceedings of the Royal Society of London A: Mathematical, Physical andEngineering Sciences, 223(1155):446–468, 1954. ISSN 0080-4630. →pages 8[129] G. Taylor. Deposition of a viscous fluid on the wall of a tube. J. FluidMech., 10(02):161–165, 1961. → pages 8[130] S. Thorpe. Experiments on instability and turbulence in a stratified shearflow. J. Fluid Mech., 61(4):731–751, 1973. → pages 21[131] A. Ullmann, M. Zamir, S. Gat, and N. Brauner. Multi-holdups in co-currentstratified flow in inclined tubes. Int. J. Multiph. Flow, 29(10):1565 – 1581,2003. ISSN 0301-9322. → pages 53[132] A. Ullmann, M. Zamir, Z. Ludmer, and N. Brauner. Stratified laminarcountercurrent flow of two liquid phases in inclined tubes. Int. J. Multiph.Flow, 29(10):1583–1604, 2003. → pages 35[133] A. Ullmann, A. Goldstein, M. Zamir, and N. Brauner. Closure relations forthe shear stresses in two-fluid models for laminar stratified flow. Int. J.Multiph. Flow, 30(7–8):877–900, 2004. ISSN 0301-9322. A Collection ofPapers in Honor of Professor G. Yadigaroglu on the Occasion of his 65thBirthday. → pages 36[134] M. Ungarish. Two-layer shallow-water dam-break solutions for gravitycurrents in non-rectangular cross-area channels. J. Fluid Mech., 732:537–570, 2013. → pages 23, 29[135] M. Ungarish, Z. Borden, and E. Meiburg. Gravity currents with tailwatersin Boussinesq and non-Boussinesq systems: two-layer shallow-waterdam-break solutions and Navier–Stokes simulations. Environmental FluidMechanics, 14(2):451–470, Apr 2014. ISSN 1573-1510. → pages 23187[136] S. Vanaparthy and E. Meiburg. Variable density and viscosity, miscibledisplacements in capillary tubes. European J. Mechanics-B/Fluids, 27(3):268–289, 2008. → pages 9[137] Z. Yang and Y. Yortsos. Asymptotic solutions of miscible displacements ingeometries of large aspect ratio. Phys. Fluids, 9:286–298, 1997. → pages 9[138] C. Yih. Instability due to viscosity stratification. J. Fluid Mech., 27:337–352, 1967. → pages 22[139] J. Znaien, Y. Hallez, F. Moisy, J. Magnaudet, J. P. Hulin, D. Salin, and E. J.Hinch. Experimental and numerical investigations of flow structure andmomentum transport in a turbulent buoyancy-driven flow inside a tiltedtube. Phys. Fluids, 21(11):115102, 2009. → pages 10[140] J. Znaien, F. Moisy, and J.-P. Hulin. Flow structure and momentumtransport for buoyancy driven mixing flows in long tubes at different tiltangles. Phys. Fluids, 23:035105, 2011. → pages 10[141] E. E. Zukoski. Influence of viscosity, surface tension, and inclination angleon motion of long bubbles in closed tubes. J. Fluid Mech., 25(4):821–837,1966. → pages 8188Appendix AResearch methodologyA.1 Displacement flow-loopThe experiments are carried out in a transparent acrylic pipe with an internaldiameter of dˆ = 19.05mm and length of Lˆ = 4m. The pipe is made of threeseparate pieces. The main section is made of two pieces, each 1.6m long, are putinside fish-tanks and are carefully connected together to avoid disturbance of thefluid flow. which are put inside fish-tanks and are carefully connected together toavoid disturbance of the fluid flow. The acrylic fish-tanks, are filled with a glycerol-water solution to correct for the light refraction due to the acrylic walls. Initially,the pipe is filled with the displaced fluid and a gate-valve separates it from a shorter80cm long pipe, which is initially filled with the displacing fluid. The pipes arefitted in a frame that provides vertical and horizontal supports for the fish-tanks atseveral points along their length and can be tilted to any angle between horizontal(β = 90◦) and vertical (β = 0◦). Additionally, there are internal supports insideeach fish-tank to fix the acrylic pipe and prevent it from bending. All supports aredesigned to allow for visual access to the flow from both front and top.The displacing and displaced fluids are prepared in separate buckets and pumpedinto two acrylic tanks. The tanks are pressurized to 10psi using compressed air, andare used to pump the displaced fluid before starting the experiment, and provide thecontinuous flow of the displacing fluid into the pipe during the experiment. Thisallows feeding the pipe at all angles without the disturbances caused by pumping the189 Cam 1 Cam 2Fluid 1 Fluid 2Frame Fish-tank supportsGate valve MirrorEntrance pipeInternal supportsMain pipeFigure A.1: A simple schematic of the experimental set-up.fluids directly to the pipe. The back pressure is held constant and the flow rate foreach experiment is set by a needle-valve at the outlet. Several solenoid valves andtubes are used in the setup to allow for filling the tanks, feeding the pipe, runningthe experiment, draining and cleaning the flow loop.The flow-loop used in the current study is based on the same set-up usedby K. Alba [1] for iso-viscous displacement flow experiments. However beforestarting the experiments the flow-loop was completely revamped with a numberof improvements in mind. It was noticed that due to temperature changes in theComplex Fluids Lab the pipe was bent on several points. The internal supports inthe fish-tanks were added in the new design to overcome this issue and ensure thepipe would remain straight. No bending in the pipe were observed after completionof the experiments. Another issue in the older set-up was gradual deposition ofink on the pipe walls, which degraded the image quality over time. To avoid this,the pipe walls were cleaned after each set of experiments by circulating water andscrubbed using micro-fiber cloths on a regular basis. The image acquisition and190Figure A.2: An image of the displacement flow-loop set-up inComplex FluidsLab at UBC.processing was also improved which will be discussed below.A.1.1 Flow visualization & image processingTo visualize the flow, black ink is added to the displaced fluid. The pipe is backlitusing LED stripes and light diffusers are used to make the lighting uniform. Twohigh-speed, black andwhite cameras (one for each fish-tank) are used to take imagesof the flow throughout the experiments. First surface mirrors are used to provide atop view of the pipe as well as a side view. We use a rotameter and a needle valveto measure and control the mean flow rate in each experiment.Image processing is conducted using custom scripts written inMATLAB. In theimage processing step the light intensity at each pixel of the photos are translated toa concentration value between 0 and 1. After calibration we find that the transmittedlight intensity varies with concentration as I(C) = ψ exp(αC), where ψ and α arephysical constants. The relation is valid up to a maximum concentration value of1910 200 400 600 800 1000 1200456789Figure A.3: a) Average light intensity as a function of ink concentration. Theink-water solution becomes saturated at 800mg/L. b) The top twoimages are calibration images for the black and clear fluids. The thirdimage an unprocessed snapshot from an experiment. The last imageis processed using the calibration images. The red broken line is thedepth-averaged concentration.Cmax = 623mg/L. In our experiments we use lower concentration for the pureblack (displaced) fluid. At each pixel the concentration can be calculated asC−CminCmax −Cmin =log I(C)− log I(Cmin)log I(Cmax)− log I(Cmin), (A.1)where Cmin and Cmax are concentration values for the transparent (displacing) andthe black (displaced) fluids and I(C) is the is the light intensity correspondingto concentration of C. This relation allows us to determine local normalizedconcentration value without knowing ψ and α. Using the images of the puretransparent and the pure black fluids as reference images, we can translate the lightintensity along the pipe to normalized concentration, average over the depth of thepipe.To achieve highest contrast possible, the concentration of the black ink wasgradually increased until it became saturated and the light intensity vs. concentra-tion relation deviated from (A.1). The maximum concentration of ink was foundto be 800mgL−1 (see Fig. A.3a). Although the cameras are able to write 12-bitimages, i.e., 212 = 4096 gray-scale levels, in practice the difference between thelight intensity of the pure black fluid and pure white fluid were found to use around1923000 levels at most. The following steps were taken to achieve high-quality imagesand reliable data from post-processing:• In the labView software the auto-exposure was turned off under the camerasettings. This ensures that the exposure of the images do not vary fromcalibration to experiment images.• The camera settings were set such that the images were not clipped at eitherend of histogram, meaning no completely black or white pixels existed in thepipe section of the images.• Several iterations on the ink concentration and camera settings were requiredto maximize the dynamic range of the images with the clear and black fluids.• In the calibration process several images were stored from clear and blackfluids before each experiment. The images were then averaged to minimizeillumination noise.A.1.2 Fluid preparationIn previous displacement flow experiments [1, 116], density differencewas achievedusing NaCl, while higher viscosities were obtained using glycerol solutions. How-ever, we found that keeping the density difference small (∆ρˆ/ρˆ < 0.01) while achiev-ing significant viscosity ratio using glycerol was problematic. Small amounts ofglycerol change the density of water significantly, which would require adding largeamounts of NaCl to keep the density difference small. Instead, we found that itwould be more practical to densify the fluids using glycerol and use xanthan gumto achieve a viscosity contrast. A scale with 0.001g accuracy was used to measureglycerol and xanthan gum powder. The solutions were mixed for 10-15 minutesand the densities were measured using Anton Paar - DMA 35N density meter witha resolution of 0.0001gcm3. A thermometer with ±0.1◦ resolution was used torecord the temperature of the fluids.The rheological behaviour of xanthan gum solutions was measured with aMalver Bohlin Gemini HR Nano rheometer. A smooth cone-and-plate geometrywith 40mm diameter, 4◦ cone angle and 30 µm gap at the cone tip, was used for193100 101 10210−310−210−1γ˙(1/s)µ(γ˙)(Pas) 100mg/L200mg/L300mg/L400mg/L1000mg/LFigure A.4: Rheometry data for different concentrations of xanthan gum inwater. All solutions have 3% Glycerol.Xanthan concentration (mgL−1) 100 200 300 400 1000κˆ (mPasn) 3 7 10 28 80n 0.85 0.77 0.7 0.6 0.51Table A.1: Effect of xanthan gum concentration on shear-thinning fluid rhe-ology. Fitted κˆ and n values are based on µˆ = κˆ Ûγn−1.all rheometry and the temperature was controlled by a Peltier system. A shear rateramp was applied, varying over the range of 0.001−100s−1, with 100 data pointsand 400s sweep time. The xanthan gum solutions are modeled as power law fluids,with the effective viscosity written as µˆ = κˆγn−1, where κˆ is the consistency andthe power-law index is n < 1. We use the strain rate range 1−100s−1 to fit the fluidconsistency, κˆ, and power-law index, n, from a log-log plot of the effective viscosityversus strain rate. This range of shear rate covers that found in our experiments. Foreach xanthan gum solution, 3 different rheometry measurements were completed.Sample rheometry data for different concentrations are presented in Fig. A.4. Thecorresponding consistency and power-law index values are listed in Table A.1.194In our experiments we consistently prepare and use two xanthan gum solutionswith concentrations 100 and 300mgL−1. The shear-thinning effects in the lowconcentration solutions are minimal (κˆ ≈ 3mPasn and n ≈ 0.85) and the higherconcentration is also only weakly shear-thinning (κˆ ≈ 10mPasn and n ≈ 0.7). ForNewtonian fluids m = µˆL/µˆH is defined as the viscosity ratio. For shear-thinningfluids we use an effective viscosity defined asµˆe,k = κˆk(3nk +14nk)nk (8Vˆ0dˆ)nk−1, (A.2)for each fluid and use m = µˆe,L/µˆe,H as the viscosity ratio. The mean effectiveviscosity used for Reynolds numbers (3.1 and 3.2) is defined as ˆ¯µ = (µˆe,L µˆe,H )1/2.With these definitions and the range of flow rates given in Table 3.1, in Newtoniandisplacements the effective viscosity of the more viscous fluid is typically µˆe ≈2.7mPas at Fr = 0.4 and µˆe ≈ 1.9mPas at Fr = 4.3. For the shear-thinningexperiments, the effective viscosity is about µˆe ≈ 8.5mPas at the lowest Fr andµˆe ≈ 4.2mPas at the highest.A.2 OpenFOAM simulationsThe numerical simulations are implemented and solved using OpenFOAM. Thesolver used is based on the twoLiquidMixingFoam case in the OpenFOAM li-brary. The governing equations are the full Navier-Stokes equations, and the twoincompressible liquids are modeled using Volume of Fluids (VOF) method, firstproposed by Hirt and Nichols [57]. In this method the volume of a fluid in a cellis computed as CVcell, with C being the phase fraction in the cell and Vcell is thevolume of the cell. The momentum and continuity equations are∂ρU∂t+∇ · ρUU = ∇p+∇ ·2µS+ ρg, (A.3)∇ ·U = 0, (A.4)195where S = [(∇U)+ (∇U)T ]/2 is the mean rate of strain tensors and ρ and µ are themixture density and viscosity found fromρ = Cρ1 + (1−C)ρ2, (A.5)µ = Cµ1 + (1−C)µ2, (A.6)withC ∈ [0,1] being the fluid 1 phase fraction. The transport equation for the phasefraction C is∂C∂t+∇ ·CU+∇ · [C(1−C)Ur ] = 0. (A.7)Note that we have set the diffusion term as zero. The molecular diffusion coefficientwas also set to zero, giving infinite Schmidt number (Sc). The diffusion in thenumerical solution is only due to mesh size and numerical schemes. The additionalterm ∇ · [C(1−C)Ur ] in (A.7) is a compression term, with Ur being the relativevelocity between the two-phases, also called as compression velocity. It can beadded to the classical transport equation to compress the interface and result in asharper interface between the two phases. Inclusion of C(1−C) ensures this termis only active at the interface.A.2.1 Transport modelsFor the rheology of the fluids, constant viscosity and power-law models were usedfor Newtonian and shear-thinning fluids respectively. For Newtonian fluids we setconstant values for dynamic viscosity µˆH ,L of the fluids. For shear-thinning fluids,the viscosity can be modelled as power-law µˆ = κˆγn−1.A.2.2 Geometry & meshFigure A.6 shows the mesh used in a cross-section of the pipe. The mesh wasgenerated by creating five blocks using the blockMesh utility. In the coarsestcase the block in the center has 20× 20 cells, and the four blocks have 20× 12cells, which get gradually finer at the wall, resulting in 1360 cells at each cross-sectional slice. Non-orthogonality of the mesh can introduce significant errors inthe solution. Maximum non-orthogonality of the mesh used was smaller than 20,which is acceptable. The axial mesh size of δ xˆ = 1 ∼ 1.5mm or δx = 0.05 ∼ 0.08196Lˆdˆ = 19.05mmFigure A.5: Mesh topology for simulations in a pipe. Pipe diameter is thesame as the experimental set-up.was used so that long aspect ratios are avoided.A.2.3 Initial & boundary conditionsThe boundary conditions used are as follows. For velocity, no-slip conditions wereset at the pipe wall. At the inlet fully-developed Poiseuille profile was used and atthe outlet zero-gradient condition. For phase fraction (C), constant value of 1 wasused at the inlet and zero-gradient everywhere else. For the initial condition, we setC = 1 at a length Li from the inlet, so that the initial interface between the two fluidswould be at x = Li. Since no back-flow occurs in density-stable displacements thislength was a few diameters from the inlet, typically Li = 5d.A.2.4 SolversGradients are computed using Gauss linear scheme, and for divergence terms weuse Gauss linearUpwind grad(U) for U and limitedLinear01 for C. Two197(a) (b) (c)Figure A.6: The computational grid used in our the pipe cross-section. Thenumber of cell faces in the cross-sections are 1360, 3300 and 4800,respectively.sub-cycles are used for the advection/diffusion equation, with compressive coeffi-cient (cAlpha) of cα = 1. Although using a non-zero cα has significant effect ininterface sharpness when the fluids are immiscible, we found it does not affect theresult in our miscible flows and diffusion at the interface is dominated by the meshsize. The momentum equation is solved using the PIMPLE algorithm with momen-tum predictor and non-orthogonal corrector. The PIMPLE algorithm is a combina-tion of the pressure-implicit split-operator (PISO) and the semi-implicit method forpressure-linked equations (SIMPLE) algorithm. The detail of these methods canbe found in the OpenFOAM user-guide provided on http://www.openfoam.com.The explicit Euler method is used for time marching. The method is first orderaccurate, compared to second order accuracy of Crank-Nicolson method, and isnot unconditionally stable. Therefore the time-step size is limited by the Courantnumber limit. However it has two advantages: it does not require writing thegradients in each time-step and hence uses less storage, and it is bounded. Thetime-step ∆t is dictated by the CFL number. After some iterations, we found themaximum allowable CFL of 0.1 for both momentum and phase fraction solversworks best. Higher CFL values (up to 0.8) were used without numerical difficulty,however to avoid high degree of numerical diffusion we kept the CFL number below0.1 for all cases presented here.The simulations were run in parallel on a custom-built 24-core workstation.Running the code in parallel significantly reduces the CPU time, and the gain in ef-198ficiency increaseswith the number of cells. Thewhole domainwas decomposed into24 sub-domains, each assigned to a single processor. The decomposition and recon-struction of the solution is easily implement in OpenFOAM using decomposeParand reconstructPar utilities.A.3 Numerical solution of two-layer modelIn this section we present the numerical method used to solve the two-layer model.By adding numerical viscosity to the system, we change the hyperbolic equationsof (2.24) to the advection-diffusion equations of (2.64). We follow the method usedin [59] and instead of solving the full equation we separate (2.64) into a advection-source equation and a diffusion equation. The advection-source equation then issame as (2.24)∂∂tV+ ∂∂zF(V) = S(V), (A.8)and the diffusion equation is∂∂tV = E ∂2∂z2V. (A.9)Second-order Strang splitting method [74] is then used to solve (A.8) and (A.9)alternatingly. The method is as follows:1. At time-step n, (A.9) is solved half a time-step to get the solution V˜n+1/2.2. (A.8) is solved for a full time-step with V˜n+1/2 as the initial condition to getthe solution Vn+1/2.3. Finally, the solution at time-step n+ 1 is found by solving (A.9) for a halftime-step using Vn+1/2 as the initial condition.This splitting method is second-order accurate and unconditionally stable, i.e., itdoes not add additional stability restrictions.A.3.1 Advection EquationThe discretized form of (A.8) isVn+1i −Vni∆t+Fni+1/2−Fni−1/2∆x= Sni , (A.10)199where Vi is cell-averaged value at cell i and Fi±1/2 is the flux at boundary betweencells i and i±1. Note that we have changed the coordinate axis in the pipe directionfrom z in the model to x in numerical analysis for convention. The famous first-order central scheme of Lax and Friedrichs (LxF) [74] with explicit forward Eulermethod for time marching has the formVn+1i =12(Vni−1 +Vni+1) − r2[Fni+1 +Fni−1], (A.11)where r = ∆t/∆x. This method is stable if the CFL condition is metCFL = |λ |r ≤ 1. (A.12)It can be shown the numerical diffusivity of the LxF method is νLxF = ∆x2/2∆t =∆x/2r . Although numerical diffusion damps the instabilities, making the solutionstable and helping with the ill-posed problem mentioned in previous sections, theLxF method introduces much more diffusion than is actually required, and givesnumerical results that are typically badly smeared unless a very fine grid is used.Therefore, the robust Total Variation Diminishing (TVD) scheme of Kurganov et al.[68] is used instead. The method used is a second-order central difference schemewith small numerical viscosity. Its advantage over other methods is its ability tocapture shocks in the solution without needing the costly Riemann solvers. Notethat by separating the advective and diffusive parts of the model we intend tointroduce a controlled artificial diffusivity. Thus, using a high-resolution methodwith small numerical diffusivity is desirable when solving the advection equation.In this scheme the fluxes are written asFni±1/2 =12[FRi±1/2 +FLi±1/2]− ai±1/22[VR,ni±1/2−VL,ni±1/2], (A.13)200where FRi±1/2 = F(VR,ni±1/2) and FLi±1/2 = F(VL,ni±1/2). VR,ni±1/2 and VL,ni±1/2 areVR,ni+1/2 = Vni+1−∆x2(Vx)ni+1, (A.14)VL,ni+1/2 = Vni +∆x2(Vx)ni , (A.15)VR,ni−1/2 = Vni −∆x2(Vx)ni , (A.16)VL,ni−1/2 = Vni−1 +∆x2(Vx)ni−1, (A.17)with (Vx)nk being a minmod flux limiter(Vx)nk = minmod{θVni −Vni−1∆x,Vni+1−Vni−12∆x,(1− θ)Vni+1−Vni∆x}. (A.18)Note that 0 ≤ θ ≤ 1 and local propagation wave speeds ai±1/2 are found fromani±1/2 = max{ρ(A(VR,ni±1/2)), ρ(A(VL,ni±1/2))}, (A.19)with ρ(A) := maxj |λj(A)| being the spectral radius of the Jacobian. It can beshown that numerical viscosity of this method is of order of ∼ ∆x3(a(V)Vxxx)x/8,compared to O(∆x2/∆t) of the LxF scheme. The multivariable minmod functionin (A.18) is defined byminmod(x1,x2, ...) =minj{xj}, xj > 0 ∀ jmaxj{xj}, xj < 0 ∀ j0, otherwise.(A.20)A.3.2 Diffusion EquationThe diffusion half-steps are solved implicitly using the Crank-Nicolson scheme.This FTCS method is second-order in both time and space and is unconditionallystable. The discretized form of (A.9) using this scheme becomesVn+1i = Vni +∆t2∆x2E[Vn+1i+1 −2Vn+1i +Vn+1i−1 +Vni+1−2Vni +Vni−1]. (A.21)2010 10 20 3000.20.40.60.810 0.5 1 1.500.20.40.60.81a bFigure A.7: Effect of mesh size on results for a displacement flow withRet cos β = 50, Fr = 1, β = 60◦ and m = 1. Mesh sizes are ∆x = 0.05(#), ∆x = 0.1 (), ∆x = 0.2 (O) and ∆x = 0.5 (M). In all cases artificialdiffusivity is η = 10−3. A large mesh size smears any shocks in the solu-tions and result in smaller frontal shock height and higher front velocity.With a finer mesh, instabilities with shorter wavelengths are resolved,however the artificial diffusivity keeps the problem well-posed.If the only non-zero components of E are η1 = E11 and η2 = E22, the two diffusionequations for each component of V (wH and α in our model) are decoupled andcan be written as QnVn+1 = Rn, where Q is a tridiagonal matrix. Though theresulting equations can be solved using the Thomas algorithm we use the umfpack1 MATLAB solver for sparse matrices.A.3.3 Convergence CheckAs discussed earlier, by adding artificial diffusion to the model we want to filterout instabilities with wavelengths smaller than 1. However, achieving an exactcutoff wavelength is not simple and out of the scope of this work. Instead, knowingthat the cutoff wavelength depends on displacement parameters, mesh size ∆x andartificial diffusivity η, we used one ∆x and η in all of our numerical solutionsfor consistency. To ensure that the results are convergent, we solved a number of1umfpack is a set of routines for solving unsymmetric sparse linear systems (http://www.suitesparse.com/).2020 10 20 3000.20.40.60.81Figure A.8: Effect of artifical diffusivity on the results at t = 18. The dis-placement parameters are the same as Fig. A.7. In all cases the meshsize is ∆x = 0.1. Artificial diffusivities are η = 10−3 (#), η = 10−4 (O)and η = 10−5 (M). Diffusivity of η = 10−3 does not smear the shock butdecreases the amplitude of the instabilities. Reducing η beyond 10−4does not change the results since the artifical viscosity becomes smallerthan the numerical diffusivity.unstable displacement flows with shocks in the solution using successfully smallerη and ∆x. One example is presented in Fig. A.7 and Fig. A.8, showing the effectsof varying ∆x and η, respectively. By experimenting with different mesh sizes anddiffusivities, we found that using ∆x = 0.1 and η = 10−3 works very well in terms ofcomputational cost and cutting off short wavelength instabilities without smearingor over-stabilizing the flows.203Appendix BDimensionless GroupsParameter Definition Descriptionβ(◦) Inclination angle from verticalAtρˆH − ρˆLρˆH + ρˆLAtwood numberFrVˆ0(Atgˆdˆ)1/2 Froude numberReˆ¯ρVˆ0dˆˆ¯µReynolds numberRet cos βˆ¯ρ(Atgˆdˆ)1/2dˆˆ¯µcos β Modified Reynolds number (≡ Recos β/Fr)χ2∆ρˆgˆdˆ2ˆ¯µVˆ0cos β Ratio of buoyant to viscous forces (≡ 2Recos β/Fr2)mµˆ2µˆ1Viscosity ratioPeVˆ0dˆDˆmPéclet numberTable B.1: The important dimensionless groups used in this thesis. SubscriptsH,L denote heavy and light fluids, and subscripts 1,2 refer to displacingand displaced fluids, respectively.204
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Displacement flow of miscible fluids with density and viscosity contrast Etrati, Ali 2018
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Title | Displacement flow of miscible fluids with density and viscosity contrast |
Creator |
Etrati, Ali |
Publisher | University of British Columbia |
Date Issued | 2018 |
Description | We study downward displacement flow of buoyant miscible fluids with viscosity ratio in a pipe, using experimental, numerical and mathematical approaches. Investigation of this problem is mainly motivated by the primary cementing process in oil and gas well construction. Our focus is on displacements where the degree of transverse mixing is low-moderate and thus a two-layer, stratified flow is observed. An inertial two-layer model for stratified density-unstable displacement flows is developed. From experiments it has been observed that these flows develop for a significant range of parameters. Due to significant inertial effects, existing models are not effective for predicting these flows. The novelty of this model is that the inertia terms are retained, and the wall and interfacial stresses are modelled. With numerical solution of the model, back-flow, displacement efficiency and instability onset predictions are made for different viscosity ratios. The experiments are conducted in a long pipe, inclined at an angle which is varied from vertical to near-horizontal. Viscosity ratio is achieved by adding xanthan gum to the fluids. At each angle, flow rate and viscosity ratio are varied at fixed density contrast. Density-unstable flows regimes are mapped in the (Fr, Re cosß/Fr)-plane, delineated in terms of interfacial instability, front dynamics and front velocity. Amongst the many observations we find that viscosifying the less dense fluid tends to significantly destabilize the flow, for density-unstable configuration. Different instabilities develop at the interface and in the wall-layers. The results are compared to the inertial two-layer model. In density-stable experiments we mostly focus on the effects of viscosity ratio on displacement efficiency and stability of wall-layer. Unique instabilities appear in the case of shear-thinning displacements. Displacement efficiency decreases with increasing viscosity ratio, flow rate and inclination angle. Finally, a number of three-dimensional parallel numerical simulations are completed in the pipe geometry, covering both density-stable and unstable flows. Unsteady Navier-Stokes equations are solved and the Volume of Fluid (VOF) method is used to capture the interface between the fluids. The results give us great insight into several features of these flows that were not available from experiments or 2D simulations. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2018-08-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0371195 |
URI | http://hdl.handle.net/2429/66894 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2018-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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