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From displacement to mixing in a slightly inclined duct 2011

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From Displacement to Mixing in a Slightly Inclined Duct by Seyed Mohammad Taghavi B.Sc., K.N.Toosi University of Technology, 2005 M.Sc., University of Tehran, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemical and Biological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) November 2011 c© Seyed Mohammad Taghavi 2011 Abstract This thesis studies buoyant displacement flows with two miscible fluids in pipes and 2D channels that are inclined at angles (β) close to horizontal. De- tailed experimental, analytical and computational approaches are employed in an integrated fashion. The displacements are at low Atwood numbers and high Péclet numbers, so that miscibility effects are mostly observable after instability and via dispersive mixing. For iso-viscous Newtonian displacements, studying the front velocity variation as a function of the imposed flow velocity allows us to identify 3 distinct flow regimes: an exchange flow dominated regime characterized by Kelvin-Helmholtz-like instabilities, a laminarised viscous displacement regime with the front velocity linearly increasing with the mean imposed flow rate, and a fully mixed displacement regime. The transition between the first and the second regimes is found to be marked by a stationary layer of displaced fluid. In the stationary layer the displaced fluid moves in counter- current motion with zero net volumetric flux. Different lubrication/thin- film models have been used to predict the flow behaviour. We also succeed in characterising displacements as viscous or inertial, according to the ab- sence/presence of interfacial instability and mixing. This dual characteri- sation allows us to define 5-6 distinct flow regimes, which we show collapse onto regions in the two-dimensional (Fr, Re cosβ/Fr)-plane. Here Fr is the densimetric Froude number and Re the Reynolds number. In each regime we have been able to offer a leading order approximation to the leading front velocity. A weighted residual method has also been used to include the effect of inertia within the lubrication modelling approach, which allows us to predict long-wave instabilities. We have extended the study to include the effects of moderate viscosity ratio and shear-thinning fluids. We see many qualitative similarities with the iso-viscous studies. Predictive models are proposed (and compared with experiments and simulations) for the viscous and inertial regimes. Having a significant yield stress in the displaced fluid leads to completely new phenomena. We identify two distinct flow regimes: a central-type dis- placement regime and a slump-type regime for higher density differences. In ii Abstract both regimes, the displaced fluid can remain completely static in residual wall layers. iii Preface In this preface, we briefly explain the contents of the papers that are pub- lished or submitted for publications from the current thesis. We also mention the relative contributions of collaborators and co-authors in the papers. • S.M. Taghavi, T. Seon, D.M. Martinez and I.A. Frigaard. Buoyancy- dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1-35 (2009). This paper provides an analytical solution to a miscible displacement problem in a plane rectangular geometry under conditions of viscous dominance in the presence of strong buoyancy, and when dispersive effects are not present. Under these assumptions, use of the lubrica- tion approximation allows one to solve the kinematic problem with relatively simple manipulations even for fluids of relatively complex rheology. In a close collaboration with I.A. Frigaard, I developed the mathematical model and carried out the numerical solutions for the model. I.A. Frigaard and myself are the primary authors of this paper. T. Seon and D.M. Martinez provided useful comments. They also read the draft and gave corrections. • S.M. Taghavi, T. Seon, D.M. Martinez and I.A. Frigaard. Influence of an imposed flow on the stability of a gravity current in a near hor- izontal duct. Phys. Fluids 22, 031702 (2010). In this publication we report on experimental results concerning the influence of a mean flow superimposed to a classical lock-exchange flow in a nearly horizontal pipe. Three flow regimes are found: (i) for very low imposed flows, the previous lock-exchange results are recovered; (ii) for intermediate imposed flows, a laminarised regime was found; (iii) for very high imposed flows a turbulent regime was observed as expected. I constructed the experimental apparatus and conducted the experiments while T. Seon supervised them. T. Seon and I were primary authors of this study, and I.A. Frigaard and D.M. Martinez supervised the research and provided guidance. iv Preface • S.M. Taghavi, T. Seon, K. Wielage-Burchard, D.M. Martinez and I.A. Frigaard. Stationary residual layers in buoyant Newtonian dis- placement flows. Phys. Fluids 23 044105 (2011). This work deals with the displacement of two miscible fluids of dif- ferent densities in a tilted duct (i.e. pipe and plane channel) with the two fluids initially in a gravitationally unstable configuration. In this work we study the the transition between the first and second regimes (discussed in the previous paper) which is controlled by either the buoyant interpenetration or the imposed flow. We observed that, for some flow rates, the interface between the two fluids is stationary, indicating a zero net flow of the displaced fluid. I conducted the ex- periments supervised by T. Seon. K. Wielage-Burchard assisted with code development through writing the initial version of the computa- tional code. I developed the analytical model, which was proposed by I.A. Frigaard. I wrote this paper in collaboration with T. Seon and I.A. Frigaard. This research was supervised by I.A. Frigaard and D.M. Martinez, who also contributed through several helpful discussions • S.M. Taghavi, K. Alba, T. Seon, K. Wielage-Burchard, D.M. Mar- tinez and I.A. Frigaard. Miscible displacements flows in near-horizontal ducts at low Atwood number. Submitted for publication. In this extensive study we consider buoyant displacement flows with two miscible fluids of equal viscosity in the regime of low Atwood num- ber and in ducts that are inclined close to horizontal. We show that three dimensionless groups largely describe these flows: Fr (densimet- ric Froude number), Re (Reynolds number) and β (duct inclination). We demonstrate that the flow regimes in fact collapse into regions in a two-dimensional (Fr; Re cosβ/Fr)-plane. I.A. Frigaard and I wrote this paper together; the other authors read the draft and provided useful comments and corrections. I conducted the experiments and simulations. I developed the analytical model in collaboration with I.A. Frigaard. T. Seon supervised the experiments and K. Wielage- Burchard helped with code development. K. Alba assisted in devel- oping the weighted residual model approach presented in this paper. I.A. Frigaard and D.M. Martinez supervised the research. • S.M. Taghavi, K. Alba, M. Moyers-Gonzalez and I.A. Frigaard. In- complete fluid-fluid displacement of yield stress fluids in near-horizontal pipes: experiments and theory. Accepted for publication in J. Non-Newton. Fluid Mech. v Preface The paper is a primarily experimental study of displacement of a yield stress fluid from an inclined tube in the situation that the yield stress is strong relative to typical viscous forces. This results in an interesting balance between inertia and buoyancy in yielding the fluid. The main finding is that the type of displacement front observed can be one of two types (central or slump) and that this division depends primarily on the ratio of Reynolds number to densimetric Froude number (also known as the Archimedes number). It is notable that this particu- lar group does not depend on the mean displacement velocity. I.A. Frigaard and I wrote this paper together; the other authors read the draft and provided comments. I conducted the experiments and was assisted by K. Alba. M. Moyers-Gonzalez collaborated through code development of the finite element method used in this paper; I ran the code and produced the results. I.A. Frigaard developed the simple analytical model, which I solved numerically; he also supervised the entire research. • S.M. Taghavi, K. Alba and I.A. Frigaard. Buoyant miscible displace- ment flows at moderate viscosity ratios and low Atwood numbers in near-horizontal ducts. Accepted for publication in Chem. Eng. Sci. In this work, we present results from a study of buoyant miscible dis- placements flows at moderate viscosity ratios in near-horizontal pipes and plane channels. We show that small viscosity ratios lead to more efficient displacements, as is intuitive. In each geometry we find a mix of viscous and inertial flows, in broadly the same pattern as for the iso-viscous displacements studied extensively in our previous works. Predictive models are proposed for the viscous regime, in the case of the plane channel, and for the inertial exchange flow regime, in both geometries. We also study displacement flows with shear-thinning flu- ids, over a more restrictive range of parameters. We show that with an appropriate definition of the effective viscosity the scaled front ve- locities fit well with the results from the Newtonian displacements, in both pipe and plane channel geometries. I.A. Frigaard and I wrote this paper together and K. Alba read the draft and provided useful com- ments. I conducted the experiments and simulations and developed the analyses. K. Alba assisted with the shear-thinning fluid experi- ments. I.A. Frigaard supervised the research. vi Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Problem of study . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Fundamental interest and applications . . . . . . . . 3 1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Primary cementing background . . . . . . . . . . . . . . . . . 6 2.1.1 Industrial relevance (to Canada) . . . . . . . . . . . . 7 2.1.2 Physical process description . . . . . . . . . . . . . . 8 2.1.3 Process challenges . . . . . . . . . . . . . . . . . . . . 11 2.1.4 Studies of primary cementing displacement . . . . . . 13 2.1.5 Engineering design software . . . . . . . . . . . . . . 16 2.1.6 Summary of industrial literature . . . . . . . . . . . . 17 2.2 Associated fundamental problems . . . . . . . . . . . . . . . 18 2.2.1 High Pe miscible displacements . . . . . . . . . . . . 19 2.2.2 Instability and transition to turbulence . . . . . . . . 21 2.2.3 Gravity currents . . . . . . . . . . . . . . . . . . . . . 25 2.2.4 Taylor dispersion . . . . . . . . . . . . . . . . . . . . 36 2.2.5 Effects of Rheology . . . . . . . . . . . . . . . . . . . 37 vii Table of Contents 2.2.6 Summary of fundamental literature . . . . . . . . . . 43 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4 Research objectives . . . . . . . . . . . . . . . . . . . . . . . 46 3 Research methodology . . . . . . . . . . . . . . . . . . . . . . . 50 3.1 Experimental technique . . . . . . . . . . . . . . . . . . . . . 50 3.1.1 Experimental setup . . . . . . . . . . . . . . . . . . . 50 3.1.2 Visualization and concentration measurement . . . . 51 3.1.3 Velocity measurement . . . . . . . . . . . . . . . . . . 56 3.1.4 Fluids characterisation . . . . . . . . . . . . . . . . . 57 3.1.5 Experimental results validation . . . . . . . . . . . . 61 3.2 Computational technique . . . . . . . . . . . . . . . . . . . . 62 3.2.1 Code benchmarking . . . . . . . . . . . . . . . . . . . 65 4 Preliminary experimental results . . . . . . . . . . . . . . . . 69 4.1 Observation of 3 different regimes . . . . . . . . . . . . . . . 69 4.2 Stabilizing effect of the imposed flow . . . . . . . . . . . . . 71 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Lubrication model approach for channel displacements . . 76 5.1 Two-fluid displacement flows in a nearly horizontal slot . . . 77 5.1.1 Constitutive laws . . . . . . . . . . . . . . . . . . . . 79 5.1.2 Buoyancy dominated flows . . . . . . . . . . . . . . . 80 5.1.3 The flux function q(h, hξ) . . . . . . . . . . . . . . . . 84 5.1.4 The existence of steady traveling wave displacements 86 5.2 Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.1 Examples of typical qualitative behaviour . . . . . . . 90 5.2.2 Long-time behaviour . . . . . . . . . . . . . . . . . . 92 5.2.3 Flow reversal and short-time behaviour . . . . . . . . 95 5.3 Non-Newtonian fluids . . . . . . . . . . . . . . . . . . . . . . 100 5.3.1 Shear-thinning effects . . . . . . . . . . . . . . . . . . 100 5.3.2 Yield stress effects . . . . . . . . . . . . . . . . . . . . 104 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6 Stationary residual layers in Newtonian displacements . . 112 6.1 Pipe displacements . . . . . . . . . . . . . . . . . . . . . . . 113 6.1.1 Experimental observations . . . . . . . . . . . . . . . 113 6.1.2 Lubrication model . . . . . . . . . . . . . . . . . . . . 119 6.1.3 Experimental and theoretical comparison . . . . . . . 124 6.2 Plane channel geometry (2D) . . . . . . . . . . . . . . . . . . 126 viii Table of Contents 6.2.1 Lubrication model . . . . . . . . . . . . . . . . . . . . 126 6.2.2 Numerical overview . . . . . . . . . . . . . . . . . . . 127 6.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . 128 6.3 Simple physical model . . . . . . . . . . . . . . . . . . . . . . 135 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7 Iso-viscous miscible displacement flows . . . . . . . . . . . . 142 7.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.1.1 Viscous and inertial flows . . . . . . . . . . . . . . . . 144 7.1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2 Displacement in pipes . . . . . . . . . . . . . . . . . . . . . . 145 7.2.1 Basic flow regimes observed . . . . . . . . . . . . . . 145 7.2.2 Lubrication/thin film model . . . . . . . . . . . . . . 153 7.2.3 Comparison of experimental results and the lubrica- tion model . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2.4 The exchange-flow dominated range . . . . . . . . . . 156 7.2.5 Overall classification of the flow regimes . . . . . . . 157 7.2.6 Engineering predictions and displacement efficiency . 161 7.2.7 Dispersive effects . . . . . . . . . . . . . . . . . . . . 164 7.3 Displacement in channels . . . . . . . . . . . . . . . . . . . . 166 7.3.1 Exchange flow results . . . . . . . . . . . . . . . . . . 167 7.3.2 Displacement flow results . . . . . . . . . . . . . . . . 170 7.3.3 Quantitative prediction of the front velocity . . . . . 176 7.3.4 Overall flow classifications . . . . . . . . . . . . . . . 181 7.4 Inertial effects on plane channel displacements . . . . . . . . 187 7.4.1 A weighted residual lubrication model . . . . . . . . . 187 7.4.2 Inertial effects on front shape and speed . . . . . . . 190 7.4.3 Flow stability . . . . . . . . . . . . . . . . . . . . . . 192 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8 Effects of viscosity ratio and shear-thinning . . . . . . . . . 200 8.1 Displacement experiments in an inclined pipe . . . . . . . . . 201 8.1.1 Range of experiments . . . . . . . . . . . . . . . . . . 201 8.1.2 Newtonian displacement results . . . . . . . . . . . . 202 8.1.3 Shear-thinning displacement flows . . . . . . . . . . . 208 8.1.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.2 Displacement simulations in a channel . . . . . . . . . . . . . 216 8.2.1 Newtonian displacement results . . . . . . . . . . . . 218 8.2.2 Shear-thinning displacement results . . . . . . . . . . 223 ix Table of Contents 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9 Effects of yield stress . . . . . . . . . . . . . . . . . . . . . . . 228 9.1 Scope of the study . . . . . . . . . . . . . . . . . . . . . . . . 228 9.2 Selection of fluids . . . . . . . . . . . . . . . . . . . . . . . . 230 9.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.3.1 The transition between central and slump displace- ments . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.3.2 Central-type displacements . . . . . . . . . . . . . . . 233 9.3.3 Axial flow computations . . . . . . . . . . . . . . . . 239 9.3.4 Slump-type displacements . . . . . . . . . . . . . . . 242 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 10 Conclusions and perspectives . . . . . . . . . . . . . . . . . . 255 10.1 Dynamics of the flow . . . . . . . . . . . . . . . . . . . . . . 255 10.1.1 Flow regimes . . . . . . . . . . . . . . . . . . . . . . . 255 10.1.2 Effects of viscosity ratio and shear-thinning . . . . . . 258 10.1.3 Effects of yield stress . . . . . . . . . . . . . . . . . . 260 10.1.4 Other contributions . . . . . . . . . . . . . . . . . . . 261 10.2 Industrial recommendations . . . . . . . . . . . . . . . . . . 261 10.3 Future perspective . . . . . . . . . . . . . . . . . . . . . . . . 263 10.3.1 Main limitations of the current study . . . . . . . . . 263 10.3.2 LIF, UDV and PIV techniques . . . . . . . . . . . . . 265 10.3.3 Vertical or inclined pipe displacement flows . . . . . . 266 10.3.4 3D numerical simulations . . . . . . . . . . . . . . . . 267 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Appendices A Computing the flux function q(h, hξ) . . . . . . . . . . . . . . 282 A.1 Existence of a velocity solution . . . . . . . . . . . . . . . . . 285 B Monotonicity of q with respect to b . . . . . . . . . . . . . . 286 C Flux functions for 3-layer lubrication model . . . . . . . . . 287 D The coefficients R1...R5 . . . . . . . . . . . . . . . . . . . . . . 288 x List of Tables 2.1 Typical ranges of fluid properties and flow parameters in pri- mary cementing . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Typical ranges of non-dimensional parameters for iso-viscous Newtonian displacements in the pipe . . . . . . . . . . . . . . 11 7.1 Experimental plan. . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2 Numerical simulation plan. . . . . . . . . . . . . . . . . . . . 167 8.1 Experimental range for Newtonian displacements . . . . . . . 201 8.2 Experimental plan for shear-thinning displacements, all con- ducted at β = 85 ◦. . . . . . . . . . . . . . . . . . . . . . . . . 202 8.3 Numerical simulation parameters for Newtonian displacements performed for β = 83, 85, 87 & 89 ◦ and At = 3.5× 10−3. . . 216 8.4 Numerical simulation parameters for shear-thinning displace- ments performed for β = 85 ◦ and At = 3.5× 10−3. . . . . . . 216 9.1 Composition and properties of the displaced fluid used in our experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 xi List of Figures 1.1 Schematic of displacement geometry . . . . . . . . . . . . . . 2 2.1 Schematic of a simplified primary cementing process . . . . . 9 2.2 Principle of the development of Kelvin-Helmholtz instability . 22 2.3 The growth of instabilities at the interface of layer of water and salt water . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Flow of cold air in warm air; shadow pictures showing the profile of a front of a gravity current . . . . . . . . . . . . . . 26 2.5 A schematic diagram of a gravity current . . . . . . . . . . . 27 2.6 Experimental results of a full depth lock-exchange . . . . . . 28 2.7 The dimensionless net energy flux and the Froude number . . 29 2.8 Illustration of three regimes observed through variation of front velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.9 Variation of the normalized velocity V̂f/V̂t as a function of V̂ν cosβ/V̂t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.10 Images of the concentration and swirling strength . . . . . . . 35 2.11 Schematic of the different possible characteristic axial velocity profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.12 A typical interface evolution Yi . . . . . . . . . . . . . . . . . 42 2.13 Schematic illustration of the two types of streamline behavior in displaced fluid . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1 Schematic (top) and real (bottom) views of the experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Schematic view of experimental set-up. . . . . . . . . . . . . . 53 3.3 Variation in logarithmic scale of light intensity across the tube 54 3.4 An image taken by camera #1 of a section of the pipe and corresponding luminous intensity . . . . . . . . . . . . . . . . 55 3.5 Experimental profiles of normalized interface height, h(x̂, t̂) . 56 3.6 Variation of the effective viscosity η̂ with shear rate ˆ̇γ . . . . 60 3.7 Example flowcurve for a visco-plastic solution . . . . . . . . . 61 3.8 Schematic of the computational domain . . . . . . . . . . . . 63 xii List of Figures 3.9 Computational concentration field evolution obtained for β = 85 ◦, At = 3.5× 10−3, ν̂ = 1 (mm2.s−1), V̂0 = 15.8 (mm.s−1), (Re = 300) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.10 Spatiotemporal diagram of the average concentration . . . . . 66 4.1 Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for β = 83 ◦, At = 10−2, µ̂ = 10−3 (Pa.s) . . . . . 70 4.2 Three snapshots of video images taken for different mean flow and showing the flow stability . . . . . . . . . . . . . . . . . . 71 4.3 Sequence of images showing the initial bump shape spread out by the Poiseuille velocity gradient . . . . . . . . . . . . . 73 4.4 Illustration of stabilizing effect of the imposed flow on the waves observed at the interface . . . . . . . . . . . . . . . . . 74 5.1 Schematic of displacement geometry . . . . . . . . . . . . . . 77 5.2 Schematic of displacement types considered . . . . . . . . . . 82 5.3 Examples of q for 2 Newtonian fluids . . . . . . . . . . . . . . 85 5.4 Examples of HL displacements . . . . . . . . . . . . . . . . . 91 5.5 Use of the equal areas rule (5.38) in determining the front height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.6 Front heights for a Newtonian fluid HL and Parameter regime in the (m,χ)-plane . . . . . . . . . . . . . . . . . . . . . . . . 95 5.7 Examples of front shapes in the moving frame of reference for a HL displacement . . . . . . . . . . . . . . . . . . . . . . . . 96 5.8 Profiles of h(ξ, T ) for T = 0, 1, .., 9, 10, with parameters χ = 50, m = 0.1, illustrating flow reversal . . . . . . . . . . . 97 5.9 The similarity solution and comparison with the numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.10 Examples of HL displacements for 2 power law fluids, Bk = 0, χ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.11 Front heights and velocities, plotted against m for a HL dis- placement of 2 power law fluids . . . . . . . . . . . . . . . . . 102 5.12 Profiles of h plotted against ξ/T at T = 10 . . . . . . . . . . 105 5.13 Front heights and velocities, plotted against m, nk = 0 . . . . 106 5.14 Plots of ∂q∂h showing the front positions for parameters . . . . 107 5.15 Maximal static wall layer thickness . . . . . . . . . . . . . . . 108 5.16 Maximal static wall layer Ystatic = 1−hmin when a power-law fluid displaces a Herschel-Bulkley fluid . . . . . . . . . . . . . 109 5.17 An example of sudden movement of static layer . . . . . . . . 110 xiii List of Figures 6.1 Sequence of images showing the stationary upper layer . . . . 114 6.2 Four snapshots of video images taken at different mean flow rates and illustrating the different regimes . . . . . . . . . . . 115 6.3 Spatiotemporal diagrams of the variation of the light . . . . . 117 6.4 Ultrasonic Doppler Velocimeters profiles . . . . . . . . . . . . 118 6.5 Schematic views of the distribution of the two fluids . . . . . 119 6.6 Contours of q(h, 0) and the contour ∂q∂h(h, 0) = 0 (bold black line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.7 Profiles of h(ξ, T ) for T = 0, 1, .., 9, 10, with χ = χc . . . . . 123 6.8 The experimental results in a pipe over the entire range of control parameters . . . . . . . . . . . . . . . . . . . . . . . . 125 6.9 Contours of q(h, 0) and the contour ∂q∂h(h, 0) = 0 (bold black line), in a plane channel displacement . . . . . . . . . . . . . 127 6.10 Sequence of concentration field evolution obtained for β = 87 ◦, ν̂ = 2× 10−6 (m2.s−1), At = 3.5× 10−3 . . . . . . . . . . 129 6.11 Spatiotemporal diagram of the average concentration . . . . . 130 6.12 The velocity profiles corresponding to Fig. 6.10 for a channel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.13 The velocity profile close to the pinned point (with the axial position x̂/D̂ = 26.25) . . . . . . . . . . . . . . . . . . . . . . 132 6.14 Four possible conditions for a viscous buoyant channel flow . 133 6.15 Classification of our simulation results in a channel . . . . . . 134 6.16 Schematic variation of the velocity and V̂0/(V̂ν cosβ) plotted versus (D̂/X̂bff ) tanβ for 2 series of experiments . . . . . . . . 136 7.1 Sequence of images showing propagation of waves along the interface for V̂0 = 40 (mm.s−1) . . . . . . . . . . . . . . . . . 147 7.2 Contours of axial velocity . . . . . . . . . . . . . . . . . . . . 148 7.3 Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for different values of density contrast and viscos- ity at two inclination angles . . . . . . . . . . . . . . . . . . . 149 7.4 Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for different values of density contrast and viscos- ity at β = 85 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.5 A sequence of snapshots from experiments with increased im- posed flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.6 Examples of spatiotemporal diagrams and corresponding UDV measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xiv List of Figures 7.7 Numerical examples of pipe flow displacements based on the lubrication model and variation of the front speeds (solid line) and heights (broken line) . . . . . . . . . . . . . . . . . . . . 155 7.8 Normalized front velocity, V̂f/V̂ν cosβ, plotted against nor- malized mean flow velocity, V̂0/V̂ν cosβ, for the full range of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.9 Normalized front velocity as a function of normalized mean flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.10 Classification of our results for the full range of experiments in the first and second regimes . . . . . . . . . . . . . . . . . 160 7.11 Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for β = 85 ◦, At = 3.5× 10−3, ν = 1 (mm2.s−1) . . 162 7.12 Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for β = 83 ◦, At = 10−2, ν = 1 (mm2.s−1) . . . . . 163 7.13 Comparison between the ratio V̂0/V̂f and the value of the displacement efficiency . . . . . . . . . . . . . . . . . . . . . . 165 7.14 Variation of the normalised stationary front velocity V̂f/V̂t as a function of the inertial Reynolds number Ret cosβ = V̂ν cosβ/V̂t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.15 Variation of the downstream front velocity V̂f as a function of mean flow velocity V̂0 for different inclination angles . . . . 169 7.16 Sequence of concentration field evolution obtained for β = 87 ◦, At = 10−3, ν̂ = 1 (mm2.s−1), V̂0 = 26.3 (mm.s−1) . . . . 171 7.17 Panorama of concentration colourmaps for displacements with ν = 1 (mm2.s−1), each taken at t̂ = 25 (s) . . . . . . . . . . . 172 7.18 Panorama of velocity profiles . . . . . . . . . . . . . . . . . . 173 7.19 Sequence of concentration field evolution obtained for β = 87 ◦, ν̂ = 1 (mm2.s−1), each taken at t̂ = 25 (s) . . . . . . . . 174 7.20 Sequence of concentration field evolution obtained for At = 3.5× 10−3, ν̂ = 1 (mm2.s−1) and V̂0 = 26.3 (mm.s−1) . . . . . 175 7.21 Schematic of the displacement geometry . . . . . . . . . . . . 177 7.22 Results for contours in the 3-layer model for χ = 10 in a h−yi map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.23 Normalized front velocity, V̂f/V̂ν cosβ, as a function of nor- malized mean flow velocity, V̂0/V̂ν cosβ, for the full range of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.24 Normalized front velocity as a function of normalized mean flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 xv List of Figures 7.25 Classification of our results for the full range of simulations in the first and second regimes . . . . . . . . . . . . . . . . . 183 7.26 Front velocity V̂f as a function of mean flow velocity V̂0 for a viscous regime displacement . . . . . . . . . . . . . . . . . . . 185 7.27 Variation of the front velocity V̂f as a function of V̂0 for a sequence of inertial regime displacements . . . . . . . . . . . 186 7.28 Front velocity and shape influences at χ = 0 . . . . . . . . . . 191 7.29 Experimental profiles of normalized h(x̂, t̂) . . . . . . . . . . . 192 7.30 Marginal stability curves for the long-wave limit . . . . . . . 194 7.31 Examples of the spatiotemporal evolution of the interface . . 196 7.32 Stability diagram indicating stable flows (¤) and unstable flows (•) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.1 Experimental results for Newtonian displacements; variation of front velocity V̂f as a function of mean flow velocity V̂0 for At = 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Experimental results for Newtonian displacements; variation of front velocity V̂f as a function of mean flow velocity V̂0 for At = 10−3 at β = 85 ◦ . . . . . . . . . . . . . . . . . . . . . . 204 8.3 Experimental results for Newtonian displacements: contours of front velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.4 Normalized front velocity, V̂f/V̂ν cosβ, plotted against nor- malized mean flow velocity, V̂0/V̂ν cosβ . . . . . . . . . . . . . 206 8.5 Values of normalized front velocity, V̂f/V̂ν cosβ, plotted in a plane of viscosity ratio m versus normalized mean flow veloc- ity, V̂0/V̂ν cosβ . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.6 Normalized front velocity, V̂f/V̂t, as a function of normalized mean flow velocity V̂0/V̂t = Fr . . . . . . . . . . . . . . . . . 208 8.7 Schematic of general behavior in displacements in which one of the fluids is shear-thinning . . . . . . . . . . . . . . . . . . 210 8.8 Experimental results for shear-thinning displacements; varia- tion of front velocity V̂f as a function of mean flow velocity V̂0 at β = 85 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.9 Variation of front velocity V̂f as a function of mean flow ve- locity V̂0 for At = 3.5× 10−3 at β = 85 ◦ . . . . . . . . . . . . 212 8.10 Normalized front velocity, V̂f/V̂ν cosβ, plotted against nor- malized mean flow velocity, V̂0/V̂ν cosβ . . . . . . . . . . . . . 213 xvi List of Figures 8.11 Values of normalized front velocity, V̂f/V̂ν cosβ, plotted in a plane of viscosity ratio m versus normalized mean flow veloc- ity, V̂0/V̂ν cosβ . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.12 Experimental spatiotemporal diagrams obtained to illustrate stabilizing and destabilizing effect of the imposed flow . . . . 217 8.13 Panorama of concentration colourmaps and velocity profiles for displacements with viscosity ratio greater than 1 . . . . . 219 8.14 Panorama of concentration colourmaps and velocity profiles for displacements with viscosity ratio less than 1 . . . . . . . 220 8.15 Simulation results for Newtonian displacements; variation of front velocity V̂f as a function of mean flow velocity V̂0 . . . . 221 8.16 Comparison between the critical value of χ . . . . . . . . . . 222 8.17 Normalized front velocity, V̂f/V̂ν cosβ, plotted against nor- malized mean flow velocity, V̂0/V̂ν cosβ . . . . . . . . . . . . . 223 8.18 Normalized front velocity, V̂f/V̂ν cosβ = 2Vf/χ, from our nu- merical experiments for all viscosity ratio simulations . . . . . 224 8.19 Simulation results for shear-thinning displacements; variation of front velocity V̂f as a function of mean flow velocity V̂0 . . 225 8.20 The critical value of χc predicted by the lubrication model at long times for m = 1 . . . . . . . . . . . . . . . . . . . . . . . 225 8.21 Normalized front velocity, V̂f/V̂ν cosβ = 2Vf/χ . . . . . . . . 226 9.1 Classification of our experiments . . . . . . . . . . . . . . . . 232 9.2 Central displacement for β = 83 ◦, At = 3 × 10−3, V̂0 = 32 (mm.s−1) with Carbopol solution A . . . . . . . . . . . . . 234 9.3 Variation of a) C̄(ŷ) and b) C̄(x̂) in the rectangular region . . 235 9.4 Wavelength content (power spectrum) of C̄(x̂) versus inverse wavelength 1/Λ̂ and reconstruction of interface through in- verse of DFFT . . . . . . . . . . . . . . . . . . . . . . . . . . 236 9.5 Central displacement for β = 85 ◦, At = 4 × 10−3 and V̂0 = 44 (mm.s−1), with Carbopol solution C . . . . . . . . . . . . 237 9.6 An example of central displacement . . . . . . . . . . . . . . 238 9.7 Contours of the maximal static layer thickness (1 − λi,min), in the BN -φB plane . . . . . . . . . . . . . . . . . . . . . . . 239 9.8 2D computational results with the parameters of the experi- ment shown in Fig. 9.2 . . . . . . . . . . . . . . . . . . . . . . 241 9.9 Variation of measured front velocity V̂f with V̂0 for a sequence of experiments with Carbopol solution C . . . . . . . . . . . . 242 9.10 Displacement of Carbopol C for β = 85 ◦, At = 10−2 at V̂0 = 26 (mm.s−1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 xvii List of Figures 9.11 Displacement of Carbopol solution C for β = 85 ◦, At = 10−2: a) & b) show data for V̂0 = 42 (mm.s−1) . . . . . . . . . . . . 245 9.12 An example of slump-like displacement for which the second front stops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 9.13 Normalized static layer depth dstatic . . . . . . . . . . . . . . 247 9.14 2D computational solution with a horizontal interface . . . . 248 9.15 Maximal static layer depth dmax and fraction of total flow rate flowing in the lower layer . . . . . . . . . . . . . . . . . . 249 9.16 Unsteady slump-like displacement for β = 85 ◦, At = 10−2, V̂0 = 36 (mm.s−1) with Carbopol solution C . . . . . . . . . . 250 9.17 Velocity profiles, w(z, y), obtained though 2D computation . 251 9.18 Variation of Reh versus Re for 3 sets of experiments in slump type displacement . . . . . . . . . . . . . . . . . . . . . . . . 252 xviii Acknowledgements First and foremost, I express my great gratitude to my supervisors Prof. Ian Frigaard and Prof. Mark Martinez for incomparable quality of their supervision during the years of my Ph.D. program. I wish to express my deepest thanks to Prof. Ian Frigaard who was always supportive, available, helpful and kind. Working with him truly made these years of research rewarding, pleasant and unique. In addition to plenty of valuable scientific skills, he has taught me many of his superior qualities such as discipline and modesty. I really appreciate his immense knowledge that he shared with me, his impressive patience that he had with me all the time and his strong support in all aspects. I truly owe him a lot and I do not even know how to express my gratitude. I would like to thank Prof. Mark Martinez for his generous support, touching kindness and helpful guidance. He was always enthusiastic about my work and without him my research would never have been as efficient and enjoyable. Working with him was a real pleasurable opportunity. Now I would like to thank a good friend, who was also my excellent supervisor through the experimental part of this study. Dr. Thomas Seon and I had numerous precious discussions, which really helped me understand physical explanations behind the apparent complexity of the phenomena studied. Merci Thomas-J’espre que tu sera toujours heureux et fructueux! I want to thank Dr. Kerstin Wielage-Burchard, who has greatly con- tributed to the progression of this work through her help with numerical code development. I sincerely wish her success and happiness in life. I want to specially thank my great friend Mr. Kamran Alba, who always supported me. He helped me a lot during the progress of my research. His crucial contribution in experimental and analytical part of this study cannot be described in words. Kamran, I will never forget good times we have spent together for research, which at the same time strengthened our friendship. Thank you for everything! I thank Mr. Nicolas Flamant from Schlumberger Oilfield Services com- pany, who provided the opportunity to an internship at Design and Produc- tion research center in Paris. During my 4 month work under his leadership, xix Acknowledgements I became familiar with the industrial aspects of the problem studied. It was a great privilege! I had the pleasure of working with two interns for performing experi- ments. Ms. Krista Thielmann assisted us for 4 months in summer 2009 and Mr. Saman Gharib also worked with us during 8 months from fall 2009 to winter 2010. I wish these two nice, active and talented fellows the best of all. This research was supported financially by Schlumberger and NSERC. This support is gratefully acknowledged. I deeply thank my family whose support and encouragement gave me strength throughout this endeavor. I specially want to thank my lovely little sister, Nazi, who has always been a great source of love to me. I extend special thanks to people who are outside the research group but have largely contributed to the success of this thesis. I thank Dr. Anthony Wachs for reading my thesis and his constructive suggestions. I indeed thank Mr. Amirabbas Aliabadi, who is my excellent friend. His comments about my presentations and also on this dissertation were unexplainably useful. I thank my great life-long friends, Mr. Hamid Javadi and Mr. Mohammad- Amin Alibakhshi, for their support and all people that I have forgotten ... for their help and confidence in me throughout these years. xx Dedication This dissertation is affectionately dedicated to my mother Zahra. Her con- tinuous support, strong encouragement, and constant love have always sus- tained me throughout my life journey. Thank you mom ... and I love you so much! xxi Chapter 1 Introduction 1.1 Problem of study This thesis investigates the forced displacement of one miscible fluid by an- other of different density, initially placed in a density unstable configuration in near-horizontal ducts (pipes and plane channels). Although buoyancy is a significant driving force for all flows we study, there is also a net imposed flow in the downward direction, along the duct. We study the effects of con- stant imposed flows (with mean velocity V̂0), small density ratios (quantified by the Atwood number, At), inclination angles (β), viscosity ratios between the two fluids (m), and the rheology of the fluids. The diameter or width (D̂) of the duct is small compared to its length (L̂). The inclination angle β remains close to horizontal where we expect to find more viscous flows. The fluids used in this study are generalized Newtonian fluids which include Newtonian fluids, shear thinning fluids with a power-law index n and shear thinning fluids with a yield stress τ̂Y . Fig. 1.1 shows a schematic view of our problem geometry.1 We shall see that such problems are common in oceanography, hydrol- ogy, petroleum or chemical engineering, but our main motivation comes from complex displacement flows present in many oil industry processes, concerned with either well construction (drilling, cementing, fracturing) or production (pipelining), as well as in other process industries. Laminar flows often occur in these processes, due to either high viscosities or other process constraints. Non-Newtonian fluids are also prevalent. In many situations it is not feasible to physically separate fluid stages as they are pumped and two practical questions are: (i) to what degree does the fluid mix across the duct; (ii) what is the axial extent along the duct of the mixed region (meaning that in which we find both fluids present)? 1In this thesis, we adopt the convention of denoting dimensional quantities with ˆ symbol (e.g. the pipe diameter is D̂) and dimensionless quantities without. 1 1.1. Problem of study β D̂ 0V̂ Interface Light Heavy Figure 1.1: Schematic of displacement geometry: a heavy fluid displaces a light fluid in a slightly inclined duct with transverse dimension D̂. Direction of the imposed flow, with mean velocity V̂0, is depicted by the arrow. The interface shape between the two fluids is illustrative only. The mixed region in (ii) could consist either of distinct fluid streams that overlap over some length of duct, or could be partially or fully mixed across the duct cross-section. Density unstable displacements are treated as for these flows buoyancy acts to spread the fluids along the duct, i.e. the mixed region is longer than for density stable flows, which means that estimates of the length are more critical. The aim of our study is to present results of an extensive study, targeted at understanding (i) & (ii), which combines the use of experimental, numerical and analytical techniques. With some generality, we can say that two fluid flow problems are com- plex and have not been understood in depth. However in the literature there exist many valuable studies classifying different effects in particular flows, e.g. the effects of small density ratios and inclination angles in the absence of an imposed flow. In addition, there are many works related to displacement flows with viscosity ratios and density differences in vertical ducts and in the absence of inertia. One can also find scattered studies on the influence of rheological parameters. However, buoyant displacement flows in inclined ducts at non-zero Reynolds numbers are not at all comprehensively studied. Based on the studies of miscible exchange flows, we expect distinctly different flows at duct inclinations near to horizontal than in ducts that are close to vertical. This thesis focuses on ducts that are slightly inclined to the horizontal direction. We consider small At flows (At = ρ̂Heavy−ρ̂Lightρ̂Heavy+ρ̂Light ), but with significant buoyancy forces, and study the effects of gradually increasing 2 1.1. Problem of study the imposed flow rate. As well as buoyancy, inertia and (bulk) viscous forces, we study the effects of different rheological properties in the 2 fluids. 1.1.1 Fundamental interest and applications Industrial displacement flows often involve both density and rheological differences between fluids. With buoyancy, there are a number of dis- placement studies in vertical ducts, both for miscible and immiscible flu- ids [84, 91, 92, 126], but here we focus exclusively on near-horizontal inclina- tions which are phenomenologically different, in that near-stratified viscous regimes are more prevalent. Motivation for our study comes from various op- erations present in the construction and completion of oil wells, (e.g. primary cementing, see [103], drilling, gravel-packing, fracturing). These processes often involve displacing one fluid with another or with a sequence of differ- ent fluids. The geometries are typically pipe, annular or duct-like, all with long aspect ratios. Large volumes are pumped so that fluids may be consid- ered separated, i.e. we have a 2-fluid displacement, not an n-fluid displace- ment. A very wide range of fluids are used. Significant density differences of up to 500 (kg/m3) can occur, shear-thinning and yield stress rheologi- cal behaviours are widely found and are often the dominant non-Newtonian effects, (more exotic non-Newtonian effects may also be present). Many different types of industrial displacement flows arise. Turbulent displacing regimes are typically more effective, but are not always possible due to pro- cess constraints; here our flows are laminar. A second distinction comes in the volume of displacing fluid that is used. In some processes an essentially continuous stream can be pumped through the duct, e.g. water in turbo- machinery, and there are few time restrictions. In other processes such as primary cementing, due to either disposal issues or cost of the fluids (e.g. cement slurry), it is desirable to fully displace the in-situ fluid (e.g. drilling mud) with more or less a single “duct volume” of fluid, i.e. we are replacing the in-situ fluid with another. Generally our study highlights this particular displacement regime. The advent of horizontal oil well drilling dates back 4 decades, when several wells were drilled at Norman Wells in Canada, to produce from a reservoir below the MacKenzie River [103]. Horizontal wells can be employed to reach inaccessible oil and gas reservoirs, such as under cities, water, and rugged terrains. These wells give the potential to overcome many challenges imposed by reservoir geometry, fluid characteristics, economic conditions, or environmental constraints. They are also economically of interest as they can produce on average about four times more than vertical wells, primarily 3 1.2. Thesis outline due to increased production area. Thus at present, hundreds of horizontal wells are drilled each year. Environmental considerations are also an issue that horizontal drilling could address under special circumstances. Similar to conventional cementing, mud displacement is a key element to obtain a good primary cement job. In general, displacement flows in close to horizontal ducts are compli- cated to analyze. The problem complexities include the effects of a large number of flow parameters, as well as different configurations and non- Newtonian behaviors. Depending on contributions of these parameters, dif- ferent types of displacement and/or mixing flow can occur, and depending on balances among participating forces, different flow regimes (e.g. iner- tial, viscous) are possible. Thus, it is hard to predict the degree of mixing between two fluids and accurately design fluid volumes needed, fluid proper- ties, and flow rates. In addition, low efficiency displacement flows can lead to contamination of the fluids. This can have a significant impact on well pro- ductivity, destruction of the near well ecosystem, environment pollution and safety hazard. Therefore, there is a strong industrial motivation to better understand these flows. The application of this knowledge in process design would lead to reduced environmental impact and increased productivity. 1.2 Thesis outline The outline of this thesis is as follows. The next chapter (§2) reviews related important papers found in the engineering and scientific literature. At the end of this chapter, we highlight the deficiencies in the literature and also identify the main physical mechanisms related to our study. In addition, we will have a better image about the important fundamental questions that we will attempt to answer throughout this work. We will then lead to a research objective. In Chapter 3 we explain the research methodology in- cluding experimental procedures and devices and computational procedures. In all cases, we describe the methods used for analyzing different data and extract desirable information from them. In Chapter 4 we present the pre- liminary results of our experimental approach. We qualitatively discuss the effect of imposed flow, tilt angle and density ratio for Newtonian miscible displacement flows. Briefly, in this chapter we define 3 different flow regimes. This chapter builds the foundation for the following chapters by providing basic definitions of these different flow regimes observed. All the chapters following §4 are focused on variant aspects of the different regimes explained in this chapter. Chapter 5 is devoted to our simple mathematical analysis, 4 1.2. Thesis outline which analytically studies in depth the second regime. Chapter 6 discusses the transition between the first and second regimes. This transition is as- sociated with an interesting feature of the flow, whereby the displaced fluid layer remains stationary within the pipe (or channel) and the displacing fluid passes underneath. In Chapter 7 we present a clear picture of all the regimes observed in buoyant miscible Newtonian displacements. The argu- ment includes detailed experimental, analytical and numerical discussions. We will also briefly comment on the stability of the flow and its transition. In Chapter 8 we observe the effect of viscosity ratios and shear-thinning especially on the second regime and partially on the first regime. Chapter 9 characterizes the effect of a yield stress on the displacement flows. This thesis concludes in Chapter 10. 5 Chapter 2 Background The present chapter is organized as follows. Firstly, we start with describing the related engineering background of our problem in §2.1. We take as our principal example the primary cementing process. We describe the process and its relevance, then review the industrial literature (loosely speaking cur- rent guidelines or recommended best practices). We close the first section by a summary of the engineering background and discuss the main shortcom- ings. In the second section of this chapter (§2.2), we introduce associated fundamental studies that can pave the road for better understanding our flow problem. Describing these studies is guided by three main goals: (i) What are the limiting cases of our displacement flows? (ii) Can previous research help identify basic mechanisms that contribute towards explaining more complicated features in our specific problem? (iii) What aspects of buoyant displacement flows are still unknown or need to be more investigated? We subtly seek this last goal through describing the previous studies. The second section is closed by a summary of fundamentals in the literature, where we explicitly describe features of displacement flow knowledge that are lacking in the current literature. Section §2.3 briefly concludes the chapter. Section §2.4 presents our research objective. 2.1 Primary cementing background Primary cementing is performed at least once on every well constructed in the world. The process objective is to hydraulically seal oil and gas wells. This increases well productivity, prevents formation fluids from leaking to surface, and lowers the risk of severe environmental and safety consequences. 6 2.1. Primary cementing background 2.1.1 Industrial relevance (to Canada) Canada is the 4th-largest producer of natural gas and the 6th largest pro- ducer of crude oil in the world. The upstream sector is the largest single private sector investor in Canada. Approximately 5,141 oil wells and 3,431 gas wells were drilled in Canada in 2010 [1]. Over the last five years, around 63,250 oil and gas wells have been completed in Canada [1]. Oil and gas industry are also of significant economical importance to Canada. For ex- ample, in 2009, the net cash expenditure of the petroleum industry was around $46.2 billion. Significant contributions are also made to federal and provincial taxes by the industry. The world’s 3 largest oilfield services companies, which all have branches in Canada, are Schlumberger, Halliburton and Saipem. Schlumberger is one of the most technologically focused oilfield service companies in the world and is a key player in Canada. As well as having a large global market share operationally, Schlumberger has a suite of engineering software tools that are used to design different wellbore operations, converting physical understanding into useable engineering practice. This company operates in around 80 countries and has 110,000 employees worldwide. Halliburton was founded in 1919, specialising in cementing oil well walls in Texas, USA. To- day Halliburton has 50,000 employees and operates in around 70 countries. The company provides technical products and services for oil and gas explo- ration and production. Saipem, founded in late 1950s, has made its name handling the oilfield services for a number of challenging projects both on and offshore. It has over 30,000 employees operating in all the major oil and gas producing nations. Although the multi-nationals have a large market share, there also exist Canadian oil well cementing companies, most of which operate in Western Canada, e.g Trican Well Service Ltd and Magnum Cementing Services Ltd. Trican operates in four continents and has corporate headquarters in Cal- gary, Alberta, Canada. This company has performed an annual average of more than 9,200 cementing jobs over the past three years. Magnum has recently started to offer primary cementing services. Although the process objective of primary cementing is to hydrauli- cally seal oil and gas wells, there is strong evidence that this is not always achieved. Since the mid-1990s, the occurrence of leaking wells, also known as “Surface Casing Vent Flows” (SCVF), has received much attention in the industry, see e.g. [46, 104]. Loosely speaking these are wells that show some pressure at surface in the annulus. These wells are not necessarily leaking, as they can also be shut-in and suspended, but some certainly do leak. Equally, 7 2.1. Primary cementing background suspended wells can still have fluids that percolate through the near-surface strata and adversely affect ecosystems. Unfortunately there are few preven- tative solutions but many oil companies are selling post-treatments. Various physical mechanisms may be responsible. This problem is particularly ev- ident in Western Canada where a large proportion of the wells are shallow gas wells. Some statistics have estimated that in Western Canada alone up to 18,000 instances of SCVF have been reported. In some cases, these wells have been required to be shut-in or suspended [107]. Some other reports suggest that around 15% of the wells in Western Canada have SCVF or gas migration that requires testing or repair [28]. In 2002, Dusterhoft et al. [46] surveyed 3 areas in Alberta, Canada, and reported that in Tangleflags 10.5%, in Wildmere 25%, and in Abbey around 80% of wells are leakers. SCVF’s occur elsewhere in the world, although perhaps less well doc- umented. Although the causes are not precisely known, one thing that is clear is that the primary cementing job has failed. Some of the possible causes of failure have fluid mechanic origins and this is a major motivation for further research into the physics of fluid-fluid displacement processes. 2.1.2 Physical process description After a new wellbore is drilled to a desired depth, the drillpipe and bit are removed from the wellbore. A steel casing or liner is run into the well until it reaches the bottom of the well. During the operation, the drilling mud used to remove formation cuttings during drilling the well is still in the wellbore. In primary cementing the casing is cemented into place through pumping a sequence of fluids from surface down the inside of the casing to bottom hole, returning up the annular space between the rock formation and the outer cylinder, see Fig. 2.1. The annulus is initially full of drilling mud (as shown in Fig. 2.1a) that must be removed from the annular space and replaced with a cement slurry (Fig. 2.1c), which later solidifies (Fig. 2.1d). To clean drilling mud from the annulus, which can be eccentric, the cement slurry is preceded by a chemical wash and/or a spacer fluid (Fig. 2.1b). Further process details can be found in [103]. The rheologies and densities of the spacer and cement slurries can be designed in order to aid in displacement of the annulus drilling mud, within the limits of maintaining well security [15]. The fluid volumes are designed so that the cement slurries fill the annular space to be cemented. Drilling mud follows the final cement slurry to be pumped and the circulation is stopped with a few meters of cement at the bottom of the inside of the casing, see Fig. 2.1d. The final part of cement inside the casing is drilled out as the well 8 2.1. Primary cementing background Drilling Mud Wash Cement Slurry (a) (b) (c) (d) Figure 2.1: Schematic of a simplified primary cementing process in an ide- alized case where no mixing occurs between successive fluid stages: (a) the pipe and annulus are initially full of the drilling mud; (b) & (c) a wash or spacer is pumped in the casing followed by one or more cement slurries; (d) cement is allowed to set. 9 2.1. Primary cementing background Q̂ (l/min) ρ̂ (kg/m3) κ̂ (Pa.sn) n τ̂Y (Pa) 300− 3000 900− 2200 0.003− 3 0.1− 1 0− 20 Table 2.1: Typical ranges of fluid properties and flow parameters in primary cementing. Q̂, ρ̂, κ̂, n, τ̂Y respectively denote flow rate, density, consistency, power-law index, and yield stress. These data are collected from Ref. [103]. proceeds. The completed well often has a telescopic arrangement of casings and liners [15, 103]. A liner is a casing that extends downwards from just above the previous casing. In the present day, it is routinely feasible to construct wells with hor- izontal extensions in the 7 − 10 (km) range. Drilling fluids are typically 100 − 600 (kg/m3) lighter than cement slurries. Drilling fluids and cement slurries are usually non-Newtonian and often possess a yield stress. Typi- cally, well inner diameters can start at anything up to 50 (cm) and can end as small as 10 (cm) in the producing zone. Extremes occur outside of these ranges and obviously diameters depend on the local conditions and intended length of the well. Casings and liners are assembled from sections that are typically of length roughly 10 (m) each. The gap between the outside of the casing and the inside of the wellbore is typically 2 (cm). Table 2.1 shows typical ranges of fluid properties and flow parameters in primary cementing From data presented in Table 2.1, we can give typical ranges of non- dimensional parameters for iso-viscous Newtonian displacements in the pipe, as shown in Table 2.2. Inclination angle β can be essentially anything. The Atwood number, At, can increase up to 0.5. The Reynolds number, is al- ways significant, O(10) and larger. Flows are both turbulent and laminar. The Reynolds number quantifies the importance of inertial effects to viscous ones. The densimetric Froude number, Fr, which represents the ratio be- tween inertial forces to buoyant forces, can vary in the range 0.1− 50. The combination Re/Fr2, which shows the ratio between buoyant stresses to viscous stresses, is another non-dimensional parameter that will be referred to in the following chapters, e.g. in Chapters 6 and 7. This parameter can cover a very wide range as seen in Table 2.2. Looking at these non-dimensional groups, we realize that buoyancy is always important, and that flows can be laminar or turbulent at all incli- nations. In reality, we also have other non-dimensional groups2 involved: 2Although elasticity can have importance in some situations, in general the shear rhe- ology is believed to dominate the flows of interest. Hence considering inelastic fluids is 10 2.1. Primary cementing background β ◦ At = ρ̂Heavy−ρ̂Lightρ̂Heavy+ρ̂Light Re ≡ ρ̂V̂0D̂ µ̂ Fr ≡ V̂0√AtĝD̂ Re/Fr 2 0− 90 0.001− 0.5 40− 40000 0.1− 50 0.1− 4× 106 Table 2.2: Typical ranges of non-dimensional parameters for iso-viscous Newtonian displacements in the pipe. The viscosity is denoted by µ̂. viscosity ratio between the fluids (m), ratio of advective to diffusive mass transport (represented by Péclet number, Pe), and various rheological pa- rameters e.g. power-law index (n) and dimensionless yield stress (Bingham number, B). Our dimensional analysis reveals that at least 10 dimensionless parameters govern these displacement flows (note that in considering a dis- placement we must include rheological parameters in a minimum of 2 fluids). None of the parameters is universally negligible. Considering the number and the ranges of the non-dimensional parameters discussed, we conclude that it is simply not possible to study these displacement flows fully in an experimental setting. 2.1.3 Process challenges A large number of problems arise in cement placement and mud removal; see [87, 96, 98, 103, 123]. Two general problems that have a clear fluid mechanic origin are that (i) drilling mud is not completely removed from the annulus (ii) and that the cement slurry is contaminated by the other fluids. In either case, the hydraulic seal of the well is compromised, the well pro- ductivity is diminished, and the environmental and safety hazards of gas leakage to the surface are present. Occasionally cementing companies employ a mechanical plug to avoid mixing. The plug is inserted between the pumped fluid stages and bursts under high pressure when it reaches the bottom of the casing, where it comes to rest. There are machines that allow multiple plugs to be used, separating multiple fluid stages that are pumped, but these are not always available and not popular due to the cost and complexity. In reality in many situations, it is not possible to use any mechanical barrier to separate the fluids. Thus, unless a mechanical plug is used it is not practically possible reasonable, from the perspective of modelling. 11 2.1. Primary cementing background to totally prevent mixing at the interface between two adjacent fluid stages that are circulated down the pipe (at initial stage) and up the annulus (at final stage). Mixing or by-passing of fluid stages in the casing has two consequences: (i) large scale contamination of the cement slurry before it enters the annulus so that it either does not set or sets in the wrong position due to chemical incompatibility; (ii) dilution of additives. Additives are used in the chemical wash to im- prove cleaning of the mud from the walls and are also added to the cement slurries to counter the effects of gas migration. Gas migra- tion occurs during setting of cement, as the cement begins to form a self-supporting structure, which reduces the hydrostatic pressure and allows gas invasion. There are many factors that can directly or indirectly impact the pri- mary cementing process. These are wellbore geometry, mud and cement properties, the pump rate, to name but a few. It is not clear how exactly these parameters can affect the process, especially when applied in combi- nation with one another. In this process, there are also many other sources of uncertainty and imprecision: • Since water is heavy to transport, local supplies might be used with uncertain mineral composition. This is combined with further un- certainty due to receiving solids (chemicals) from different suppliers, perhaps storing in imperfect conditions prior to transporting to the rigsite. Different mixing conditions are also possible due to human error in the execution. All these factors combine to mean that the ac- tual fluid properties at the rigsite might vary considerably from either design values or test values in a field lab. • Process design is based on volumes required to fill the annulus. How- ever, the size of the drilled well may be uncertain, due to drilling into a weak formation and parts of the wellbore being washed out during drilling. It is rare that a calliper is run to determine size. • Exact temperatures at different depths in the wellbore are not mea- sured. Often the procedure involves estimating temperature from the formation temperature in a well that is geographically close by and geo- logically similar. The consequence is that fluid temperature-dependent properties may be different from design values. 12 2.1. Primary cementing background • Properties of the fluid in the well at the beginning of the cementing process are not usually measured. Even if the properties of the original drilling mud were known, the mud has been circulated many times around the well and its polymeric properties (e.g. shear-thinning) will have degraded to some extent. Also the mud will have varying degrees of fine solids suspended in it from the drill cuttings. Finally, the mud has typically sat in the well for a period of hours while the casing is assembled and run into the well. The mud can partially dehydrate in this period and other thixotropic effects can occur, see Ravi et al. [118, 119] for details. 2.1.4 Studies of primary cementing displacement It should be emphasized that the final aim of the primary cementing is to replace mud around the casing in the annulus configuration of the wellbore. Thus the majority of studies in the engineering community have concerned annular displacements. Numerous empirical, computational and analytical studies have been conducted to shed light on primary cementing process; see [16, 33, 103, 122, 127, 147] for general developments. The first research on cement placement process dates back 70−80 years, when some basic key factors affecting primary cement job failures were rec- ognized. For instance in an early work, Jones and Berdine [83] used a large-scale simulator to propose effective ways to displace mud in the an- nulus including fluid jets, scrapers or scratchers, casing reciprocation, and possibly pumping acid ahead of the cement slurry. An important pilot-scale study was carried out by Howard and Clark [76] who found that flow regime of the displacing fluid can affect the mud displacement efficiency3. They claimed that higher Reynolds number of the displacing fluid with the flow in transitional or turbulent regimes can create better displacement. Using essentially a hydraulic approach, Mclean et al. [98] proposed design rules for primary cementing. Extensions of their work have led to whole systems of design rules for laminar displacements, [30, 82, 96, 103], also based on 3This is a commonly used parameter for defining the ability of a given fluid to displace another. There is no universal definition of this parameter but one common measure is given in [103]. Assume our duct (or annulus) is filed with a displaced fluid initially at rest at t = 0. When the displacement process starts, the displacing fluid enters the duct. At any time t > 0 during the process, displacement efficiency can be defined as the fraction of duct (or annulus) volume occupied by the displacing fluid. In this chapter, we introduce displacement efficiency only to provide a rough idea of how successful a displacement process is. The definition of displacement efficiency that we use is given in Chapter 5, where we relate displacement efficiency to the front velocity of the displacing fluid. 13 2.1. Primary cementing background hydraulic reasoning. In general, these rules set state that there should be a hierarchy of the fluid rheologies pumped, (i.e. each fluid should generate a higher frictional pressure than its predecessor), and that there should be a hierarchy of the fluid densities, (each fluid heavier than its predecessor) [15]. Whilst such approaches may contain a number of physical facts, the level of fundamental understanding is low and predictions made are gener- ally conservative. They were also proposed at a time when nearly all wells were drilled vertically. During 1970-90s, some studies, mostly focused on the annulus displace- ments in vertical wells, introduced new rule-based systems for better cement- ing job designs. These works were also based on modeling of laminar flow displacements. Some details regarding the design recommendations can be found in [11, 50, 72, 103, 140, 168]. For example, two common suggestions are given below [103]: • Everything else being equal, and at least for the case of low flow rates, the upward displacement of a dense fluid by a lighter one leads to an unstable phenomenon known as buoyant plume (the same phenomenon is observed for downward displacement of a light fluid by a dense fluid). In contrast, for the upward displacement of a heavier displacing fluid, buoyancy forces have a tendency to flatten the interface and stimulate efficient displacement. • Everything else being equal, if a thick fluid displaces a thin one in the laminar flow, the displacement efficiency is higher than the reverse scenario, which is believed to create an unstable interface (see also Hooper & Grimshaw [72]). The above statements are qualitative, and do not take into account the combined effect of density and rheology. In another study, Flumerfelt [50] presented an approximate solution for the displacement of a shear-thinning fluid by another in laminar flow. Beirute and Flumerfelt [11] developed the solution for to a more general non-Newtonian model. In both cases there are mass conservation errors in the eventual models. Beirute & Flumerfelt [11] recommenced the following: • The density ratio can play a predominant role if the flow rate is not too large. • Displacement efficiency increases with increasing effective viscosity ra- tios but the sensitivity to this parameter is not as important as to the 14 2.1. Primary cementing background density ratio. The displacement of a more viscous fluid by another one (with usually the same density) leads to viscous fingering, where the less viscous fluid penetrates into the more viscous one. This ef- fect results in a bad displacement. Thus, the opposite case of more viscous fluid displacing a less viscous one is likely to present higher displacement efficiency. • For some shear-thinning fluids, better displacement efficiencies are ob- tained when the power-law index of the displacing fluid is lower than that of the displaced fluid. • Yield stresses, specially when present in the displaced fluid, are very critical. Better displacement is usually achieved when dimensionless yield stress values of the displacing fluid exceeds that of the displaced fluid. • Reducing mud density and viscosity will probably always result in improved efficiency. Zuiderwijk [168] used a power-law model and performed a large number of high efficiency mud displacement tests and suggested that: • Well-treated mud (i.e. mud with power-law index close to unity) has been observed to be more easily displaced by a very thin cement slurry at higher velocities; • At low velocities, better displacement is obtained with cement slurries having a higher viscosity than the mud. Starting in the early 1990s, multi-dimensional analyses focused on com- puting the entire or a short section of annular flow. The first analysis of nar- row eccentric annular flows of visco-plastic fluids was carried out by Walton & Bittleston [157] and Szabo & Hassager [142] but only for flows of a sin- gle fluid in 2 spatial dimensions. There are also more recent computational studies such as a 2D representation of annulus of Bittleston et al. [15] and a 3D model of King et al. [89]. Three-dimensional Newtonian displacements in eccentric annular geometries have been also computed in [143]. More recent models and computations can be found in [20, 21, 101, 102]. There are relatively a very few studies which consider downward displace- ment inside the casing. Allouche et al. [4], Frigaard et al. [55], Gabard [57], Gabard & Hulin [58] and Frigaard et al. [53] are some examples which have considered displacements in long axial ducts, i.e. two-dimensional slots and 15 2.1. Primary cementing background axisymmetric flows in pipes. The main reason why there is little literature for downward displacement flows inside the casing are: (i) the use of plugs (as explained in 2.1.3); (ii) attention has been paid to the annulus as that is where the ultimate mud dispalcement should take place. Much less attention has been paid to the characterization of the fluid conditions (degree of mixing and thus properties) when entering the annulus. 2.1.5 Engineering design software In industrial cementing there also exist generic simulators, most of which are actually based on single phase hydraulics models, considering no mixing between stages of the pumped fluids. The majority of the industry still de- signs cement jobs using 1D simulations that do not currently consider flow parameters and fluid rheology beyond calculation of frictional pressures and fluid volumes. Here we review descriptions of more generic and sophisticated simulators (from oil well services companies) which take into account dif- ferent flow parameters and in general deal with cement/mud displacement processes. Halliburton has developed Displace 3DTM simulator which uses ad- vanced computational fluid dynamics and it is claimed to dynamically model multiple aspects of displacement of wellbore fluids during cementing [105]. This simulator, which has a 3D visualization interface, is designed in order to consider a fully 3D wellbore environment. Fluid interface evolution is visible and this simulator is capable of predicting fluid contamination up to some degrees. Calculating the mixing interface lengths and the top of cement locations is another interesting feature. For the fluids, the modeling approach used is a generalized Herschel Bulkley model, which can somehow safely reduce the fluid complexity and standardize the problem for indus- trial purposes. The simulator developers argue that their simulator can help engineers and operators make better decisions about the cementing. This avoids cement job failure and improves well integrity, and also reduces rig time costs. Unfortunately there are almost no actual details published of what is contained in the underlying physical model. WELLCLEAN II Simulator is a two-dimensional numerical simulator developed at Schlumberger oil service company [106]. This simulator also uses computational fluid dynamics physics and, with some details, monitors the process of cement placement. The goal is to have a prediction of the efficiency of mud removal. Different features include careful consideration 16 2.1. Primary cementing background of well geometry, inclination from vertical to horizontal, interface trajectory, fluid properties, volumes, pump rates and casing centralization. The other feature is simulation of fluid placement in both laminar flow and turbu- lent flows to produce maps of fluid velocity and flow regimes. Rheological description of fluids is expressed through a Herschel-Bulkley model. Trican has designed a simulator (Cement Simulator) to predict pressures and flow regimes at various points in a wellbore [159]. The simulator mod- els conventional, reverse circulation and foam cement jobs. This simulation software captures events of a primary cementing job and, in particular, cal- culates pressures, mud removal, and fluid flow regimes at zones of interest. No details are available regarding the physical models used. 2.1.6 Summary of industrial literature We now summarize our engineering literature review. (i) Generally speaking, most recommendations for a better displacement are qualitative. (ii) Many studies in literature provide narrow data regarding ranges of pa- rameters that displacing/displaced fluids can have. This may strongly affect their result interpretation. Further progress in the area of well cementing process can be achieved through combination of experimen- tal and theoretical studies to cover a wide range of non-dimensional parameters. These studies are largely lacking in the literature. (iii) A limitation that strongly hinders further research is the difficulty of model validation against field data. Most of the time, the fluid prop- erties, as mixed in the wellbore, are not measured. Monitoring and recording actual controlled cementing job data is neither easy nor com- mon. Thus, only a few percent of real field measurement data can be considered reliable and treated as experimental results. In this sense, proposed mathematical models must be validated with controlled aca- demic laboratory experiments with accurate designs and standard flu- ids. (iv) The majority of industry literature concerns the annular displacement flow, but this will be impractical if the fluids are already mixed or contaminated by the time the annulus is reached. Although less rele- vant industrially, the physical process in the annulus depends on the physics of the downward displacement. For example, a common design 17 2.2. Associated fundamental problems rule states that heavy fluids displace better in the upward direction in the annulus. However, ensuring a stable density difference in the an- nulus means that the downward displacement in the casing (pipe) will be density unstable. Therefore, it is possible that cement reaching the end of the pipe and entering the annulus is altered and highly contam- inated; this can lead to the job failure. (v) The range in terms of non-dimensional groups and expected flow phe- nomena is too wide for any single study. This cannot be the aim of a thesis to understand all of this; instead later we will define some sub- set of typical parameters to be the focus. In particular we will focus mainly on laminar flows. (vi) The laminar flows of importance contain all of: (a) significant buoyancy; (b) significant inertia; (c) different inclinations; (d) viscous effects; (e) interesting rheologies on some of the fluids. 2.2 Associated fundamental problems In this section we review the scientific in a number of areas that are closely related to our problem. This helps to frame the fundamental mechanisms present in our flow problem. With some generality, although our flow prob- lem is complex it can be better understood by considering a combination of simpler problems and mechanisms. These basic mechanisms have been deeply investigated and can be found in the literature. We first review those previous studies that appear particularly important for our problem. We will then comment on where there are significant deficiencies in the litera- ture and our knowledge. More specifically we consider the following: • Laminar flows in the processes that we study typically have high Péclet numbers (Pe À 1) and long aspect ratios (δ ¿ 1), but commonly δPe & 1. In the absence of instability and dispersive mixing, these flows exhibit sharp interfaces, qualitatively similar to immiscible dis- placements. We present an overview of high Pe regimes in §2.2.1. 18 2.2. Associated fundamental problems • Our displacement flows are naturally vulnerable to interfacial instabili- ties. We review the most relevant mechanisms in §2.2.2. On increasing the imposed flow, for very large V̂0, we logically expect the flow to ex- perience a transition and finally to fall into a turbulent regime. We comment on this at the end of §2.2.2. • An alternative way of viewing the background to our problem is as a variant of a confined gravity currents. When V̂0 → 0, we inevitably expect to recover the results of a confined gravity current. A detailed review of the most significant experimental, analytical and computa- tional studies of gravity currents is given in §2.2.3. • A limit arising in high Pe flows is the Taylor-dispersion regime, which for our case can be found only in long pipes at long times; this disper- sive regime is explained in §2.2.4. • As previously stated, studies on the effect of rheology on displacement flows are somewhat scattered and less deep. In §2.2.5, we will pro- vide short descriptions of most related works considering rheological parameters. 2.2.1 High Pe miscible displacements Displacement of one fluid with another can be regarded as an archetypical flow, occurring in many industrial settings, which is made more complex to understand when there are density differences between the fluids. Many practical processing situations involving aqueous liquids in laminar duct flows with diameters D̂ ∼ 10−2 (m) and mean velocities V̂0 . 0.1 (m/s) necessarily fall in to the category of high Pe flows, conservatively in the range 103−107. For such flows the laminar Taylor-dispersion regime [6, 144] (explained in §2.2.4) is strictly found only for duct lengths L̂À D̂Pe, which are arguably less common in processing geometries, even though D̂/L̂ ¿ 1 is usual. Thus, in an industrial setting probably the most relevant laminar regime is the non-dispersive high Péclet number regime, where the ducts have long aspect ratio, but still lie well below the Taylor dispersion regime. This high Péclet number regime has been studied analytically, computationally, and experimentally in [24, 112, 116, 163] in the case of Newtonian iso-density displacements (and typically low Re). These studies show that, provided that the displacement flow remains stable, sharp interfaces persist over wide ranges of parameters for dimensionless times (hence lengths) smaller than 19 2.2. Associated fundamental problems the Péclet number. At longer times (lengths) the dispersive limit is at- tained. For fixed lengths and increasing Péclet number (while remaining laminar) the flow is comparable to an immiscible displacement (with zero- surface tension). The dispersive limit of miscible iso-density displacements has been considered by Zhang & Frigaard [167], also for a range of simple non-Newtonian fluids. In an experimental paper (accompanied by a corresponding simulation paper of Chen & Meiburg [24]), Petitjeans and Maxworthy [112] investi- gated the miscible displacement of glycerine by a glycerine-water mixture which had a lower viscosity. They measured the amount of the fluid left on the capillary tube wall (M) as a function of the Pe and also the viscosity ratio; another functionality they investigated was of a parameter showing the importance of viscous to gravitational effects (F ). They also found the asymptotic value ofM for large Pe when the viscosity ratio tends to infinity. They pointed out an interesting argument that displacement flows at infi- nite capillary number can be in fact interpreted as immiscible flows with zero surface tension. Similarly infinite Pe can be seen as a miscible flow with zero diffusion. Therefore they stated that it is possible to identify the interface between two immiscible fluids with that between two miscible fluids without molecular diffusion. Thus, the asymptotic values ofM should have the same value for both the immiscible and miscible displacement flows. This value interestingly agrees with that found in experimental results of immiscible displacements (i.e. Taylor [146] found M = 0.56 but the corrected value for M found by Cox [31] was 0.6) as well as the corresponding results of the numerical simulation results (in [24]). The asymptotic value ofM is reached for Pe in order of 10,000 in the experiments whereas for the simulations this limit was observed at Pe = 1600. For Pe greater than 1000, they reported the observation of sharp interface. For large Pe all the curves of M (for different values of F ) tend to the same asymptotic value depending only on the viscosity ratio and independent of the buoyancy force due to the density difference. For small Pe however, the behaviour of M depends on the tube diameter and orientation. For example for F > 0, M increases by decreas- ing Pe. In contrast, when F < 0 the opposite trend happens even for the horizontal pipe. Unable to find any trustworthy value for the diffusion coef- ficient between glycerine and a known glycerine-water mixture, they chose to measure the average diffusion coefficient in a separate experiment. In a theoretical and experimental study, Lajeunesse et al. [91] considered a Hele-Shaw cell with downward vertical displacement of two Newtonian flu- ids in a density stable configuration (i.e. the lighter fluid above the heavier one) for large Pe. They observed a well-defined interface between the two 20 2.2. Associated fundamental problems fluids for which the transverse average concentration profile has features of a kinematic wave. The important variables in their symmetric displacement were the viscosity ratio and a normalized flow rate number (or a ratio show- ing buoyancy forces to viscous ones). Based on a discussion about existence of internal or frontal shocks, they characterised three different domains in a map of the flow rate number versus the viscosity-ratio. They also analysed the stability of the flow and found critical values for the flow parameters at which the 2D flow developed 3D structures. In the same paper, they also conducted a similar study for an axisymmetric displacement in a tube. 2.2.2 Instability and transition to turbulence Flow instabilities are closely associated with the subject studied in this the- sis. Not only do we study flows at increasingly large flow rates but also we consider density and viscosity differences between the fluids. Each of these effects considered alone can be a source for instabilities to develop. On the other hand the study of hydrodynamic instability is very evolved and con- siders a broad range of flows, many of which are close to ours. Consequently, the related literature can only be reviewed selectively. Hydrodynamic stability deals with the stability and instability of mo- tion of fluids. The fundamental problems of hydrodynamic stability were expressed and formulated in the 19th century, particularly by Helmholtz, Kelvin, Rayleigh, and Reynolds (see [45] for detail). One of the first scien- tists to study such problems was Osborne Reynolds. He described his classic series of experiments in a well known paper published in 1883 (see [120]). This paper helps us to qualitatively explain the transition from laminar flow to turbulence with some certainty. The Reynolds number (Re = ρ̂Û D̂/µ̂) describes the relative importance of inertial to viscous forces. For a Poiseuille flow in a pipe, when the Reynolds number is sufficiently small, both large and small perturbations eventually decay (i.e. roughly for Re ≤ 2000). Above this, the flow is believed to be unstable to perturbations of sufficiently large finite amplitude. Practically, these perturbations are usually introduced into the flow at the inlet or by pipe wall irregularity. They rapidly grow to an extent that nonlinearity becomes strong and large eddies and/or turbulent spots form. As the Reynolds number is further increased, the threshold amplitude of perturbations needed to create the instability decreases. For large values of Reynolds numbers turbulence occurs due to the unavoidable presence of perturbations of small amplitude. Therefore the flow becomes random, strongly three-dimensional, very non-axisymmetric and strongly nonlinear everywhere in the flow [45]. 21 2.2. Associated fundamental problems Kelvin-Helmholtz instability The Kelvin-Helmholtz instability theory is a way to predict the onset of linear instability in stratified layers of fluids with different densities which are in relative motion. Let us consider the basic flow of two incompressible inviscid fluids in horizontal parallel infinite streams (two-dimensional in the (x̂, ẑ)-plane) of various velocities and densities. One of the streams is above the other (see Fig. 2.2). By using a simple linearized stability theorem for two incompressible inviscid fluids, it is not too difficult to obtain the necessary and sufficient condition for the linear instability [45]:√ k̂2 + l̂2ĝ(ρ̂21 − ρ̂22) < k̂2ρ̂1ρ̂2(Û1 − Û2)2, (2.1) where k̂, l̂ are the wavenumbers in x̂ and ŷ directions respectively (note that ŷ axis is perpendicular to the paper). Therefore the flow is always unstable (to modes with sufficiently large k̂, that is, to short waves) provided that Û1 6= Û2. When the heavier fluid is placed below, condition 2.1 for Kelvin- Helmholtz instability signifies an imbalance between the destabilizing effect of inertia and the stabilizing effect of buoyancy. It should be noted that this simple model of analysing Kelvin-Helmholtz instability is only a first attempt at understanding the mechanism behind this instability. This model does not include important features of the instability, such as the effects of viscosity and nonlinear effects of inertia.  x z U2 U1 g ^ ^ ^ ^ ^ Figure 2.2: Principle of the development of Kelvin-Helmholtz instability for fluid layers moving with different velocities. A small deformation of the interface is magnified if condition 2.1 is satisfied. 22 2.2. Associated fundamental problems Reynolds also mentioned some experiments on Kelvin-Helmholtz insta- bility [120]. Later, Thorpe in a series of interesting papers (e.g. [149–152]) advanced Reynolds’ experiment and clearly identified the Kelvin-Helmholtz instability. He proposed a technique to produce stratified shear flows in a controlled laboratory setting. Thorpe [151] was able to measure the thresh- old and growth rate of instability for miscible layers of brine (i.e. salt water) and water. He observed the development of the disturbances to finite ampli- tude, transition to turbulence and also the resulting turbulence. By drawing a comparison between experiments and theory he concluded that the insta- bility arises from the Kelvin-Helmholtz mechanism. Some of his results are illustrated in Fig. 2.3 which shows the development of Kelvin-Helmholtz in- stabilities. In Fig. 2.3a, the two fluids begin to accelerate. At this moment the fluids are perfectly separated and the density gradient at their interface is very high. The velocity gradient increases until the interface is destabilized and the characteristics waves of the Kelvin-Helmholtz instability appear, as observed in Fig. 2.3c. Figure 2.3d shows that the amplitude of these waves increases and they finally unfurl inducing transverse mixing between the two fluids (see Figs. 2.3e and f). As a consequence the transverse concentration gradient decreases. A further decrease in the concentration gradient can result in a stable parallel flow since the velocity gradient is no longer large enough to create the instabilities at the interface. It should be noted that increasing the density contrast can have opposing effects on the stability of the interface. It first of all increases the longitu- dinal pressure gradients, increases the velocity gradient and consequently can trigger instabilities through the Kelvin-Helmholtz mechanism. On the other hand increasing the density contrast promotes transverse pressure gra- dients, which helps to create a stable stratified flow and stabilizes the growth of waves. Multi-layer flow instability Much of our study concerns regimes for front propagation. However, we also expect to observe some instabilities at the interface as it elongates. In such flows, since axial variations are very slow, the flow on any particular cross-section is not distinguishable from a miscible multi-layer flow. There are only a limited number of studies associated with instability of such flows, e.g. [63, 64] and [125, 126]. There is also extensive literature on the instability of immiscible parallel multi-layer flows, dating from the classical study of [166]. Explanations of the physical mechanisms that govern this type of instability for Newtonian fluids have been offered by Hinch [70] and 23 2.2. Associated fundamental problems Figure 2.3: The growth of instabilities at the interface of layer of water and salt water (colored). The density difference is 7.95 × 10−2 (g/cc). Molecu- lar diffusion has acted for 30 (min) on the tube horizontally before it was inclined at 4.4 ◦. The first shot was taken 3.35 (s) after tilting the tube and the time interval between successive frames is 0.35 (s) (from Thorpe [151]). Charru & Hinch [23]. In the context of miscible multi-fluid flows there is less work on shear instabilities; note also that the term multi-layer is ill-defined if the fluids can mix. Linear stability studies generally assume a quasi-steady parallel base state. Ranganathan & Govindarajan [117] and Govindarajan [62] analysed the stability of miscible fluids of different viscosities flowing through a chan- nel in a three-layer Poiseuille configuration. They obtained instabilities at high Schmidt numbers and low Reynolds numbers, resembling those of [166]. In Couette flow it appears that the stability characteristics of the miscible flow are predicted by those of the immiscible flow with zero surface tension; see [49]. However, for core annular flow this is not the case; see [129]. Recent studies have considered convective instabilities in miscible multi- layer flows, both experimentally by d’Olce et al. [42–44] and computation- ally/analytically by Selvam et al. [130]. Sahu et al. [125, 126] have recently considered the onset of convective instabilities in 3-layer plane channel flows. Amaouche et al. [5] have recently proposed a weighted-residual-based ap- proach for two-layer weakly inertial flows in channel geometries (see also Mehidi & Amatousse [99]). In their study they extensively compared sta- 24 2.2. Associated fundamental problems bility predictions of their simplified thin-film model against those from an Orr-Sommerfeld equation approach, showing good agreement specially in the long wavelength regime. Turbulent entrainment Although in the current thesis we do not really focus on turbulent aspects of our flows, we should mention an important turbulent effect which con- cerns us, namely turbulent entrainment in stratified flows. In probably the best known work related to this subject, Ellison and Turner [48] considered the motion of a relatively thin turbulent layer embedded in stratified flows. The turbulent region grows with distance downstream as the non-turbulent region (initially at rest in their experiment) becomes entrained in it. This entrainment indicates a flow of the non-turbulent surrounding fluid into the turbulent layer, and therefore a relatively small mean velocity perpendicular to main flow is created. They developed an analysis that assumed that the entrainment is proportional to the velocity scale of the layer. They called the constant of proportionally the entrainment coefficient and found it ex- perimentally as a function of only the overall (averaged) Richardson number (Ri), considering the Boussinesq approximation. In fact their analysis could be equally written in the form of the densimetric Froude number, which essentially represents the same physical concept (Ri ∝ 1Fr ). Their theory shows that the layer reaches an equilibrium state where Ri does not vary with distance downstream and there exists a balance between gravitational force on the layer and the drag due to entrainment together with friction on the wall. They showed that the entrainment coefficient quickly decreases when Ri increases. 2.2.3 Gravity currents A more general motivation for our work arises since buoyancy-driven flows of miscible Newtonian fluids over near-horizontal surfaces occur frequently in the oceanographic, meteorological and geophysical contexts (see [12, 139]) i.e. gravity currents. Such flows are driven by buoyancy, but the physical mechanisms that limit the flow may be inertia or viscosity depending on the geometric configuration, the mean flow and the type of fluids. Most frequently these flows have been studied in unconfined geometries (e.g. [13, 14, 39, 75, 138]). Slightly closer to our study are those of lock-exchange flows in tanks (open channels). Such flows are typically studied in a regime where vis- 25 2.2. Associated fundamental problems cous effects are unimportant and buoyancy forces are balanced by inertia. The velocity is essentially constant in each interpenetrating stream. The mathematical approach for studying these flows dates back to the work of Benjamin [12]. See Shin et al. [138] and references therein for an overview and critical appraisal. Recently Birman et al. [13] have studied gravity cur- rents in inclined channels. These are high Re flows, vulnerable to interfacial instabilities, (loosely of Kelvin-Helmholz type), and local mixing. Typically the edges of gravity currents are not well-defined, due to local instability and mixing. More recently, due to the importance of these flows in the industrial world, confined geometries such as a vertical pipes (see [7, 37, 38]) or inclined pipes (see [131–135]) have been considered. There are also more geophysically oriented studies at low Reynolds number, e.g. [10, 80, 141]. Many of these involve fluids of different viscosity as well as density. Most of these studies in confined geometries (ducts) focus on the exchange flow configuration where there is zero net flow along the duct. Figure 2.4: Flow of cold air in warm air; shadow pictures showing the profile of a front of a gravity current. The temperature difference between the fluid is (a) 0.5 ◦, (b) 1.5 ◦, (c) 4 ◦, (d) 7 ◦, (e) 15 ◦, (f) 35 ◦ and (f) (from Ref. [128]). In a gravity current, the flow of the interpenetrating front usually has an important role in the development of the flow. For example this front can in a sense limit the gravity current and change the dynamics in the flow. Figure 2.4 shows an example of changes in the front profile in a gravity current induced by a flow of cold air in warm air. The temperature increases from a very low value in Fig. 2.4a 0.5 ◦ up to 35 ◦ in Fig. 2.4f. The latter corresponds to a density difference of 1%. In this context, increasing the density contrast can be interpreted as the effect of increasing the Reynolds 26 2.2. Associated fundamental problems number of the flow, which can significantly change the shape of the front. For low Reynolds numbers (e.g. Fig. 2.4a), viscous forces dominate the buoyancy forces, the forehead is small and there is little mixing. When the Reynolds number increases (e.g. Fig. 2.4f), the size of the front also increases, shear instabilities appear and induce some mixing with the ambient fluid. Gravity currents produced by lock-exchange Shin et al. [138] presented a theory supported by experiments on gravity cur- rents in a lock-exchange flow configuration. Their geometry was a horizontal rectangular channel. The geometry they considered is shown in Fig. 2.5 with current of density ρ̂2, propagating with constant velocity Û into fluid of den- sity ρ̂1. The depth of the current far behind the front where the interface is flat is denoted by ĥ. Previously for the same geometry, Benjamin [12] considered a frame of reference moving with the current and developed a hydraulic theory for the steady propagating front. He showed that assum- ing the energy flux entering and leaving the control volume were equal, the current would occupy half the channel thickness. To achieve this, he applied the equations of continuity and Bernoulli along the interface between two layers of fluids. For such a flow the Froude number is: FH = Û√ 2AtĝĤ . (2.2) ^ ^ ^ ^ ^ ^ Figure 2.5: A schematic diagram of a gravity current in a frame moving with current [138]. 27 2.2. Associated fundamental problems From the above equation, one can obtain the front velocity as Û = 0.7 √ AtĝĤ according to the theory of Benjamin (which gives FH = 1/2). Interestingly as we will see later despite the strong differences that a config- uration can impose, Seon et al. [132] also found the same relation for their experiments, performed in a tilted pipe. Shin et al. [138] experimentally verified this Benjamin’s theory. In Fig. 2.6 we observe good agreement be- tween the experiment and theoretical potential flow solution (dashed line) despite the mixing between the two fluids as well as the dissipation due to turbulence and viscous stress (especially close the walls). This result sug- gests that the propagation speed of gravity current can be determined by considering only the equilibrium of pressure at the front, employing inviscid flow theory and neglecting the dissipation of energy downstream. Figure 2.6: Experimental results of a full depth lock-exchange with Ben- jamin’s [12] potential flow solution (dashed line). This image is extracted from reference [138]. Although gravity current measurements provide satisfactory agreement with the theory of Benjamin, Shin et al. [138] noticed that in fact the current speed is not a sensitive test to whether the current is energy conserving. This is illustrated in Fig. 2.7a and Fig. 2.7b, which show (dimensionlessly) the theoretical variations of the Froude number and the net energy flux vs the normalized interface hight, respectively. It can be clearly seen that for maximum dissipation (Ė) we have FH = 0.527, while we can also have FH = 0.5 when energy is conserved. The difference in speeds between a current with maximum dissipation (ĥ/Ĥ = 0.347 and FH = 0.572) and the energy conserving current with (ĥ/Ĥ = 0.5 and FH = 0.5) is difficult to be experimentally determined. It should also be noted that the gravity currents which occupy less than half of the channel are not energy conserving. In this work Shin et al. [138] also presented a detailed study of the heights of gravity currents produced by lock exchange and found that the front Froude number in a deep ambient is equal to 1 rather previously accepted value of √ 2. They concluded that the dissipation effect of turbulence and mixing is negligible when the Reynolds 28 2.2. Associated fundamental problems 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 ĥ/Ĥ F H a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 ĥ/Ĥ ∆ Ė b) Figure 2.7: a) The Froude number FH plotted against the dimensionless height of the current ĥ/Ĥ. b) The dimensionless net energy flux Ė plotted against the dimensionless height of the current ĥ/Ĥ [138]. number is high enough and that the speed reduction compared to that of a conserving energy current is only a few percent. Lock-exchange flows in sloping channels Birman et al. [13] studied a lock-exchange channel flow problem of two fluids of different densities. They carried out high resolution simulations accom- panied by complementary experiments. Their simulations reveal that the flow initially experiences a quasi-steady phase with a constant front veloc- ity which persists up to a dimensionless time of O(10). This front velocity increases with tilt angle from horizontal and reaches a plateau for the range 30 ◦ < β < 50 ◦ (with a maximum at β = 40 ◦). This finding was also supported by experiments performed in a different geometry (i.e. a circu- lar pipe in the experiments of Seon et al. [132]) which surprisingly provided similar qualitative results. The flow afterwards undergoes a transition phase in which the front is unsteady and large. By using their numerical simulations and corresponding experiments, they noted the important role of the front in controlling the dynamics of flow. They suggested that during the early stages the front velocity is governed by local dynamics in the frontal region, but in later stages the mechanism governing the front velocity is different. They used this argument to justify the observed transition. In an inclined channel, the bulk flow behind the front in fact experiences a continuous acceleration caused by the longitudinal gravity vector component. This acceleration helps the flow in the stratified 29 2.2. Associated fundamental problems layers behind the front move faster than the front. At early times, the fluid added to the front from behind only results in enlargement of the front, while its velocity does not vary. However at longer times, the front becomes so large that it cannot maintain its constant velocity anymore. At this moment the so-called transition is triggered. The authors quantified the transition time for different slopes and density contrasts and found that the higher the density contrast, the earlier the transition time for the denser current. They also put forward a two-layer and also a three-layer conceptual model introduced by Thorpe [149] to compare with the simulations. The model is focused on the stratified region that connects the downward and upward current fronts. They found that for early dimensionless times, the two-layer model provides better results while in the later stages the three- layer model, which includes the mixed region, seems to be more compelling. They also used the models mentioned above to estimate upper and lower bounds for the transition time and observed good agreement with the sim- ulation results. Gravity currents in confined geometry In the absence of an imposed mean flow, a detailed experimental study of buoyancy driven miscible flows in inclined pipes has been carried out by Seon et al. [131–135]. In these studies the pipe is closed at the ends so that an exchange flow results. Seon et al. [132] experimentally characterised the velocity of the interpenetrating fronts of light and heavy fluids, as a function of viscosity, density ratio and inclination angle. For different inclinations of the pipe from horizontal to vertical they observed three flow regimes: in- creasing front velocity, constant front velocity and decreasing front velocity. These three regimes are shown in Fig. 2.8. In the first regime segregation and mixing effects control the front velocity. In the second regime, the front velocity is independent of inclination angle and fluid viscosity, controlled by the balance between inertia and buoyancy. For the first and the second regimes, they obtained a correlative formulation based on characteristic vis- cous and inertial velocities. In the last regime, found close to horizontal, the fluids are separated into two parallel counter-current streams. The near-horizontal regime is studied in more detail by Seon et al. [134], who found a small critical value of inclination, above which the front veloc- ity is fully controlled by inertia. When the inclination is below this critical value, the front velocity is initially controlled by inertia but later by viscosity. As soon as viscous effects start to control the front velocity, it gradually de- creases towards a steady-state value, which is proportional to the sine of the 30 2.2. Associated fundamental problems 20 15 10 5 0  Vfront 806040200 β  (mm/s) 2 1 3 ^ Figure 2.8: Illustration of three regimes observed through variation of front velocity as a function of the inclination angle β for At = 4 × 10−3, µ̂ = 10−3 (Pa.s). The insets are pictures of a 30 (cm) long section of tube just above the gate valve in the corresponding flow domains. The dashed lines qualitatively represent the boundaries between the domains and the dotted lines are only guides for the eye. This figure is extracted from [132]. inclination angle, from horizontal. This final velocity thus tends to zero for a horizontal tube. They also showed that the fluid concentration/interface profiles depend on the reduced variable x̂/t̂, i.e. spreading diffusively. In viscous regimes for near horizontal pipes the transverse gravitational com- ponent suppresses the development of instabilities, so that there is no mixing between the fluids and the interface remains clear. This shift from an ini- tial inertial-buoyancy balance to a viscous-buoyancy balance was also found by Didden and Maxworthy [39] and Huppert [79], who considered viscous spreading of gravity currents with an imposed flow. In the absence of an imposed mean flow there is some subtlety in the transition between strictly horizontal ducts and slightly inclined ducts. Buoyancy acts both via the slope of the duct and the slope of the interface, relative to the duct axis. When the interface elongates the latter effect of buoyancy diminishes but the former effect remains present. We should note that for our study there is a third driving force, that of the imposed flow, which does not diminish over 31 2.2. Associated fundamental problems time. Thus, the distinction between strictly horizontal ducts and slightly inclined ducts is not so critical as in the work of Seon et al. [135]. 1.0 0.8 0.6 0.4 0.2 0.0 6005004003002001000 Vt Vfront ^ ^ ^ Vνcos Vt ^ β Figure 2.9: Variation of the normalized velocity V̂f/V̂t as a function of V̂ν̂ cosβ/V̂t for the set of data points related to different experiments in the range [At, µ̂] ∈ [4× 10−4 − 3.5× 10−2, 10−3 − 4× 10−3]. This figure is extracted from [132]. In the exchange flow context, the driving force is the buoyancy and the physical mechanisms that limit the flow are either inertia or viscosity, depending on the geometric configuration and the type of fluids ([132]). Two characteristic velocities can be defined. Firstly, a viscous velocity scale V̂ν : V̂ν = AtĝD̂ ν̂ (2.3) when buoyancy and viscous term are balanced. Secondly, an inertial velocity 32 2.2. Associated fundamental problems scale V̂t: V̂t = √ AtĝD̂ (2.4) when buoyancy and inertia terms are balanced. Here At is the Atwood number, defined as the ratio of the difference of densities of the two fluids by their sum, ĝ is the acceleration due to gravity, D̂ is the diameter of the pipe and ν̂ is the common kinematic viscosity of the fluids, defined with the mean density. Exchange flows have been classified as either inertial or viscous according to which effect is dominant in balancing buoyancy forces. Seon et al. [132] showed that inertial exchange flows are found in pipes if V̂ν cosβ V̂t = Ret cosβ ' 50, (2.5) and viscous exchange flows otherwise. Here β measures the inclination of the pipe from vertical. Figure 2.9 depicts the relevance of the characteristic velocities V̂t and V̂ν through plotting the normalized stationary (long time) front velocity V̂f/V̂t as a function of the ratio V̂ν cosβ/V̂t. It is clearly observed that for V̂ν cosβ/V̂t < 50, all data points corresponding to different values of all control parameters collapse onto a single linear variation (i.e. V̂f ' 0.0145V̂ν cosβ), implying that the front velocity is controlled by viscous dissipation in the fluid bulk. For V̂ν cosβ/V̂t > 50 the points are close to a horizontal line corresponding to V̂f ' 0.7V̂t, implying the the flow is inertial in this domain (although mixing is weak). Effects of channel geometry Hallez and Magnaudet [67] used a Direct Numerical Simulation (DNS) tech- nique to observe the evolution of concentration and flow fields in buoyant mixing of miscible fluids in tilted channels, for the pure exchange flow. They were mostly interested in estimating the effect of the channel geometry and considered different geometries including a two-dimensional (2D) channel, a square channel and finally a three-dimensional (3D) pipe. They reported key differences in the flow structure among these geometries when in the inertial regime. They claimed that the striking differences between the flow dynamics are as a result of vortices, which are strong, coherent and persis- tent over long times in 2D. In contrast, in a 3D geometry the vortices tend to be stretched and are accordingly much weaker. The comparison of their results with those of [132–134] shows quite reasonable agreement. We are now going to review this related article in more depth. 33 2.2. Associated fundamental problems Buoyant currents in the confined geometry experience different phases. While in the initial acceleration phase and also the first slumping phase, 2D simulations might provide a good estimation of the flow evolution. However, they fail to predict the flow behaviour in the long-time whenever the effects of geometry are significant. For the initial slumping phase of all the ge- ometries studies, at intermediate tilt angles with respect to vertical, Hallez and Magnaudet [67] observed that the Froude number (defined by them as Fd = V̂f/ √ ĝD̂(ρ̂1 − ρ̂2)/ρ̂1) gradually increases with angle and after a plateau region it decreases until the channel is close to horizontal. However, the front velocity in a 3D cylindrical pipe is larger than in both 2D channels and 3D square channels at all the inclinations studied. At long times, after the initial slumping phase, viscous forces come to play an important role and the front velocity decreases with time. In Fig. 2.10, we observe the illustration of characteristic snapshots of concen- tration field (top image in each set) for different geometries. These photos show the flow after relatively long times in a channel highly deviated from vertical. The corresponding swirling strength is also depicted (bottom image in each set). Swirling strength can be seen as an indicator of instantaneous vorticity and defined as an imaginary part of conjugate eigenvalues of the vorticity gradient tensor. This figure clearly shows that for the intermediate tilt angles of 60−80 ◦ (Figs. 2.10a and b) for the 3D flow, vortices caused by the Kelvin-Helmholtz instabilities (produced by shear between the layers) are not sufficiently strong to cut the channel of light (heavy) fluid feeding the front with pure fluid. Hence, the concentration difference at the front has its maximum value and the front velocity is independent of the inclina- tion angle (V̂f = 0.7V̂t). In contrast, for lower tilt angles (Figs. 2.10d and e), the shear overcomes the segregation effect (created by the transverse grav- ity component) and Kelvin-Helmholtz rolls are this time sufficiently strong to only temporary cut the thin channels of pure fluid. Therefore, the local concentration at the tip of the light (or heavy) fluid is formed a by a balance between the mixing at the front and feeding of pure fluid from behind. The cutting phenomenon results in local mixing being able to decrease the value of concentration jump at the tip which consequently enforces a reduction in front velocity. At this moment the vortices stretch and break; this causes the feeding to start afresh. In 2D simulations, the Kelvin-Helmholtz vortices are relatively more coherent. This results in having vortices with higher intensity which can persist over a longer time. Therefore in this case a chain of vortices can be created and developed in the flow to periodically cut the mechanism of pure 34 2.2. Associated fundamental problems Figure 2.10: Images of the concentration (top in each set) and swirling strength (bottom in each set) taken from [67] for the three different ge- ometries at two different inclination angles with respect to vertical. The first 3 sets are at β = 60 ◦ and the second 3 sets are at β = 20 ◦. Each view is taken in the vertical or central plane of the geometries: (a) and (d) cylindrical pipe; (b) and (e) square channel; (c) and (f) 2D channel. fluid feeding. An example of these chain vortices can be observed in body of mixing region shown in Fig. 2.10c. These vortices are now on the threshold of breaking the channel of the pure fluid. Interestingly, provided that the vortex chain in the flow is sufficiently strong, it can even tear off the head of 35 2.2. Associated fundamental problems the gravity current and isolate it from the main body of the flow. This head then acts as an isolated drop of light (heavy) fluid and slowly diminishes by diffusion effects or twisting in its own wake. This phenomenon never occurs in the 3D pipe flow simulations since reconnection of the front to the mixing region behind it is always the case in this configuration. 2.2.4 Taylor dispersion The detailed study of miscible displacement flows and mixing in pipes and channels (§2.2.1) is relatively recent, compared to the study of dispersive regimes which dates to the 1950s; see [144] and [6]. In simple words, in a laminar flow the Taylor dispersion is a regime in which although there is no efficient mixing structures in the flow, dispersion can occur only because of the contribution of molecular diffusion and the velocity gradient. What is known as the Taylor dispersion is a shearing process of a passive tracer in- jected into a flow driven by a pressure gradient along a duct, i.e. a Poiseuille flow. We briefly review this effect, analyzed for the first time in [144] (for a laminar flow) and [145] (for a turbulent flow). If at initial time t̂ = 0 a line of constant concentration dye is placed transverse to the Poiseuille flow, in the absence of molecular diffusion it would be quickly stretched into a parabola under the effect of the velocity profile. After a time t̂, the tracer would be distributed over a distance ∆x̂ that increases linearly with time, implying ∆x̂ ∼ V̂0t̂ with V̂0 the mean speed. For a sufficiently long time, molecular diffusion perpendicular to the axis of the pipe limits this effect of stretching and transversely homogenizes the distribution of the tracer. The characteristic time T̂ needed for the tracer to diffuse across the pipe is T̂ ∼ D̂2/D̂m. Assuming that we are in frame of reference that moves with mean speed of the flow, the longitudinal dispersion (for the mean concentration Ĉm) can be expressed by: ∂Ĉm ∂t̂ = D̂T ∂2Ĉm ∂x̂2 (2.6) where D̂T is the Taylor dispersion coefficient. According to (2.6) D̂T ∼ ∆x̂2/t̂. If we assume that we have reached the time required for statistically stationary dispersion regime where the transverse concentration distribution has been homogenized, the two characteristic times will have the same order, and setting T̂ ∼ t̂ we deduce: D̂T ∼ V̂ 2 0 D̂ 2 D̂m (2.7) 36 2.2. Associated fundamental problems The pre-factor above was found by Taylor [144]. Aris [6] also found a first order correction and derived the relationship by a different method. It may seem counter-intuitive that D̂T reversely varies with the diffusion coefficient. In fact, the effect of transverse diffusion homogenizes the dis- tribution of the tracer and limits the effects of the velocity gradient which is the dispersive mechanism. Using (2.7) it can be shown that the Taylor dispersion coefficient is proportional to D̂mPe2 (with Pe = V̂0D̂/D̂m). We should highlight here that even though in our investigation we are in the high Pe regime (which would consequently have a large dispersion coeffi- cient), the Taylor dispersion regime can be found only after very long times. This is not in our timescale of interest. Apart from the consideration of a long timescale to achieve the Taylor dispersion regime we also need a tube with a length L̂, sufficiently long for the tracer to spread in the transverse direction during the process timescale. For a developed turbulent flow, the spreading of the concentration profile remains linear with time which is similar to the effect of velocity gradient in the laminar case. However, the mechanism of transverse mixing is not by molecular diffusion but is by fluctuations of the turbulent velocity field. For a given mean speed, the diffusion coefficient is lower for a turbulent flow than for a laminar flow. This reflects the fact that transport by transverse velocity fluctuations in a turbulent flow is much more effective than by molecular diffusion in a laminar flow. The transverse homogenization is faster and the resulting longitudinal diffusion coefficient is much lower. Finally, we should mention that in flows of the type we study, although in a laminar regime (according to the imposed flow Reynolds number) the dominant dispersive effect is not always related to molecular diffusion. For example, in the experiments of Seon et al. [131, 135] transverse mixing is very efficient at inclinations away from horizontal and is due to turbulent flow driven by buoyancy. In these inclinations dispersion (i.e. macroscopic diffusion coefficient) in the axial direction is larger than closer to horizon- tal angles for which viscous effects become important and mixing in the transverse direction is less efficient. 2.2.5 Effects of Rheology The literature for non-Newtonian fluid displacements in ducts is obviously less developed than that for Newtonian displacements. By far the largest body of work concerns Hele-Shaw geometries, where there are several nu- merical, experimental and analytical studies of viscous fingering with non- Newtonian fluids; see [29, 90, 124, 162] and [95] as examples. Gas-liquid dis- 37 2.2. Associated fundamental problems placements in tubes have been studied for visco-plastic fluids by Dimakopou- los & Tsamopoulos [40, 41] and by De Sousa et al. [35]. The focus here is typically on residual layers in steady state displacements. The flow around the displacement front is multi-dimensional. Other multi-dimensional dis- placement flows with generalised Newtonian fluids have been studied, nu- merically and analytically by Allouche et al. [4], Frigaard et al. [55], as well as experimentally by Gabard [57] and Gabard & Hulin [58]. These are all iso-density viscous-dominated displacements of miscible fluids in the high Pe regime. Gabard and Hulin [58] investigated iso-density miscible displacements in which a more viscous fluid is displaced by a Newtonian fluid. In their experimental investigation the geometry used was a vertical tube. They observed the effect of rheology of the displaced fluid and the flow velocity on the transient residual film thickness during the displacement process. They showed that in displacements of shear-thinning fluids with non-zero shear- stress, the residual thickness decreases (28−30% of the radius) compared to the known residual thickness value in the displacements of Newtonian fluids (38%). For yield stress fluids the residual thickness is even further decreased (24− 25%). They suggested that the 3D flow field close to the displacement front can play an important role in forming the residual film thickness. Their numerical simulations confirmed the approximated thicknesses of the reported experimental values. In addition, they showed that a downstream thickness reduction is achieved by development of instabilities; this reduction is enhanced when either viscosity ratios are lower or when displacement flow rates are larger. In [19] a novel reactive miscible displacement technique was studied. Instead of viscosifying the entire displacing fluid in order to improve dis- placement efficiency, the authors engineered a reaction to take place at the front when the two fluids mixed. The effect of the reaction was to locally viscosify the fluid mixture, with the idea of using this high viscous plug to improve displacement. The method in [19] did indeed produce enhanced displacement efficiencies, but not by the anticipated mechanism: instead via locally destabilizing the flow. Moving slightly further, many authors have considered displacement flows of non-Newtonian fluids from capillar- ies, driven by a gas flow, e.g. [35, 40, 41, 69, 85, 86, 113–115]. A variety of methods have been used (experimental, analytical, computational). These studies are often focused at extending the classical results of [31, 146] into non-Newtonian regimes. Phenomenologically these studies are far from the regimes we study. In the Hele-Shaw geometry, Lindner et al. [95] studied the Saffman- 38 2.2. Associated fundamental problems Taylor (viscous fingering) instability in a Hele-Shaw cell while including yield stress fluids. They observed a yield stress dominated regime at low velocity and a viscous dominated regime when the velocity was higher. The former regime shows branched patterns because in simple words each finger does not really feel the presence of walls or other fingers due to the fluid’s yield stress. In the viscous dominated regime, yield stress does not play an important role and the finger can find the Hele-Shaw cell. Their observations were confirmed by a linear stability analysis. They also conducted experi- ments with foams presenting very different results due the wall slip. Other investigations of viscous fingering (with stability analyses) include [29] and the earlier Darcy-flow analogues of [108–110]. Slightly less related to our study are studies of viscous spreading of thin layers fed with an imposed flow at a source. These arise in particular in the context of lava dome formation and spreading (i.e. non-Newtonian fluid); see [66]. Frequently, the models and experiments used to understand these phenomena are complicated with thermal effects, which then bear little re- semblance to our work. However, Balmforth et al. [8, 9] have studied lava dome formation in an isothermal setting and with visco-plastic fluids of the type considered here. Although the lubrication/thin-film modeling is simi- lar, these flows are unconstrained single fluid flows in which the flux function is typically determined analytically and hence mathematical progress is sim- pler. In contrast to the amount of computational work, there are relatively few experimental studies of displacement of yield stress fluids by other fluids. Experimental studies involving two fluid flows of yield stress fluids in the pipe geometry include Crawshaw & Frigaard [32] and Malekmohammadi et al. [97] who have studied the exchange flow problem (i.e. buoyancy driven flow in a closed ended pipe). The focus of these studies is stopping the motion using the yield stress of one of the fluids. Huen et al. [78] and Hormozi et al. [73] have studied core-annular flows, using a yield stress fluid for the outer lubricating layer and a range of different Newtonian and non- Newtonian fluids for the core. The start-up phase of these experiments is displacement-like, although the final steady state is a multi-layer flow. Finally, a number of authors have considered the displacement of yield stress fluids by a gas. De Souza Mendes et al. [36] investigated the displace- ment of viscoplastic flows in capillary tubes experimentally through gas in- jection. They showed that below a certain critical flow rate, the visco-plastic liquid is completely displaced by the displacing fluid. However above this critical flow rate small lumps of unyielded liquid will remain on the walls. For increased values of imposed flow rate a smooth liquid layer of uniform 39 2.2. Associated fundamental problems thickness forms. They reported that the thickness of this layer increases with the dimensionless flow rate. There have also been extensive compu- tational studies of these flows [35, 40, 41, 148]. Finally, there is a limited amount of analytical work concerning bubble propagation/displacement in Hele-Shaw geometries; see [3]. Static wall layers in the displacement of two visco-plastic fluids There are many industrial processes in which it is necessary to remove a gelled material or soft-solid from a duct. Examples include bio-medical applications (mucus [77, 100], biofilms [26, 158]), cleaning of equipment and food processing [25, 27], and most relate to our problem, oil well cementing and waxy crude oil pipeline restarts. The main feature of a yield stress fluid is that the fluid does not deform until a critical shear stress is exceeded locally. Therefore, when these fluids fill ducts and are displaced by other fluids, there is a tendency for the yield stress fluid to remain stuck to the duct walls and in particular in parts of the duct where there are constrictions or corners. This type of feature was first recognised in the context of oil well cementing by Mclean et al. [98], who identified potential bridging of a static plug of mud on the narrow side of an eccentric annulus. Avoidance of this feature has since been an ingredient of industrial design rules for oilfield cementing [30, 82], and latterly also simulation based design models, [15, 111]. Further features of oilfield cementing are discussed in [103], but geometries of our study are simpler. In waxy crude oil pipeline restarts (see [22, 34, 136, 154]) a large pressure is applied at one end of the pipe, to break the gel of the waxy oil. The waxy state has formed due to a drop in temperature below the wax appearance temperature, often related to stopping the pipeline for maintenance or other issues. Temperature is not particularly important in the restart process itself [156]. It is common to displace the in situ oil with a much lighter and Newtonian oil (often this is the same oil at higher temperature). In the displacement it is possible for static residual layers to form on the walls of the pipeline; see also [56, 155]. The phenomenon of a static wall layer in a plane channel was first con- sidered by Allouche et al. [4] who studied symmetric displacement of two visco-plastic fluid flowing inside a plane channel (with Cartesian coordinates (x, y)). They addressed the question of what conditions are needed for this layer to exist and, if so, what its thickness is. For the existence of this layer it is necessary that the yield stress of the displaced fluid exceeds that of the displacing fluid. They were mainly interested in flows for which a (fixed) 40 2.2. Associated fundamental problems flow rate is imposed and the displacing fluid is heavier than the displaced fluid. They argued that under such conditions and for a combination yield stresses of the displacing and displaced fluids, four qualitatively different velocity profiles could exist as shown in Fig. 2.11. The focus of their work was on the unique velocity profile of Fig. 2.11a in which the fluid adjacent to the wall does not move (i.e. static wall layer). In this case the yield stress of the displacing is not exceeded at the wall, or anywhere within the layer, and the condition of no-slip means that the layer is static. Figure 2.11: Schematic of the different possible characteristic axial velocity profiles when displacing fluid is heavier than the displaced fluid; U is the velocity and 1 and 2 denote the displacing and displaced fluid respectively; Yi is the interface position [4]. They found a critical ratio of the yield stresses (ϕy) versus the Bingham number of displacing fluid (B1) above which the static layer cannot exist. Below the critical ratio, the maximum static residual layer thickness hmax can be found as a function of only ϕy, B1 and a third parameter ϕB, giving the ratio between the buoyancy stress and the yield stress of the displaced fluid. This concept can be explained by considering a displacement at fixed flow rate when the front has evolved into a quasi-parallel multi-layer flow. In the case of static residual wall layers, the whole imposed flow rate has to pass through the mobile layer of the displacing fluid. Assuming now that 41 2.2. Associated fundamental problems for the fixed flow rate the thickness of the static residual layer is increased, the shear stresses in the displacing fluid layer should also increase. These stresses are conveyed at the interface to the static residual layer. For any residual layer the shear stresses eventually increase to such an extent that finally the shear stress at the wall exceeds the yield stress of the displaced fluid and consequently the fluid yields and starts to move. This limit is denoted by the maximal layer thickness. An example of a typical interface evolution showing the static residual wall layer is plotted in Fig. 2.12. Figure 2.12: A typical interface evolution Yi plotted every 200 timesteps in an axial displacement showing the static residual wall layer; because of symmetry only half of the channel is shown [4]. Allouche et al. [4] also presented 2D simulations of transient displace- ments mainly focused on the static layer concept for a limited range of parameters. They showed that the computed static thickness was signifi- cantly less than hmax. However, they showed that when the front moves in steady motion, the layer thickness can be well approximated by the re- circulation layer thickness hcirc. They defined this thickness as the (static) layer thickness at which a steadily advancing interface would move at the same speed as the center line velocity of the flow in downstream. They anal- ysed the streamline configuration close to a steadily advancing displacement front and argued that in a steady displacement viewed in a moving frame of reference, for h < hcirc there is a recirculatory region in the channel center, in front of the interface (see Fig. 2.13). Such a recirculation would increase local visco-plastic dissipation and suggested that the flow tends to avoid this situation to minimise dissipation. They finally introduced the the following relation for the thickness of static residual layer: hstatic = min {hmax, hcirc} (2.8) 42 2.2. Associated fundamental problems Figure 2.13: Schematic illustration of the two types of streamline behavior in displaced fluid: (a) no recirculation; (b) with recirculation [4]. Frigaard et al. [55] extended the approach of Allouche et al. [4], show- ing that in a steady displacement flow with a uniform static wall layer the thickness of the layer and the shape of the interface are non-unique for the steady displacement problem and consequently must result from transient aspects of the flow. The concept of maximal static wall layers was further explored in [54]. More recently in [161], an extensive computational study of static layer thickness in iso-density fluid displacements (Newtonian fluid displacing Bingham fluid) was performed, including the effect of flow rate oscillations. This has shed further light on the effects of the main 3 di- mensionless parameters (Reynolds number, Bingham number and viscosity ratio), in the absence of density differences. 2.2.6 Summary of fundamental literature Through reviewing the scientific literature, we understand that buoyant mis- cible displacement flows are associated with various fundamental problems. These displacement flows are naturally complex due to the presence of many 43 2.2. Associated fundamental problems parameters in the flow, corresponding to a number of competing physical effects. In this perspective, we have seen that relatively few studies directly address the specific problem that concerns us. However different behaviors in our flows have many aspects in common with phenomena which have been the subject of extensive literature. These are namely high Pe regime, instabilities (in particular Kelvin-Helmholtz type), gravity currents in con- fined geometries and the Taylor dispersion. We also find that the literature on non-Newtonian displacement flows is relatively poor in comparison to its Newtonian counterpart. However, the concept of static wall layers in these displacement flows has been studied before and can be related to our study. Based on our literature review in this chapter, we can note a number of areas where basic knowledge lacks for buoyant displacement flows: (a) Newtonian flows: Buoyant exchange flows have been previously stud- ied in depth for different inclination angles (β) and density ratios (At). However, there are no extensive studies examining the effects of adding a mean imposed flow (V̂0) to the buoyant exchange flow. Understat- ing the combined influence of these three important parameters i.e. the mean imposed flow velocity, V̂0, the density difference, At, and near- horizontal inclination angles, β, has not been in depth before. There- fore, proper regime classifications even for Newtonian flows predicating the behavior of the flow do not currently exist. In this research area, producing reliable data to shed light on flow characteristics by employ- ing experimental, analytical and computational approaches is of major importance. (b) Effects of viscosity ratio and shear-thinning: Including a viscos- ity ratio in iso-density displacement flows has a long history in liter- ature. However, buoyant displacement flows when a viscosity ratio is present have not been deeply studied. Quantifying the effects of in- creasing/decreasing the viscosity of the displacing/displaced fluid is ex- tremely important especially because, practically speaking, the indus- trial buoyant displacement flows often involve a viscosity ratio. Shear- thinning fluids are in fact viscous fluids with variation of viscosity with shear stress. Therefore, they can present the effects of displacements of variable viscosity ratios. There are not many research works which thoroughly investigate the effects of presence of shear-thinning effects in buoyant displacement flows in close to horizontal geometries. (c) Effects of yield stress: Considering the small number of studies relat- ing to Newtonian buoyant displacement flows in slightly inclined ducts, 44 2.3. Conclusions it is expected that the effects of a yield stress for buoyant flows is even less investigated. In this context, the case where the displaced fluid has a yield stress is of more interest and practical concern. Static residual wall layers are common for these flows. Nonetheless, it is felt that firstly appearance or non-appearance of these static residual layers should be studied in more depth. Second, the thickness of these layers should be quantified versus the flow parameters. 2.3 Conclusions In this chapter we have reviewed both engineering and scientific backgrounds of our displacement flow problem. The engineering background overview clearly demonstrates the complexity of the problem that industry is faced with. Although the main factors accountable for low efficiency displacements during primary cementing operations have been long identified, an overall deep understanding of the problem is yet to be achieved. So far, the modeling and experimental analyses have only managed to propose simple qualitative guidelines for improving primary cement jobs. Therefore, it is fair to say that there is no consensus as a whole on the displacement flow subject. On the other hand, although previous fundamental scientific studies can be of enormous help in understanding of the basic mechanisms of misci- ble buoyant displacement flows, there are still many aspects to explore. In particular the combined effects of many parameters involved are to be de- termined. Below in §2.4 we state our research objectives. One area we do not deal with is turbulent displacements. From the industrial perspective one rea- son for this is that turbulent displacements are typically quite effective, so there is less motivation to improve them. From the scientific perspective we should also regard laminar displacements as the more likely to lead to long regions of “mixed” fluid. As mentioned before, by “mixed” we mean a part of the pipe where more than one fluid is present (i.e. even if the fluids are largely in 2 separate layers). The reason for this is that in a fully turbulent flow the fluid concentration is typically fairly uniform on each cross-section and spreads axially relative to the mean displacement front via turbulent dispersion. This is a diffusive process, which means that the mixed region spreads proportional to the square root of time. We can estimate the dis- persivity via Taylor’s analysis [145], at least in rough order of magnitude. On the other hand in laminar regimes the extent of the “mixed” region is determined by considering the difference in speeds between the fastest and 45 2.4. Research objectives slowest propagating fronts. The fronts often propagate at constant speed (after initial transients) and the difference in front speeds is typically pro- portional to the imposed mean velocity. This means that the “mixed region” grows linearly in time. This linear growth can be considered as a worst case scenario as it is possible for instability and mixing to slow the interpene- tration of fluid layers. Therefore, our study of laminar displacement flows, and at inclinations where they are most likely to be found, is effectively a consideration of the worst case. 2.4 Research objectives The scientific and technical aims of the current thesis are to provide reliable knowledge to better understand buoyant miscible displacement flows, such as those found in well construction processes, by means of an extensive study involving a variety of tools. There are a number of non-dimensional parameters governing and formulating these displacement flows. The key objective is to quantify the effects of each parameter. This essentially starts from examining the effects of the density difference, mean imposed flow velocity, tilt angle, viscosity ratio and rheological parameters (i.e. the power law index and the yield stress). It seems impossible to cover a wide range for each of these parameters. However, we have tried to cover as wide ranges for theses parameters as possible where time, practicality, and budget allowed. Our literature review showed that there are extensive studies regarding classical buoyant miscible Newtonian flows where the basic mechanisms are known. However, the effects of the different controlling parameters espe- cially the imposed flow velocity (V̂0) is not well investigated. The related literature reveals that depending on different parameters, the fluids can be mixed across the duct or stay separated. Based on the available literature, we a priori expect that fluids segregation is usually the case in a close to horizontal configuration and mixing usually comes into view as the pipe or channel is drastically inclined. In this work, we aspire to advance our knowledge of the displacement flows through three activities: (i) Scaled laboratory displacement flow experiments are sought in a cir- cular pipe with orientations close to horizontal. A realistic range of fluid properties and flow parameters are considered. It is of interest to quantify the evolution of the interface between the fluids when they are separated. It is interesting for us to measure the velocity of interpen- etrating fluids and also measure 1D local velocity profiles in a center 46 2.4. Research objectives line across the pipe. Comments are made on mixing between the fluids when it appears. Efforts are devoted to produce experimental corre- lation predicting the behaviors of the flow when possible. The effects of each parameter are determined on the displacement flow and flow regime diagrams are produced. In terms of parameters, the following items are considered: • An experimental apparatus is constructed with a long pipe (i.e. 4 (m)). This pipe is initially filled with two different fluids. The typical experiment consists of a displacement at fixed flow rate and a fixed inclination angle; this forms the bulk of the exper- imental work. There is a possibility to change the tilt angle to investigate the effects of β. • There is possibility to accurately vary the value of the mean im- posed flow added to the control volume to study the effects of V̂0. • Sets of experiments are carried out in which the properties of dis- placing and displaced fluids are modified. It is natural to start with Newtonian fluids and firstly change the density ratio to ob- serve the influence of At. • Viscosities of displacing and displaced fluids are changed to pro- vide an understanding of the effects a viscosity ratio between the two fluids (i.e. m). • Experiments with shear-thinning fluids (i.e. fluids whose viscosi- ties decrease as shear rates increase) are conducted. The be- haviour of these fluids is simply described by a consistency, κ̂, and a power-law index, n. • Finally, our experimental work ends with including yield stress (τ̂Y ) effects usually in the displaced fluids. (ii) In a simplified mathematical study, the focus is on a limiting parameter regime that appears to be tractable semi-analytically and which also has practical relevance. Our mathematical models are mostly lubrica- tion style where the inertial effects are neglected. These models are developed for channel (generalised Newtonian) and pipe (Newtonian) flow displacements. Through this, it is our aim to investigate the ef- fects of the flow parameters on the limits of the buoyant displacement flows (e.g. viscous limit). In this analytical approach the following items are considered: 47 2.4. Research objectives • The study is proceeded non-dimensionally to produce predictive models for channel and pipe flows. For Newtonian iso-viscous flows, hence, our main focus is on a combination of 3 dimension- less parameters which are Re, Fr and β (while neglecting the effects of Pe and At). A viscosity ratio adds another dimension- less parameter (m) to the problem. For non-Newtonian flows, the power law index n and the Bingham number B (i.e. a ratio be- tween yield and viscous stress) are varied. It is our aim to obtain non-dimensional front velocities, interface heights, displacement efficiencies, static layer thickness etc. These analytical (math- ematical) results are then interpreted and directly or indirectly compared to the experimental and simulation results. • Our main focus is on a priori expected viscous regimes in nearly horizontal angles. • Our lubrication style model consists of a 2-layer channel flow model for non-Newtonian fluids. For iso-viscous Newtonian flows both 2-layer and 3-layer channel flows and also 2-layer pipe flows are studied. • Finally, the effects of including weak inertial terms into the lu- brication model of the 2-layer channel flows are considered. In addition, the instabilities involved in the model are studied. (iii) 2D flow simulations with reasonable accuracy are computed over a similar range of parameters compared to our experiments. The re- sults of the simulations are compared with those of the experimental and analytical approaches. It is also our strong desire to provide de- tailed descriptions of the displacement flows and provide flow regime diagrams. To achieve this purpose, the aim is at the following items: • For our computations, code PELICANS developed at IRSN (the French Nuclear Safety Research Institute) is used. The code is firstly benchmarked and validated at sensible mesh resolutions for our usage. • Naturally, the range of dimensional and non-dimensional parame- ters considered is firstly similar to those of the experiments. Then the parameter coverage is extended to include a wider range. • Similarities and differences between the computational study and its corresponding experimental investigation are reported. When 48 2.4. Research objectives similar behavior is observed, the results are compared qualita- tively with those of the experiments as well as quantitatively with those of the analytical models. In this work, for different buoyant displacement flows, the attempt is to provide physical arguments and also formulate appropriate balance equa- tions. As much as the details of these flows are exciting for us, it is of inter- est to generate flow regime maps to give predictions of the flow behaviours for a wide range of non-dimensional parameters. In these flow regime dia- grams, compact experimental and computational results are included. Also, simplified and predictive analytical results are superimposed on the same diagrams. Thus, the key flow features that may occur are described; this is accompanied with established supporting analyses. 49 Chapter 3 Research methodology In this thesis we employ experimental, analytical and numerical approaches to better understand miscible displacement flows, in the presence of buoy- ancy and in ducts that are inclined close to horizontal. Although we mostly consider Newtonian fluids (with or without a viscosity ratio), we also investi- gate generalized Newtonian fluids with both shear thinning and yield stress rheological features. A large part of the effort in this research is devoted to producing high quality experimental and computational data, which is lacking in the literature. In this chapter we describe the following: 1. The experimental techniques: two miscible fluids of different densities are initially placed in an unstable configuration in a long pipe, inclined close to horizontal. A fixed flow rate is applied at the upper end of the pipe. 2. The computational techniques: two miscible fluids of different densities are initially placed in an unstable configuration in a long plane channel, inclined close to horizontal. A fixed flow rate is applied at the upper end of the plane channel. The data from these 2 approaches is analyzed and combined with the analysis of simpler mathematical models. The methodology of the modelling approach is explained later, as the models are developed (e.g. see Chapter 5). 3.1 Experimental technique 3.1.1 Experimental setup Views of the experimental apparatus are given in Figs. 3.1 and 3.2. Our experimental study was performed in a 4 (m) long, 19.05 (mm) diameter, transparent pipe with a gate valve located 80 (cm) from one end. The pipe was mounted on a frame which could be tilted to a given angle. Initially, the lower section of the pipe was filled with a lighter fluid coloured with a small 50 3.1. Experimental technique amount of (black India) ink, and the upper part by a transparent denser solution. The pipe was fed by gravity from an elevated tank. The reason for using gravity as the driving force was to avoid disturbances induced by a pump. The imposed flow rate was controlled by a valve and measured by both a rotameter (Omega, variable-area type) and a magnetic flowmeter (Omega, low-flow type), located downstream of the pipe. The role of the gate valve was to initially separate the fluids. Its mechanism consisted of two distinct parts which were positioned at the upper and lower sections of the pipe. These two parts were clamped together by four sets of long bolts and nuts. Although very rigid, the mechanism allowed the free (reciprocation- like) movement of a thin metal plate, in which a hole with the same size as the pipe diameter was pierced. Note that, despite the necessary precautions taken to minimize disruption of the flow at the valve, the movement (and the resulting shear stress) could slightly disturb the flow (and affect the shape of the interface) at the very beginning of the experiment (i.e. very short time). Also, we measured the inclination angle using an electronic inclinometer (SmartTool) with a digital display and a resolution of ±0.1 ◦. 3.1.2 Visualization and concentration measurement Our main measurement method was based on quantitative image analysis, extracting information regarding large-scale features of the flow such as the front velocity. We were also interested in studying patterns of variation of the average concentration in the pipe cross section (and in a few cases along it). We measured the timewise light intensity passing through the pipe with digital cameras. After computer image processing, this allowed us to obtain the concentration evolution profiles along the tube, averaged concentration profiles on the cross section, and also the spatiotemporal diagrams of changes in these profiles. Note that in this method the measured concentrations were always already integrated along the path of light rays through the pipe. The imaging system consisted of 2 low noise high-speed digital cameras with images recorded at a frame rate of typically 2 or 4 (Hz). These cameras were able to distinguish 212 = 4096 gray-scale levels. The large number of gray levels that was distinguished allowed to analyze a wide range of con- centrations. Each of these cameras usually covered 160 (cm) of the lower section of the pipe but in some cases one of them was used to cover the upper section of the pipe (above the gate valve). In order to help the visu- alization of the phases, the pipe was illuminated from behind by a light box containing 6 fluorescent light tubes filtered through a diffusive paper giving a homogeneous light. Light absorbtion calibration was carried out for both 51 3.1. Experimental technique Flowmeter Drain Gate valve Jack 3.2 m 0.8 m UDV Camera UDV probe Elevated tank Elevated tank Light box Gate valve Pipe Flowmeter Rotameter UDV probe UDV Jack Drain Figure 3.1: Schematic (top) and real (bottom) views of the experimental apparatus. 52 3.1. Experimental technique V Dyed light fluid Transparent heavy fluid Gate valve (open) β Drain Tank UDV probe^ 0 Figure 3.2: Schematic view of experimental set-up. cameras. Fig. 3.3 depicts the variation in logarithmic scale of light intensity across the pipe versus the ink concentration. This calibration plot implies that the transmitted light intensity varies with the concentration following formula I(C) = ΨexpαC , where Ψ and α are physical constants, till a maximum value CMAX which depends on the fluid property and the pipe diameter. Here in our case, we found CMAX = 623 (mg/l). In our experiments, the concentration of the ink in the fluid had to be lower than its maximum satu- rated value and was typically chosen in the range 500−550 (mg/l) to satisfy the optical law perviously mentioned. In this range, relative concentration of dyed fluid with the black India ink is govern by: C − Cmin Cmax − Cmin = log I(C)− log I(Cmin) log I(Cmax)− log I(Cmin) Where I(Cmin) represent the intensity measured without dye (Cmin = 0). This relation allowed us to determine the local normalized concentration, without having to know the calibration constants Ψ and α. The images of the light intensity along the pipe (with typical sizes of 1400× 34 pixels for camera #1 and 1600 × 38 pixels for camera #2) were then translated to normalized concentration maps using reference images previously taken for each of the two pure solutions (0 for the colored light fluid and 1 for the transparent heavy fluid). This measurement method enabled us to obtain images of the normalized concentration, averaged over the depth of pipe. 53 3.1. Experimental technique 0 200 400 600 800 1000 10 100 1000 Concentration of ink (mg/l) A ve ra ge  li gh t i nt en sit y of  th e pi pe CMAX Figure 3.3: Variation in logarithmic scale of light intensity across the tube (in gray levels), depending on the amount of black India ink averaged over 11 pixels across the pipe and 1200 pixels along the pipe. The dotted line corresponds to 623 (mg/l) ink and determines the maximum concentration above which the change of light intensity can no longer be considered to vary exponentially as a function of the concentration of the ink. In order to find the best possible region of optical measurements, we took an image of the pipe filled with the dyed fluid with the concentration Cmax. Then we obtained the average longitudinal light intensity (line by line) measured by camera #1. Fig. 3.4a shows this image and the region where the longitudinal averaging was carried out. Fig. 3.4b demonstrates the average light intensity distribution. We notice that there is an area where edge effects are significant. For camera #1, we limited therefore the range of the practical image processing to 11 pixels, from pixel 11 to 22, in the transverse direction of the pipe. Similarly for camera #2, we found that the practical range of interest had 14 pixels in the transverse direction. Note that along the pipe, 1 pixel ' 1.14 (mm) for camera #1 and 1 pixel ' 1.07 (mm) for camera #2 for most of the experiments. During each experiment and after opening the gate valve, images were obtained at regular time intervals, which enabled us to create spatiotemporal 54 3.1. Experimental technique Longitudinal direction of the pipe (pixel) Tr an sv er se  d ire ct io n of  th e pi pe  (p ixe l) 200 400 600 800 1000 5 10 15 20 25 30 a) Gate valve 100 200 300 400 500 9 11 13 15 17 19 21 23 Average light intensity distribution along the pipe acquired by the camera #1 Tr an sv er se  d ire ct io n of  th e pi pe  (p ixe l) b) Figure 3.4: a) An image taken by camera #1 of a section of the pipe filled with dyed fluid with concentration Cmax; the dashed rectangle shows the region used for the longitudinal averaging in the plot on the right. b) The corresponding luminous intensity longitudinally averaged in the rectangular region. The dashed lines bound the limits of interest to 11 pixels, from pixel 11 to 22, where the effect of the pipe curvature is negligible. diagrams of the concentration profiles along the length of the pipe. The displacement fronts were marked on these diagrams by a sharp boundary between the different relative concentrations of the fluids (the boundary was identified through an edge detection method). The front velocities were obtained from the slope of this boundary (see e.g. Fig. 3.5b). When there is no mixing between the fluids, the normalized concen- tration across the pipe can be interpreted as the normalized height h(x̂, t̂) of the interface at each time, an example of which is shown in Fig. 3.5a. This figure shows a sequence of interface evolution in time obtained for β = 87 ◦, At = 3.6 × 10−3, ν̂ = 1 (mm2.s−1) (i.e. the kinematic viscosity) and V̂0 = 19 (mm.s−1). The distances are measured with respect to the position of the gate vale. The time interval between interface profiles is 2 (s) while the first interface (on the left) corresponds to 6 (s) after opening the gate valve. After a short transition, the interfaces converge to become self-similar; the interface front velocity is constant at long time as the in- set implies. Fig. 3.5b, for the same parameters, depicts the spatiotemporal diagrams where average concentration across the pipe is shown in a plane with distance (x̂) and time (t̂). Using this diagram, it is easy to observe the movement of the front and obtain the front velocity. The front velocity V̂f is equal to the slope of the boundary marked on this diagram. For this case the front velocity is found to be V̂f = 29 (mm.s−1). In the experiments 55 3.1. Experimental technique a) 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 0.8 1 x̂ (mm) h (x̂ ,t̂ ) 0 20 40 60 0 0.2 0.4 0.6 0.8 1 x̂/t̂ (mm/s) h (x̂ /t̂ ) x̂ (mm) t̂ (s ) 1600140012001000800 600 400 200 0 10 20 30 40 50 b) Figure 3.5: a) Experimental profiles of normalized interface height, h(x̂, t̂), for t̂ = 6, 8, .., 48, 50 (s), with β = 87 ◦, At = 3.6× 10−3, ν̂ = 1 (mm2.s−1) and V̂0 = 19 (mm.s−1). The inset shows experimental profiles of normalized h(x̂/t̂) for the same experiments; b) the corresponding spatiotemporal dia- gram obtained for the same parameters, where the slope of the dashed line represents the front velocity. that we have carried out, the slope of this sharp boundary was essentially constant (within the experimental uncertainty) after a short length (> D̂) below the gate valve (this means that the boundary was a linear line). 3.1.3 Velocity measurement As a complementary, we used a velocimetry technique to measure local velocity profiles, which can help to understand the flow dynamics from a different view. We measured the velocity profile somewhere downstream of the flow (usually at 80 (cm) below the gate valve), using an ultrasonic Doppler velocimeter DOP2000 (model 2125, Signal Processing SA) with 8 (MHz), 5 (mm) (TR0805LS) transducers (with a duration of 0.5 (µ.s)). This velocimetry technique well suits our experimental needs since it does not require transparent medium. The measuring volume has a cylindrical shape and its axial resolution in our fluids is around 0.375 (mm) and the lateral resolution is equal to the transducer diameter (5 (mm)) slightly vary- ing with depth. The slightly diverging ultrasonic beam enters the fluids by passing through a 3.175 (mm)-thick plexiglass pipe wall. This technique is based on the pulse-echo technique and allows measurement of the flow ve- locity projection on the ultrasound beam, in real time [65]. This projection gives only the axial component of velocity. The instrument sends a series 56 3.1. Experimental technique of 4-cycles of short bursts and records the echoes back scattered from the particles suspended in fluids. Through the time elapsed between the pulse and the received echo, the distance of the particles from the transducers can be computed; meanwhile the associated Doppler frequency shift gives the value of the velocity at each distance. Reflection effects at the lower wall of the pipe affect the velocity measurement locally, making it hard to measure a zero velocity at the lower wall. For a typical acquisition time of the velocity profiles, 120 (m.s) per pro- file was set while no real time filtration of signals was applied during the recording process. For the tracer, we used polyamid seeding particles with a mean particle diameter of 50 (µm) with volumetric concentration equal to 0.2 (g.l−1) in the both fluids. Considering the trade off between a good signal to noise ratio and also small ultrasonic signal reflections [18], the probe was mounted at an angle in the range 67− 78 ◦ relative to the axis of the pipe. Since the probe was mounted outside the pipe, our measurement technique was completely non-intrusive. We also assumed that the density difference of the fluids used in our experiment is sufficiently small to neglect the differences in the speed of sound in the fluids. 3.1.4 Fluids characterisation Most of the experiments were conducted using water as the common fluid, with salt (NaCl) as a weighting agent to densify the displacing fluid. The fluid densities were measured by a high accuracy portable density meter (An- ton paar, DMA 35N) with a resolution of 0.0001 (g/cm3). To ensure that we had a temperature balance between the two fluids, their temperatures were measured using a high resolution thermometer (Omega mini thermocouple, ±0.1◦C) just before each experiment. To achieve higher viscosity, glycerol solutions were prepared by diluting pure glycerol with water. To provide shear-thinning effects, low percentage Xanthan-water solutions were used. To make a fluid with a yield stress, we used Carbopol solutions. All exper- iments reported in this thesis were density unstable, i.e. heavy fluid in the upper part of the pipe displacing a less dense fluid below. Shear-thinning solution preparation For our shear-thinning solutions we used Xanthan gum powder. Xanthan is a polysaccharide used as a food additive and rheology modifier. Xanthan solutions were mixed at concentrations of 0.3% and less, where the solu- tions are relatively inelastic. In preparation we first weighted the Xanthan 57 3.1. Experimental technique powder then (when needed) gradually added the powder to water while the mixer blade was slowly rotating. This negates the tendency of Xanthan to accumulate. Solutions were then mixed for 24 hours before the experiment was performed. Since the Xanthan concentration were relatively low, the rheology was found to be insensitive to mixing times of this length. Equally, the rheometry results were not sensitive to the blade shape or its rotation speed, usually set between 100−400 (rpm). Samples were taken before each experiment and the rheology was measured at a consistent time after each experiment (although again sensitivity was minimal). Yield stress solution preparation For our viscoplastic fluids we used Carbopol R© EZ-2 polymer (Noveon Inc). Carbopol is widely used as a thickener, stabilizer and suspending agent. It is utilized in a broad range of personal care products, pharmaceuticals and cleaners. Carbopol polymers are high molecular weight acrylic acid chains (usually cross-linked) and are available as powders or liquids. The rheology of Carbopol is largely controlled by the concentration and pH of the solution. Once mixed with water, Carbopol makes an acidic solution with no yield stress. The yield stress is developed at intermediate pH on neutralising with a base agent (in our case NaOH). The neutralised solution is fairly transparent and has the same density as water (for low concentrations). We first weighted Carbopol powder, and then gradually added it to wa- ter while the blade was rotating and stirring the whole solution. In contrast to Xanthan powder, Carbopol molecules do not tend to accumulate in wa- ter thus making it easier to mix. We always mixed Carbopol with water in a consistent way (in our case 24 hours). However, since the Carbopol concentration needed in our experiments was not too high, this mixing time was found to have negligible effect on rheometry. It was also found that when Carbopol concentration was low, the rheometry results (similar to those of Xanthan solutions) were not sensitive to the blade shape or its rotating speed. The Carbopol-water solution is acidic (e.g. pH = 4 for a concentration 0.12 % (wt/wt) corresponding to an approximate yield stress of 3 (Pa) once neutralized) and does not have any yield-stress. In order to form the gel we added Sodium-hydroxide, NaOH. Note that for a given Carbopol solution, the neutralisation takes place over a limited range of NaOH concentration. In other words, the weight/weight ratio of Carbopol to Sodium-hydroxide at which the neutralisation happens (thus forming the gel) is almost a constant (in our case around 3.5). If too much (and/or too low) NaOH is added then the solution transforms to liquid phase again. 58 3.1. Experimental technique The pH of the neutralised gel would fall in the range of 6− 8 which makes it safe for human-related applications (e.g. hair gel). When adding NaOH to Carbopol-water solution we were very cautious about mixing and par- ticularly the blade speed. A careless mixing could introduce significant air bubbles into the gel-like solution. Once trapped inside the solution it is not easy to free up the air bubbles due to the fluid yield stress and its high vis- cosity. If there are air bubbles trapped in gel-like Carbopol solution, using a vacuum pump might help get rid of them. In most of our cases the process at which NaOH was being added to Carbopol-water solution took about 10 minutes for a 35-liter solution. The mixer was then turned off and the samples were taken for rheometry. The Carbopol solution is thixotropic (i.e. rheological properties change with time). In this case the gel-like Carbopol solution loses its viscosity over time. Therefore we had to carry out the rheometry right after each experiment. The samples were taken before each experiment and the rheometry would be done in a consistent way (timewise) after the experiment. A rheological model that fits well the shear behaviour of Carbopol solu- tion is the Herschel-Bulkley model: τ̂ = τ̂Y + κ̂ˆ̇γ n . (3.1) This includes the simpler Bingham, power law and Newtonian models and is defined by three parameters: a fluid consistency index κ̂, a yield stress τ̂Y , and a power law index n. Rheology measurement All the rheological measurements were performed using a Bohlin digital controlled shear stress-shear rate rheometer. A smooth cone-and-plate ge- ometry of 40 (mm) cone diameter, 60 (mm) plate diameter, 4 ◦ cone angle and 150 (µm) gap at the cone tip, was used for rheometry. Fluid samples were first loaded on the bottom plate. The top plate was then lowered to the desired gap height of 150 (µm) by squeezing the extra paste out from between the plates. The excess paste at the plate edges was neatly trimmed with cotton sticks. Identical loading procedures were followed in all the tests. Temperature was being controlled by NESLAB heater/cooler (NES- LAB instruments Inc., Newington, NH, U.S.A.) based on water circulation under the rheometer’s plate. For yield stress measurements, we also had to add a tiny layer of sand paper (with 400 grit roughness) to both cone and plate to be able to read the yield stress; otherwise the sample would slip. 59 3.1. Experimental technique Determining the rheology of Xanthan solution was carried out in a usual fashion. For our shear rheology we applied a strain rate ramp varying over the range 0.1 − 100 (s−1). Xanthan solutions used are modeled as power- law fluids (τ̂ = κ̂ˆ̇γ n ). We used the strain rate range 10 − 100 (s−1) to fit the fluid consistency index, κ̂, and power-law index, n, from a log-log plot of the effective viscosity versus strain rate. Eliminating the very low shear rates ensures high repeatability and is characteristic of the experimental wall shear rate range. In order to ensure that Xanthan solutions have been prepared correctly and to crosscheck the rheometry measurements, the ef- fective viscosity of different Xanthan-water solutions versus the shear rate is compared in Fig. 3.6 against the rheometry results of Gabard & Hulin [58]. The comparison shows good agreement taking into account that data from [58] is for Xanthan-water solutions plus 70 % glycerol while our data is for pure Xanthan-water solutions. The small difference between the our results and data from [58] is probably due to presence of glycerol in their solutions. 10−1 100 101 102 10−2 10−1 100 ˆ̇γ (s−1) η̂ (P a .s ) Figure 3.6: Variation of the effective viscosity η with shear rate ˆ̇γ for Xanthan-water solutions of various concentrations. The data points cor- respond to 0.3 % (•), 0.2 % (N) and 0.15 % (H); filled data points are our measurement while hollow data points refer to rheometry of Xanthan-water solutions (+ 70 % glycerol) reported by Gabard & Hulin [58]. 60 3.1. Experimental technique For yield stress fluids, we determined the yield stress through the shear stress value at the global maximum of the viscosity. Afterwards, we sub- tracted the yield stress value from the remaining data and then we found the best fit to a power law curve. The practical range of shear rate used to obtain repeatable results for determining κ̂ and n to fit in the power law model was 10 − 100 (s−1). The error for the yield stress value of the Car- bopol solution was in the range 5 − 27 % and for the consistency (κ̂) and the power law index (n) was always below 7 % and 12 % respectively. An example flowcurve from the rheometer measured data compared with the curve fitted from Herschel-Bulkley model is shown in Fig. 3.7. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 ˆ̇γ (1/s) τ̂ (P a ) Figure 3.7: Example flowcurve for a visco-plastic solution with Carbopol=0.12 % (wt/wt) and NaOH=0.0343 % (wt/wt). The rheological properties of the Carbopol solution are described by the Herschel-Bulkley model, τ̂ = τ̂Y + κ̂ˆ̇γ n : the solid line shows the curve fit with parameters τ̂Y = 3.05 (Pa), n = 0.60 and κ̂ = 8.24 (Pa.sn). 3.1.5 Experimental results validation We first calibrated our apparatus against exchange flow results of Seon et al. [132, 135] for different Atwood numbers at β = 85 ◦ and β = 87 ◦. The 61 3.2. Computational technique errors in measured front velocity were always below 2% for the cases studied and the experiments had a high degree of repeatability. 3.2 Computational technique In place of physical experiments, we have carried out a number of numerical simulations of 2D displacements in an inclined plane channel. The geometry and notation are as represented in Fig. 3.8. The computations are fully iner- tial, solving the full 2D Navier-Stokes equations with phase change modelled via a scalar concentration, c. The system for two Newtonian fluids of equal viscosity is given as: [1 + φAt] [ut + u · ∇u] = −∇p+ 1 Re ∇2u+ φ Fr2 eg, (3.2) ∇ · u = 0, (3.3) ct + u · ∇c = 1 Pe ∇2c. (3.4) Here eg = (cosβ,− sinβ) and the function φ(c) = 1−2c interpolates linearly between 1 and −1 for c ∈ [0, 1]. The 4 dimensionless parameters appearing in (3.2) are the angle of inclination from vertical, β, the Atwood number, At, the Reynolds number, Re, and the (densimetric) Froude number, Fr. These are defined as follows: At ≡ ρ̂1 − ρ̂2 ρ̂1 + ρ̂2 , Re ≡ V̂0D̂ ν̂ , F r ≡ V̂0√ AtĝD̂ . (3.5) Here ν̂ is defined using the mean density ρ̂ = (ρ̂1 + ρ̂2)/2 and the common viscosity µ̂ of the fluids. In (3.4) appears a 5th dimensionless group, the Péclet number, Pe, defined by: Pe ≡ V̂0D̂ D̂m , (3.6) with D̂m the molecular diffusivity (generally assumed constant for simplic- ity in our work). In our computations, the effect of molecular diffusion is neglected. This neglect is due to the large Péclet number that correspond to our experimental flows, for which we typically have a well defined interface. The equations (3.2)-(3.4) have been discretised using a mixed finite ele- ment/finite volume method. The Navier-Stokes equations are solved using 62 3.2. Computational technique Galerkin finite element method, where the divergence-free condition is en- forced by an augmented Lagrangian technique [60]. We use a fixed time step for the Navier-Stokes equations, advancing from time step N to N +1. Regarding the implementation of the nonlinear terms, we use a semi-implicit method. The convective velocity is approximated at time step N while the linear spatial derivatives of the velocity are approximated implicitly at time step N + 1. The pressure is approximated at time step N + 1. The computations are carried out on a structured rectangular mesh, with linear elements (Q1) for the velocity and constant elements (P0) for the pressure discretisation. The concentration equation (3.4) uses a finite volume method, in which the concentration is approximated at the cen- tre of each regular mesh cell. The advective terms are dealt with via a MUSCL scheme (Monotone Upstream-centered Schemes for Conservation Laws). These are essentially slope-limiter methods for reducing oscillations close to discontinuities; see e.g. [160] and [94] for more description. On each (Navier-Stokes) time step a splitting method is used to advance the concen- tration equation over a number of smaller sub-timestep. This time advance is explicit and a CFL (Courant-Friedrichs-Lewy) condition is implemented for the sub-timesteps to ensure numerical stability. x eu 0 ˆˆ V= D̂ L̂ x̂ ŷ on walls ,0ˆ =u ĝ β Figure 3.8: Schematic of the computational domain, the geometry and the notation. The initial interface starts within the channel, with c = 0 upstream and c = 1 downstream. Typically we choose the channel thickness equal to our experimental pipe diameter (D̂ = 19.05 (mm)). For the channel length, we typically have L̂ = 100× D̂. The present numerical algorithm is implemented in C++ as an appli- cation of PELICANS. PELICANS is an object oriented platform developed 63 3.2. Computational technique a) x̂ (mm) t̂ (s ) 500 1000 1500 2000 10 20 30 40 50 60 70 80 90 100 b) Figure 3.9: Computational concentration field evolution obtained for β = 85 ◦, At = 3.5 × 10−3, ν̂ = 1 (mm2.s−1), V̂0 = 15.8 (mm.s−1), (Re = 300). a) Sequence of images from top to bottom are shown for t̂ = 0, 10, 20, 30, 40, 50, 60 (s). b) Spatiotemporal diagram of the aver- age concentration variations (blue and red colors represent heavy (c = 0) and lighter (c = 1) fluids respectively) along the channel. The heavy broken line shows the temporal evolution of the leading front and its slope is the leading front velocity (V̂f = 22.4 (mm.s−1)). at IRSN (the French Nuclear Safety Research Institute), to provide a gen- eral framework of software components for the implementation of partial differential equation solvers. PELICANS is distributed under the CeCILL license agreement (http:// www.cecill.info/ licences/ Licence CeCILL V2- en.html). PELICANS can be downloaded from https:// gforge.irsn.fr/ gf/ project/ pelicans/. Although the equations could have been implemented in a commercial CFD (Computational Fluid Dynamics) code, these codes are often over-stabilised and give little access to the detailed implementation. As boundary conditions for our simulations, we impose no-slip and zero flux of c at the solid walls. A plane Poiseuille flow is imposed at the inflow, along with c = 0. Outflow conditions are imposed at the channel exit. The initial interface starts within the channel, with c = 0 upstream and c = 1 downstream. The initial velocity is u = 0. We have usually selected a range of dimensional parameters that is similar in scope to those of our pipe flow experiments. After running each simulation, the front velocities were calculated from the spatiotemporal plot of c, i.e. mimicking the experimental procedure. For example Fig. 3.9 shows computational concentration field evolution obtained for a typical simulation with parameters β = 85 ◦, At = 3.5 × 10−3, ν̂ = 64 3.2. Computational technique 1 (mm2.s−1), V̂0 = 15.8 (mm.s−1), (equally Re = 300). In Fig. 3.9a we observe the sequence of images and in Fig. 3.9b we show the corresponding spatiotemporal. The slope of the heavy dashed line in Fig. 3.9b represents the front velocity. Mesh refinement was carried out until successively calculated front ve- locities on meshes differed by 1−4%, (over the range of physical parameters explored). For the meshes in most of the computations we used 28 cells across the channel, refined slightly towards the walls, and 400 cells along the length of the channel. However, we have conducted a number of sim- ulations with (e.g. up to twice as much) finer mesh resolution producing only a little difference in the measured front velocities, within the limits of our desired accuracy. We acknowledge that the meshes used are relatively coarse, but note that the principal information being extracted from the simulations is bulk information, e.g. spatiotemporal plots and front speeds. These features are less sensitive to refinement, which would be advisable if e.g. flow instabilities and mixing were to be directly studied. 3.2.1 Code benchmarking Various simple test problems have been implemented. The code has also been benchmarked against representative numerical and experimental stud- ies. For example, we have compared our simulation results with those of Fig. 2 and Fig. 3 in [126], (for β = 60 ◦, Re = 200, V̂0/ √ ĝD̂ = 0.316, µ̂2/µ̂1 = 2, ρ̂2/ρ̂1 = 1.5); see Fig. 3.10. We find close agreement with the computed front velocity and also observe similar qualitative behavior in the displacement flow behind the front. In private communications with Sahu & Matar (from the Chemical Engineering Department at Imperial College Lon- don) we have also benchmarked our code for near-horizontal channels. We have compared two sets of displacement flows (ν̂ = 1 (mm2.s−1), At = 10−3), at β = 83 ◦ and β = 87 ◦. In each case we have studied a sequence of in- creasing imposed flow (Re = 50−500). For low-moderate Reynolds numbers (i.e. Re ≤ 300), we found our results matching well with theirs. For higher Reynolds numbers we found the onset of small interfacial waves, occurring in our simulations at slightly higher values of the imposed flows than with their code. We have also compared our results with those of Hallez & Magnaudet [67] for exchange flow in a 2D channel. The emphasis in [67] is on the initial slumping phase (which is also inertial) and on quantifying the details of mixing and instability. They have consequently considered shorter channel lengths (32 × D̂) and shorter computational times than we have. By com- 65 3.2. Computational technique 0 10 20 30 40 50 0 10 20 30 40 50 x t Figure 3.10: Spatiotemporal diagram of the average concentration varia- tions (blue and red colors represent heavy (c = 0) and lighter (c = 1) fluids respectively) along the channel for β = 60 ◦, Re = 200, V̂0/ √ ĝD̂ = 0.316, µ̂2/µ̂1 = 2, ρ̂2/ρ̂1 = 1.5. The heavy broken line shows the temporal evo- lution of the leading front from Fig. 2 in [126]. Axes x and t are non- dimensionalised using D̂ and D̂/V̂0 respectively. parison, we are concerned with displacement flows, long time flow behaviour and estimating global features such as the front velocity. Our typical com- putational channel length exceeds 100× D̂ and we have significantly coarser meshes. We have however performed a number of simulations for channel ex- change flow configurations to compare with [67] over the range β = 60−90 ◦, and captured all the main trends and qualitative behaviors reported in their work. For example, we observe the strong influence of vortices periodically cutting the channels of pure fluid which feed the advancing fronts and help to maintain constant front velocity (see §2.2.3). In near-horizontal channels we have observed an initial inertial phase during which the front velocity re- mains approximately constant. Afterwards, viscous effects come to play and front velocity decreases and attains a final velocity, depending on balance between viscous and permanent/logitudinal buoyancy forces. Also similar to [67], for a wide range of inclinations (β = 60 − 90 ◦) for At = 4 × 10−3, 66 3.2. Computational technique ν̂ = 1 (mm2.s−1) and D̂ = 20 (mm), we have compared the densimetric Froude number during the initial slumping phase in a channel flow. On increasing the angle from horizontal, we have observed a slight increase in the front velocity and found a constant plateau of modified Froude number versus tilt angle between β = 70 − 80 ◦; see e.g. Fig. 5 in [67]. Although we have good qualitative agreement with [67], some quantitative differences exist. For example, our front velocities were 10 − 15% lower than values reported by Hallez & Magnaudet [67]. These authors actually commented that their front velocities were larger than expected (see Figs. 4 or 7 in their work), by comparison e.g. with the corresponding experiments performed by Seon et al. [131–133, 135]. This difference is at least partly attributed to the short timescale of the numerical experiments in [67], i.e. for a few cases in [67] the computational runs were extended, giving markedly better comparisons with experimental values. Apart from these comparisons, the same code has been used extensively in [73] where it has been benchmarked against the recent experiments of [42, 43], in which miscible core-annular Newtonian flows of differing viscosi- ties develop pearl and mushroom shaped instabilities. Good quantitative comparisons were made. There is numerical diffusion present in solution of (3.2)-(3.4). Imple- menting molecular diffusion within (3.4) was also tested, i.e. by adding (1/Pe)∇2c to the right hand side. However, for the mesh sizes we have used it was found that for Pe ≥ 105 there was no discernible difference in results, i.e. numerical diffusion is dominant. This range of Pe easily includes the experimental range. It is interesting that for some of our simulations we do get substantial mixing and this signifies that the cause of the mixing is primarily dispersion via secondary flows and instability To summarize, our code has produced similar results to the available computational and experimental studies. These are complex flows with few precise analytical solutions to benchmark against. In comparison to our code, we must acknowledge that there are numerically more sophisticated codes in current usage, e.g. [13, 67, 68]. If we wanted to study inertial and unstable regimes in detail, higher resolution and/or development and usage of such a code could be advisable. However, here our principal aim is extraction of bulk flow features (such as front velocity) over a range of parameters for which our code is adequate. Usage of our particular code is also partly influenced by its flexibility to be extended to non-Newtonian multi-fluid flows, which is the eventual aim of the study of these flows (al- though we do not present any results from non-Newtonian fluid simulations in this thesis). Here other researcher in our laboratory has also made some 67 3.2. Computational technique progress, e.g. [73, 74, 161]. 68 Chapter 4 Preliminary experimental results4 Over the course of the thesis a large number of experiments (and computa- tions) were performed, in a wide range of parameters. In a typical experi- mental sequence, all parameters were fixed and experiments were performed at successively increasing mean imposed flow velocity V̂0. Some qualitative features were commonly found across all experimental sequences. In this chapter we give a quick overview of the main experimental observations and key qualitative characteristics of the flows. Specifically, we identify several commonly observed flow regimes. We demonstrate that the superposition of a pressure-driven flow on an exchange flow strongly influences the front velocity and the physical mecha- nisms that dissipate energy. The front velocity V̂f is presented as a function of the mean flow velocity, V̂0, in three different flow regimes. An interesting finding of this work is that a transition of the flow from inertia-dominated behaviour to viscous-dominated behaviour, was observed with increased en- ergy introduced into the system (via V̂0). 4.1 Observation of 3 different regimes We present the results of a typical experimental sequence, as V̂0 is increased from zero, in Fig. 4.1 for β = 83 ◦, At = 10−2 and µ̂ = 10−3 (Pa.s). Phenomenologically, we observe 3 distinct behaviours as V̂0 is increased from zero. (i) As V̂0 → 0, we observe an exchange-flow dominated regime: the imposed flow has only a slight influence on the dynamics of the exchange flow. For the case depicted in this figure, we are in the inertial regime [132], since Ret cosβ = 101 > 50, and the flow develops some shear instabilities at the interface (see §2.2.3 for details and definitions). (ii) In the second 4A version of this chapter has been published: S.M. Taghavi, T. Seon, D.M. Martinez and I.A. Frigaard. Influence of an imposed flow on the stability of a gravity current in a near horizontal duct. Phys. Fluids 22, 031702 (2010). 69 4.1. Observation of 3 different regimes Vf (mm/s) V0 (mm/s) Vf = 1.3V0 +20 (mm/s) Vf (mm/s) 1 2 3 200 150 100 50 0 200150100500 250 0 200 400 600 800 0 2 4 6 8 10 12 16 10 3 Re 3 ^ ^ ^ ^ ^ × Figure 4.1: Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for β = 83 ◦, At = 10−2, µ̂ = 10−3 (Pa.s). The dashed line is a linear fit of data points in the mean flow dominated regime whereas the dotted line shows the slope of the final mean flow regime (V̂f ∼ V̂0). The inset displays the data but for higher mean flow values, and as a function of Reynolds (number based on V̂0), the dashed square represents the range of the main plot. The insets are pictures of a 20 (cm) long section of tube, 80 (cm) below the gate valve in the corresponding flow domains. regime, the balance between pressure gradient and dissipative forces still exists but the mean flow becomes stronger than the buoyancy driven flow, and so controls its dynamic. The main feature here is a linear relationship between V̂f and V̂0 (here V̂f/V̂0 ≈ 1.3). We have conducted a large number of experiments at various At and ν̂ in this regime. We have observed that the slope V̂f/V̂0 does not vary significantly with At and ν̂. In particular we emphasize that this linear relationship is found for cases for which the first regime may be either inertial or viscous. These observations will be discussed quantitatively and in more depth in Chapter 7. (iii) For V̂0 À 0, we observe a second linear regime, with V̂f ≈ V̂0. This third regime is displayed partially on the main curve and more completely in the inset of Fig. 4.1. It is defined by the buoyancy forces becoming negligible compared 70 4.2. Stabilizing effect of the imposed flow (a) (b) (c) Figure 4.2: Three snapshots of video images taken for different mean flow and showing the flow stability induced by the Poiseuille flow. These images are obtained for β = 83 ◦, At = 10−2, µ̂ = 10−3 (Pa.s) and mean flow veloci- ties: (a) V̂0 = 9 (mm.s−1), (b) V̂0 = 71 (mm.s−1) and (c) V̂0 =343 (mm.s−1) (the corresponding buoyant velocity is V̂ V̂0=0f = 31 (mm.s −1)). The field of view is 700 × 19 (mm) and taken 30 centimeters below the gate valve. The images are taken at: (a) 33 (s), (b) 12 (s), and (c) 5 (s) after opening the valve. to the imposed pressure gradient. The third regime occurs when the imposed flow is turbulent (Re ≥ 3000, see inset). As a result, the two fluids mix (see inset) and are completely displaced (V̂f ≈ V̂0). We also observe a transitional zone between the second and third regimes in Fig. 4.1. 4.2 Stabilizing effect of the imposed flow We now focus on an interesting finding of our work, i.e. the influence of the imposed flow on the stability of the system. To illustrate this we show in Fig. 4.2 images from the flows of Fig. 4.1, for three different representative mean imposed flow velocities. Fig. 4.2 displays images of the 70 (cm) long section of the tube, tilted at β = 83 ◦, taken 30 (cm) below the gate, (out of view on the left hand side), for the same density contrast and viscos- ity. The heavier transparent fluid is moving downward, i.e. from left to right. In Fig. 4.2a we observe an inertial gravity current where, behind the front, pseudo-interfacial shear instabilities (Kelvin-Helmholtz like) develop and induce a little mixing between the two fluids transversally across the section. This low mean flow case (V̂0 = 9 (mm.s−1)) is in the first regime (see Fig. 4.1) where the flow is driven by a balance between buoyancy and inertia, (since here Ret cosβ > 50, see §2.2.3 for details). In Fig. 4.2b with an increased imposed flow we observe a stable flow in which there are no Kelvin-Helmholtz instabilities at the interface. Consequently there is no mixing between the two fluids. Moreover, the front height is small and the slope of the interface with respect to the pipe axis is constant and weak. 71 4.2. Stabilizing effect of the imposed flow We infer that the velocity field is quasi-1D and is therefore under conditions where the lubrication approximation becomes valid; the flow dissipates its energy by viscosity. Compared to Fig. 4.2a, this behaviour appears quite counter intuitive since more energy is being injected into the system as V̂0 is greater than in the previous case. As the mean flow approaches a Poiseuille flow, the flow is inherently stable in this range of Reynolds number. This demonstrates the key observation of this chapter: even though the Reynolds number is increased, the imposed flow stabilizes the initial inertial exchange flow by making the streamlines quasi-parallel. Furthermore, as stability re- sults from a quasi-parallel approximation, a small perturbation can break this fragile geometry and induce the propagation of a local burst along the interface. When such a burst appears, it induces transverse mixing. Finally, if the mean flow velocity (see Fig. 4.2c) is further increased, i.e. much higher than the buoyant velocity, the flow reaches the third regime where buoyancy forces are negligible. In this case, the stretched interface combined with the transverse mixing induced by the turbulent mean flow results in a complete displacement. The two pure fluids are separated by a mixing zone. If we consider the pure exchange flow in this configuration, Seon et al [135] showed that this exchange flow can become viscous by using a lubri- cation approximation argument. However, in this case, this quasi-parallel approximation is usually not valid everywhere. The front usually appears in the form of an inertial “bump”, with a velocity equal to √ Atĝĥf , where ĥf (height of the front) adapts itself to maintain a front velocity equal to the viscous bulk velocity. Such a viscous exchange flow with an inertial bump is displayed on the top image of Fig. 4.3. This sequence displays a 45 (cm) section of the tube, a few centimeters below the gate valve (out of view on the left hand side). The images are plotted every ∆t̂ = 0.5 (s), and this sequence corresponds to an experiment conducted at β = 87 ◦, where the mean flow (V̂0 = 77 (mm.s−1)) was imposed after the first image. We observe in this sequence that the inertial bump disappears under the effect of the mean flow. Indeed, the top of the bump seems to move faster than its base, or in other words, the Poiseuille velocity gradient spreads the initial shape of the bump out. The lubrication approximation, which could not be valid at the front for the exchange flow configuration, is now valid everywhere due to the mean flow (except perhaps very close to the front). Indeed, the only way for the inertial bump to disappear is to be subjected to a laminar flow in this region and this can only be achieved when the streamlines in this region are parallel. Therefore, the flow is now dominated by the Poiseuille flow and 72 4.3. Summary Figure 4.3: Sequence of images showing the initial bump shape spread out by the Poiseuille velocity gradient. This sequence is obtained for β = 87 ◦, At = 10−2, µ̂ = 10−3 (Pa.s). and the mean flow (V̂0 = 77 (mm.s−1)) is imposed after the first image (top one). The field of view is 452 × 20 (mm) and taken a few centimeters below the gate valve. The sequence starts 7 (s) after opening the gate valve and the time interval between images is ∆t̂ = 0.5 (s). the buoyancy driven flow becomes a correction. In order to have a better image of the stabilizing effect of the imposed flow, Fig. 4.4 illustrates snapshots of an experimental sequence of increasing the imposed flow for β = 85 ◦, At = 10−2 and µ̂ = 10−3 (Pa.s). In this figure we clearly observe the decay in the amplitude of the interfacial waves propagating at the interface. 4.3 Summary To summarise, these experiments have allowed us to quantify the influence of an imposed flow on the well-studied buoyant exchange flow configuration. We have observed 3 distinct regimes as a function of V̂0. • In the first regime, defined for a low mean flow, the dynamics is gov- erned by the balance between buoyancy forces and dissipative forces, 73 4.3. Summary −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 −0.1 0 0.1 100 140 180 220 260 300 −0.1 0 0.1 t̂ (s) Figure 4.4: Illustration of stabilizing effect of the imposed flow on the waves observed at the interface for β = 85 ◦, At = 10−2 and µ̂ = 10−3 (Pa.s). Y-axis is h(t̂) −∑300 (s) t̂=100 (s) h(t̂), where h(t̂) is the normalized concentration across the pipe at each time averaged over 20 pixels (22.7 (mm)) measured 80 (cm) below the gate valve. From top to bottom we show images for V̂0 = 38, 42, 44, 49, 61 (mm.s−1). which depends on the fluid properties and can be either viscous or inertial. • In the second regime, defined for higher values of the mean flow, the front velocity varies linearly with the imposed flow velocity. We will show that this result is in a good agreement with theoretical/analytical work presented in Chapter 5 in the case of a laminar flow between parallel plates and in Chapter 7 for a pipe geometry. • In the imposed flow dominated regime (i.e. the second regime) the im- posed flow stabilizes the unstable buoyant flow by making the stream- lines more parallel. In other words, it tends to decrease the inertial 74 4.3. Summary term in the governing Navier-Stokes equations. We have seen that this inertial term, which was not negligible at the front for the laminar ex- change flow (e.g. presence of the inertial bump), is removed by a suffi- ciently strong imposed flow. A different way of viewing this is to note that when V̂0 → 0, the instabilities at the surface of the current are due to the shear created by the exchange flow (due to buoyancy). If a mean flow is imposed, the relative influence of buoyancy decreases compared with that of the pressure gradient: the velocity gradient at the surface will decrease whereas the stratification remains unchanged. Thus, the local gradient Richardson number (loosely speaking Ri = StratificationShear ) increases and the flow becomes more stable. Obviously, both expla- nations require quantifying. In order to partly quantify the decrease of the inertial term, we have carried out local velocity measurements using the Ultrasonic Doppler Velocimetry (UDV) technique, for which the results will be presented in Chapter 7. • On the other hand, it is expected that higher buoyancy forces would not stabilize the flow. Indeed in this case, the mean flow required to stabilize the buoyant flow may itself be unstable, and so the flow would transition from an unstable buoyancy dominated regime to a turbulent pressure-driven regime. • Finally, in the third regime, defined when the buoyancy forces are negligible, the mean flow is turbulent. The two fluids are displaced at the mean flow velocity and a mixing zone separates the two pure fluids. In this turbulent regime, we can expect that for a suitably strong mean flow and over long enough time-scale, the mixing zone will spread diffusively governed by turbulent Taylor dispersion [145]. Thus, V̂f ∼ V̂0 may not be strictly valid in this regime for longer times. The occurrence of the above 3 regimes and the transitions between vis- cous dominated and inertially dominated flows frames much of the work presented in this thesis. 75 Chapter 5 Lubrication model approach for channel displacements5 As we have seen in Chapter 4, as the displacement flow rate is increased from zero we enter a regime that is dominated by the imposed flow, where the front velocity increases approximately linearly with the imposed mean velocity. Frequently these flows are viscous dominated and the interface elongates progressively as the front proceeds. This is a classical configuration where it is common to adopt a thin-film or lubrication approach to modelling the flow. This type of model is easiest to develop for a 2D plane channel displacement, rather than the 3D pipe flow. This is the approach that we develop in this chapter. In outline we proceed as follows: • A 2D plane channel that is considered, inclined close to horizontal and with a single elongated interface separating two generalised Newtonian fluids of different density. • We simplify the Navier-Stokes equations and derive a lubrication/thin film approximation. • A semi-analytical solution is found for the flux function that drives the interface propagation problem. • We analyse the flux function and show that there are no steady trav- eling wave solutions to the interface propagation equation. • At short times, diffusive effects of the interface slope are dominant and there is an exchange flow, relative to the mean flow. We find a short-time similarity solution governing this initial counter-current flow. 5A version of this chapter has been published: S.M. Taghavi, T. Seon, D.M. Martinez and I.A. Frigaard. Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1-35 (2009). 76 5.1. Two-fluid displacement flows in a nearly horizontal slot • At longer times we analyse the hyperbolic part of the thin film model, which allows us to predict the propagation speeds of the displacement (at long times). • We explore the effects of viscosity ratio, inclinations and rheological properties on the front height and front velocity, which also define the displacement efficiency. • For displaced fluids with a yield stress it is possible for the displaced fluid to remain static on the wall of the channel. We analyse the maximal static layer thickness. The simplification of the plane channel allows us to develop the lubrica- tion model for a very wide range of fluid types. Later in this thesis (Chap- ters 6 and 7) we develop a similar model for iso-viscous Newtonian fluids in a pipe. In Chapter 8 we analyse the plane channel model further, for Newtonian and power law fluids of different viscosities. Thus, the methods of this chapter have much wider application. β D̂ 0V̂ Light Heavy ŷ x̂ ˆ ˆˆ ˆ( , )y h x t= Figure 5.1: Schematic of displacement geometry. 5.1 Two-fluid displacement flows in a nearly horizontal slot We consider a two-dimensional region between two parallel plates, separated by a distance D̂, that are oriented at an angle β ≈ pi/2 to the vertical. The 77 5.1. Two-fluid displacement flows in a nearly horizontal slot slot is initially filled with fluid 2, which is displaced by fluid 1, injected at x̂ = −∞ with a mean velocity V̂0. Cartesian coordinates (x̂, ŷ) are as shown in Fig. 5.1. Both fluids are assumed to be generalised Newtonian fluids, with rhe- ologies described below, and although the fluids are miscible we consider the large Péclet number limit in which no significant mixing occurs over the timescales of interest. The dimensionless equations of motion, valid within each fluid region Ωk, k = 1, 2, are: φkRe [ ∂u ∂t + u ∂u ∂x + v ∂u ∂y ] = −∂p ∂x + ∂ ∂x τk,xx + ∂ ∂y τk,xy + φk cosβ St , (5.1) φkRe [ ∂v ∂t + u ∂v ∂x + v ∂v ∂y ] = −∂p ∂y + ∂ ∂x τk,yx + ∂ ∂y τk,yy − φk sinβ St , (5.2) ∂u ∂x + ∂v ∂y = 0. (5.3) Here u = (u, v) denotes the velocity, p the pressure, and τk,ij is the ij-th component of the deviatoric stress in fluid k. The parameter φ1 ≡ 1, and the 3 dimensionless parameters appearing above are the density ratio φ2, the Reynolds number, Re, and the Stokes number, St, defined as follows. φ2 = φ ≡ ρ̂2 ρ̂1 , Re ≡ ρ̂1V̂0D̂ µ̂1 , St ≡ µ̂1V̂0 ρ̂1ĝD̂2 . (5.4) Here ρ̂k is the density of fluid k, µ̂1 is a viscosity scale for fluid 1 and ĝ is the gravitational acceleration. Further dimensionless parameters will appear in constitutive laws, defining the deviatoric stresses. In order to derive (5.1)- (5.3) we have scaled distances using D̂, velocities with V̂0, time with D̂/V̂0, pressure and stresses with µ̂1V̂0/D̂. On the walls of the slot the no-slip condition is satisfied. Due to the scaling adopted, we have ∫ 1 0 u dy = 1. (5.5) in each cross-section. The slot is assumed infinite in x, with the interface between fluids initially localised close to x = 0. We shall consider flows that are buoyancy dominated, in which the heavier fluid lies at the bottom of the slot, separated from the lighter upper fluid by an interface that we denote 78 5.1. Two-fluid displacement flows in a nearly horizontal slot by y = h(x, t) and assume to be single-valued. Across the interface, velocity and stress are continuous. The interface is simply advected with the flow, satisfying a kinematic condition. 5.1.1 Constitutive laws The fluids are assumed to be generalised Newtonian fluids. In particular we are interested to understand shear thinning and yield stress effects. A suitable model that incorporates these effects is the Herschel-Bulkley model, which incorporates also the simpler Bingham, power law and Newtonian models. Constitutive laws for the Herschel-Bulkley fluids are: γ̇(u) = 0 ⇔ τk(u) ≤ Bk, x ∈ Ωk, (5.6) τk,ij(u) = [ κkγ̇ nk−1(u) + Bk γ̇(u) ] γ̇ij(u) ⇔ τk(u) > Bk, x ∈ Ωk. (5.7) where the strain rate tensor has components: γ̇ij(u) = ∂ui ∂xj + ∂uj ∂xi , (5.8) and the second invariants, γ̇(u) and τk(u), are defined by: γ̇(u) = 1 2 2∑ i,j=1 [γ̇ij(u)]2 1/2 , τk(u) = 1 2 2∑ i,j=1 [τk,ij(u)]2 1/2 . (5.9) Herschel-Bulkley fluids are described by 3 dimensional parameters: a fluid consistency κ̂, a yield stress τ̂Y and a power law index, n. The parameter κ1 = 1 and κ2 is the viscosity ratio m: m ≡ µ̂2 µ̂1 = κ̂2[V̂0/D̂]n2−1 κ̂1[V̂0/D̂]n1−1 , (5.10) where µ̂2 is a viscosity scale for fluid 2. Note that in the case of 2 Newtonian fluids, µ̂k = κ̂k. The Bingham numbers Bk are defined as: Bk ≡ τ̂k,Y κ̂1[V̂0/D̂]n1 . (5.11) 79 5.1. Two-fluid displacement flows in a nearly horizontal slot 5.1.2 Buoyancy dominated flows The objective of our study is to understand a particular limit of (5.1)-(5.3), in which inertia is not considered to be dominant and the interface orients approximately horizontally along the axis of the slot: moderate Re, β ≈ pi/2 and φ ∼ O(1). The ratio of buoyancy to viscous forces is given by the parameter |φ − 1|/St. We suppose that |φ − 1|/St À 1 so that the interface elongates over some (dimensionless) length-scale δ−1 À 1. To define this length-scale we assume that the dynamics of spreading of the interface, relative to the mean flow, will be driven by buoyant stresses which have size: |ρ̂1 − ρ̂2|ĝ sinβD̂ in the y-direction. These stresses, which act across the interface where there is a density difference, translate into axial stresses according to the slope of the interface. If the slope of the interface has size D̂/L̂, the stress that acts to spread the flow axially has size |φ − 1|ρ̂1ĝ sinβD̂2/L̂. This tendency to spread is resisted by viscous stresses within the fluids, of size µ̂1V̂0/D̂, which dissipate the energy injected by buoyancy. By matching these two terms, we can obtain the characteristic spreading length in this regime : |φ− 1|ρ̂1ĝ sinβD̂2/L̂ = µ̂1V̂0/D̂ ⇒ L̂ = |φ− 1|ρ̂1ĝ sinβD̂ 3 µ̂1V̂0 (5.12) Thus, the ratio between the axial length-scale and channel width is: δ−1 = L̂ D̂ = |φ− 1|ρ̂1ĝ sinβD̂2 µ̂1V̂0 = |φ− 1| sinβ St (5.13) Following standard methods, see e.g. [93], we re-scale as follows δx = ξ, δt = T, δp = P, v = δV, and arrive at the following reduced system of equations, in each fluid region Ωk, k = 1, 2: δφkRe [ ∂u ∂T + u ∂u ∂ξ + V ∂u ∂y ] = −∂P ∂ξ + ∂ ∂y τk,ξy + φk cosβ St +O(δ2), δ3φkRe [ ∂V ∂T + u ∂V ∂ξ + V ∂V ∂y ] = −∂P ∂y − δφk sinβ St +O(δ2), ∂u ∂ξ + ∂V ∂y = 0. 80 5.1. Two-fluid displacement flows in a nearly horizontal slot To aid interpretation of our model results, note that the time and length variables, (T, ξ), are related to the dimensional time and length by: |ρ̂1 − ρ̂2|ĝ sinβD̂3 µ̂1V̂0 ξ = x, |ρ̂1 − ρ̂2|ĝ sinβD̂3 µ̂1V̂ 20 T = t (5.14) Note that we have used D̂/V̂0 to scale t, which is the usual convective timescale based on the mean velocity and D̂. Therefore, the scale related to the slow time variable, T , corresponds to the time taken to travel the characteristic spreading length L̂ at mean velocity V̂0. We now consider the limit δ → 0 with Re fixed: 0 = −∂P ∂ξ + ∂ ∂y τk,ξy + χ φk |1− φ| , (5.15) 0 = −∂P ∂y − φk|1− φ| , (5.16) where χ = cotβ/δ. The parameter χ measures the relative importance of the slope of the channel to the slope of the interface, in driving buoyancy related motions. We wish to consider channels that are close to horizontal, where the slopes of both the channel and the interface may be of comparable importance. Thus, we assume χ is an order 1 parameter, i.e. we consider inclinations β = pi/2+O(δ). For χ > 0 the slope of the channel is “downhill”, in the direction of the flow, and for χ < 0 the flow is uphill. Note that for larger χ the model does not necessarily break down, but effectively we have chosen the wrong scaling as the effect of the channel slope is dominant. Before proceeding, we observe that there are 2 qualitatively different types of displacement flows. (i) HL (heavy-light) displacement: fluid 1 is heavier than fluid 2, and the lower layer of fluid is consequently fluid 1. Parameters are: (nH , κH , BH , nL, κL, BL) = (n1, 1, B1, n2,m,B2). (ii) LH (light-heavy) displacement: fluid 1 is lighter than fluid 2, and the lower layer of fluid is consequently fluid 2. Parameters are: (nH , κH , BH , nL, κL, BL) = (n2,m,B2, n1, 1, B1). These are illustrated schematically in Fig. 5.2. We do not consider mechan- ically unstable configurations, i.e. heavy fluid over light fluid. 81 5.1. Two-fluid displacement flows in a nearly horizontal slot Heavy0V̂ 0V̂ Light Light Heavy a) b) Figure 5.2: Schematic of displacement types considered: a) Heavy fluid displaces Light fluid, (HL displacement); b) Light fluid displaces Heavy fluid, (LH displacement). We integrate (5.16) across both fluid layers to give the pressure: P (ξ, y, T ) =  P0(ξ, T ) + χ φH |1− φ|ξ − φH |1− φ|y y ∈ [0, h], P0(ξ, T ) + χ φH |1− φ|ξ − φH − φL |1− φ| h− φL |1− φ|y y ∈ [h, 1], (5.17) where P0(ξ, T ) is defined by: P0(ξ, T ) = P (ξ, 0, T )− χ φH|1− φ|ξ, with φH = ρ̂H/ρ̂1 for the heavier fluid, φL = ρ̂L/ρ̂1 for the lighter fluid. On substituting into (5.15), we arrive at: 0 = −∂P0 ∂ξ + ∂ ∂y τH,ξy, y ∈ (0, h), (5.18) 0 = −∂P0 ∂ξ + ∂ ∂y τL,ξy − χ+ ∂h ∂ξ , y ∈ (h, 1). (5.19) 82 5.1. Two-fluid displacement flows in a nearly horizontal slot In the lubrication approximation, the leading order strain rate compo- nent is γ̇ξy = ∂u∂y , and the leading order shear stress τk,ξy is defined in terms of γ̇ξy via the following leading order constitutive laws: ∂u ∂y = 0 ⇔ |τk,ξy| ≤ Bk, x ∈ Ωk, (5.20) τk,ξy = κk ∣∣∣∣∂u∂y ∣∣∣∣nk−1 + Bk∣∣∣∣∂u∂y ∣∣∣∣  ∂u∂y ⇔ |τk,ξy| > Bk, x ∈ Ωk. (5.21) Thus, for given h and ∂h∂ξ , (5.18) & (5.19) define an elliptic problem for u(y). Boundary conditions for u(y) are u = 0 at y = 0, 1. At the interface, y = h, u is continuous and τH,ξy = τL,ξy, representing stress continuity. These 4 conditions are sufficient to determine u for given ∂P0∂ξ . The pressure gradient is determined by the additional constraint that (5.5) is satisfied. For now we assume that the solution of this problem may be computed and we note that the dependence of u on (ξ, T ) enters only via h(ξ, T ), which satisfies ∂h ∂T + u ∂h ∂ξ = V. (5.22) Combining the kinematic equation with the divergence free constraint leads, in the usual manner, to the equation: ∂h ∂T + ∂ ∂ξ q(h, hξ) = 0, (5.23) where q(h, hξ) is defined as: q(h, hξ) = ∫ h 0 u(y, h, hξ) dy. (5.24) The remainder of our study concerns behaviour of solutions to the system (5.23) & (5.24). As boundary conditions, for a HL displacement we have that h(ξ, T )→ 1, as ξ → −∞; h(ξ, T )→ 0, as ξ →∞, (5.25) as the channel is assumed full of pure fluid 1 and fluid 2 at the two ends of the channel. As initial conditions we note that an initial profile in the unscaled variables h(x, t = 0) = h0(x) is transformed to h(ξ, T = 0) = h0(ξ/δ). Since h0 should be compatible with the far-field conditions we have that as δ → 0, h(ξ, 0)→ 1−H(ξ), (5.26) 83 5.1. Two-fluid displacement flows in a nearly horizontal slot where H(ξ) is the usual Heaviside function. In other words, in terms of ξ, the initial change in h is localised to ξ = 0. For a LH displacement this is reversed, i.e. h(ξ, T )→ 0, as ξ → −∞; h(ξ, T )→ 1, as ξ →∞, (5.27) h(ξ, 0) = H(ξ), (5.28) since the far-field pure fluids are reversed. 5.1.3 The flux function q(h, hξ) In the general case, finding the flux function q(h, hξ) requires computation, and this is addressed in Appendix A. For the particular case of a Newtonian fluid the analytical solution may be found trivially. Denoting b = χ − hξ, for a HL displacement we find: q(h; b,m) = qA(h;m) + bqB(h;m). (5.29) where qA(h;m) and qB(h;m) represent the advective and buoyancy-driven components of the flux q(h; b,m): qA(h;m) = 3mh2(mh2 + (h+ 3)(1− h)) 3[(1− h)4 + 2mh(1− h)(h2 − h+ 2) +m2h4] (5.30) qB(h;m) = [h3(1− h)3(mh+ (1− h))] 3[(1− h)4 + 2mh(1− h)(h2 − h+ 2) +m2h4] . (5.31) For a LH displacement, the flux function is given by: q(h; b,m) = qA(h; 1/m) + bqB(h; 1/m). (5.32) Examples of computed q are given in Fig. 5.3. For all examples, these functions have been computed using the procedure described in Appendix A, with the results compared against (5.29) in the case of Newtonian fluids, to verify the numerical method. We observe that the curves for m = 0.1 and m = 10 in Fig. 5.3a, (with b = 0), show a reflective symmetry, as do those for b = ±10 in Fig. 5.3b, (with m = 1). Note also that in Figs. 5.3a & b, the flux functions are relevant to both HL and LH displacements, but with m replaced by 1/m in the case of LH displacements. This apparent symmetry between HL and LH displacements is not obvious. Note that although the fluxes are mathematically identical for the same b, in fact b = χ − hξ will not be the 84 5.1. Two-fluid displacement flows in a nearly horizontal slot a) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 h q b) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 h q c) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 h q d) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 h q Figure 5.3: Examples of q for 2 Newtonian fluids: a) b = 0 and different m; HL displacement withm = 0.1 (◦),m = 1 (O),m = 10 (¤); LH displacement with m = 10 (◦), m = 1 (O), m = 0.1 (¤); b) m = 1 and different b; HL or LH displacements with b = −10 (◦), b = 0 (O), b = 10 (¤). Examples of q for 2 non-Newtonian fluids in HL displacement: c) b = 1, m = 1, B2 = 1, nk = 1, B1 = 0 (◦), B1 = 5 (/), B1 = 10 (.), B1 = 20 (¤); d) b = 1, Bk = 1, nk = 1, m = 0.1 (◦), m = 1 (O), m = 10 (¤). same since hξ will have different sign between the two displacement types. In addition, m is the ratio of displaced to displacing fluid viscosity, which changes with the displacement type. In other words, replacing m with 1/m and switching from HL to LH does give the same q, but does not give the same “shape” of interface (meaning that we replace h with 1− h, since the LH displacement front slumps along the top of the channel). Instead the HL and LH interfaces are the same shape for the same m in the case of a horizontal channel χ = 0, (see later Figs. 5.4c & d, and will be the same shape for small inclinations if we retain the same m and replace χ with −χ. This does not therefore contradict observations from lubrication-type 85 5.1. Two-fluid displacement flows in a nearly horizontal slot models of iso-density displacements with central finger-like interfaces, where the cases m and 1/m also produce markedly different results. Figs. 5.3c & d illustrate non-Newtonian effects on q in HL displacements. In Fig. 5.3c we observe that as the heavy fluid yield stress, B1, is increased q = 0 in some interval of small h. For these thin layers the yield stress fluid remains static. In Fig. 5.3d we see that the effects of viscosity ratio m is broadly similar for non-Newtonian and Newtonian fluids. For the examples shown q increases monotonically with little apparent effect of varying the parameters. This is however not always the case, as we have presented only a limited subset of the 6 parameters, mostly of O(1). With more slightly extreme parameter combinations it is not difficult to find q that are non-monotone for example. We shall see later that most of the qualitative information concerning the long-term behavior of the solution is contained in ∂q∂h , for which the differences are significant. 5.1.4 The existence of steady traveling wave displacements One of the most important practical questions in considering this displace- ment flow is whether or not (5.23) & (5.24) admit steady traveling wave so- lutions. This determines whether or not the displacement can be effective. In this section we demonstrate that, regardless of fluid type and of rheo- logical differences between fluids, it is impossible for there to be a steady traveling wave solution. Having discounted this possibility, in later sections we turn to a qualitative description of the solutions for different fluid types. First let us note that the slope of the interface hξ acts always to spread the interface. To see this note that following the construction of the previous section, we may write q(h, hξ) = q(h, b) where b = χ−hξ. Formally we may write (5.23) as ∂h ∂T + ∂q ∂h ∂h ∂ξ = −∂q ∂b ∂b ∂ξ = ∂q ∂b ∂2h ∂ξ2 , (5.33) from which we see that the interface spreads diffusively provided that q(h, b) increases with b. We prove the following result in Appendix B. Lemma 5.1.1 q(h, b) is non-decreasing for all b. Now we examine the condition for there to be a steady traveling wave solution. Since fluid 1 is injected at mean speed 1, the only steady speed that needs be considered is unity. Shifting to a moving frame of reference, say z = ξ − T , we see that if the solution is steady in this frame, h = h(z), 86 5.1. Two-fluid displacement flows in a nearly horizontal slot we must have that d dz [ h− q(h, χ− dh dz ) ] = 0, and since q = 0 at h = 0, this implies that: h = q(h, χ− dh dz ), (5.34) must be satisfied for all h ∈ [0, 1] if there is to be a steady traveling wave solution. For a HL displacement we impose the further conditions that h(z) decreases monotonically from 1 to 0 with z. For a LH displacement these conditions are reversed: h(z) increases monotonically from 0 to 1 with z. Using lemma 5.1.1, with b = χ− dhdz we see that the following is true. Lemma 5.1.2 For a HL displacement, a necessary condition for there to be steady traveling wave solution is that q(h, χ) ≤ h for all h ∈ [0, 1]. For a LH displacement, a necessary condition for there to be steady traveling wave solution is that q(h, χ) ≥ h for all h ∈ [0, 1]. This follows directly since for a HL displacement we require that dhdz ≤ 0 so that q(h, b) ≥ q(h, χ). If this condition is not satisfied we would therefore be unable to find a solution to (5.34). Similarly for the LH displacement. Following the procedures in [21] we can in fact show that the conditions of lemma 5.1.2 are in fact sufficient as well as necessary. Finally, we shall show that the conditions of lemma 5.1.2 are in fact never satisfied. We focus only on the HL displacement, the LH displacement being treated similarly. We consider solutions u(y) to the system ∂ ∂y τH,ξy = −f, y ∈ (0, h), ∂ ∂y τL,ξy = χ− f, y ∈ (h, 1), for any of the constitutive laws, with no slip at the walls and continuity of stress and velocity at y = h, plus the flow rate constraint (5.5), which determines f . We fix χ and consider h = 1 − ², noting first that both the velocity solution and f(h) will vary smoothly with h. For any h ∈ [0, 1] we note that the shear stress throughout the light fluid layer is given by: τL,ξy(y;h) = τL,ξy(1;h) + (1− y)(f(h)− χ), 87 5.1. Two-fluid displacement flows in a nearly horizontal slot and as h→ 1, we have τL,ξy(y;h) ∼ τL,ξy(1; 1)− ²∂τL,ξy ∂h (1; 1) + ²(f(1)− χ) +O(²2). Thus, the velocity gradient within the light fluid layer is given by: ∂u ∂y = ∂u ∂y (τL,ξy(1; 1)) +O(²), where the algebraic relation for the velocity gradient comes directly from the constitutive laws. Hence we may straightforwardly compute the flux in the lighter fluid layer: qL(²) = ∫ 1 h u(y) dy ∼ −² 2 2 ∂u ∂y (τL,ξy(1; 1)) +O(²3). Now when h = 1 the channel is full with the heavy fluid, and the pressure gradient corresponds to the Poiseuille flow solution, say f(1) = fH(1) > 0, which can be easily calculated. The stress at the upper wall is thus −0.5fH(1) and since the shear stress is continuous we have: τL,ξy(1; 1) = −0.5fH(1) < 0 ⇒ qL(²) ∼ −² 2 2 ∂u ∂y (−0.5fH(1)) > 0. Since via the flow rate constraint we have that the total flux is equal to unity, we have that q(h, χ) ∼ 1 + (1− h) 2 2 ∂u ∂y (−0.5fH(1)) > h, as h→ 1. (5.35) Consequently for an HL displacement the necessary conditions of lemma 5.1.2 are always violated sufficiently close to h = 1, regardless of fluid type and rheological differences. Similarly, we can show that for a LH displace- ment the necessary conditions of lemma 5.1.2 are always violated sufficiently close to h = 0, regardless of fluid type and rheological differences. This leads to the following result. Lemma 5.1.3 There are no steady traveling wave solutions to (5.23). Remarks: 88 5.2. Newtonian fluids • This is the key theoretical result of the chapter. It is perhaps surprising that for no combination of rheology or density differences are we able to achieve a “perfect” displacement, (under the assumptions of the lubrication displacement model). This changes the focus of the study. Firstly, in order to achieve a good displacement, we are driven to study those parameter combinations that give the best efficiency, close to 100%. Secondly, if we wish to improve the efficiency we need consider phenomena that might do this, other than those accounted for in this simplistic model, e.g. hydrodynamic instability & mixing, or the short- time dynamics in the interfacial region before the interface slumps. • For a Newtonian fluid displacement, we might find this result rather more directly as the solution may be computed. For example, in [135] the simpler problem of 2 Newtonian fluids of identical viscosity in an inclined pipe is considered, in the absence of a mean imposed flow. No traveling wave solutions are found. Here however, the mean flow re- sults in a different structure to the flux functions q, i.e. for Newtonian fluids the advective and buoyant components, qA and qB, are present whereas only qB is present in [135], (also with an algebraically different form). For non-Newtonian fluids the division of the flux into qA and qB is not possible, due to nonlinearity. Thus, we have to work with qualitative properties of the fluxes for such fluids. While we might an- ticipate from results such as [135] that no traveling waves solutions to (5.23) can be found, from a physical perspective addition of a constant volume flux (i.e. a displacement) makes this a natural and legitimate question. • Although we have focused on Herschel-Bulkley fluids for definiteness, the same results could be demonstrated for any of the popular gen- eralised Newtonian models, e.g. Carreau fluids, Cross model, Casson model, etc.. . 5.2 Newtonian fluids We commence with an analysis of Newtonian fluid displacements. Although the industrial applications discussed in Chapter 1 and Chapter 2 typically involve non-Newtonian fluids, many of the qualitative behaviours are ex- hibited in a Newtonian fluid displacement. Analysis of the Newtonian fluid case not only provides simplification in terms of the number of dimensionless parameters, i.e. (m,χ), but also since q is given by the analytical expression 89 5.2. Newtonian fluids (5.29) numerical solution is considerably faster. For non-Newtonian fluids, each evaluation of q requires numerical solution of the nested iteration de- scribed in Appendix A. The convection-diffusion equation (5.23) was discretized in the conser- vative form, second order in space and first order in time; and afterward, integrated straightforwardly by using a Lax-Wendroff scheme in which an artificial dissipation was added to the equation to compensate for the desta- bilizing effects of the known anti-diffusion due to the first order time dis- cretization. The only unsatisfactory aspect of the method applied was a small amount of smoothing close to the sharp front tip of the interface. This feature was found to be consistent with time since the flux function and added dissipation vanish in both walls. 5.2.1 Examples of typical qualitative behaviour Example computed HL displacements are shown in Fig. 5.4. The results at long times are not found to be particularly sensitive to the initial condition, which we have taken as a linear function of ξ: typically h(ξ, T = 0) = ∓ξ±0.5 for HL and LH displacements, respectively. When we have wished to study the early-time evolution of the interface, we steepen the initial profile, e.g. in Fig. 5.4a the initial condition is h(ξ, T = 0) = −ξ+0.05. Figs. 5.4a & b plot the solution for m = 1, χ = 0, (i.e. equal viscosities in a perfectly horizontal channel). In the early times, T ∈ [0, 1] we observe that the interface develops quickly into a slumping profile; see Fig. 5.4a. Over longer times, the solution consists of 2 segments: an advancing front of apparently constant shape moving at constant speed and a region at the top which is stretched, the top of the interface simply not moving. The longer time profiles of h may be conveniently plotted against ξ/T , in which variable the interface profiles collapse to a single similarity profile as T → ∞; see Fig. 5.4b. To clarify interpretation of figures such as Fig. 5.4b, the x-axis of the final similarity profile gives the speed of the interface at different heights: vertical lines correspond to segments of the interface that advance at steady speed. Note that the first interface profile in Fig. 5.4b, for T = 1, effectively shows h(ξ, T ) at T = 1, and in this we may observe that the top of the interface is pinned to the upper wall at the initial position, ξ = −0.5. The convergence at the upper wall as T → ∞ simply follows ξ/T = −0.5/T , and the interface itself does not move, as evidenced in Fig. 5.4a over shorter times. Thus, the apparent discrepancy between the last interface profile of Fig. 5.4a and the first interface profile of Fig. 5.4b is simply due to the different initial conditions. 90 5.2. Newtonian fluids a) −0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 ξ h(ξ , T ) b) −1 −0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 ξ /T h c) −0.1  0.1 0.3 0.5 0.7 0.9 0.11 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ /T h 0 0.5 1.1 1.5 0 0.5 1 d) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ /T h 0 0.5 1.0 1.5 0 0.5 1 Figure 5.4: Examples of HL displacements: a) h(ξ, T ) for T = 0, 0.1, .., 0.9, 1, parameters χ = 0, m = 1; b) h(ξ/T ) for T = 1, .., 9, 10, parameters χ = 0, m = 1. Examples of HL displacements: c) h(ξ/T ) for χ = 0: m = 0.1 (◦), m = 1 (O), m = 10 (¤); d) h(ξ/T ) for m = 1: χ = −10 (◦), χ = 0 (O), χ = 10 (¤). The inset figures in c & d show the results of LH displacements for the same parameters. This qualitative behaviour is similar for other parameters and indeed convergence to the “final” similarity profile is relatively quick, occurring over an O(1) timescale (in T ). For our other results we present only the interface at T = 10, which is always very close to the final similarity profile. Fig. 5.4c shows the final shape for 3 different values of viscosity ratio m and Fig. 5.4d shows the final shape for 3 different values of the inclination parameter χ. For larger m the height of the steadily moving front, say hf , is smaller. This is intuitive, since increasing m corresponds to an increasingly less viscous fluid displacing a more viscous fluid. The interface above the steadily moving front also transitions from convex to concave curvature asm is increased, further emphasizing the extending finger. Similarly, for χ > 0 91 5.2. Newtonian fluids the heavy fluid flows downhill through the lighter fluid and hf is accordingly smaller in this configuration. The inset figures in Fig. 5.4c & d show the analogous LH displacements for the same parameters. The effects of m are identical with those for the HL displacement, (since m is the ratio of in- situ fluid viscosity to displacing fluid viscosity). The effect of varying χ is however reversed: χ > 0 retards unsteady spreading for an LH displacement and χ < 0 promotes unsteady spreading. 5.2.2 Long-time behaviour We have seen in Fig. 5.4 that the interface tends to evolve on an O(1) timescale into a shape that consists of 2 parts: (i) a front region that re- mains approximately constant but advances at steady speed; (ii) a stretched region, in which the interface is continually extended, as t → ∞. For the HL displacement the steadily moving front occupies the lower part of the channel, and for the LH displacement the front advances along the upper wall. In place of computations, we would like to directly compute this long- time behaviour. In what follows below we focus for simplicity on the HL displacements. We commence with the upper stretched region. If we denote the steady front height and speed by hf and Vf , respectively, we observe that at long times the slope of the interface is approximately ∂h ∂ξ ∼ −1− hf VfT → 0, as T →∞. Therefore, as T → ∞, we have that b = χ − ∂h∂ξ → χ, and the interface motion in the stretched region is governed approximately by: ∂h ∂T + ∂ ∂ξ q(h, χ) = 0, (5.36) which is hyperbolic rather than parabolic. The interface in this region ad- vances with speed Vi(h) given by: Vi(h) = ∂q ∂h (h, χ). Thus, the total area of fluid flowing behind the interface in the interval [hf , 1] at long times is T ∫ 1 hf Vi(h) dh = T [1− q(hf , χ)]. 92 5.2. Newtonian fluids Furthermore, at the front height hf the interface speed should equal the front velocity Vf , i.e. ∂q ∂h (hf , χ) = Vi(hf ) = Vf . (5.37) The total area of fluid behind the interface is T and since the area of fluid flowing behind the interface in the interval [0, hf ] is approximately TVfhf , we have the following relationship: Tq(hf , χ) = T − T [1− q(hf , χ)] = TVfhf = Thf ∂q ∂h (hf , χ), from which: q(hf , χ) = hf ∂q ∂h (hf , χ). (5.38) Equation (5.38) is an equation for the front height hf . This is instantly recognisable as the same condition that must be satisfied in the case of a kinematic shock, in order to conserve mass. Therefore, note that the long time behaviour is that determined by the underlying hyperbolic conservation law. An example of the use of the equal areas rule (5.38) to determine the front height is shown in Fig. 5.5a. In Fig. 5.5b we plot h against ξ/T for T = 1, .., 9, 10, showing that hf does indeed represent the moving front, which has the same speed as indicated in Fig. 5.5a. Although for most of the parameters we have considered, there is a single propagating front, some parameters result in a double front. Loosely speaking, for Newtonian fluids this appears to arise at more extreme parameter values when physical effects are somehow opposing one another. An example is shown in Figs. 5.5c & d. In this illustration, the competing effects are buoyancy, driven by the downhill slope, which acts to spread the interface, and the viscosity ratio, which acts to sharpen the front. Fig. 5.6a shows calculated front heights for HL and LH displacements6 for different values of χ and m. We observe that higher viscosity ratios tend to have a lower front height, which simply means that in order to have a more efficient displacement, the displacing fluid should be more viscous in comparison to the displaced fluid. Increasing χ tends to reduce efficiency for the HL displacement but increase efficiency for the LH displacement. Via repeated computations of q for different (m,χ) we are able to delineate the 6To interpret this figure for the LH displacement The front height hf for LH displace- ment is defined as the distance from the top wall to the stretched part of the interface. 93 5.2. Newtonian fluids a) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 h ∂q ∂h b) −1 −0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 ξ /T h c) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 h ∂q ∂h d) −1 −0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 ξ /T h Figure 5.5: Use of the equal areas rule (5.38) in determining the front height: a) a single front height, χ = 10, m = 8; b) h plotted against ξ/T for T = 1, .., 9, 10, parameters χ = 10, m = 8, broken horizontal line indicates the front height determined from (5.38); c) two front heights, χ = 10, m = 0.08; d) h plotted against ξ/T for T = 1, .., 9, 10, parameters χ = 10, m = 0.08, broken horizontal lines indicate the front heights. regime in the (m,χ)-plane in which multiple fronts are found, see Fig. 5.6b. Within the shaded region of Fig. 5.6b, note that some parameter values give front speeds that are negative, i.e. there is a backflow driven by buoyancy. The expression (5.38) for the front height is exactly the same equation that would be solved for computing a kinematic shock for the hyperbolic conservation law. It is, however, important to emphasize that the front is not a shock since diffusive effects are always present for h ∈ (0, 1). Having determined hf from (5.38) and then Vf from (5.37), we may shift to a moving frame of reference z = ξ−VfT and seek a steadily traveling solution to (5.23), which satisfies: d dz [ hVf − q(h, χ− dh dz ) ] = 0, ⇒ hVf − q(h, χ− dh dz ) = 0. (5.39) 94 5.2. Newtonian fluids a) 10 −1 100 101 0.5 0.6 0.7 0.8 0.9 1 m hf b) 10 −1 100 101 0 10 20 30 40 50 m χ Figure 5.6: a) Front heights for a Newtonian fluid HL displacement with χ = −10 (O), χ = −5 (¤), χ = 0 (.), χ = 5 (◦), χ = 10 (/). This figure also gives the front heights for a Newtonian fluid LH displacement with χ = 10 (O), χ = 5 (¤), χ = 0 (.), χ = −5 (◦), χ = −10 (/). For the LH displacement the front height is measured down from the top wall; b) Parameter regime in the (m,χ)-plane in which multiple fronts (shaded area). Elsewhere there is only a single front. Equation (5.39) must be solved numerically for h ∈ (0, hf ). Example shapes are shown in Fig. 5.7. Fig. 5.7a shows HL displacement front shapes for 2 Newtonian fluids for different values of viscosity ratio at χ = 0. Fig. 5.7b shows HL displacement front shapes for 2 Newtonian fluids for different values of χ at m = 1. These are the same parameters as for the transient displacements in Figs. 5.4c & d. Observe from (5.39) as h→ h−f that, since hf is determined from (5.38) and Vf from (5.37), we must have: q(h, χ− dh dz )→ q(hf , χ) as h→ h−f , which implies that dhdz → 0 as h→ h−f , as can be seen in Fig. 5.7. Evidently, as T → ∞ the stretched region of the interface also aligns horizontally, so that the long-time solution is smooth at hf . 5.2.3 Flow reversal and short-time behaviour The model results presented so far have been derived under lubrication scal- ing assumptions, with the length-scale determined by dominant buoyancy effects, compatible with the assumed stratification. Our study of the long- time behaviour has revealed only forward propagating fronts, which of course 95 5.2. Newtonian fluids a) −0.4 −0.3 −0.2 −0.1 0 0.1 0 0.2 0.4 0.6 0.8 1 z hf (z) b) −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0 0.2 0.4 0.6 0.8 1 z hf (z) Figure 5.7: Examples of front shapes in the moving frame of reference for a HL displacement, computed from equation (5.39): a) χ = 0, m = 0.1 (¤),m = 1 (◦),m = 10 (.); b) χ = −10 (O), χ = −5 (◦), χ = 0 (.), χ = 5 (¤), χ = 10 (/). Compare with transient computations in Figs. 5.4c & d. are more common since a positive flow rate is imposed. If the channel is hor- izontal then, as the front advances and the slope of the interface decreases, the driving force to oppose the mean flow also diminishes. Thus, we cannot expect flow reversal in a horizontal channel at long times. On the other hand, with an inclined channel there is a constant buoyancy force that may either reinforce or oppose the mean flow. For example, with a HL displacement at fixed positive inclination, χ > 0, buoyancy acts to push the lighter fluid against the mean flow direction. For sufficiently large χ and small viscosity ratio, we observe that the lighter fluid may be driven backwards against the flow, resulting in a sustained flow reversal. An example of this is shown in Fig. 5.8. Flow reversal may also be observed in other situations. The most obvious of these is the case T ¿ 1, since for short times large interface slopes may mean that gravitational spreading may dominate the imposed flow. Since our model is anyway an asymptotic reduction of the full equations in which T effectively represents a long-time relative to the advective timescale over the channel width, the limit T → 0 is one in which the underlying assump- tions of the model break down. Nevertheless, the problem for T ¿ 1 is mathematically well-defined and of physical interest. To study this limit, we shift to the steadily moving frame of reference z = ξ−T , recall that b = χ−hξ, and consider (5.23) for an HL displacement, 96 5.2. Newtonian fluids −15 −10 −5 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 ξ h(ξ , T ) Figure 5.8: Profiles of h(ξ, T ) for T = 0, 1, .., 9, 10, with parameters χ = 50, m = 0.1, illustrating flow reversal. which becomes: ∂h ∂T + ∂ ∂z [ qA(h;m) + (χ− ∂h ∂z )qB(h;m)− h ] = 0, (5.40) where qA(h;m) and qB(h;m) represent the advective and buoyancy-driven components of the flux q(h; b,m), which is defined by (5.29) for 2 Newtonian fluids, i.e. Introducing η = z/ √ T this becomes: 1 2 η dh dη − √ T d dη (qA − h+ χqB) + qB d 2h dη2 + ∂qB ∂h ( dh dη )2 = 0, (5.41) Therefore, provided that T ∼ 0 we may seek a similarity solution satisfying: 1 2 η dh dη + qB d2h dη2 + ∂qB ∂h ( dh dη )2 = 0, (5.42) or in conservative form: 1 2 η dh dη = d dη ( −qB dh dη ) . (5.43) 97 5.2. Newtonian fluids Since qB(h;m) vanishes at both h = 0 and h = 1, it is clear that there is some singular behaviour in h(η) at these points. Thus, it is more comfortable to work with the function η(h). The boundary conditions are, η(0) = η0 and η(1) = η1, where η0 & η1 are unknown at this stage. Physically we expect that η0 > 0 & η1 < 0 as the spreading of the interface is caused by gravitational slumping. A Taylor expansion reveals that η(h) ∼ η0 +O(h3) as h → 0, with similar asymptotic behaviour as h → 1, i.e. η′(h) → 0 quadratically at both ends of the interval. We integrate equation (5.43) as follows: 1 2 ηdh = d ( −qB dh dη ) (5.44)∫ h 0 1 2 η dh = ∫ h 0 d ( −qB dh dη ) = −qB(h)dh dη + qB(0) dh dη = −qB(h)dh dη , (5.45) (note qB(h)→ 0 as h→ 0 with order h3). Now taking h→ 1 and using the asymptotic behaviour qB(h) ∼ (1− h)3, we have: 1 2 ∫ 1 0 η dh = 0. (5.46) Let us now define g(h) such that η = g′. Therefore, g(h)− g(0) = ∫ h 0 ηdh, (5.47) and from equation (5.46), we see that g(1) = g(0). For convenience, we set g(0) = 0 so that (5.45) may be written as: g′′g = −2qB. (5.48) We use the initial condition g(0) = 0 and g′(0) = η0. We then integrate forward, with respect to h and iterate on η0 via a shooting method to satisfy g(1) = 0. This numerical procedure appears to work well. Figure 5.9a plots the similarity solutions η(h) for various m. Note that the solution is not sym- metric with respect to m. For the heavy-light displacement the heavy fluid viscosity is 1 and the light fluid viscosity is m. Buoyancy effects have no bias between the fluids, but the more viscous fluids evidently resist motion. Thus, we see that for large m the axial extension η0 − η1 is smaller than for small m. This effect might have been removed had we scaled viscosity 98 5.2. Newtonian fluids a) −0.5 −0.3 −0.1 0.1 0.3 0.5 0 0.2 0.4 0.6 0.8 1 η h(η) m = 100 m = 0.01 b) −0.4 −0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1 η h(η) T = 0.001 T = 0.1 Figure 5.9: a) The similarity solution h(η) for m = 0.01, 0.1, 1, 10, 100; b) comparison of the similarity solution with the numerical solution of (5.40) for m = 1, at T = 0.001, 0.01, 0.1. with an appropriate mean value. A symmetrical shape is of course found at m = 1. These solutions have been compared with the solution of PDE equation (5.40) as T ∼ 0 and agree well for short times. An example is shown in Fig. 5.9b for the case m = 1. Mathematically, these solutions serve primarily to demonstrate that for short times, (e.g. after opening a gate valve in an experiment), buoyancy dominates and an exchange flow should occur, relative to the mean displacement. For smaller mean velocities the parameter δ → 0 and the dimensional time period over which buoyancy dominates extends to infinity, ensuring compatibility with exchange flow studies, for which there is zero net flow rate and hence a flow reversal in each layer. To explore this analogy further, let us fix β = pi/2, in which case we may note that the similarity variable η is defined in terms of dimensional variables by: η = z T 1/2 = x̂− V̂0t̂ t̂1/2 √ µ̂1 |ρ̂1 − ρ̂2|ĝD̂3 . We may compare this with the analysis in [135] for exchange flows in hor- izontal pipes, wherein diffusive similarity profiles are found for Newtonian fluids of the same viscosity. We may note that the scaling |ρ̂1− ρ̂2|ĝD̂3/µ̂1 is the same as the (V̂νD̂)1/2 that scales the similarity variable x̂/t̂1/2 in [135], (see equation (27) and §VII.B in this paper). However, although this is the same viscous-buoyancy balance driving the diffusive spreading in both cases, here we have the additional criterion that T 1/2 ¿ 1, and we have seen 99 5.3. Non-Newtonian fluids numerically that the diffusive regime does not last for longer times. This criterion can be written dimensionally as: t̂V̂0 D̂ ¿ 1 V̂0 |ρ̂1 − ρ̂2|ĝD̂2 µ̂1 = L̂ D̂ . The most simplistic interpretation therefore is that t̂V̂0 ¿ L̂ , i.e. the dis- tance advected during the time considered must be much less than the char- acteristic slump length, (dimensionlessly, we require that z ¿ 1). Alter- natively the left-hand side is the ratio of advected distance to the channel width, whereas the quantity in the middle is the ratio of the viscous velocity scale to the advective velocity scale. Finally, observe that the short time diffusion is measured in a frame of reference moving with the mean velocity. The criterion t̂V̂0 ¿ L̂ also means that the moving frame has not moved very far relative to the stationary frame in which the usual exchange flow analysis takes place. 5.3 Non-Newtonian fluids We turn now to results for non-Newtonian fluids. Primarily we shall be concerned with long-time results since the short-time behaviour does not yield simple analytical results in the form of similarity solutions. The reason for this becomes clear if we consider for example a Poiseuille flow of a power law fluid. The strain rate in the fluid is proportional to the pressure gradient to the 1/n-th power, and hence the areal flow rate also. In a two-layer flow of the type we have, the short-time behaviour is dominated by that part of q(h, hξ) driven by the pressure gradient due to the slope of the interface. However, the flux in fluid layer k is proportional to |hξ|1/nk and the two fluxes are coupled via the flow rate constraint. Thus, it is immediately obvious that there can be no single similarity variable unless the two fluids happen to have the same shear-thinning index. In this case the similarity variable is η = z/tn/(n+1). Although of mathematical interest, the practical interest is limited. 5.3.1 Shear-thinning effects We commence by considering only shear-thinning effects, Bk = 0, and shall also focus only on HL displacements. Figs. 5.10a & b show the final similarity profiles of the interface for m = 1 and χ = 0, i.e. the only effects are the relative values of the two power law indices. We observe that for fixed nH 100 5.3. Non-Newtonian fluids the front height increases as nL decreases. Conversely, for fixed nL the front height decreases as nH decreases. Both effects are essentially predictable, in that with all other parameters fixed (or neutralised in the case of inclination, χ = 0), varying the power law indices makes one fluid progressively less or more viscous. a) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ/T h b) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ /T h c) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ/T h d) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ/T h Figure 5.10: Examples of HL displacements for 2 power law fluids, Bk = 0, χ = 0: a) h for m = 1, nH = 1: nL = 1/2 (◦), nL = 1/3 (O), nL = 1/4 (¤); b) h for m = 1, nL = 1: nH = 1/2 (◦), nH = 1/3 (O), nH = 1/4 (¤); c) h for nH = 1/4, nL = 1, m = 0.1 (O), nH = 1, nL = 1/4, m = 10 (¤); d) h for m = 0.1, nH = 1: nL = 1 (◦), nL = 1/2 (O), nL = 1/4 (¤). All interfaces plotted at T = 10. Less obvious effects are found when the “bulk” viscosity of one fluid is for example large but has smaller power law index than the other fluid. For example, should nH = 1/4, nL = 1, m = 0.1 provide a better displacement than nH = 1, nL = 1/4, m = 10? Typically in industrial settings one is un- able to choose the rheological properties of the fluids. These displacements are shown in Fig. 5.10c and we see that in fact the latter case displaces 101 5.3. Non-Newtonian fluids a) 10 −1 100 101 0 0.2 0.4 0.6 0.8 1 m hf   b) 10 −1 100 101 0.6 0.7 0.8 0.9 1 m hf   c) 10 −1 100 101 0 0.2 0.4 0.6 0.8 1 m hf d) 10 −1 100 101 0.6 0.7 0.8 0.9 1 m hf e) 10 −1 100 101 1 1.5 2 2.5 3 m Vf f) 10 −1 100 101 1 1.1 1.2 1.3 1.4 m Vf Figure 5.11: Front heights and velocities, plotted against m for a HL dis- placement of 2 power law fluids, Bk = 0; a) hf for nL = 1, nH = 1/4; b) hf for nH = 1, nL = 1/4; c) hf for nL = 1, nH = 1/2; d) hf for nH = 1, nL = 1/2; e) Vf for nL = 1, nH = 1/4; f) Vf for nH = 1, nL = 1/4. For all plots χ = −10 (O), χ = −5 (¤), χ = 0 (.), χ = 5 (◦), χ = 10 (/), and the heavy broken line indicates multiple fronts. 102 5.3. Non-Newtonian fluids better. Often shear-thinning behaviour can be brought about by the addi- tion of a relatively small amount of a polymer additive. In cases when the displacement is anyway reasonable, due to a viscosity ratio m < 1, shear thinning effects can result in displacements that are close to 100% efficient. An example of this are shown in Fig. 5.10d, where for m = 0.1, nH = 1 we show the effects of decreasing nL. Note that as nL → 0, the light fluid effectively slips at the upper wall and we are able to have a steady traveling wave displacement. The analysis of interface motion at long times is identical to that for the Newtonian fluid displacements of the previous section. The long-time behaviour can be analyzed over a wide range of parameters by direct treat- ment of the flux function q. We present a range of parametric results below. Until now we have given only the front height, hf . However, in displace- ment experiments it is usually easier to estimate the front speed Vf from captured images, especially when the interface is diffuse. The front speed is calculated straightforwardly for Newtonian displacements, but for non- Newtonian fluids this is more laborious. A slightly different interpretation of the front speed is as an indicator of displacement efficiency. No single measure or definition is universal, e.g. for finite length ducts it is common to present quantities such as the volume fraction displaced after 1 volume of displacing fluid has been pumped, or alternatively after an infinite volume has been pumped. Here we define: Displacement Efficiency = 1 Vf . (5.49) At long times this approximates the area fraction behind the front that is displaced at time T . An alternative interpretation is as the breakthrough time, i.e. the time at which displaced fluid is first seen at unit length down- stream. Examples of variations in front height and speed, for different χ and m, as either nH or nL is reduced, are shown in Fig. 5.11. Essentially the displacement efficiency increases as the displacing fluid becomes less shear- thinning, as would be expected, and as the inclination increases. As with Newtonian displacements, for certain parameter ranges the long-time be- haviour is characterised by two steady fronts, with the lower front moving faster. Parameters for which this happens are indicated in Fig. 5.11 by the heavy broken line. It can be observed that the transition from 1 front to 2 fronts can be either smooth or sudden. Later we illustrate in detail how these different transitions occur. For 2 Newtonian fluids the occurrence of multiple fronts is relatively easy to identify, as there are essentially only 2 103 5.3. Non-Newtonian fluids effects that compete: viscosity and buoyancy, see Fig. 5.6. However, for power law fluids we may have fluid combinations that are either more or less viscous than each other, for different shear rates, and these effects are then complemented with effects of different channel inclinations. Thus, the possible combinations of effects are vastly increased and it is hard to map out regions in parameter space where multiple fronts exist. Flow reversal oc- curs in HL displacements for large values of χ > 0 and for suitable viscosity ratios. For example, in Fig. 5.11b at small m for χ = 10, the heavy broken line indicates 2 moving fronts, but one front has negative speed, (hence the decrease in efficiency). The jump in Fig. 5.11b, (at small m for χ = 10), in fact indicates a transition from 2 fronts to 3 fronts: 2 moving forward and 1 moving backwards! 5.3.2 Yield stress effects We turn now to yield stress fluids and for simplicity we set nk = 1, i.e. these are Bingham fluids. Such fluids are in any case shear-thinning, due to the yield stress, but no additional power law behaviour is considered. We start by examining the effects of a single yield stress on a Newtonian displace- ment, (for m = 1, χ = 0), by increasing either BH or BL. Again only HL displacements are considered. Fig. 5.12 shows the interfaces at T = 10, plotted against ξ/T for each of these cases. It can be observed that increas- ing BH improves the displacement, due to the enhanced effective viscosity, Fig. 5.12a. Similarly, increasing BL makes the displacement less efficient, see Fig. 5.12b. The new physical phenomena observed in Fig. 5.12b for larger BL, is the possibility to have a static wall layer. Observe that for BL = 20 the interface at T = 10 has not displaced the light fluid in the upper part of the channel. This will be attached to the upper wall in a HL displacement and to the lower wall in a LH displacement. This type of phenomena has been observed and studied before, both as part of a transient displacement flow and as a static situation, see e.g. [4, 54]. We discuss static wall layer solutions further in §5.3.2. The long-time analysis of solutions is qualitatively similar to that dis- cussed earlier. Examples showing the effects of χ and m on the front height and speed are shown in Fig. 5.13. General effects of varyingm, χ and Bk are mostly in line with our physical intuition, i.e. effects that make the displac- ing fluid more viscous usually (but not always) improve the displacement. However, for parameter ranges where some ambiguity exists, this type of computation determines which effects dominate. We also observe the same 104 5.3. Non-Newtonian fluids a) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ/T h b) −0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 0 0.2 0.4 0.6 0.8 1 ξ/T h Figure 5.12: Profiles of h plotted against ξ/T at T = 10: a) χ = 0, nk = 1, BL = 0,m = 1, BH = 1(◦), BH = 5(O), BH = 20(¤); b) χ = 0, nk = 1, BH = 0,m = 1, BL = 1(◦), BL = 5(O), BL = 20(¤). range of different solution types as before when the parameters are varied, i.e. transitions from single to multiple fronts that may be smooth or sudden. To clarify how transitions occur between single and multiple fronts, (e.g. in Fig. 5.13 and similar figures previously), Fig. 5.14 illustrates the two different type of transition, by showing ∂q∂h(h, hξ = 0) at values of m just above and below the critical values at which transition occurs. In Figs. 5.14a & b we observe that the smooth transition typically corresponds to a change in the shape of ∂q∂h(h, hξ = 0) from unimodal to bimodal (or vice versa). We have two fronts and as a process parameter is changed the slower front sim- ply disappears. The sudden transition, illustrated in Figs. 5.14c & d, is due to a change in the actual front height when switching between branches of a bimodal ∂q∂h(h, hξ = 0). We have two fronts and as a process parameter is changed the slower front increases in speed, eventually overtaking the faster front, thus combining into one front. Note that there is no jump in the front speed, (see Fig. 5.13b). We have simply plotted the height of the fastest moving front, as this is the front that is most relevant for the displacement efficiency. The static wall layer The defining novel feature of a yield stress fluid displacement is the pos- sibility for residual fluid to remain permanently in the channel, i.e. even asymptotically as T → ∞ a fraction of fluid 2 may not be displaced. The origin of the static residual layer has a straightforward physical explanation. The lubrication displacement model that we study is based on an underlying 105 5.3. Non-Newtonian fluids a) 10 −1 100 101 0.6 0.7 0.8 0.9 m hf   b) 0.1 1 10 1 1.1 1.2 1.3 m Vf c) 10 −1 100 101 0.5 0.6 0.7 0.8 0.9 m hf d) 10 −1 100 101 0.5 0.6 0.7 m hf Figure 5.13: Front heights and velocities, plotted against m, nk = 0; a) HL displacement hf versus m for BL = 0, BH = 5, b) HL displacement Vf versus m for BL = 0, BH = 5, c) HL displacement hf versus m for BH = 0, BL = 5, d) HL displacement hf versus m for BH = 0, BL = 20. Parameters: χ = −10(O), χ = −5(¤), χ = 0(.), χ = 5(◦), χ = 10(/) for all plots. Broken heavy line indicates multiple fronts. parallel flow of 2 fluids. If the wall stress created by the displacing fluid, flowing at unit flow rate through the channel, does not exceed the yield stress of the displaced fluid, it follows that there could be a static residual layer on the wall. It can also be argued that there exists a uniquely defined maximal static layer thickness, either physically or mathematically: see [4, 54]. On following a similar procedure to that of [4], we may show that the maximal residual wall layer thickness depends only on the following param- eters, (for a HL displacement): nH , B̃1 = BH κH , ϕY = BH BL , ϕb = χ BL (5.50) The parameter B̃1 is a rescaled Bingham number, relevant to the displac- 106 5.3. Non-Newtonian fluids a) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 h ∂q ∂h b) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 h ∂q ∂h c) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 h ∂q ∂h d) 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 h ∂q ∂h Figure 5.14: Plots of ∂q∂h showing the front positions for parameters: nk = 1: a) BH = 1, BL = 0, χ = 10, m = 0.1, multiple fronts; b) m = 0.2, single front; c) χ = 0, BH = 5, BL = 0, m = 2.3, multiple fronts; d) χ = 0, BH = 5, BL = 0, m = 2.4, single front. ing fluid; ϕY is simply the yield stress ratio and ϕb measures the ratio of buoyancy stress due to the slope of the channel and the yield stress of the displaced fluid. The critical condition for the existence of any static wall layer is independent of the buoyancy ratio, ϕY . Fig. 5.15 shows the variation in maximum static wall layer Ystatic with the parameters ϕY and 1B̃1 for 3 fixed values of the ratio ϕb. The shaded area marks the limit where no static wall layers are possible. As nH decreases, the contours become increasingly parallel to the vertical axis, which implies that the layer thickness is becoming independent of B̃1 = BH/κH . As ϕb increases from negative to positive the static layer thickness is increasing. The limit BH → 0 must be treated separately. Straightforwardly, we find that Ystatic depends on nH , χ̃ = χ/κH and B̃2 = BL/κH . Fig. 5.16 shows the variation in maximum static wall layer with the parameters χ̃ and B̃ for 4 107 5.3. Non-Newtonian fluids ϕY 1 B̃1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 a) ϕY 1 B̃1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 b) ϕY 1 B̃1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 c) ϕY 1 B̃1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 d) ϕY 1 B̃1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 e) ϕY 1 B̃1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 f) Figure 5.15: Maximal static wall layer thickness Ystatic(nH , B̃1, ϕY , ϕb), with contours spaced at intervals ∆Ystatic = 0.1: a) ϕb = −2, nH = 1; b) ϕb = −2, nH = 0.2; c) ϕb = 0, nH = 1; d) ϕb = 0, nH = 0.2; e) ϕb = 2, nH = 1; f) ϕb = 2, nH = 0.2. different fixed values of the power law index nH . An interesting consequence of Fig. 5.16 is that for a small change in e.g. yield stress, it appears that 108 5.4. Summary we may transition from having no static layer to having a finite static layer! An example illustration of this is given in Fig. 5.17. Although there is a discontinuity in the thickness of static layer, there is no discontinuity in the physical process, i.e. the layers of fluid that move do so very slowly as the static layer criterion is violated. 1 B̃2 χ̃ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 a) 1 B̃2 χ̃ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 b) 1 B̃2 χ̃ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 c) 1 B̃2 χ̃ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 d) Figure 5.16: Maximal static wall layer Ystatic = 1− hmin when a power-law fluid displaces a Herschel-Bulkley fluid, with contours spaced at intervals ∆Ystatic = 0.1: a) nH = 1; b) nH = 1/2; c) nH = 1/3; d) nH = 1/4. 5.4 Summary The main contributions and results of this chapter are as follows: • We have derived a 2-layer lubrication/thin film model and developed a semi-analytical solution method to find the flux function. • We have shown that there are no steady traveling wave solutions to the displacement problem, in the lubrication/thin film limit. 109 5.4. Summary a) 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 ξ h(ξ, T ) b) 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 ξ h(ξ, T ) Figure 5.17: An example of sudden movement of static layer corresponding Fig. 5.16d: a) h(ξ, T ) for T = 0, 1, .., 9, 10, parameters χ = 10, m = 1, nH = 1/4, nL = 1, B1 = 0, B2 = 2; b) h(ξ, T ) for T = 0, 1, .., 9, 10, parameters χ = 10, m = 1, nH = 1/4, nL = 1, B1 = 0, B2 = 1. • At short times, diffusive effects of the interface slope are dominant and there is an exchange flow relative to the mean flow. We have found a short-time similarity solution governing this initial counter-current flow. • At longer times the interface propagates in a number of fronts (mov- ing at steady speeds), joined together by interface segments that are stretched between the fronts. The front heights and speeds can be directly computed. • We have explored the effects of viscosity ratio, inclinations, and other rheological properties. (i) More efficient displacements are generally obtained with a more viscous displacing fluid. (ii) Modest improvements in displacement efficiency may also be gained with slight positive inclination in the direction of the density dif- ference. (iii) Fluids that are highly shear-thinning may be displaced at high efficiencies by more viscous fluids. (iv) Generally, a yield stress in the displacing fluid increases the dis- placement efficiency and yield stress in the displaced fluid de- 110 5.4. Summary creases the displacement efficiency, eventually leading to com- pletely static residual wall layers of displaced fluid. (v) The maximal layer thickness of these static layers can be directly computed from a 1D momentum balance and indicates the thick- ness of static layer found at long times. (vi) The maximal static layer thickness increases with the yield stress of the displaced fluid, or with a mild buoyancy difference opposing the flow. It decreases with buoyancy difference in the direction of the imposed flow and with increases in the effective viscosity of the displacing fluid. 111 Chapter 6 Stationary residual layers in Newtonian displacements7 In Chapter 4 we identified 3 regimes in our displacement flows, as a function of the mean flow velocity (V̂0) increasing from zero: (i) an exchange flow dominated regime; (ii) a laminarised viscous displacement regime; (iii) a fully mixed displacement regime. This chapter presents an in-depth study into the physics of the transition between the first two of these regimes. At the outset it is not obvious how this transition should be defined. This chapter represents the coalescence of three ideas. a) Experimental sequences such as in Chapter 4 has focused on changes in the front speed as V̂0 is increased from zero. We could identify the transition with respect to the curve V̂f vs V̂0. b) An interesting phenomenon was observed during many of our exper- iments in this range. The layer of displaced fluid remained at the top of the pipe (diameter D̂) during the entire duration of the experiment, appar- ently stationary for very long times (t̂ & 103D̂/V̂0). We have termed this a stationary residual layer. c) Whereas Chapter 4 has focused on the behaviour of the leading dis- placement front in a typical experimental sequence, this has ignored what happens upstream. In an exchange flow we have zero net flux and there is an equal flow of heavy fluid downstream as there is of light fluid upstream. For the displacement flow, denoting the flow rates of heavy and light fluids through a given cross-section by Q̂ and Q̂l, respectively, we always have piD̂2V̂0 4 = Q̂+ Q̂l. (6.1) 7A version of this chapter has been published: S.M. Taghavi, T. Seon, K. Wielage- Burchard, D.M. Martinez and I.A. Frigaard. Stationary residual layers in buoyant New- tonian displacement flows. Phys. Fluids 23 044105 (2011). 112 6.1. Pipe displacements At V̂0 = 0 we have Q̂l = −Q̂ < 0 in the exchange flow. As V̂0 is increased, the net buoyancy force available to resist motion in the imposed flow direction remains constant, but the imposed flow creates viscous stresses which act on the lighter fluid layer at the interface and drag the lighter fluid along the duct. The viscous drag increases with V̂0 and eventually we expect to attain a transition where Q̂l = 0, and thereafter Q̂l > 0. Could we target the upstream flow and represent the transition between (i) and (ii) by where Q̂l = 0? It turns out that all 3 of these ideas are to some extent correct and equivalent. An outline of this chapter follows. Section 6.1 presents the results of our study in the pipe geometry. The experimental observations are pre- sented, focusing particularly at the region upstream of the gate valve. This is followed by development of a lubrication/thin film model for the pipe geometry. This model is used to make quantitative predictions that are in reasonable agreement with our experimental data. In the second part of this chapter (section 6.2) we study the same phenomenon, but in the sim- pler plane channel geometry. Here the lubrication model leads directly to analytic predictions of the stationary layer. These predictions are compared with results from fully 2D computations of the displacement flows in this regime. An excellent agreement is found. In §6.3 we outline a simple physi- cal model based only on a momentum balance, that is able to give the same qualitative behaviours as the more complex models. The chapter concludes with a discussion and summary. 6.1 Pipe displacements 6.1.1 Experimental observations Before giving a broad description of our general results, we describe in detail the experimental observation that motivated our deeper investigation. In systematically increasing the mean flow velocity V̂0 from zero we came across flows in which the downstream layer of in-situ fluid remained apparently stationary and uniform at the top of the pipe, while the displacing fluid traveled underneath. Fig. 6.1 displays an example of such a flow in the configuration where a heavy fluid (transparent) is injected to displace the lighter fluid (black), which is initially filling the inclined pipe. The displacement is from left to right. The leading front of the heavy fluid slumps underneath the light fluid at the start of the displacement (first image). We observe that 25 (s) after 113 6.1. Pipe displacements (450s) (25s) (5s) (250s) T(s)0 450 D ^ ^ Figure 6.1: Sequence of images showing the stationary upper layer. This sequence is obtained for 5, 25, 250 and 450 (s) after opening the gate valve. The field of view is 1015 × 20 (mm) and taken right below the gate valve. For this experiment the pipe is tilted at 85 ◦ from vertical. The normalized density contract is At = 10−2, the viscosity is µ̂ = 10−3 (Pa.s) and the mean flow velocity is V̂0 = 38 (mm.s−1). The figure below the sequence is a spatiotemporal diagram of the variation of the light intensity in the transverse dimension, averaged over 20 pixels along the pipe in the region marked on the pipe above, with a time step of ∆t̂ = 0.5 (s). It shows the variation of the layer height with time. the beginning of the process the two fluids are stratified along the length of the pipe. Since only the transparent fluid is injected, it is obvious that the two layers have different mean velocities and intuitively we would not expect this configuration to remain stationary. However, looking at the next two images (250 (s) and 450 (s)) we observe that the upper layer retains the same thickness. The image at the bottom of Fig. 6.1 is a spatiotemporal diagram of the light intensity across the pipe (averaged over the small square marked on the fourth image). The horizontal scale is time and vertical scale is the pipe diameter. At t̂ = 0 the image is all black because the pipe is full of black fluid. After around 15 (s) the heavy fluid arrives in this part of the pipe and we observe on the spatiotemporal diagram the two layers with the transparent fluid below the black fluid. The thickness of the layers stays constant until the end of the experiment, about seven minutes. The surprising feature of this observation was the longevity of the upper layer, outliving the duration of our experiment. During the time of the ex- periment in Fig. 6.1, five times the volume of the pipe have flowed through the pipe. Alternatively, the layers are constant for ∼ 103 times the advec- tive timescale D̂/V̂0 ≈ 0.5 (s). Also unexpected, but found only after our analysis, was that the interfacial velocity (i.e. wave speed of the interface) is zero so that the stationary layer is not simply a consequence of the flow 114 6.1. Pipe displacements becoming near-parallel. (b) (a) (d) (c) 63 cm Gate valve 22 cm x V0 ^ ^ y^ Figure 6.2: Four snapshots of video images taken at different mean flow rates and illustrating the different regimes. The heavy transparent fluid flows downward under the combined effects of buoyancy (∆ρ̂) and pressure gradient (V̂0). The light black fluid has different behaviors (flows upward or downward) depending on the control parameters values. These images were obtained at β = 85 ◦, At = 10−2, µ̂ = 10−3 Pa.s. The mean flow velocities were: (a) V̂0 = 29 (mm.s−1), (b) V̂0 = 38 (mm.s−1), (c) V̂0 = 42 (mm.s−1), and (d) V̂0 = 61 (mm.s−1). The field of view is 1015× 20 mm, and contains the gate valve (wide black stripe) and a pipe support (thin black stripe). The images are taken at: (a) 150 (s), (b) 290 (s), (c) 365 (s), (d) 75 (s) after opening the valve. We turn now to a more general description of our results. On closer investigation it became evident that as V̂0 was increased from zero, the most obvious changes in the flow occurred above the gate valve with the trailing front, rather than below with the leading front (which typically was quickly advected out of the 4 (m) pipe). The trailing or upstream front (meaning upstream of the mean flow) exhibited 4 different characteristic behaviours. Fig. 6.2 illustrates these 4 behaviours in a 1015 (mm) long section of the pipe, tilted at β = 85 ◦, for a sequence of displacements at the same density difference (At = 10−2) but at different V̂0. In each image the heavier transparent fluid moves downward from left to right, the black part at the right of each image is the gate valve and in the middle is a bracket supporting the pipe. In Fig. 6.2a the lighter fluid is moving upward against the imposed flow and the front moves steadily upstream without stopping. This picture has been taken a few seconds after the tip of the trailing front reached the upper end of the experimental pipe. The low mean velocity (V̂0 = 29 (mm.s−1)) allows a counter-current flow similar to the exchange flow, except that the back flow moves slower. We describe such flows as sustained back flows, i.e. there is a sustained upstream flow which advects the trailing front con- 115 6.1. Pipe displacements tinually upstream against the mean flow. In Fig. 6.2b with an increased imposed flow (V̂0 = 38 (mm.s−1)) we observe that the trailing front moves initially upstream against the flow, but then stops moving. This picture has been taken 60 (s) after the front stopped when it is stationary (290 (s) after the beginning of the experiment). This is the same experiment as in Fig. 6.1, for which the thickness of the upper layer in the downstream part of the pipe remains constant for a long time. We classify such flows as stationary interface flows. In the next image (Fig. 6.2c), with a slightly higher mean velocity (V̂0 = 42 (mm.s−1)) the trailing front moves upstream and stops, but closer to the initial position. The front stays in this position for a while but is eventually displaced downstream. We classify this behaviour as a temporary back flow, i.e. there is a flow backwards against the mean flow which initially advects the trailing front upstream but the back flow is not sustained over long times. Finally, if the mean velocity is further increased (Fig. 6.2d), the trailing front between clear and dark fluid is simply displaced downstream. We call this high mean flow case an instantaneous displacement, (V̂0 = 61 (mm.s−1)). For a more in-depth look at the transition between the stationary inter- face and the instantaneous displacement regimes, we display spatiotemporal diagrams of the back flows corresponding to Figs. 6.2b & c in Figs. 6.3a & b, respectively. These spatiotemporal diagrams are realized along a line in the upper part of the pipe section, where the back flow rises. The vertical scale depicts time (500 (s) in each figure) and the horizontal scale denotes distance along the pipe, from just above the gate valve. The instantaneous front ve- locities are determined from the local slope of the boundaries separating the black regions of the diagram (back flow zones) from the gray regions (transparent fluid). We observe in Fig. 6.3a that the back flow starts with a constant velocity and then slows down until it stops. It does not move significantly until the end of the experiment (except for small longitudinal oscillations). As the interface of the upper layer is stationary this demon- strates that throughout this period we have a balance between the pressure driven flow and the buoyant flow. For a slightly increased imposed flow we observe in Fig. 6.3b the temporary back flow regime. The back flow stops closer to the gate valve and starts to be displaced downward before the end of the experiment. Longer times are not shown on this figure but the back flow is displaced until its original position (the gate valve) and beyond. These behaviors will be interpreted in the next section. A closer inspection of Fig. 6.3a at long times shows a small deviation of the boundary from vertical, smaller but in the same direction as Fig. 6.3b. This may indicate 116 6.1. Pipe displacements Gate valve 500s 63 cm 63 cm x x t ^ ^ Pipe support(a) (b) Gate valvePipe support ^ Figure 6.3: Spatiotemporal diagrams of the variation of the light intensity along a line parallel to the pipe axis in the upper section of the pipe. The vertical scale is time (∆t̂ = 0.5 (s) and 500 (s) for both) and the horizontal scale is the distance along the pipe above the gate valve (see Fig. 6.2). The orientation of the x̂ axis is the same as in Fig. 6.2: downward. These diagrams correspond to the experiments: (a) Fig. 6.2b (V̂0 = 38 (mm.s−1)) and (b) Fig. 6.2c (V̂0 = 42 (mm.s−1)). the slow onset of temporary back flow. Fig. 6.4 displays transverse profiles of the longitudinal velocity (paral- lel to the pipe axis) averaged over time for 3 regimes: sustained back flow (Fig. 6.4a), stationary interface (Fig. 6.4b), and instantaneous displacement (Fig. 6.4c). These are measured below the gate valve along a line passing through the centre of the pipe. The vertical scale represents the distance from the upper wall and horizontal scale the longitudinal velocity compo- nent, with positive values measured in the flow direction. The horizontal dashed line shows the position of the interface. Close to the lower wall there are instrumental errors: an oblique dashed line has been added to artificially 117 6.1. Pipe displacements −40 −20 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 û(ŷ) (mm/s) D̂ − ŷ (m m ) a) −40 −20 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 û(ŷ) (mm/s) D̂ − ŷ (m m ) b) −40 −20 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 û(ŷ) (mm/s) D̂ − ŷ (m m ) c) Figure 6.4: Ultrasonic Doppler Velocimeters profiles for the same series of experiment as Fig. 6.2: (a) V̂0 = 29 (mm.s−1) (see Fig. 6.2a) sustained back flow regime, profiles averaged between 60 and 120 (s), (b) V̂0 = 38 (mm.s−1) (see Fig. 6.2b) stationary interface regime, profiles averaged between 240 and 300 (s), (c) V̂0 = 74 (mm.s−1) instantaneous displacement regime, profiles averaged between 120 and 240 (s). The vertical scale represents the distance from the upper wall (ŷ measuring distance from the lower one) and the horizontal scale the corresponding value of the longitudinal flow velocity. The horizontal dashed line shows the position of the interface. The vertical dashed line shows the zero velocity. The oblique dashed line close to the lower wall has been added to guide the eye where the profiles are distorted by instrumental error. complete the profile to the wall, where the velocity is zero. First of all, we observe that the 3 figures show a downward global net flow, due to the mean flow. By looking specifically at each regime, we 118 6.1. Pipe displacements observe that in the sustained back flow regime (Fig. 6.4a corresponding to the experiment of Fig. 6.2a) the velocity at the interface is small. Almost the entire upper layer moves upstream. In the stationary interface regime (Fig. 6.4b corresponding to the experiment of Fig. 6.2b) we observe that the fluid velocity at the interface is positive, but that both positive and negative velocities are found in the upper layer. Therefore, although the interface is apparently stationary the fluid in the upper layer is not motionless but moves in a counter-current recirculatory motion. The displacing fluid is observed to pass underneath the upper layer and so we expect that the net flow rate through the upper layer should be very close to zero. This measurement is averaged along a transverse axis positioned centrally in the pipe cross- section. Although plausibly close to zero, the measurements are not precise enough to evaluate this zero net flow condition. Additionally there are variations in the z-direction which would need estimating or measuring. Finally, in the instantaneous displacement regime (Fig. 6.4c) the lighter fluid has been mostly displaced leaving only a very thin residual layer. The above constitutes a description of the distinct flow regimes observed in our experiments, as V̂0 is varied. One could say that we have essentially 3 regimes, with the stationary layer apparently representing a transition state between flows with sustained back flow and those that displace. Below in §6.1.2 we derive a simple model that predicts similar flow regimes and transitions. In §6.1.3 we present the comparison between the predictions of this model and the classification of our experiments. 6.1.2 Lubrication model • β g y x D ^ ^ ^ X bf^ f ^ Ω H (heavy) Ω L (light) y=h(x,t) ^ ^ ^ ^ Cross-section h(x,t) ^ ^ ^ z ^ 0V ^ Figure 6.5: Schematic views of the distribution of the two fluids in two perpendicular vertical planes of the pipe (diametrical and transversal). The notation is that used in the models. 119 6.1. Pipe displacements Our experimental observations suggest that (after the initial few seconds of our displacements and away from the tips of the leading/trailing fronts) most of the flow occurs within regions where the fluids are separated by interfaces that are aligned approximately with the pipe axis. It is there- fore very natural to develop a thin-film/lubrication style model for the pipe displacement flow. The procedure is more or less standard and we follow largely that of our previous chapter (§5). At each axial position x̂ the flow is assumed stratified with interface denoted ŷ = ĥ(x̂, t̂); see the geometry illustrated schematically in Fig. 6.5. The leading order equations are the momentum balances: 0 = −∂p̂ ∂ẑ , (6.2) 0 = −∂p̂ ∂ŷ − ρ̂kĝ sinβ, (6.3) 0 = −∂p̂ ∂x̂ + µ̂ [ ∂2ŵ ∂ẑ2 + ∂2ŵ ∂ŷ2 ] + ρ̂kĝ cosβ, (6.4) (ẑ, ŷ) ∈ Ωk k = H,L and the incompressibility condition, ∇· û = 0. At the walls û = 0, and both velocity and traction vectors are continuous at the interface. For the flows considered a mean flow V̂0 is imposed by pumping in the positive x̂-direction. Thus, the additional constraint piD̂2V̂0 4 = ∫ ΩH ⋃ ΩL w dẑdŷ, (6.5) is satisfied by the solution. We eliminate p̂ and derive the evolution equation for ĥ: ∂ ∂t̂ Â(ĥ) + ∂ ∂x̂ Q̂ = 0, (6.6) where Â(ĥ) is the area occupied by the heavier fluid, Â(ĥ) = |ΩH |, and Q̂ = ∫ ΩH ŵ dẑdŷ. (6.7) The flux consists of a superposition of Poiseuille and exchange flow compo- nents: Q̂ = Q̂(ĥ, ĥx̂) = 2V̂0 ∫ Â(ĥ) ( 1− 4 ẑ 2 + ŷ2 D̂2 ) dẑdŷ + piD̂2 8 F0V̂ν ( 1− 4(D̂/2− ĥ) 2 D̂2 )7/2( cosβ − sinβ∂ĥ ∂x̂ ) 120 6.1. Pipe displacements where V̂ν = At.ĝ.D̂2/ν̂ and F0 is given in [135] as F0 = 0.0118. The exchange flow component has been estimated (see [135]) by extrapolating from the value at ĥ = D̂/2 and from asymptotic expressions for ĥ ∼ 0 and ĥ ∼ D̂. In §5 we defined dimensionless parameters, δ and χ via δ = µ̂V̂0 [ρ̂H − ρ̂L]ĝ sinβD̂2 = V̂0 2V̂ν sinβ , χ = cotβ δ = 2V̂ν cosβ V̂0 , (6.8) and scaled the system using a length-scale L̂ = D̂/δ in the x̂-direction, with L̂/V̂0 as timescale. Here we adopt the same scalings and also scale Â(ĥ) with piD̂2/4, Q̂ with piD̂2V̂0/4 and (ĥ, ŷ, ẑ) with D̂. The resulting dimensionless equations are ∂ ∂T α(h) + ∂ ∂ξ q(h, hξ) = 0, (6.9) where h ∈ [0, 1] is now dimensionless, α(h) = 4Â(ĥ)/piD̂2 is the area fraction occupied by the heavy fluid: α(h) = 1 pi cos−1(1− 2h)− 2 pi (1− 2h) √ h− h2 q(h, hξ) = 32 pi ∫ α(h) ( 1 4 − x2 − y2) dxdy + F0[χ− hξ] 4 ( 1− (1− 2h)2)7/2 , T and ξ are the dimensionless time and length variables, respectively. Although the algebraic form of (6.9) differs from that analysed for the plane channel, we expect similar behaviour. Let us first consider the down- stream behaviour. At long times the interface is expected to elongate (as shown in §5), which negates the effect of the slope of the interface in all regions except local to the advancing front. The behaviour is approximated by the hyperbolic part of (6.9), i.e. setting q = q(h, 0). We have observed (Fig. 6.2) that the interface remains stationary for the duration of the ex- periment, with constant flow rate of displacing heavy fluid. In the context of (6.9), considered at long times, this implies that the interfacial speed is zero and the flux, q(h, 0) = 1. The interfacial speed Vi is simply the characteristic speed: Vi = ∂q ∂α (h, 0) = ∂q ∂h (h, 0) [ dα dh (h) ]−1 , 121 6.1. Pipe displacements and since α is monotone with respect to h, the condition Vi = 0 implies that ∂q∂h(h, 0) = 0. Note that q(h, 0) depends on the single parameter χ. In Fig. 6.6 we plot contours of q(h, 0) and the contour ∂q∂h(h, 0) = 0, against (h, χ). The intercept of q(h, 0) = 1 and ∂q∂h(h, 0) = 0 occurs at a critical χ = χc = 116.32.. and for h = 0.72.., indicating that there is a unique interface height and value of χ for which stationary interfaces may occur. 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 11 h χ 0.0 9.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 Figure 6.6: Contours of q(h, 0) and the contour ∂q∂h(h, 0) = 0 (bold black line). The intercept of q(h, 0) = 1 and ∂q∂h(h, 0) = 0 occurs at χ = 116.32.. and h = 0.72... Considering now the trailing front, for the plane channel displacement large values of χ resulted in a second front propagating upstream against the flow, i.e. a back flow. Large χ corresponds to a weak imposed flow relative to the buoyancy driven exchange flow component. For the pipe flow, assuming again that the long time behaviour is dominated by the hyperbolic part of (6.9), the equations determining the back flow front speed Vf < 0 and front height hf , are simply: [1− α(hf )]Vf = [1− q(hf , 0)], (6.10) Vf = ∂q ∂h (hf , 0) [ dα dh (hf ) ]−1 (6.11) For sufficiently large χ this expression has solutions Vf < 0. We now observe that if we take the limit Vf → 0 we enforce also q(hf , 0) = 1. It follows that 122 6.1. Pipe displacements the conditions for the stationary interface are identical with those determin- ing whether or not equation (6.9) has a sustained back flow: for χ > χc we have sustained back flow and for χ < χc there is no sustained back flow. Strictly speaking, both statements relate to longtime behaviour of (6.9). 0 5 10 15 20 0 0.25 0.5 0.75 1 ξ h (ξ ,T ) −0.4 −0.2 0 0.8 0.9 1 ξ h f (ξ ) Figure 6.7: Profiles of h(ξ, T ) for T = 0, 1, .., 9, 10, with χ = χc. The broken line shows the theoretical stationary h = 0.72.. at χ = χc. The inset shows the extension of the stationary frontal region. In Fig. 6.7 we plot h(ξ, T ) for the critical χ = χc, obtained by solving (6.9) numerically using the same method as in Chapter 5 for the plane channel. The trailing front at the top of the pipe is stationary as expected, while the leading front moves down the pipe. Although over long times the interface is stretched out between the stationary trailing front and the advancing leading front, the fronts are not shocks (since diffusive effects are always present). By solving for each h ∈ (hf , 1) the nonlinear equation q(h, hξ) = 0, we may find the steady interface slope, hξ(h). This can be integrated to find the shape of the steady profile for h > hf . In this frontal region, buoyancy driven by the slope of the interface (acting to smooth the interface) is in 123 6.1. Pipe displacements balance with the buoyancy force driving fluid back up the inclined pipe. The frontal profile is illustrated in the inset of Fig. 6.7. The numerical integration has been stopped when h is within 1% of hf . 6.1.3 Experimental and theoretical comparison The analysis of the previous section suggests that stationary interfaces can occur for each inclination angle β only at a critical balance V̂ν cosβ ≈ 58.16V̂0, (6.12) (recall χ = 2V̂ν cosβ/V̂0). In the lubrication model context, sustained back flows are only found upstream for smaller values of V̂0 than in (6.12), whereas downstream the interface speed becomes positive for larger V̂0 than in (6.12) and instantaneous displacement ensues. Temporary back flows are not strictly covered by the long time analysis of the lubrication model. Our experiments have been performed over the ranges: V̂0 ∈ 0− 80 (mm.s−1), At ∈ [10−3 − 10−2], β ∈ [83 ◦ − 87 ◦]. To give an overall perspective of the different flow regimes and where they occur, Fig. 6.8 presents the classification of our flows for the full range of experiments. We observe that the sustained back flow regime is clearly separated from the instantaneous displacement regime. Between these two regimes we find stationary layers and temporary back flows. We must ac- knowledge potential errors in making the classifications depicted in Fig. 6.8. For example, sustained back flow experiments are terminated when the back flow exits the upper end of the pipe (due to ensuing mixing) but in a longer pipe could reverse and become temporary. Equally, the stationary interface case is clearly a marginal transition between sustained and temporary back flows. With a finite duration experiment (with other restrictions and errors) it is difficult to definitively classify a displacement as stationary. The bold line illustrates the analytical prediction (6.12). Given the potential uncer- tainty in classifying experiments and in the approximation of the exchange flow component of q, the prediction offered by this linear relation (6.12) is surprisingly good. We also note that in those experiments that we have classified as stationary interface flows the stationary layer occupies approxi- mately 30% of the pipe at the top, which corresponds well to the theoretical stationary h = 0.72.. at χ = χc. It is worth commenting that we have plotted our results in dimensionless velocity coordinates, with both V̂0 and V̂ν cosβ scaled with the inertial scale 124 6.1. Pipe displacements 0 1 2 3 4 5 6 0 20 40 60 80 100 120 V̂0 V̂t V̂ν cos β V̂t Sustained Back Flow Stationary Interface Temporary Back Flow Instantaneous Displacement Figure 6.8: The experimental results in a pipe over the entire range of control parameters (V̂0 is in the range 0−80 (mm.s−1), At is in the range 10−3−10−2, β is in the range 83 ◦−87 ◦). The heavy line represents the prediction of the lubrication model for the stationary interface: 58.16V̂0 = V̂ν cosβ. V̂t. This of course does not affect the relation between V̂0 and V̂ν cosβ which is exemplified in Fig. 6.8, but may appear strange for phenomena that are essentially viscous. This choice can be understood better in the context of previous work. The vertical axis, shows the competition between viscous and inertial forces in balancing buoyancy, in the absence of any imposed flow. As discussed in §2 pure exchange flow studies [135] have suggested that for V̂ν cosβ ' 50V̂t the exchange flow is governed by an inertia-buoyancy balance (and viscous-buoyancy below this value). Our experiments cover this range and clearly the viscous prediction from (6.12) still is apparently relevant in what might be thought of as the inertial regime. The explanation for this comes from Chapter 4 in which we have shown that the imposition of a mean flow results in the streamlines becoming progressively aligned with the pipe axis, even in this inertial regime. The consequent stabilization as the flow rate (Reynolds number) is increased is somewhat counterintuitive. We can view (6.12) as being derived from the instantaneous displacement 125 6.2. Plane channel geometry (2D) configuration in the regime V̂ν cosβ ' 50V̂t, and provided that V̂0 is large enough the flows are sufficiently laminar and non-inertial for the validity of the model. 6.2 Plane channel geometry (2D) In the preceding section we have considered the pipe geometry, which is well suited to experiment. Our attempts to quantify the stationary layer phenomenon via (6.12) are reasonable given experimental errors and the degree of approximation necessary for semi-analytical theories. To confirm our explanation more fully we could turn to computational simulation, but in the pipe geometry this investigation would require fully 3D computations, which are exceedingly expensive computationally in pipes of long aspect ratio. Instead therefore we turn to a 2D plane channel geometry in order to confirm our understanding of the stationary layer. This geometry allows for faster computations and more precise asymptotic approximations. The channel has height D̂ and is oriented similarly to the pipe, close to horizontal. Again a heavy fluid displaces a lighter fluid in the downwards direction. 6.2.1 Lubrication model The lubrication/thin-film approach is analogous to that developed for the pipe, leading to a dimensionless evolution equation for the interface height, y = h(ξ, T ): ∂h ∂T + ∂ ∂ξ q(h, hξ) = 0. (6.13) This has been derived and extensively studied in Chapter 5 for a wide range of fluid types. We focus only on the analysis relevant to the current situation. In parallel with the earlier analysis of (6.9) we may compute a critical value of χ and h for which the entire flux passes through the lower layer and for which the interface speed is zero. Contours of q(h, 0) and the contour ∂q ∂h(h, 0) = 0 are plotted in (h, χ)-space in Fig. 6.9, from which we find χc = 69.94 for the plane channel at an interface height h = 0.707. Note here that h ∈ [0, 1] as we have scaled with the height D̂ of the channel. The relation χ = χc again provides a predictor of the stationary interface, which we now test against 2D computational solutions. 126 6.2. Plane channel geometry (2D) 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 1 1 1.1 h χ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 20 40 60 80 100 Figure 6.9: Contours of q(h, 0) and the contour ∂q∂h(h, 0) = 0 (bold black line), in a plane channel displacement. The intercept of q(h, 0) = 1 and ∂q ∂h(h, 0) = 0 occurs at χ = χc = 69.94 and h = 0.707. 6.2.2 Numerical overview We have carried out a number of numerical simulations of 2D displacements in an inclined plane channel. The geometry and notation are as represented in Chapter 3. Our computations are fully inertial, solving the full 2D Navier Stokes equations. The phase change is modelled via a scalar concentration, c, which is advected with the flow, i.e. molecular diffusion is neglected. This neglect is due to the large Péclet numbers that correspond to our experimental flows, for which we typically have a well defined interface. The Navier Stokes equations are made dimensionless using the channel height D̂ as lengthscale and V̂0 as velocity scale. The model equations are: [1 + φAt] [ut + u · ∇u] = −∇p+ 1 Re ∇2u+ φ Fr2 eg, (6.14) ∇ · u = 0, (6.15) ct + u · ∇c = 0. (6.16) Here eg = (cosβ,− sinβ) and the function φ = φ(c) interpolates linearly between −1 and +1 for c ∈ [0, 1]. The 2 additional dimensionless parameters appearing above are the Reynolds number, Re, and the (densimetric) Froude 127 6.2. Plane channel geometry (2D) number, Fr, defined as follows. Re ≡ V̂0D̂ ν̂ , Fr ≡ V̂0√ AtĝD̂ . (6.17) Here ν̂ is defined using the mean density ρ̂ = (ρ̂H + ρ̂L)/2, and the mean static pressure gradient has been subtracted from the pressure before scaling. We see that for small At the flow is essentially governed by the 3 parameters β, Re & Fr. For t > 0, no slip boundary conditions are satisfied at the solid walls (zero flux for c) and outflow conditions imposed at the channel exit. At the inflow the heavy fluid concentration is imposed (c = 0), and the velocity u is represented by a fully established Poiseuille profile. The initial interface position is some way down the channel and our initial velocity field is stationary: u = 0 at t = 0. We have selected a range of parameters that resembles that of our pipe flow experiments. Thus, we will describe the simulations in the following section with reference to V̂0, V̂ν and V̂t, as these are more natural from the experimental perspective. The mapping between parameters is simply: Re ≡ V̂0V̂ν V̂ 2t , F r ≡ V̂0 V̂t . (6.18) When considering the lubrication model predictions: χ = 2 cosβV̂ν V̂0 = 2Re cosβ Fr2 . (6.19) Unlike the pipe flow, we have limited our computational study to param- eters for which the pure exchange flow (V̂0 = 0) is in the viscous regime. The reason for this restriction is that in general the stabilizing effect of the imposed flow (as shown in §4) does not affect the channel exchange flow in the same way as it affects a pipe exchange flow. For the pure exchange flow Hallez and Magnaudet [67] have reported key differences in the flow structure for pipe and plane channel geometries when in the inertial regime. We will further comment on this in the next chapter. 6.2.3 Numerical results Fig. 6.10 gives an example of a displacement that is typical of those found close to the stationary interface regime, (parameters β = 87 ◦, ν̂ = 2× 10−6 (m2.s−1), At = 3.5 × 10−3, V̂0 = 9.5 (mm.s−1), [Re = 90, Fr = 0.37]). The 128 6.2. Plane channel geometry (2D) upper image in Fig. 6.10 depicts the initial condition for the concentration field at t̂ = 0 (s). The subsequent images (from top to down) show the evolution of the concentration field at t̂ = 25, 50, 100, 200, 300 (s). Although we observe that for t̂ > 0 the trailing front initially moves backwards against the mean flow, for t ≥ 100 (s) the front appears stationary with the top of the interface seemingly pinned to the upper wall. We observe that downstream the height of the interface is h ≈ 0.7, which is in good agreement with the analytical prediction from the lubrication model. Note that although numerical diffusion is well limited by the MUSCL scheme, dispersion due to (physical) secondary flows is not restricted. This accounts for the grey regions in Fig. 6.10. 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 x̂/D̂ Figure 6.10: Sequence of concentration field evolution obtained for β = 87 ◦, ν̂ = 2× 10−6 (m2.s−1), At = 3.5× 10−3, V̂0 = 9.5 (mm.s−1), [Re = 90, Fr = 0.37]. The images are shown for t̂ = 0, 25, 50, 100, 200, 300 (s) (from top to bottom). To have a better understanding of the different regimes in typical station- 129 6.2. Plane channel geometry (2D) x̂ (mm) t̂ (s) 500 1000 1500 2000 100 200 300 1 2 3 Figure 6.11: Spatiotemporal diagram of the average concentration variations (white and black colors represent heavy and lighter fluids respectively) along the channel for β = 87 ◦, ν̂ = 2 × 10−6 (m2.s−1), At = 3.5 × 10−3, V̂0 = 9.5 (mm.s−1), [Re = 90, Fr = 0.37]. Vertical scale: time; horizontal scale: distance along the channel. Dashed lines have slopes equal to velocities estimated for the leading and the trailing fronts. The stationary slope (1) shows that the front velocity is constant. Dashed line (2) is the initial inertial velocity for the trailing front, which is followed by a decreasing viscous velocity. Dashed line (3) is vertical, which implies that the back flow velocity (of the lighter fluid) is zero (near stationary). ary flows in a channel Fig. 6.11 displays the spatiotemporal diagram of the average concentration along the channel for the same parameters as used in Fig. 6.10. In this diagram the contrast has been slightly increased for illus- trative purposes. We observe three characteristic behaviours. The slope of dashed line (1) represents the constant velocity of the leading front traveling towards downstream. The velocity of the trailing front (traveling upstream) 130 6.2. Plane channel geometry (2D) 20 22 24 26 28 30 32 34 36 38 20 22 24 26 28 30 32 34 36 38 20 22 24 26 28 30 32 34 36 38 20 22 24 26 28 30 32 34 36 38 20 22 24 26 28 30 32 34 36 38 20 22 24 26 28 30 32 34 36 38 x̂/D̂ Figure 6.12: The velocity profiles corresponding to Fig. 6.10 for a channel flow at t̂ = 0, 25, 50, 100, 200, 300 (s) (from top to bottom). is not constant with time. Initially the trailing front flows backwards with constant velocity shown by the slope of dashed line (2). As the front elon- gates the velocity starts to decrease. We infer that inertial effects control the initial back flow velocity; the corresponding initial viscous velocity, which is proportional to the slope of the interface, would be too large (infinite at t̂ ∼ 0). During the first acceleration when the interface between the two motionless fluids starts to move, the back flow is accelerated by buoyancy up until it attains approximately the inertial velocity. At this point inertia prevents the fluid from accelerating faster. When the trailing front stretches beyond a characteristic length the viscous velocity becomes lower than the inertial velocity. At this point the back flow can dissipate its energy in the bulk by viscosity. Thereafter the trailing front velocity starts to decrease and after a transient phase (between the lines (2) and (3) in Fig. 6.11) it reaches its limiting/final velocity (i.e. equal to zero). The dashed line (3) is almost vertical which implies that the trailing front velocity remains close to zero. The flow is in the stationary regime. Note that this is essentially 131 6.2. Plane channel geometry (2D) the same picture that we have observed experimentally. Fig. 6.12 shows the velocity profiles corresponding to Fig. 6.10 at t̂ = 0, 25, 50, 100, 200, 300 (s). The figure shows the region between 20 < x̂/D̂ < 38. The initial interface is located at x̂/D̂ = 25. As expected, for t̂ < 100 (s) we see a counter-current flow in the longitudinal direction, with net flow equal to the imposed flow rate. In this time frame we transition from an initially inertially limited flow to a viscously limited flow. 0 1 2 0 0.2 0.4 0.6 0.8 1 û/V̂0 ŷ/D̂ Figure 6.13: The velocity profile close to the pinned point (with the axial position x̂/D̂ = 26.25) corresponding to Figs. 6.10 & 6.12 for a channel flow at t̂ = 300 (s): illustrating the counter-current inside the stationary (lighter/black) fluid. Dashed line represents the local height of the interface. Fig. 6.13 illustrates a single velocity profile at t̂ = 300 (s) at an axial position close to the pinned point, where the interface meets the upper wall. The local interface height (h ≈ 0.775) is shown by the dashed line, which is higher than the interface height downstream (h ≈ 0.7). We can observe the counter-current flow inside the stationary upper layer. Fig. 6.14 displays the four archetypical regimes for different imposed flow velocities. In Fig. 6.14a, we see that for a low imposed flow the velocity of the 132 6.2. Plane channel geometry (2D) downstream front is constant at all times. The upstream front initially has a constant (inertially limited) velocity, which gradually decreases and finally reaches a constant buoyant velocity, allowing the lighter fluid to keep rising (sustained back flow). At a larger mean imposed flow velocity, Fig. 6.14b is the stationary interface regime. A further increase in the imposed flow (Fig. 6.14c) leads to an upstream front which advances, stops and then recedes down the pipe. This corresponds to the temporary back flow regime. Finally, for a sufficiently strong imposed flow (Fig. 6.14d), there is no back flow from the beginning of the displacement process. An instantaneous displacement is achieved. 0  1000 2000 0 50 100 150 x̂ (mm) t̂ (s ) 0  1000 2000 0 50 100 150 x̂ (mm) t̂ (s ) 0  1000 2000 0 50 100 150 x̂ (mm) t̂ (s ) 0  1000 2000 0 50 100 150 x̂ (mm) t̂ (s ) b) c) a) d) Figure 6.14: Four possible conditions for a viscous buoyant channel flow when an imposed flow is present: the parameters are β = 89 ◦, ν̂ = 10−6 (m2.s−1), At = 10−2; a) V̂0 = 16.8 (mm.s−1) [Re = 323, Fr = 0.39], b) 18.9 (mm.s−1) [Re = 363, Fr = 0.44], c) 21.0 (mm.s−1) [Re = 403, Fr = 0.49], d) 78.6 (mm.s−1) [Re = 1509, Fr = 1.82]. Fig. 6.15 shows the collected results of our simulations: V̂0 is in the 133 6.2. Plane channel geometry (2D) range 2 − 30 (mm.s−1), At is in the range 10−3 − 10−2, ν̂ is in the range 10−6 − 2× 10−6 (m2.s−1), β belongs to the range 85 ◦ − 89 ◦. Each simula- tion has been classified from the spatiotemporal plot as exhibiting one of the four characteristic behaviours. The bold line in Fig. 6.15 illustrates the an- alytical prediction of the stationary interface, for which V̂ν cosβ ≈ 34.97V̂0, (i.e. χ = χc = 69.94). We observe that there is good agreement between the lubrication model prediction and the stationary interfaces obtained by nu- merical simulation. This suggests that the two layer model considered in the lubrication approximation is useful for predicting the long time behaviour of buoyant channel displacements. In addition, the simulations represent- ing the temporary back flows are clearly separated from those showing the instantaneous displacement flows. The transition between temporary back flows and the instantaneous displacement flows seems to be governed by a balance between the imposed pressure gradient (roughly speaking V̂0) and the characteristic inertial velocity (V̂t). For viscous flow in a channel, this transition (V̂0/V̂t = γ) lies somewhere in the range γ = 0.6 − 0.8, probably with minor dependency on the inclination angle β. 0 0.5 1 1.5 2 0 5 10 15 20 25 V̂0 V̂t V̂ν cos β V̂t Sustained Back Flow Stationary Interface Temporary Back Flow Instantaneous Displacement Figure 6.15: Classification of our simulation results in a channel. The heavy line represents the prediction of the lubrication model for the stationary interface: 34.97V̂0 = V̂ν cosβ. 134 6.3. Simple physical model In a more inclined channel, where the flows become inertial, we anticipate that there could be an increase in the value of γ. Experimental observations for an exchange flow in a pipe reveal that this increase can be up to 40% with respect to the horizontal, see [135]. Although the precise value of γ is of interest, it should be noted that this does not affect the long time behaviour of the flow/interface. Indeed whether or not the back flow is temporary or the displacement is instantaneous, the displacing fluid eventually washes out the displaced fluid as long as V̂0/V̂ν cosβ is large enough. 6.3 Simple physical model We close our result section by showing that many qualitative features of our experiments can be predicted by a simplified conceptual model, along the lines of that presented by Seon et al. [135] for pure exchange flows. The objective is to describe the speed of propagation of the trailing front (V̂ bff ) and its position (X̂bff ) as they move backwards against the imposed flow, see Fig. 6.5. First of all, it is clear that the only driving mechanism to push the lighter fluid up the channel is buoyancy. Except at early times, the flows above the gate valve appear quasi-parallel (e.g. Figs. 6.2b&c), which suggests that the driving buoyancy force is balanced by viscosity. An appropriate velocity scale that reflects this balance is V̂ν . Buoyancy acts both axially along the pipe (∝ cosβ) and perpendicular to the pipe axis (∝ sinβ). The latter trans- verse component acts only when the interface between the fluids is tilted with respect to the pipe axis (i.e. if ∂ĥ/∂x̂ 6= 0) and is then proportional to − sinβ∂ĥ/∂x̂. The second force affecting the back flow comes from the imposed flow that determines a net pressure gradient pushing fluids down- wards, along the pipe. Since the fluids are Newtonian we may assume that this force scales approximately linearly with V̂0. Therefore, on summing the different driving forces we might postulate that: V̂ bff = V̂ν cosβ ( Ka −Kt∂ĥ ∂x̂ tanβ ) −KmV̂0, (6.20) where the coefficients Ka, Kt & Km reflect the relative influences of axial buoyancy, transverse buoyancy and the mean flow, respectively. For the case V̂0 = 0 (exchange flow) this is the model of Seon et al. [135], who estimate Ka & Kt from their experiments. We see that the second term in (6.20) decreases in size as the trail- ing front propagates, reducing the slope of the interface. Therefore, V̂ bff 135 6.3. Simple physical model Inertial Viscous γVt 0 ^ Xbff ^ KaVν cosβ ^ KmV0 ^ = 0 : stopping lengthV bf f ^ Vbf+ KmV0f ^ ^ 0.2 0.24 0.28 0.32 0.36 0.0105 0.0115 0.0125 0.0135 V0 ^ KaVν cosβ ^ D tanβ X bf f ^ ^ (a) (b) Figure 6.16: (a) Schematic variation of the velocity V̂ bff +KmV̂0 as a function of distance X̂bff from the gate valve (continuous line) in a viscous regime and for β 6= 90 ◦. The short dashed line represents the final viscous velocity V̂ bff + KmV̂0 = KaV̂ν cosβ. The dotted line marks the boundary between the transient inertial regime and the viscous regime. We also represent the case γV̂t > KmV̂0 > KaV̂ν cosβ, using the long dashed line, to underline the stopping length condition. The arrows on the curve show the trend of the evolution of the velocity with time. (b) V̂0/(V̂ν cosβ) is plotted versus (D̂/X̂bff ) tanβ for 2 series of experiments at different angles β: 83 ◦ (¤) and 85 ◦ (◦), and same density contrast and viscosity (At = 10−2, µ̂ = 10−3 (Pa.s). The experiments plotted here are either in the temporary back flow regime or in the stationary interface regime, and X̂bff represents the position where the front stops (maximal X̂bff ). The dashed line is a guide for the eye to show the common linear curve. 136 6.3. Simple physical model decreases with distance (and time) as is shown schematically in the vis- cous regime indicated in Fig. 6.16a. Equation (6.20) can be turned into a crude differential equation for X̂bff by approximating the interface slope with ∂ĥ∂x̂ ≈ −D̂/X̂bff , which leads to: dX̂bff dt̂ = V̂ bff = V̂ν cosβ ( Ka +Kt D̂ X̂bff tanβ ) −KmV̂0, (6.21) At short times (and distances) the model (6.21) would predict an infinite front velocity, which is not physically possible. In practice, in this early period of the flow V̂ bff will be limited by inertial effects rather than viscous effects. We may expect this balance to persist until the viscous front velocity, determined from (6.21), falls below a value that is related to the inertial velocity scale, V̂t. We see for example that in Fig. 6.3 the front velocity is indeed initially constant before it decreases. This cut-off behaviour is illustrated schematically in Fig. 6.16a. Consequently, we may modify (6.21) as follows: dX̂bff dt̂ = min { γV̂t, V̂ν cosβ ( Ka +Kt D̂ X̂bff tanβ )} −KmV̂0, (6.22) where γ is a further coefficient to be determined. Although simplistic we believe (6.22) contains the essential elements of the trailing front dynamics. Evidently V̂ bff decreases with time as the front propagates, in all cases. Let us consider some different possible be- haviours. First, let us suppose that the imposed flow is weak, so that KmV̂0 < KaV̂ν cosβ. The flow has a transient phase during which the in- terface slope decreases and the speed also, but the buoyancy force is strong enough to maintain a sustained back flow: V̂ bff → KaV̂ν cosβ −KmV̂0 and the front advances steadily up the pipe. Secondly, suppose that the imposed flow is stronger, so that KmV̂0 > KaV̂ν cosβ, but that KmV̂0 < γV̂t (case represented in Fig. 6.16a). The transient phase of the back flow elongates the interface so that the slope decreases until there is a perfect balance: V̂ν cosβ ( Ka +Kt D̂ X̂bff tanβ ) = KmV̂0, (6.23) corresponding to the stationary regime. Rearranging this shows that the 137 6.4. Discussion stopping lengths Xf satisfy: V̂0 V̂ν cosβ = 1 Km ( Ka +Kt D̂ X̂bff tanβ ) . (6.24) Note that at larger V̂0 the transient phase of the back flow will be reduced and stopping length too. We might also expect that this delicate balance be affected over longer times by changes in the interface profile below the gate valve, allowing the trailing front to recede down the pipe, (which is not taken into account in our model). This is the temporary back flow regime. In this simple conceptual model, the stationary interface regime and the temporary back flow regime are both characterised by a stopping length, determined from (6.24), which is the maximum height attained. The fully stationary layer is simply a marginal state that is theoretically present, but not easily observable. Finally, for still larger V̂0, say KmV̂0 > γV̂t, we expect no back flow and the instantaneous displacement regime is entered. In Fig. 6.16b we have plotted V̂0/(Vν cosβ) against (D̂/X̂ bf f ) tanβ for 2 series of experiments at different angles. Only those experiments are plotted that were characterised as a temporary back flow or stationary interface and X̂bff is taken as the maximal measured front distance above the gate valve. We observe that the 2 series collapse approximately onto the same linear curve, as predicted by (6.24). This supports the assumptions made regarding the driving forces of the buoyant back flow in the presence of a mean flow. In principle, this also allows us to determine directly the β- independent coefficients Ka/Km & Kt/Km, via linear regression, and to use the model in equation (6.24) predictively. However, to be more confident in determining Ka/Km & Kt/Km we would need to conduct more experiments for a wider range of At, ν̂ and D̂. The purpose of the model is instead to show that the types of behaviour observed qualitatively can be attributed to a fairly simple force balance. 6.4 Discussion We have observed an interesting displacement flow phenomenon in which a buoyant displacement flow retains a stationary upper layer of displaced fluid for the duration of our experiment. The same feature was observed in our plane channel displacement simulations. Some aspects of this flow are obvious. For example, as we increase the imposed flow rate from zero we do expect to reach a flow rate for which the upper layer has zero flow. Less 138 6.4. Discussion obvious is that the flow structure should remain stationary, i.e. the layer thickness of the lighter fluid that is found at the transition state is one at which the interface speed is zero. The flow apparently evolves to select this interfacial position, so that the flow structures observed for V̂0 close to the transition persist over very long timescales (as described in detail in §6.1). We have found 2 parallels to this phenomenon in the literature. Hup- pert & Woods [81] have considered a range of porous media flows driven by density difference, using a lubrication approximation. Part of their study considers two-layer exchange flows between reservoirs and amongst the so- lutions investigated there exist those for which the flow in one layer is zero. There are many differences between porous media flows and those governed by the Navier-Stokes equations. In the present context we note that the main differences are that in porous media flows of Huppert & Woods [81] zero flow in one fluid layer means the velocity is everywhere zero in that layer and the modified pressure gradient is also zero. In the Navier-Stokes context (current work) there is a positive pressure gradient driving the light fluid layer backwards against the flow and the velocities are non-zero within the stationary layer. In looking simply at lubrication-type models with an imposed flow those based on underlying Hele-Shaw (or porous media) me- chanics allow steady state interface propagation at the imposed velocity [21] whereas those based on the Navier-Stokes systems do not. Our study has revealed that the stationary residual layer phenomenon marks the transition between flow parameters that displace fully and those that do not. Observations of the upstream region above the gate valve allow us to categorize the displacements as one of 4 different states: (a) sustained back flow, (b) stationary interface, (c) temporary back flow, (d) instantaneous displacement. The stationary residual layers observed downstream coincide with the stationary interface regime observed for the upstream/trailing front. The same 4 states observed experimentally in the pipe are found computationally in 2D computational simulations of plane channel displacements. Instantaneous displacements and sustained back flow regimes can also be found at long times in thin-film/lubrication style models of these flows. 139 6.4. Discussion The transition between states is the stationary layer, which is predicted by the lubrication model, at critical conditions: 58.16V̂0 = V̂ν cosβ (6.25) for the pipe geometry and 34.97V̂0 = V̂ν cosβ (6.26) for a plane channel geometry. In the context of Chapter 4 where we have studied flow rate effects on the downstream front velocity, the stationary layer flows studied mark the boundary between the exchange flow dominated regime and the regime where the downstream front velocity (V̂f ) increases linearly with V̂0, for which the imposed flow becomes increasingly dominant. The transition between temporary back flows and instantaneous dis- placements appears to be characterised by a condition V̂0 = γV̂t, with γ = 0.6− 0.8, for the plane channel geometry. This estimate has been made using only flow parameters for which the pure exchange flow would be viscous in the plane channel. It is interesting to reflect that although we have classified 4 different states, in our experiments and in each of the models we have used we are only able to identify 3 states definitively. For the experimental results we simply classify observed flows within the practical limits of our experiments. Thus, if the back flow exceeds the end of the pipe (above the gate valve) we classify the flow as a sustained back flow (although given a longer pipe some of these might be temporary); the stationary back flow is identified when there is a stationary residual layer still remaining at the end of the experiment. The 2D plane channel computations are limited in much the same way as the experiments, in that computational times limit the range of feasible mesh sizes, computational domains and time intervals to be investigated. The lubrication models have only been analysed in the long time limit. In this limit the model exhibits in fact only 3 states: sustained back flow, stationary back flow and instantaneous displacement. Although at short times (and distances) the model presented in Chapter 5 always has a fast initial phase where temporary back flows may exist, they are not present at long times. At short times the lubrication model assumptions are not immediately valid. This underlines the value of adopting a range of different techniques to understand the dynamics of complex flows: each technique gives different insights. 140 6.5. Summary Amongst the 4 different states classified, the stationary interface is a transition state, only marking the flow that exists at the boundary between sustained and temporary back flow regimes. This means that it would be near impossible to find exactly the correct parameters to capture this state exactly. In all likelihood, any such state would anyway finally evolve into a temporary back flow via downstream processes such as fluid entrainment (see e.g. turbulent entrainment in §2.2.2), thinning the layer below the critical thickness. Thus, it is relevant that in our study we have observed (and classified as stationary) states which are probably only close to the transition state, but nevertheless persist for the duration of our experiments (physical or numerical). It is the existence of these near-stationary states, persisting over timescales of many thousand D̂/V̂0, that have practical importance. Certainly such longevity could prove problematic for processes such as the primary cementing of near-horizontal oil and gas wells. 6.5 Summary To summarise, the main novel contributions of this chapter are as follows. • Identification and physical explanation of the stationary residual layer flow. • Classification of the flow transitions occurring upstream of the initial fluid positions. • Usage of the lubrication/thin film approach to make predictions of the critical imposed flow for which stationary residual layers occur; see (6.25) and (6.26), together with validation of these approximations with experimental and 2D simulation data. 141 Chapter 7 Iso-viscous miscible displacement flows8 The aim of this chapter is to bring together the studies of Chapters 4 − 6 with a more complete investigation of these displacement flows in the iso- viscous setting. We aim to give a complete classification of the types of flow occurring, together with predictions of their regimes and the leading front velocity, all given in appropriate dimensionless terms. We again use a combination of experimental, computational and ana- lytical methods. Fluid miscibility is relatively unimportant as we work in a high Péclet number regime at low At. Three dimensionless groups largely de- scribe these flows: Fr (densimetric Froude number), Re (Reynolds number) and β (duct inclination). Our results will show that the flow regimes in fact collapse into regions in a two-dimensional (Fr,Re cosβ/Fr)-plane. These regions are qualitatively similar between pipes and plane channels, although viscous effects are more extensive in pipes. In each regime we are able to give a leading order estimate for the velocity of the leading displacement front, which is effectively a measure of displacement efficiency. 7.1 Problem Setting We have already introduced the scenario studied throughout the thesis in the pervious chapters. In this chapter we only consider the case in which the (Newtonian) fluids have the same viscosity µ̂, are miscible and have differing densities. In general we study laminar flows. This flow may be studied from a number of different perspectives. First of all, from a modeling perspective a natural formulation involves a concentration-diffusion equation coupled to the Navier-Stokes equations. The phase change between pure fluids 1 and 2 is modeled via a scalar concentration, c. On making the Navier- 8A version of this chapter has been submitted for publication: S.M. Taghavi, K. Alba, T. Seon, K. Wielage-Burchard, D.M. Martinez and I.A. Frigaard. Miscible displacements flows in near-horizontal ducts at low Atwood number. 142 7.1. Problem Setting Stokes equations dimensionless using D̂ as length-scale, V̂0 as velocity scale, and subtracting a mean static pressure gradient before scaling the reduced pressure, we arrive at: [1 + φAt] [ut + u · ∇u] = −∇p+ 1 Re ∇2u+ φ Fr2 eg, (7.1) ∇ · u = 0, (7.2) ct + u · ∇c = 1 Pe ∇2c. (7.3) Here eg = (cosβ,− sinβ) and the function φ(c) = 1−2c interpolates linearly between 1 and −1 for c ∈ [0, 1]. The 4 dimensionless parameters appearing in (7.1) are the angle of inclination from vertical, β, the Atwood number, At, the Reynolds number, Re, and the (densimetric) Froude number, Fr. These are defined as follows: At ≡ ρ̂1 − ρ̂2 ρ̂1 + ρ̂2 , Re ≡ V̂0D̂ ν̂ , F r ≡ V̂0√ AtĝD̂ . (7.4) Here ν̂ is defined using the mean density ρ̂ = (ρ̂1 + ρ̂2)/2 and the common viscosity µ̂ of the fluids. In (7.3) appears a 5th dimensionless group, the Péclet number, Pe, defined by: Pe ≡ V̂0D̂ D̂m , (7.5) with D̂m the molecular diffusivity (here assumed constant for simplicity). It appears that 5 dimensionless parameters are required to fully describe this flow. However, commonly the Péclet number is very large as we consider lab/industrial scale flows rather than micro-fluidic devices, e.g. Pe > 106 is common. If the fluids are initially separated we expect diffusive effects to be initially limited to thin interfacial layers of size ∼ Pe−1/2. These layers may grow, via instability, mixing and dispersion, but in the many situations where the flows remain structured and partially stratified we commonly observe interfaces that are sharp over experimental timescales. Such flows are close to their immiscible fluid analogues (at infinite capillary number, i.e. vanishing surface tension), which are modeled by setting Pe = ∞ and ignoring the right-hand-side of (7.3). Secondly, we see that the direct effect of the density difference on inertia is captured by At. Supposing for example that we restrict our attention to density differences of the order of 10% (as in our experiments) we see that 143 7.1. Problem Setting At ≤ 0.05. We expect therefore that for moderate density differences the solution for At = 0 will give a reasonable approximation.9 Therefore, we see that the 5 parameters are really reduced to 3: (Re, Fr, β) in this large Pe, small At limit that is representative of many practical displacement flows. Moreover, we consider only β such that the duct inclination is close to horizontal since this range of inclinations is where viscous effects are mostly found. Thus, the overall aim of our study is to build a quantitative description of the different flow regimes found, in terms of Re and Fr, for β close to pi/2. One of the tools used will be lab-scale experiments in an inclined pipe. From an experimental perspective, there are a limited number of suitable experimental fluids (cost, ease of cleaning and mixing, rheological and optical properties, etc). To preserve consistency of the fluids used it is natural to mix a pair of fluids and then to conduct experimental sequences in which we vary the mean flow V̂0 at fixed inclination. We observe that both Re and Fr increase linearly with V̂0 in such an experimental sequence. The results of pure exchange flow studies are governed by the relative sizes of V̂ν (denoting the velocity at which buoyancy and viscous stresses balance) and V̂t (denoting the velocity at which inertial and buoyancy stresses balance). Thus, if we wish to measure the departure from the exchange flow setting as the flow rate is increased, the natural experimental description revolves around V̂0, V̂ν and V̂t, at fixed β. The relationships between these parameters and Re & Fr are: Re ≡ V̂0V̂ν V̂ 2t , F r ≡ V̂0 V̂t . (7.6) 7.1.1 Viscous and inertial flows Frequently in discussing our results below we shall refer to flows as either vis- cous or inertial. This terminology has been borrowed from Seon et al. [131– 135] and needs a few words of explanation. Firstly, since typically Re > 1, all our flows are inertial. Secondly, it is obvious that as the imposed flow V̂0 is increased, viscosity plays an increasing role in balancing the mean pressure drop, and the amount of inertia injected into the flow increases. Therefore, our usage of viscous and inertial is primarily phenomenological, in describing observed results. Where the flow remains primarily laminarised and uni-directional, with a clean interface and no evidence of instability, we 9Note also that the incompressibility condition (7.2) in fact requires small At in order to be valid for intermediate c in the case that the 2 individual pure fluids can be considered incompressible. 144 7.2. Displacement in pipes β ◦ ν̂ (mm2.s−1) At (×10−3) V̂0 (mm.s−1) Re Fr 83a 1− 2 1− 40 0− 841 0− 16021 0− 19.45 85 1− 2 1− 91 0− 80 0− 1524 0− 5.37 87 1− 2 1− 10 0− 77 0− 1467 0− 5.63 aMost of the experiments were conducted in the ranges At (×10−3) ∈ [1, 10], V̂0 ∈ 0− 110 (mm.s−1). Table 7.1: Experimental plan. refer to the flow as viscous. Where we observe two and three-dimensional regions of flow, typically associated with instability and (at least localised) mixing close to the interface, we refer to the flows as inertial. 7.1.2 Outline The main content of this chapter proceeds in 3 sections. The first section (§7.2) concerns pipe flow displacements. The main methods are experimen- tal and semi-analytical, using a lubrication/thin-film modeling approach. The second section (§7.3) presents analogous studies in a plane channel ge- ometry. Here the physical experiments are replaced with numerical experi- ments. In both geometries we obtain reasonable agreement with predictions from the semi-analytical models. The discrepancies are possibly attributable to inertial effects, which we study in §7.4. We also study the flow stability in §7.4. The chapter ends with a brief summary. 7.2 Displacement in pipes The first geometry studied is the pipe. We present here an extended set of experimental results, beyond the preliminary results in Chapter 4, and give quantitative comparisons of the displacement flow behaviour based on the lubrication approximation from §5 and extrapolation from the exchange flow studies of [132] and [135]. This culminates (§7.2.6) in a simple predic- tive model for the displacement front velocity in all observed regimes. Our experiments were conducted over the ranges shown in Table 7.1. 7.2.1 Basic flow regimes observed In a typical experiment we observe a short inertial phase following the open- ing of the gate valve. The fluids are initially at rest. When the gate valve 145 7.2. Displacement in pipes is opened the static head accelerates both fluids from rest and at the same time the density difference between fluids accelerates the fluids in oppos- ing directions. This first stage is very fast (order of seconds). We then characteristically observe two fronts emerge. The leading front is towards the bottom of the pipe and moves downstream faster than the mean flow. The trailing front is towards the top of the pipe and moves slower than the leading front (see Chapter 6 for details on the trailing front dynamics). Depending on the buoyancy forces the trailing front may move either up- stream against the mean flow (buoyancy forces dominate imposed flow) or downstream (imposed flow dominates buoyancy forces). The front may also move initially upstream and then become washed downstream over a longer time interval. The interface between these two advancing fronts is essen- tially stretched axially along the pipe. Inertia is always the main balancing force for buoyancy in the first part of the experiment, when the interface is transverse to the pipe axis, but as the flow elongates it appears that viscous forces dominate, over a wide range of flow rates. For most of our study we disregard the initial phase and concentrate on characterizing longer time dynamics. However, the time evolution from an initial acceleration phase to an inertia-buoyancy balance to a viscous- buoyancy phase, is interesting in itself. As an example, Fig. 7.1 shows a sequence of images of the interface in a typical experiment. In this flow the trailing front initially moves back upstream, but is eventually displaced at longer times. The initial displacement front shows a characteristic “inertial tip” and the initial images show evidence of interfacial instability. As the displacement progresses and the trailing front moves downstream the un- derlying axial velocity profile becomes progressively positive and the flow is progressively stabilized. As well as these spatial images, we can process data from our Ultrasonic Doppler Velocimeter (UDV) system (located 80 (cm) below the gate valve) for the same experiment. This is shown in Fig. 7.2 over the same range of times as the images in Fig. 7.1. Superimposed on the velocity map is the measured interface height at the position of the UDV. This height is interpreted from the light intensity in our images of the displacement and is effectively an area averaged concentration of dark fluid, translated into a height. In constructing this we average data over 20 pixels (22.7 (mm)) around the position of the UDV. We can observe the initial unsteadiness of the flow in both interface position and underlying velocity field. The total flow rate is fixed, so the initial period of backflow corresponds to the fastest velocities downstream in the lower layer. As the displacement progresses we see a steady decrease in maximal absolute velocity in both layers and a 146 7.2. Displacement in pipes UDV probe Figure 7.1: Sequence of images showing propagation of waves along the interface for V̂0 = 40 (mm.s−1) along a 1245 (mm) long section of the pipe a few centimeters below the gate valve. Other parameters are β = 85 ◦, At = 1.67× 10−2 and ν̂ = 1 (mm2.s−1). From top to bottom the images are taken at t̂ = 5, 15, 50, 100, 150, 350, 550, 750, 950 (s) after opening the gate valve. progressively steady interface. Considering now longer times, in §4 we reported preliminary experi- mental evidence that the flows transition between 3 distinct stages as the mean imposed flow V̂0 is increased from zero. At low V̂0, an exchange-flow dominated regime is found, as expected. This exchange flow may either be viscous (low Ret cosβ = V̂ν cosβ/V̂t) or inertial (high Ret cosβ), following [132, 135]. In the latter case the flows are characterized by Kelvin-Helmholtz like instabilities. With increasing V̂0 we observed that the flow becomes sta- ble. The speed of the leading front (say V̂f ) increases approximately linearly with V̂0, with slope larger than 1. We have termed this a viscous regime. At even larger V̂0 we find that V̂f ∼ V̂0, as the fluids effectively mix transver- sally. These 3 regimes form the framework for our understanding. Here we 147 7.2. Displacement in pipes t̂ (s)   350 400 450 500 550 600   50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 18  D̂ − ŷ (m m )   650 700 750 800 850 900 −20 0 20 40 60 80 100 120 140 Figure 7.2: Contours of axial velocity (mm.s−1) obtained from the Ultrasonic Doppler Velocimeter for the same experiment as in Fig. 7.1. The velocity readings are taken through the pipe centreline in a vertical section, with the UDV angled at 67 ◦ to the surface of the pipe. The vertical axis shows depth measured from the top of the pipe. The thick black line shows the interface height at the position of the UDV, which is averaged spatially over 20 pixels (22.7 (mm)). report a much fuller data set than in §4. In Fig. 7.3 we plot the variation of the leading front velocity V̂f as a function of mean flow velocity V̂0 for different values of density contrast and viscosity at pipe inclination angles: β = 87 ◦ and β = 83 ◦. Figure 7.4a shows similar data at β = 85 ◦. In these figures we observe mostly the first and second regimes of the displacement, i.e. an initial plateau at low V̂0 (exchange flow regime) followed by linear increase in V̂f at larger V̂0. Also shown in Figs. 7.3 & 7.4a is a secondary classification of the front motion, described in detail in §6, that relates to the behaviour of the trailing front. It was found that for low V̂0 buoyancy forces were sufficiently strong to produce a sustained upstream motion of the trailing front (a sustained back flow). On increasing V̂0 we found a marginal state in which the trailing front advanced upstream against the flow and stopped for the duration of our ex- periment (a stationary interface flow). At the same time the downstream leading front is advected from the pipe leaving behind an apparently station- ary residual layer. At larger V̂0 the trailing front moved only upstream for a finite time, eventually reversing and moving downstream (a temporary back flow). Finally, at large V̂0 the trailing front moves directly downstream (an instantaneous displacement). In each figure we have classified the displace- ments by examining the spatiotemporal diagram for the trailing front. As well as the transition from exchange flow dominated to viscous displacement regime, we also observe the transition from sustained back flow through to 148 7.2. Displacement in pipes 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 V̂0 (mm/s) V̂ f (m m / s) a) β = 87 ◦ 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 V̂0 (mm/s) V̂ f (m m / s) b) β = 83 ◦ Figure 7.3: Variation of the front velocity V̂f as a function of mean flow veloc- ity V̂0 for different values of density contrast and viscosity at two inclination angles: a) at β = 87 ◦ data correspond to At = 10−2 (¥), At = 3.6 × 10−3 (•), At = 10−3 (N) with ν = 1 (mm2.s−1) and At = 3 × 10−3 (H) with ν = 1.8 (mm2.s−1); b) at β = 83 ◦ data correspond to At = 4 × 10−2 (∗), At = 10−2 (¥), At = 3.5 × 10−3 (•), At = 10−3 (N) with ν = 1 (mm2.s−1) and At = 3.5 × 10−3 (H) with ν = 1.7 (mm2.s−1). In both plots sustained back flows and instantaneous displacements are marked by the superposed squares and circles respectively; data points without marks are either tempo- rary back flows, stationary interfaces or undetermined experiments (i.e. in- sufficient experiment time or short pipe length above the gate valve). instantaneous displacement on each data set, as V̂0 is increased. Figure 7.4b examines the second regime more closely for the data at inclination angle β = 85 ◦. In the data shown we have excluded those points classified as sustained back flows and observe that these correspond well to the viscous regime and indeed have an approximately linear variation. The dashed lines give an approximate linear fit to each data set. The inset of Fig. 7.4b shows that by normalizing with V̂ν cosβ the data in the viscous regime collapses onto a single curve, which we now explain below in §7.2.2. It is this collapse of the data onto a single curve that establishes the essential viscous nature of the flow in this regime. Further explanation is given in §7.2.2 below. Certainly one of the most interesting aspects of the longer-time behaviour in our experiments is the laminarisation as V̂0 increases, which is largely counter-intuitive. We take a more detailed look at this transition here. In Fig. 7.5 we show snapshots from a sequence of experiments performed for progressively large V̂0. In this case the pure exchange flow (V̂0 = 0) is 149 7.2. Displacement in pipes 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 180 200 V̂0 (mm/s) V̂ f (m m / s) a) β = 85 ◦ 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 V̂0 (mm/s) V̂ f (m m / s)   b) 0 0.1 0.2 0 0.1 0.2 0.3 V̂0/(V̂ν cos β) V̂ f /( V̂ ν co s β )   β = 85 ◦ Figure 7.4: Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for different values of density contrast and viscosity at β = 85 ◦. a) Sustained back flows and instantaneous displacements are marked by the superposed squares and circles respectively; data points without marks are either temporary back flows, stationary interfaces or undetermined experi- ments. b) Illustration of the imposed flow dominated regime where, com- pared to the left plot, only temporary back flows are excluded. The dashed lines are linear fits of data points for each set of increasing V̂0 (fixed At and ν̂). The inset shows normalized front velocity V̂f/V̂ν cosβ as a function of normalized mean flow velocity V̂0/V̂ν cosβ, for which the data superim- pose. The solid line is a linear fit to all the normalized data points. In both figures the data correspond to At = 9.1 × 10−2 (I), At = 1.1 × 10−2 (¥), At = 3.5 × 10−3 (•), At = 10−3 (N) with ν = 1 (mm2.s−1) and At = 3.7× 10−3 (H) with ν̂ = 1.7 (mm2.s−1). strongly inertial and in the first few snapshots we see a propagating layer of heavy fluid at the bottom of the pipe with a significant mixed layer on top. At intermediate imposed velocities we see the clear laminarisation of the flow (e.g. at V̂0 = 57 & 72 (mm.s−1)). Finally at larger V̂0 we see progressively more mixing, except now there is sufficient inertia to mix across the whole pipe cross-section. Examples of spatiotemporal diagrams related to flows in the first and second regimes are shown in Fig. 7.6a-b, (from a different sequence than Fig. 7.5). For the parameters selected the pure exchange flow is inertial. For low V̂0 = 30 (mm.s−1), the flow remains unstable. In Fig. 7.6a we can observe the initial front propagating and behind it unstable waves appear at the interface, as evidenced below the initial sharply defined dark region in 150 7.2. Displacement in pipes S B S B S B T B T B T B ID ID ID ID IDR eg im e 3 R eg im e 2 R eg im e 1 V̂0 Figure 7.5: A sequence of snapshots from experiments with increased imposed flow rate; the parameters are β = 83 ◦, At = 10−2 and ν̂ = 1 (mm2.s−1). From top to bottom we show images for V̂0 = 9, 19, 31, 44, 56, 57, 72, 108, 257, 474, 841 (mm.s−1). The figure shows a 1325 (mm) long section of the pipe a few centimeters below the gate valve. Key: SB = sustained backflow; TB = temporary backflow; ID = instanta- neous displacement. Fig. 7.6a. We observe a range of wave speeds differing slightly from the front propagation speed. No second front is observed, as for this experiment the trailing front moves backward, upstream against the flow. For an increased V̂0 = 75 (mm.s−1) the flow has become stable; see Fig. 7.6b. The slope of the line separating the black region and gray region represents the velocity of the leading front, at the lower wall. We can also discern a separating curve between the gray and white regions: the slope of this curve represents the (lower) velocity of the trailing front at the upper wall. The corresponding UDV results for the same two experiments are shown in Fig. 7.6c-d. In Fig. 7.6c we observe temporal oscillations corresponding to the flow instability. The sustained back flow is evident in the negative veloc- ity values at the top of the pipe. The stable flow is illustrated in Fig. 7.6d. The UDV images are ensemble-averaged over 15 consecutive images, corre- 151 7.2. Displacement in pipes x̂ (mm) t̂ (s ) 140012001000800600400200 0 25 50 75 100 125 150 a) Regime 1 x̂ (mm) t̂ (s ) 140012001000800 600400200 0 25 50 75 100 125 150 175 200 225 b) Regime 2 t̂ (s) D̂ − ŷ (m m )   20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 −20 0 20 40 60 80 c) t̂ (s) D̂ − ŷ (m m )   50 100 150 200 0 2 4 6 8 10 12 14 16 18 0 20 40 60 80 100 120 d) Figure 7.6: Examples of spatiotemporal diagrams and corresponding UDV measurements obtained for β = 85 ◦, At = 10−2 and ν̂ = 1 (mm2.s−1): a) & c) V̂0 = 30 (mm.s−1); b) & d) V̂0 = 75 (mm.s−1). The velocity (mm.s−1) is measured through the pipe centreline in a vertical section, with the UDV angled at 74 ◦ to the surface of the pipe, positioned at 80 (cm) below the gate valve. The vertical axis shows depth measured from the top of the pipe. Velocity contours are averaged in time over 15 consecutive velocity profiles, (1.8 (s)). sponding to a time average over a local interval of 1.8 (s). This eliminates small high frequency fluctuations, which correspond to the UDV sampling rate. If we look carefully, we can observe the presence of negative values of flow velocity towards the top of the tube. In this experiment there is no back flow of the trailing front, but this does not preclude negative velocities. These regions correspond to a temporary recirculation at this position inside the upper fluid, which persists for t̂ ≈ 125 (s), by which time the trailing front reaches the UDV probe located at x̂ = 80 (cm). After the trailing 152 7.2. Displacement in pipes front has passed a more Poiseuille-like flow is recovered. Note also that in this initial period, when negative velocities are found in the upper layer, the velocities in the lower layer must be correspondingly higher (observe the dark red region) to maintain the fixed imposed flow rate. 7.2.2 Lubrication/thin film model To explain the similarity scaling evident in our data (e.g. Fig. 7.4b), we re- sort to a lubrication/thin film style of model (assuming the immiscible limit Pe → ∞). This type of model has been developed for plane channel dis- placements in §5. Exchange flows have been studied using this type of model in [135] and in §5 we have extended this type of model to the displacement regimes studied here. For brevity, we refer to §5 for the derivation. The interface height evolution is governed by the following dimensionless equation: ∂ ∂T α(h) + ∂ ∂ξ q(h, hξ) = 0. (7.7) In this model h ∈ [0, 1] is the dimensionless interface height (scaled with the diameter), α(h) ∈ [0, 1] is the area fraction occupied by the heavy fluid (under the interface) α(h) = 1 pi cos−1(1− 2h)− 2 pi (1− 2h) √ h− h2 (7.8) and the scaled flux of fluid in the heavy layer is denoted q(h, hξ): q(h, hξ) = 32 pi ∫ α(h) ( 1 4 − x2 − y2) dxdy + F0[χ− hξ] 4 ( 1− (1− 2h)2)7/2 . (7.9) The first term is the Poiseuille component and the second term is the ex- change flow component; F0 is given by Seon et al. [132] as F0 = 0.0118. The variables T and ξ are the dimensionless time and length variables, respec- tively: T = t̂V̂0 D̂ δ, ξ = x̂ D̂ δ, (7.10) where δ = µ̂V̂0 [ρ̂H − ρ̂L]ĝ sinβD̂2 = V̂0 2V̂ν sinβ . (7.11) 153 7.2. Displacement in pipes This type of model contains the balance between viscous, buoyant and imposed flow stresses. Only a single dimensionless parameter χ remains following the model reduction: χ = cotβ δ = [ρ̂H − ρ̂L]ĝ cosβD̂2 µ̂V̂0 = 2V̂ν cosβ V̂0 = 2Re cosβ Fr2 , (7.12) which represents the balance of axial buoyancy stresses and viscous stresses due to the imposed flow. The interface slope hξ generates additional axial pressure gradients which contribute to the exchange flow component of flux in (7.9), but as the interface extends progressively longer this effect becomes irrelevant, except possibly in local regions. Thus, purely from the perspec- tive of dimensional analysis, the similarity scaling evident in Fig. 7.4b is ex- pected: it simply shows that the long-time front velocity depends uniquely on the parameter χ. Although the algebraic form of (7.7) differs from that analysed for the plane channel, we find qualitatively similar behaviour. Typically we find a short initial transient during which the interface elongates from its initial position and during which time diffusive spreading due to the presence of the term hξ in q dominates the behaviour. This is followed by the emergence of a distinct leading front, which abuts the lower wall of the pipe (including h = 0), and always propagates downstream at a speed Vf > 1. By front we mean an interval of h that moves at constant speed. At large values of χ, buoyancy is strong and a second trailing front emerges that moves upstream. As χ is reduced the trailing front speed decreases until there is no back flow (at a critical χ = χc = 116.32...). The interface displaces only in the positive direction for χ < χc. At the upper wall the interface is pinned to the wall. Figure 7.7a shows the result of solving (7.7) numerically, comparing h(ξ, T ) at T = 10 for different χ = 0, 10, 50, 200. The long time behaviour of the system is governed by the hyperbolic part of (7.7), i.e. setting q = q(h, 0). The equations determining the leading front speed (Vf ) and front height (hf ) are: α(hf )Vf = q(hf , 0), Vf = ∂q ∂h (hf , 0) [ dα dh (hf ) ]−1 , (7.13) which can be solved numerically. The variation of the front speeds and heights with χ is plotted in Fig. 7.7b. As χ→ 0 the imposed flow becomes increasingly dominant and Vf approaches a value Vf = 1.0868. For the trailing front similar conditions can be derived and solved. The transition between upstream and downstream moving trailing interface occurs at a 154 7.2. Displacement in pipes −10 −5 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 ξ h a) 0 20 40 60 80 100 0.4 0.6 0.8 1 1.2 1.4 1.6 χ h f , V f b) Figure 7.7: a) Numerical examples of pipe flow displacements based on the lubrication model solution for χ = 0 (5), χ = 10 (4), χ = 50 (◦), χ = 200 (¤); b) variation of the front speeds (solid line) and heights (broken line) with χ. critical χ = χc = 116.32... with a front height hf = 0.72.... At this value there is a stationary interface in the downstream part of the flow; see §6. The changes in Vf with χ are easy to understand mathematically for a Newtonian fluid, as the flux function q is composed of two parts, one of which is multiplied by χ. This follows simply from the principle of superposition. As χ varies we essentially interpolate (nonlinearly) between the χ = 0 and χ =∞ frontal behaviours, as determined by (7.13). 7.2.3 Comparison of experimental results and the lubrication model The superposition of the experimental data shown in the inset of Fig. 7.4b corresponds to a (near linear) variation of the normalized leading front ve- locity with χ−1. It is natural to compare the experimental front speeds with the calculated front speeds from our lubrication model. This is done in Fig. 7.8 for the full range of experimental data that fall in either exchange dominated regime or viscous dominated regimes. The bold line indicates the scaled front velocity obtained by the lubrication model, i.e. solving (7.13). The circle on the bold line indicates the theoretical balance between these two regimes, at χ = χc = 116.32... where the stationary interface is found. For values of χ < χc, instantaneous displacements in the viscous regime are found, the collapse of the data onto the theoretical curve is evident. This emphasizes that in this regime the balance is primarily between vis- 155 7.2. Displacement in pipes 10−4 10−3 10−2 10−1 100 10−3 10−2 10−1 100 V̂0/(V̂ν cos β) ≡ 2/χ V̂ f / (V̂ ν co s β ) ≡ 2 V f / χ Displacement Back flow 2 1 Figure 7.8: Normalized front velocity, V̂f/V̂ν cosβ, plotted against normal- ized mean flow velocity, V̂0/V̂ν cosβ, for the full range of experiments in the first and second regimes (limited by Re < 2300) in Table 7.1. Data points with the same symbols belong to experimental sets of increasing Reynolds number (via V̂0) for fixed At or viscosity. The heavy solid line indicates the scaled front velocity from the lubrication model. The circle indicates the theoretical transition (χ = χc = 116.32..). The thin solid line shows V̂f = V̂0, below which front velocities are not possible (denoted region 1). Region 2 represents flows with increasingly significant inertial effects. cous forces generated by the imposed flow and buoyancy. Although we have a high degree of agreement with this simple model (considering also the ex- perimental uncertainty), we note that the experimental data does generally lie just above the theoretical curve, in the viscous regime. We hypothesize that this discrepancy is an effect of inertia. Inertial effects are difficult to include in such models for the pipe geometry but we will return to this for the plane channel geometry in §7.4. 7.2.4 The exchange-flow dominated range Region 2 in Fig. 7.8 contains data from flows where inertial effects are in- creasingly significant in balancing the buoyancy-driven exchange component. Since these effects are not included in the lubrication approximation, diver- 156 7.2. Displacement in pipes gence from the theoretical front velocity curve in Fig. 7.8 is to be expected. Within this exchange-flow dominated regime we have no fully predictive model. However, as V̂0 → 0 we do recover the pure exchange flow results. In inertial exchange flows studied in pipes close to horizontal [135] found that the front velocity can be approximated by V̂f ≈ γV̂t, with γ ≈ 0.7, where V̂t = √ AtĝD̂ represents the velocity scale at which buoyancy is bal- anced by inertia. In the case that V̂t is the relevant scale as V̂0 → 0, we might consider that γ is simply the leading order term in an expansion with respect to small Fr = V̂0/V̂t, i.e. for χÀ χc we assume V̂f V̂t = f(Fr) ≈ f(0) + Frf ′(0) + Fr 2 2 f ′′(0) + ...., (7.14) with f(0) = γ ≈ 0.7. With this ansatz we rescale V̂f with V̂t for all our inertial experimental data with χ > χc, and fit the coefficients in (7.14). We find f ′(0) = 0.595 and f ′′(0) = 0.724, which are in the confidence intervals f ′(0) ∈ (0.454, 0.735) and f ′′(0) ∈ (0.478, 0.970) with confidence level 95%. Figure 7.9a shows a comparison of front velocity data in the exchange flow regime with the prediction: V̂f V̂t = 0.7 + 0.595Fr + 0.362Fr2 (7.15) The collapse of the data with respect to Fr is evident and the approximation is quite reasonable. To explore the validity of the approximation (7.15) as χ decreases, we plot in Fig. 7.9b the same data but normalised with the viscous scale (e.g. as in Fig. 7.8). The broken curves now denote (7.15), which is different for different experimental sequences. However, the curves appear to converge in this figure close to the critical value V̂0/V̂ν cosβ = 2/χc which is marked, and diverge thereafter. Note however, that in our experiments we have observed that even inertial exchange flows become viscous on increasing V̂0. Thus, above the critical V̂0/V̂ν cosβ = 2/χc our experimental sequences are fitted well by (7.13). 7.2.5 Overall classification of the flow regimes For a more global perspective on our results and in particular to exem- plify the balance with inertia we present the classification of our exper- imental results together with our flow regime predictions from both the 157 7.2. Displacement in pipes 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 V̂0/V̂t ≡ Fr V̂ f / V̂ t ≡ F rV f a) 10−4 10−3 10−2 10−1 100 10−3 10−2 10−1 100 V̂0/(V̂ν cosβ) ≡ 2/χ V̂ f / (V̂ ν co s β ) ≡ 2 V f / χ b) Figure 7.9: a) Normalized front velocity, V̂f/V̂t, as a function of normalized mean flow velocity V̂0/V̂t = Fr, (equivalently Froude number), plotted for 3 experimental sequences in the inertial regime. Data correspond to At = 9.1 × 10−2 at β = 85 ◦ (•), At = 4 × 10−2 at β = 83 ◦ (¨), At = 10−2 at β = 83 ◦ (¥), all with ν = 1 (mm2.s−1)s. The broken line shows V̂f/V̂t = 0.7 + 0.595Fr + 0.362Fr2. b) Normalized front velocity, V̂f/V̂ν cosβ, as a function of normalized mean flow velocity , V̂0/V̂ν cosβ. Data points with the same symbols belong to same experimental sequence: increasing Reynolds number through V̂0. The heavy solid line indicates the scaled front velocity from the lubrication model. The thin solid line shows V̂f = V̂0. The broken lines show our inertial exchange approximation through the simple model. lubrication model and our pure exchange flow curve fit. We plot our exper- imental results in the (V̂0/V̂t, V̂ν cosβ/V̂t)-plane. Equally, this plane is the (Fr,Re cosβ/Fr)-plane. Note that lines of constant χ correspond to linear rays through the origin, in the positive quadrant. The quantity Re cosβ/Fr is also related to a Reynolds number based on the inertial velocity scale, denoted Ret by Seon et al. [135], i.e. Ret cosβ ≡ V̂tD̂ cosβ ν̂ ≡ V̂ν cosβ V̂t ≡ Re cosβ Fr . (7.16) Figure 7.10 plots the data from the full range of our experiments, as de- scribed in Table 7.1. Each experiment has been classified according to the classifications of §6. Only data satisfying Re < 2300 has been used. The critical value χc = 116.32... corresponds to the line Re cosβ Fr = χc 2 Fr, (7.17) 158 7.2. Displacement in pipes which is also marked in Fig. 7.10. Considering pure exchange flows, accord- ing to [135] the dividing line between viscous and inertial exchange flows is at Ret cosβ = V̂ν cosβ V̂t ≈ 50, (7.18) (inertial exchange flows for larger Ret than (7.18)), which is marked in Fig. 7.10 with the heavy solid horizontal line. Finally, at sufficiently large imposed velocities we expect to transition to the mixed 3rd regime (e.g. see Fig. 7.5). A sufficient condition for this would be the onset of turbulence. Assuming that at high Re the buoyancy effects have minimal effect, we might assume transition at a nominal value Re = 2300. For different pipe inclinations these curves are marked in Fig. 7.10 with thin broken lines. At each angle the corresponding displacement data lies under the appropriate curve. With reference to Fig. 7.10 we can identify the following different flow regimes and partly quantify the behaviour within each regime. (a) Inertial exchange flow dominated regime: This regime is found for Ret cosβ & 50 and for Fr = V̂0/V̂t . 0.9. In Fig. 7.10 this regime is marked by i1. This flow is characterized by development of Kelvin- Helmholtz-like instabilities and partial mixing. Buoyancy forces are sufficiently strong for there to be a sustained back flow. The front velocity in this regime scales with Vt and is approximated reasonably well by the empirical relation (7.15). (b) Inertial temporary back flow regime: In Fig. 7.10 this regime is marked by i2 and is bounded by (7.17) and Fr = V̂0/V̂t & 0.9. On increasing the imposed flow V̂0, the destabilizing influences of inertia become progressively less efficient. The bulk flow remains generally in- ertial up until the critical stationary interface flow is encountered, along (7.17), after which the flow becomes progressively laminar. The front velocity in this regime scales with Vt and is approximated reasonably well by the empirical relation (7.15). (c) Viscous exchange flow dominated regime: For V̂0 = 0 this regime is observed for Ret cosβ =. 50 [135]. For positive V̂0 the same cri- terion appears correct. When the pure exchange flow is viscous, the corresponding displacement flows obtained by adding a small imposed flow V̂0, are also viscous at long times. This regime is marked by v1 in Fig. 7.10 and is also bounded by (7.17). In this regime inertial effects can be observed at the beginning of displacement (i.e. at short time) 159 7.2. Displacement in pipes 0 1 2 3 4 5 6 0 20 40 60 80 100 120 V̂0 V̂t ≡ Fr V̂ ν c o s β V̂ t ≡ R e c o s β F r β = 83o β = 85o β = 87o v1 v2 v3 i1 i2 χ = χc Figure 7.10: Classification of our results for the full range of experiments in the first and second regimes (Re < 2300) in Table 7.1: sustained back flow (¥, ¤), stationary interface (.), temporary back flow (J, /) and instan- taneous displacement (•). Data point with filled symbols are viscous and with hollow symbols are inertial. The horizontal bold line shows the first order approximation to the inertial-viscous transition (Ret cosβ = 50, from [135]). The dotted line and its continuation (the heavy line) represent the prediction of the lubrication model for the stationary interface, χ = χc. The vertical dashed-line is V̂0/V̂t = 0.9. The thin broken lines are only illustra- tive and show an estimate for the turbulent shear flow transition, implying to the third fully mixed regime. These are based on Re = 2300. Regions marked 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 effects are also significant local to the leading displacement front, where they usually appear in the form of an inertial bump. However, in the bulk of the flow energy is dissipated by viscosity. The front velocity can be well predicted by (7.13). (d) Viscous temporary back flow regime: These flows are found in a regime bounded by (7.17) and Fr = V̂0/V̂t . 0.9, marked by by v2 in Fig. 7.10. As with regime i2 this regime is transitionary showing a pro- gressive change from exchange-dominated to imposed flow-dominated 160 7.2. Displacement in pipes as Fr is increased. The boundary of this regime with the exchange flow dominated regime occurs along (7.17), where stationary residual layers are found; see §6. This is again a viscous regime and the front velocity can be well predicted by (7.13). (e) Imposed flow dominated regime: When the imposed velocity is sufficiently strong, for either the inertial or viscous exchange flow domi- nated regimes, the flow transitions to a laminarised state dominated by viscous effects. For the inertial exchange flow, the stabilizing effect is seen on the whole flow while in the viscous exchange flow, the stabiliz- ing effect is observed through the spreading out the inertial bump at the front. The front velocities in this regime are predicted to leading order by the lubrication/thin film model. In Fig. 7.10 this regime is marked by v3. (f) Mixed/turbulent regime: We have not studied in detail this final transition, although we consider a model problem for a simpler channel geometry later in this chapter. For the low Atwood numbers that we have mostly studied, since the imposed flow regime involves a stratified stretching of the interface along the pipe, reducing buoyancy effects, we expect that this transition should be approximately the same as for the transitional flow of a single fluid in pipe. At larger At this is less clear. Once in this regime, the front velocity is approximately equal to the imposed flow velocity for our experiments, but at longer times we would expect that dispersion is active. 7.2.6 Engineering predictions and displacement efficiency Our findings can be expressed in terms of simple predictive models for the leading front velocity as follows. (a) For Re < 2300, if Ret cosβ & 50 and χ > χc the leading front velocity is predicted by (7.15): V̂f = V̂t[0.7 + 0.595Fr + 0.362Fr2]. (b) For Re < 2300, if Ret cosβ . 50 the leading front velocity is predicted by the lubrication model: V̂f = V̂0Vf (χ), where Fig. 7.7b shows Vf (χ). (c) Re ≥ 2300, we assume the flow has mixed across the pipe: V̂f = V̂0. These models capture the leading order behaviour (after any short-timescale transients) for the range of At we have studied and for density-unstable dis- placements. To graphically illustrate the above predictions as V̂0 is increased from zero, we present two experimental sequences in Figs. 7.11 & 7.12. 161 7.2. Displacement in pipes 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 V̂0 (mm/s) V̂ f (m m / s) χ = χc Fr = 0.9 2 layer lubrication model Figure 7.11: Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for β = 85 ◦, At = 3.5 × 10−3, ν = 1 (mm2.s−1), (for the pure exchange flow Ret cosβ ≈ 42). Sustained back flows and instantaneous dis- placements are marked by the superposed squares and circles respectively. Data points without marks are either temporary back flows, stationary in- terfaces or undetermined experiments (i.e. insufficient experiment time or short pipe length above the gate valve). The thin line shows the predic- tion of lubrication model. The thick vertical line shows the prediction of stationary interface from the same model. The thick vertical broken line shows the prediction of the transition between temporary back flow and in- stantaneous displacement, through V̂0/V̂t = 0.9. The insets are pictures of a 264 (mm) long section of tube a few centimeters below the gate valve in the corresponding flow domains. In Fig. 7.11 we are in the viscous regime initially. The lubrication ap- proximation is very good for χ > χc and only begins to diverge as the stationary interface regime is passed for increasing V̂0. Looking at the inset pictures below the critical stationary interface, it is clear that the imposed flow changes the shape of the inertial bump at the displacement front. After the critical stationary interface is attained, the inset figures show that the inertial bump is absent. The divergence of V̂f from the lubrication predic- tion as V̂0 increases suggests that inertial effects in the flow are becoming 162 7.2. Displacement in pipes important. We return to this in §7.4. 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 V̂0 (mm/s) V̂ f (m m / s) χ = χc Fr = 0.9 V̂f /V̂t = 0.7 + 0.595Fr + 0.362Fr 2 Figure 7.12: Variation of the front velocity V̂f as a function of mean flow velocity V̂0 for β = 83 ◦, At = 10−2, ν = 1 (mm2.s−1). Sustained back flows and instantaneous displacements are marked by the superposed squares and circles respectively. Data points without marks are either temporary back flows, stationary interfaces or undetermined experiments (i.e. insufficient experiment time or short pipe length above the gate valve). The thin line shows the prediction of (7.15). The thick vertical line shows the prediction of stationary interface from lubrication model. The thick vertical broken line shows the prediction of the transition between sustained back flow and temporary back flow, through V̂0/V̂t = 0.9. The insets are pictures of a 1325 (mm) long section of tube a few centimeters below the gate valve in the corresponding flow domains. Fig. 7.12 shows data from a sequence in which we are initially in the inertial exchange flow regime. The solid line now shows the approximation with (7.15), which is again very good for χ > χc. For χ ≤ χc we observe that the imposed flow gradually laminarises the flow. The model (7.15) is no longer a good approximation and the viscous lubrication approximation takes over. The main interest in front velocity V̂f is in giving a measure of the displacement efficiency of the flow. There is no universal definition for the 163 7.2. Displacement in pipes displacement efficiency, but one common notion is that it should represent the fraction of the tube that is filled with displacing fluid at a given time t̂. Suppose that at t̂ the front has advanced to position L̂, beyond the gate valve. Then we have that L̂ ≈ V̂f t̂ (assuming any initial effects are short-lived) and the volume of the pipe in which the displacement has taken place up until that time is L̂piD̂2/4 ≈ V̂f t̂piD̂2/4. The volume of displacing fluid within that length of pipe is simply equal to the volume that has been pumped up until that time: V̂0t̂piD̂2/4. Therefore, the fraction of displacing fluid in this volume is simply V̂0/V̂f . If we consider only flows dominated by the imposed flow, i.e. for which the interface is clean and well defined in our experimental images, we can make a secondary estimate of the displacement efficiency by directly inte- grating the volume of the pipe occupied by the displacing fluid. This can be compared with V̂0/V̂f for verification. To compute the displaced volume we wait until the interface has developed for a sufficient time and length below the gate valve. Spatially we typically take L̂ > 75D̂. Due to the different front speeds the (growing) integration length L̂ varies between experiments: the largest integration frame we have used had a length of 1540 (mm). To reduce effects of transients, we average the interface measurements from 3 images separated by time interval 0.5 (s). An example of the comparison between V̂0/V̂f and the computed dis- placement efficiency is shown in Fig. 7.13. There is good agreement be- tween the two measurements, due essentially to the high resolution of our imaging. The efficiency increases with V̂0 to a plateau, signified by the linear viscous imposed flow dominated displacement. This primarily demonstrated that for clean interfaces we can measure consistently using either method. When there is a degree of mixing between the fluids the volume integration will clearly be vulnerable, but provided we can estimate V̂f the volume- displacement argument still holds and V̂0/V̂f gives the efficiency. 7.2.7 Dispersive effects It has been noted that in the imposed flow dominated regime there is a con- sistent discrepancy between lubrication model prediction and experimental results. A linear relation is observed between V̂f and V̂0 but the dimension- less Vf (χ) of Fig. 7.7b underestimates the slope by a few percent. Since this discrepancy increases with V̂0 one possible explanation would be to relate this to inertial effects, and we consider these later in §7.4. Instead we outline another possible cause of the discrepancy. In deriving the lubrication equation (7.7) and solving numerically we 164 7.2. Displacement in pipes 0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 V̂0 (mm/s) V̂ 0 / V̂ f , D is p la ce m en t E ffi ci en cy Figure 7.13: Comparison between the ratio V̂0/V̂f and the value of the displacement efficiency computed through the interface integral for At = 3.5 × 10−3 and ν̂ = 1 (mm2.s−1). The data points computed through the ratio V̂0/V̂f are shown by filled symbols: β = 87 ◦ (•), β = 85 ◦ (H); com- puted integrals are shown by hollow symbols: β = 87 ◦ (4), β = 85 ◦ (¤). Lubrication model prediction of the imposed velocity for the transition to imposed flow dominated regime is 11 (mm.s−1) for β = 87 ◦ and 19 (mm.s−1) for β = 85 ◦. The displacement efficiency is not defined for the data points with the imposed flow below these critical values. have assumed that the interface height is independent of the transverse co- ordinate z, across the channel perpendicular to gravity. The assumption h = h(ξ, T ) and subsequent computation is self-consistent mathematically with (7.7), but ignores whether or not an interface that is initially inde- pendent of z will remain so. Behind the evolution equation (7.7) is the kinematic equation, which states that the interface simply advects wi