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 2011Abstract This thesis studies buoyant displacement ows with two miscible uids in pipes and 2D channels that are inclined at angles (fl) 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 P¶eclet numbers, so that miscibility efiects 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 ow velocity allows us to identify 3 distinct ow regimes: an exchange ow dominated regime characterized by Kelvin-Helmholtz-like instabilities, a laminarised viscous displacement regime with the front velocity linearly increasing with the mean imposed ow rate, and a fully mixed displacement regime. The transition between the flrst and the second regimes is found to be marked by a stationary layer of displaced uid. In the stationary layer the displaced uid moves in counter- current motion with zero net volumetric ux. Difierent lubrication/thin- fllm models have been used to predict the ow 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 deflne 5-6 distinct ow regimes, which we show collapse onto regions in the two-dimensional (Fr, Re cos fl=Fr)-plane. Here Fr is the densimetric Froude number and Re the Reynolds number. In each regime we have been able to ofier a leading order approximation to the leading front velocity. A weighted residual method has also been used to include the efiect of inertia within the lubrication modelling approach, which allows us to predict long-wave instabilities. We have extended the study to include the efiects of moderate viscosity ratio and shear-thinning uids. 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 signiflcant yield stress in the displaced uid leads to completely new phenomena. We identify two distinct ow regimes: a central-type dis- placement regime and a slump-type regime for higher density difierences. In iiAbstract both regimes, the displaced uid can remain completely static in residual wall layers. iiiPreface In this preface, we brie y 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 ows 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 efiects 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 uids 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. In uence of an imposed ow 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 in uence of a mean ow superimposed to a classical lock-exchange ow in a nearly horizontal pipe. Three ow regimes are found: (i) for very low imposed ows, the previous lock-exchange results are recovered; (ii) for intermediate imposed ows, a laminarised regime was found; (iii) for very high imposed ows 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. ivPreface † S.M. Taghavi, T. Seon, K. Wielage-Burchard, D.M. Martinez and I.A. Frigaard. Stationary residual layers in buoyant Newtonian dis- placement ows. Phys. Fluids 23 044105 (2011). This work deals with the displacement of two miscible uids of dif- ferent densities in a tilted duct (i.e. pipe and plane channel) with the two uids initially in a gravitationally unstable conflguration. In this work we study the the transition between the flrst and second regimes (discussed in the previous paper) which is controlled by either the buoyant interpenetration or the imposed ow. We observed that, for some ow rates, the interface between the two uids is stationary, indicating a zero net ow of the displaced uid. 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 ows in near-horizontal ducts at low Atwood number. Submitted for publication. In this extensive study we consider buoyant displacement ows with two miscible uids 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 ows: Fr (densimet- ric Froude number), Re (Reynolds number) and fl (duct inclination). We demonstrate that the ow regimes in fact collapse into regions in a two-dimensional (Fr; Re cos fl=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 uid- uid displacement of yield stress uids in near-horizontal pipes: experiments and theory. Accepted for publication in J. Non-Newton. Fluid Mech. vPreface The paper is a primarily experimental study of displacement of a yield stress uid 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 uid. The main flnding 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 flnite 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 ows 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 ows at moderate viscosity ratios in near-horizontal pipes and plane channels. We show that small viscosity ratios lead to more e–cient displacements, as is intuitive. In each geometry we flnd a mix of viscous and inertial ows, 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 ow regime, in both geometries. We also study displacement ows with shear-thinning u- ids, over a more restrictive range of parameters. We show that with an appropriate deflnition of the efiective viscosity the scaled front ve- locities flt 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 uid experi- ments. I.A. Frigaard supervised the research. viTable 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 Efiects of Rheology . . . . . . . . . . . . . . . . . . . 37 viiTable 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 difierent regimes . . . . . . . . . . . . . . . 69 4.2 Stabilizing efiect of the imposed ow . . . . . . . . . . . . . 71 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Lubrication model approach for channel displacements . . 76 5.1 Two- uid displacement ows in a nearly horizontal slot . . . 77 5.1.1 Constitutive laws . . . . . . . . . . . . . . . . . . . . 79 5.1.2 Buoyancy dominated ows . . . . . . . . . . . . . . . 80 5.1.3 The ux function q(h; h») . . . . . . . . . . . . . . . . 84 5.1.4 The existence of steady traveling wave displacements 86 5.2 Newtonian uids . . . . . . . . . . . . . . . . . . . . . . . . . 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 uids . . . . . . . . . . . . . . . . . . . . . . 100 5.3.1 Shear-thinning efiects . . . . . . . . . . . . . . . . . . 100 5.3.2 Yield stress efiects . . . . . . . . . . . . . . . . . . . . 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 viiiTable 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 ows . . . . . . . . . . . . 142 7.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.1.1 Viscous and inertial ows . . . . . . . . . . . . . . . . 144 7.1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2 Displacement in pipes . . . . . . . . . . . . . . . . . . . . . . 145 7.2.1 Basic ow regimes observed . . . . . . . . . . . . . . 145 7.2.2 Lubrication/thin fllm model . . . . . . . . . . . . . . 153 7.2.3 Comparison of experimental results and the lubrica- tion model . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2.4 The exchange- ow dominated range . . . . . . . . . . 156 7.2.5 Overall classiflcation of the ow regimes . . . . . . . 157 7.2.6 Engineering predictions and displacement e–ciency . 161 7.2.7 Dispersive efiects . . . . . . . . . . . . . . . . . . . . 164 7.3 Displacement in channels . . . . . . . . . . . . . . . . . . . . 166 7.3.1 Exchange ow results . . . . . . . . . . . . . . . . . . 167 7.3.2 Displacement ow results . . . . . . . . . . . . . . . . 170 7.3.3 Quantitative prediction of the front velocity . . . . . 176 7.3.4 Overall ow classiflcations . . . . . . . . . . . . . . . 181 7.4 Inertial efiects on plane channel displacements . . . . . . . . 187 7.4.1 A weighted residual lubrication model . . . . . . . . . 187 7.4.2 Inertial efiects on front shape and speed . . . . . . . 190 7.4.3 Flow stability . . . . . . . . . . . . . . . . . . . . . . 192 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8 Efiects 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 ows . . . . . . . . . . . 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 ixTable of Contents 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9 Efiects of yield stress . . . . . . . . . . . . . . . . . . . . . . . 228 9.1 Scope of the study . . . . . . . . . . . . . . . . . . . . . . . . 228 9.2 Selection of uids . . . . . . . . . . . . . . . . . . . . . . . . 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 ow computations . . . . . . . . . . . . . . . . 239 9.3.4 Slump-type displacements . . . . . . . . . . . . . . . 242 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 10 Conclusions and perspectives . . . . . . . . . . . . . . . . . . 255 10.1 Dynamics of the ow . . . . . . . . . . . . . . . . . . . . . . 255 10.1.1 Flow regimes . . . . . . . . . . . . . . . . . . . . . . . 255 10.1.2 Efiects of viscosity ratio and shear-thinning . . . . . . 258 10.1.3 Efiects 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 ows . . . . . . 266 10.3.4 3D numerical simulations . . . . . . . . . . . . . . . . 267 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Appendices A Computing the ux function q(h; h») . . . . . . . . . . . . . . 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 coe–cients R1...R5 . . . . . . . . . . . . . . . . . . . . . . 288 xList of Tables 2.1 Typical ranges of uid properties and ow 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 fl = 85 –. . . . . . . . . . . . . . . . . . . . . . . . . 202 8.3 Numerical simulation parameters for Newtonian displacements performed for fl = 83; 85; 87 & 89 – and At = 3:5£ 10¡3. . . 216 8.4 Numerical simulation parameters for shear-thinning displace- ments performed for fl = 85 – and At = 3:5£ 10¡3. . . . . . . 216 9.1 Composition and properties of the displaced uid used in our experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 xiList of Figures 1.1 Schematic of displacement geometry . . . . . . . . . . . . . . 2 2.1 Schematic of a simplifled 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 proflle 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 ux 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 V^f=V^t as a function of V^” cos fl=V^t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.10 Images of the concentration and swirling strength . . . . . . . 35 2.11 Schematic of the difierent possible characteristic axial velocity proflles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.12 A typical interface evolution Yi . . . . . . . . . . . . . . . . . 42 2.13 Schematic illustration of the two types of streamline behavior in displaced uid . . . . . . . . . . . . . . . . . . . . . . . . . 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 proflles of normalized interface height, h(x^; t^) . 56 3.6 Variation of the efiective viscosity ·^ with shear rate _^ . . . . 60 3.7 Example owcurve for a visco-plastic solution . . . . . . . . . 61 3.8 Schematic of the computational domain . . . . . . . . . . . . 63 xiiList of Figures 3.9 Computational concentration fleld evolution obtained for fl = 85 –, At = 3:5£ 10¡3, ”^ = 1 (mm2.s¡1), V^0 = 15:8 (mm.s¡1), (Re = 300) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.10 Spatiotemporal diagram of the average concentration . . . . . 66 4.1 Variation of the front velocity V^f as a function of mean ow velocity V^0 for fl = 83 –, At = 10¡2, „^ = 10¡3 (Pa.s) . . . . . 70 4.2 Three snapshots of video images taken for difierent mean ow and showing the ow 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 efiect of the imposed ow 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 uids . . . . . . . . . . . . . . 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 uid HL and Parameter regime in the (m;´)-plane . . . . . . . . . . . . . . . . . . . . . . . . 95 5.7 Examples of front shapes in the moving frame of reference for a HL displacement . . . . . . . . . . . . . . . . . . . . . . . . 96 5.8 Proflles of h(»; T ) for T = 0; 1; ::; 9; 10, with parameters ´ = 50, m = 0:1, illustrating ow reversal . . . . . . . . . . . 97 5.9 The similarity solution and comparison with the numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.10 Examples of HL displacements for 2 power law uids, Bk = 0, ´ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.11 Front heights and velocities, plotted against m for a HL dis- placement of 2 power law uids . . . . . . . . . . . . . . . . . 102 5.12 Proflles of h plotted against »=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 uid displaces a Herschel-Bulkley uid . . . . . . . . . . . . . 109 5.17 An example of sudden movement of static layer . . . . . . . . 110 xiiiList of Figures 6.1 Sequence of images showing the stationary upper layer . . . . 114 6.2 Four snapshots of video images taken at difierent mean ow rates and illustrating the difierent regimes . . . . . . . . . . . 115 6.3 Spatiotemporal diagrams of the variation of the light . . . . . 117 6.4 Ultrasonic Doppler Velocimeters proflles . . . . . . . . . . . . 118 6.5 Schematic views of the distribution of the two uids . . . . . 119 6.6 Contours of q(h; 0) and the contour @q@h(h; 0) = 0 (bold black line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.7 Proflles of h(»; T ) for T = 0; 1; ::; 9; 10, with ´ = ´c . . . . . 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 fleld evolution obtained for fl = 87 –; ”^ = 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 proflles corresponding to Fig. 6.10 for a channel ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.13 The velocity proflle close to the pinned point (with the axial position x^=D^ = 26:25) . . . . . . . . . . . . . . . . . . . . . . 132 6.14 Four possible conditions for a viscous buoyant channel ow . 133 6.15 Classiflcation of our simulation results in a channel . . . . . . 134 6.16 Schematic variation of the velocity and V^0=(V^” cos fl) plotted versus (D^=X^bff ) tan fl for 2 series of experiments . . . . . . . . 136 7.1 Sequence of images showing propagation of waves along the interface for V^0 = 40 (mm.s¡1) . . . . . . . . . . . . . . . . . 147 7.2 Contours of axial velocity . . . . . . . . . . . . . . . . . . . . 148 7.3 Variation of the front velocity V^f as a function of mean ow velocity V^0 for difierent values of density contrast and viscos- ity at two inclination angles . . . . . . . . . . . . . . . . . . . 149 7.4 Variation of the front velocity V^f as a function of mean ow velocity V^0 for difierent values of density contrast and viscos- ity at fl = 85 – . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.5 A sequence of snapshots from experiments with increased im- posed ow rate . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.6 Examples of spatiotemporal diagrams and corresponding UDV measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xivList of Figures 7.7 Numerical examples of pipe ow displacements based on the lubrication model and variation of the front speeds (solid line) and heights (broken line) . . . . . . . . . . . . . . . . . . . . 155 7.8 Normalized front velocity, V^f=V^” cos fl, plotted against nor- malized mean ow velocity, V^0=V^” cos fl, for the full range of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.9 Normalized front velocity as a function of normalized mean ow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.10 Classiflcation of our results for the full range of experiments in the flrst and second regimes . . . . . . . . . . . . . . . . . 160 7.11 Variation of the front velocity V^f as a function of mean ow velocity V^0 for fl = 85 –, At = 3:5£ 10¡3, ” = 1 (mm2.s¡1) . . 162 7.12 Variation of the front velocity V^f as a function of mean ow velocity V^0 for fl = 83 –, At = 10¡2, ” = 1 (mm2.s¡1) . . . . . 163 7.13 Comparison between the ratio V^0=V^f and the value of the displacement e–ciency . . . . . . . . . . . . . . . . . . . . . . 165 7.14 Variation of the normalised stationary front velocity V^f=V^t as a function of the inertial Reynolds number Ret cos fl = V^” cos fl=V^t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.15 Variation of the downstream front velocity V^f as a function of mean ow velocity V^0 for difierent inclination angles . . . . 169 7.16 Sequence of concentration fleld evolution obtained for fl = 87 –, At = 10¡3, ”^ = 1 (mm2.s¡1), V^0 = 26:3 (mm.s¡1) . . . . 171 7.17 Panorama of concentration colourmaps for displacements with ” = 1 (mm2.s¡1), each taken at t^ = 25 (s) . . . . . . . . . . . 172 7.18 Panorama of velocity proflles . . . . . . . . . . . . . . . . . . 173 7.19 Sequence of concentration fleld evolution obtained for fl = 87 –, ”^ = 1 (mm2.s¡1), each taken at t^ = 25 (s) . . . . . . . . 174 7.20 Sequence of concentration fleld evolution obtained for At = 3:5£ 10¡3, ”^ = 1 (mm2.s¡1) and V^0 = 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 ´ = 10 in a h¡yi map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.23 Normalized front velocity, V^f=V^” cos fl, as a function of nor- malized mean ow velocity, V^0=V^” cos fl, for the full range of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.24 Normalized front velocity as a function of normalized mean ow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 xvList of Figures 7.25 Classiflcation of our results for the full range of simulations in the flrst and second regimes . . . . . . . . . . . . . . . . . 183 7.26 Front velocity V^f as a function of mean ow velocity V^0 for a viscous regime displacement . . . . . . . . . . . . . . . . . . . 185 7.27 Variation of the front velocity V^f as a function of V^0 for a sequence of inertial regime displacements . . . . . . . . . . . 186 7.28 Front velocity and shape in uences at ´ = 0 . . . . . . . . . . 191 7.29 Experimental proflles of normalized h(x^; t^) . . . . . . . . . . . 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 ows (⁄) and unstable ows (†) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.1 Experimental results for Newtonian displacements; variation of front velocity V^f as a function of mean ow velocity V^0 for At = 10¡2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Experimental results for Newtonian displacements; variation of front velocity V^f as a function of mean ow velocity V^0 for At = 10¡3 at fl = 85 – . . . . . . . . . . . . . . . . . . . . . . 204 8.3 Experimental results for Newtonian displacements: contours of front velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.4 Normalized front velocity, V^f=V^” cos fl, plotted against nor- malized mean ow velocity, V^0=V^” cos fl . . . . . . . . . . . . . 206 8.5 Values of normalized front velocity, V^f=V^” cos fl, plotted in a plane of viscosity ratio m versus normalized mean ow veloc- ity, V^0=V^” cos fl . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.6 Normalized front velocity, V^f=V^t, as a function of normalized mean ow velocity V^0=V^t = Fr . . . . . . . . . . . . . . . . . 208 8.7 Schematic of general behavior in displacements in which one of the uids is shear-thinning . . . . . . . . . . . . . . . . . . 210 8.8 Experimental results for shear-thinning displacements; varia- tion of front velocity V^f as a function of mean ow velocity V^0 at fl = 85 – . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8.9 Variation of front velocity V^f as a function of mean ow ve- locity V^0 for At = 3:5£ 10¡3 at fl = 85 – . . . . . . . . . . . . 212 8.10 Normalized front velocity, V^f=V^” cos fl, plotted against nor- malized mean ow velocity, V^0=V^” cos fl . . . . . . . . . . . . . 213 xviList of Figures 8.11 Values of normalized front velocity, V^f=V^” cos fl, plotted in a plane of viscosity ratio m versus normalized mean ow veloc- ity, V^0=V^” cos fl . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.12 Experimental spatiotemporal diagrams obtained to illustrate stabilizing and destabilizing efiect of the imposed ow . . . . 217 8.13 Panorama of concentration colourmaps and velocity proflles for displacements with viscosity ratio greater than 1 . . . . . 219 8.14 Panorama of concentration colourmaps and velocity proflles for displacements with viscosity ratio less than 1 . . . . . . . 220 8.15 Simulation results for Newtonian displacements; variation of front velocity V^f as a function of mean ow velocity V^0 . . . . 221 8.16 Comparison between the critical value of ´ . . . . . . . . . . 222 8.17 Normalized front velocity, V^f=V^” cos fl, plotted against nor- malized mean ow velocity, V^0=V^” cos fl . . . . . . . . . . . . . 223 8.18 Normalized front velocity, V^f=V^” cos fl = 2Vf=´, from our nu- merical experiments for all viscosity ratio simulations . . . . . 224 8.19 Simulation results for shear-thinning displacements; variation of front velocity V^f as a function of mean ow velocity V^0 . . 225 8.20 The critical value of ´c predicted by the lubrication model at long times for m = 1 . . . . . . . . . . . . . . . . . . . . . . . 225 8.21 Normalized front velocity, V^f=V^” cos fl = 2Vf=´ . . . . . . . . 226 9.1 Classiflcation of our experiments . . . . . . . . . . . . . . . . 232 9.2 Central displacement for fl = 83 –, At = 3 £ 10¡3, V^0 = 32 (mm.s¡1) with Carbopol solution A . . . . . . . . . . . . . 234 9.3 Variation of a) „C(y^) and b) „C(x^) in the rectangular region . . 235 9.4 Wavelength content (power spectrum) of „C(x^) versus inverse wavelength 1=⁄^ and reconstruction of interface through in- verse of DFFT . . . . . . . . . . . . . . . . . . . . . . . . . . 236 9.5 Central displacement for fl = 85 –, At = 4 £ 10¡3 and V^0 = 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 ¡ ‚i;min), in the BN -`B 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 V^f with V^0 for a sequence of experiments with Carbopol solution C . . . . . . . . . . . . 242 9.10 Displacement of Carbopol C for fl = 85 –, At = 10¡2 at V^0 = 26 (mm.s¡1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 xviiList of Figures 9.11 Displacement of Carbopol solution C for fl = 85 –, At = 10¡2: a) & b) show data for V^0 = 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 ow rate owing in the lower layer . . . . . . . . . . . . . . . . . . 249 9.16 Unsteady slump-like displacement for fl = 85 –, At = 10¡2, V^0 = 36 (mm.s¡1) with Carbopol solution C . . . . . . . . . . 250 9.17 Velocity proflles, 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 xviiiAcknowledgements 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 scientiflc 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 e–cient 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 Oilfleld 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, xixAcknowledgements 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 flnancially 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 confldence in me throughout these years. xxDedication This dissertation is afiectionately 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! xxiChapter 1 Introduction 1.1 Problem of study This thesis investigates the forced displacement of one miscible uid by an- other of difierent density, initially placed in a density unstable conflguration in near-horizontal ducts (pipes and plane channels). Although buoyancy is a signiflcant driving force for all ows we study, there is also a net imposed ow in the downward direction, along the duct. We study the efiects of con- stant imposed ows (with mean velocity V^0), small density ratios (quantifled by the Atwood number, At), inclination angles (fl), viscosity ratios between the two uids (m), and the rheology of the uids. The diameter or width (D^) of the duct is small compared to its length (L^). The inclination angle fl remains close to horizontal where we expect to flnd more viscous ows. The uids used in this study are generalized Newtonian uids which include Newtonian uids, shear thinning uids with a power-law index n and shear thinning uids with a yield stress ¿^Y . 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 ows 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 ows often occur in these processes, due to either high viscosities or other process constraints. Non-Newtonian uids are also prevalent. In many situations it is not feasible to physically separate uid stages as they are pumped and two practical questions are: (i) to what degree does the uid mix across the duct; (ii) what is the axial extent along the duct of the mixed region (meaning that in which we flnd both uids present)? 1In this thesis, we adopt the convention of denoting dimensional quantities with ^ symbol (e.g. the pipe diameter is D^) and dimensionless quantities without. 11.1. Problem of study β Dˆ 0 ˆV Interface Light Heavy Figure 1.1: Schematic of displacement geometry: a heavy uid displaces a light uid in a slightly inclined duct with transverse dimension D^. Direction of the imposed ow, with mean velocity V^0, is depicted by the arrow. The interface shape between the two uids is illustrative only. The mixed region in (ii) could consist either of distinct uid 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 ows buoyancy acts to spread the uids along the duct, i.e. the mixed region is longer than for density stable ows, 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 uid ow problems are com- plex and have not been understood in depth. However in the literature there exist many valuable studies classifying difierent efiects in particular ows, e.g. the efiects of small density ratios and inclination angles in the absence of an imposed ow. In addition, there are many works related to displacement ows with viscosity ratios and density difierences in vertical ducts and in the absence of inertia. One can also flnd scattered studies on the in uence of rheological parameters. However, buoyant displacement ows in inclined ducts at non-zero Reynolds numbers are not at all comprehensively studied. Based on the studies of miscible exchange ows, we expect distinctly difierent ows 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 ows (At = ‰^Heavy¡‰^Light‰^Heavy+‰^Light ), but with signiflcant buoyancy forces, and study the efiects of gradually increasing 21.1. Problem of study the imposed ow rate. As well as buoyancy, inertia and (bulk) viscous forces, we study the efiects of difierent rheological properties in the 2 uids. 1.1.1 Fundamental interest and applications Industrial displacement ows often involve both density and rheological difierences between uids. With buoyancy, there are a number of dis- placement studies in vertical ducts, both for miscible and immiscible u- ids [84, 91, 92, 126], but here we focus exclusively on near-horizontal inclina- tions which are phenomenologically difierent, in that near-stratifled 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 uid with another or with a sequence of difier- ent uids. The geometries are typically pipe, annular or duct-like, all with long aspect ratios. Large volumes are pumped so that uids may be consid- ered separated, i.e. we have a 2- uid displacement, not an n- uid displace- ment. A very wide range of uids are used. Signiflcant density difierences 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 efiects, (more exotic non-Newtonian efiects may also be present). Many difierent types of industrial displacement ows arise. Turbulent displacing regimes are typically more efiective, but are not always possible due to pro- cess constraints; here our ows are laminar. A second distinction comes in the volume of displacing uid 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 uids (e.g. cement slurry), it is desirable to fully displace the in-situ uid (e.g. drilling mud) with more or less a single \duct volume" of uid, i.e. we are replacing the in-situ uid 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, uid 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 31.2. 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 ows in close to horizontal ducts are compli- cated to analyze. The problem complexities include the efiects of a large number of ow parameters, as well as difierent conflgurations and non- Newtonian behaviors. Depending on contributions of these parameters, dif- ferent types of displacement and/or mixing ow can occur, and depending on balances among participating forces, difierent ow regimes (e.g. iner- tial, viscous) are possible. Thus, it is hard to predict the degree of mixing between two uids and accurately design uid volumes needed, uid proper- ties, and ow rates. In addition, low e–ciency displacement ows can lead to contamination of the uids. This can have a signiflcant 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 ows. 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 (x2) reviews related important papers found in the engineering and scientiflc literature. At the end of this chapter, we highlight the deflciencies 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 difierent data and extract desirable information from them. In Chapter 4 we present the pre- liminary results of our experimental approach. We qualitatively discuss the efiect of imposed ow, tilt angle and density ratio for Newtonian miscible displacement ows. Brie y, in this chapter we deflne 3 difierent ow regimes. This chapter builds the foundation for the following chapters by providing basic deflnitions of these difierent ow regimes observed. All the chapters following x4 are focused on variant aspects of the difierent regimes explained in this chapter. Chapter 5 is devoted to our simple mathematical analysis, 41.2. Thesis outline which analytically studies in depth the second regime. Chapter 6 discusses the transition between the flrst and second regimes. This transition is as- sociated with an interesting feature of the ow, whereby the displaced uid layer remains stationary within the pipe (or channel) and the displacing uid 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 brie y comment on the stability of the ow and its transition. In Chapter 8 we observe the efiect of viscosity ratios and shear-thinning especially on the second regime and partially on the flrst regime. Chapter 9 characterizes the efiect of a yield stress on the displacement ows. This thesis concludes in Chapter 10. 5Chapter 2 Background The present chapter is organized as follows. Firstly, we start with describing the related engineering background of our problem in x2.1. 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 flrst section by a summary of the engineering background and discuss the main shortcom- ings. In the second section of this chapter (x2.2), we introduce associated fundamental studies that can pave the road for better understanding our ow problem. Describing these studies is guided by three main goals: (i) What are the limiting cases of our displacement ows? (ii) Can previous research help identify basic mechanisms that contribute towards explaining more complicated features in our speciflc problem? (iii) What aspects of buoyant displacement ows 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 ow knowledge that are lacking in the current literature. Section x2.3 brie y concludes the chapter. Section x2.4 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 uids from leaking to surface, and lowers the risk of severe environmental and safety consequences. 62.1. 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 flve years, around 63;250 oil and gas wells have been completed in Canada [1]. Oil and gas industry are also of signiflcant economical importance to Canada. For ex- ample, in 2009, the net cash expenditure of the petroleum industry was around $46.2 billion. Signiflcant contributions are also made to federal and provincial taxes by the industry. The world’s 3 largest oilfleld services companies, which all have branches in Canada, are Schlumberger, Halliburton and Saipem. Schlumberger is one of the most technologically focused oilfleld 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 difierent 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 oilfleld services for a number of challenging projects both on and ofishore. 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 ofier 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, 72.1. Primary cementing background suspended wells can still have uids that percolate through the near-surface strata and adversely afiect 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 Tangle ags 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 uid mechanic origins and this is a major motivation for further research into the physics of uid- uid 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 uids 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 solidifles (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 uid (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 uid volumes are designed so that the cement slurries flll the annular space to be cemented. Drilling mud follows the flnal 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 flnal part of cement inside the casing is drilled out as the well 82.1. Primary cementing background Drilling Mud Wash Cement Slurry (a) (b) (c) (d) Figure 2.1: Schematic of a simplifled primary cementing process in an ide- alized case where no mixing occurs between successive uid 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. 92.1. Primary cementing background Q^ (l/min) ‰^ (kg/m3) •^ (Pa.sn) n ¿^Y (Pa) 300¡ 3000 900¡ 2200 0:003¡ 3 0:1¡ 1 0¡ 20 Table 2.1: Typical ranges of uid properties and ow parameters in primary cementing. Q^; ‰^; •^; n; ¿^Y respectively denote ow 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 uids are typically 100 ¡ 600 (kg/m3) lighter than cement slurries. Drilling uids 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 uid properties and ow 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 fl can be essentially anything. The Atwood number, At, can increase up to 0:5. The Reynolds number, is al- ways signiflcant, O(10) and larger. Flows are both turbulent and laminar. The Reynolds number quantifles the importance of inertial efiects 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 ows 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 ows of interest. Hence considering inelastic uids is 102.1. Primary cementing background fl – At = ‰^Heavy¡‰^Light‰^Heavy+‰^Light Re · ‰^V^0D^ „^ Fr · V^0pAtg^D^ 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 „^. viscosity ratio between the uids (m), ratio of advective to difiusive mass transport (represented by P¶eclet 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 ows (note that in considering a dis- placement we must include rheological parameters in a minimum of 2 uids). 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 ows 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 uid 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 uids. 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 uid 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 uid 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 uids. Thus, unless a mechanical plug is used it is not practically possible reasonable, from the perspective of modelling. 112.1. Primary cementing background to totally prevent mixing at the interface between two adjacent uid stages that are circulated down the pipe (at initial stage) and up the annulus (at flnal stage). Mixing or by-passing of uid 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 efiects 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 afiect 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 difierent suppliers, perhaps storing in imperfect conditions prior to transporting to the rigsite. Difierent mixing conditions are also possible due to human error in the execution. All these factors combine to mean that the ac- tual uid properties at the rigsite might vary considerably from either design values or test values in a fleld lab. † Process design is based on volumes required to flll 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 difierent 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 uid temperature-dependent properties may be difierent from design values. 122.1. Primary cementing background † Properties of the uid 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 flne 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 efiects can occur, see Ravi et al. [118, 119] for details. 2.1.4 Studies of primary cementing displacement It should be emphasized that the flnal aim of the primary cementing is to replace mud around the casing in the annulus conflguration 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 flrst research on cement placement process dates back 70¡80 years, when some basic key factors afiecting primary cement job failures were rec- ognized. For instance in an early work, Jones and Berdine [83] used a large-scale simulator to propose efiective ways to displace mud in the an- nulus including uid 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 ow regime of the displacing uid can afiect the mud displacement e–ciency3. They claimed that higher Reynolds number of the displacing uid with the ow 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 deflning the ability of a given uid to displace another. There is no universal deflnition of this parameter but one common measure is given in [103]. Assume our duct (or annulus) is flled with a displaced uid initially at rest at t = 0. When the displacement process starts, the displacing uid enters the duct. At any time t > 0 during the process, displacement e–ciency can be deflned as the fraction of duct (or annulus) volume occupied by the displacing uid. In this chapter, we introduce displacement e–ciency only to provide a rough idea of how successful a displacement process is. The deflnition of displacement e–ciency that we use is given in Chapter 5, where we relate displacement e–ciency to the front velocity of the displacing uid. 132.1. Primary cementing background hydraulic reasoning. In general, these rules set state that there should be a hierarchy of the uid rheologies pumped, (i.e. each uid should generate a higher frictional pressure than its predecessor), and that there should be a hierarchy of the uid densities, (each uid 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 ow 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 ow rates, the upward displacement of a dense uid by a lighter one leads to an unstable phenomenon known as buoyant plume (the same phenomenon is observed for downward displacement of a light uid by a dense uid). In contrast, for the upward displacement of a heavier displacing uid, buoyancy forces have a tendency to atten the interface and stimulate e–cient displacement. † Everything else being equal, if a thick uid displaces a thin one in the laminar ow, the displacement e–ciency 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 efiect of density and rheology. In another study, Flumerfelt [50] presented an approximate solution for the displacement of a shear-thinning uid by another in laminar ow. 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 ow rate is not too large. † Displacement e–ciency increases with increasing efiective viscosity ra- tios but the sensitivity to this parameter is not as important as to the 142.1. Primary cementing background density ratio. The displacement of a more viscous uid by another one (with usually the same density) leads to viscous flngering, where the less viscous uid penetrates into the more viscous one. This ef- fect results in a bad displacement. Thus, the opposite case of more viscous uid displacing a less viscous one is likely to present higher displacement e–ciency. † For some shear-thinning uids, better displacement e–ciencies are ob- tained when the power-law index of the displacing uid is lower than that of the displaced uid. † Yield stresses, specially when present in the displaced uid, are very critical. Better displacement is usually achieved when dimensionless yield stress values of the displacing uid exceeds that of the displaced uid. † Reducing mud density and viscosity will probably always result in improved e–ciency. Zuiderwijk [168] used a power-law model and performed a large number of high e–ciency 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 ow. The flrst analysis of nar- row eccentric annular ows of visco-plastic uids was carried out by Walton & Bittleston [157] and Szabo & Hassager [142] but only for ows of a sin- gle uid 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 152.1. Primary cementing background axisymmetric ows in pipes. The main reason why there is little literature for downward displacement ows 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 uid 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 uids. The majority of the industry still de- signs cement jobs using 1D simulations that do not currently consider ow parameters and uid rheology beyond calculation of frictional pressures and uid volumes. Here we review descriptions of more generic and sophisticated simulators (from oil well services companies) which take into account dif- ferent ow parameters and in general deal with cement/mud displacement processes. Halliburton has developed Displace 3DTM simulator which uses ad- vanced computational uid dynamics and it is claimed to dynamically model multiple aspects of displacement of wellbore uids 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 uid contamination up to some degrees. Calculating the mixing interface lengths and the top of cement locations is another interesting feature. For the uids, the modeling approach used is a generalized Herschel Bulkley model, which can somehow safely reduce the uid 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 uid dynamics physics and, with some details, monitors the process of cement placement. The goal is to have a prediction of the e–ciency of mud removal. Difierent features include careful consideration 162.1. Primary cementing background of well geometry, inclination from vertical to horizontal, interface trajectory, uid properties, volumes, pump rates and casing centralization. The other feature is simulation of uid placement in both laminar ow and turbu- lent ows to produce maps of uid velocity and ow regimes. Rheological description of uids is expressed through a Herschel-Bulkley model. Trican has designed a simulator (Cement Simulator) to predict pressures and ow 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 uid ow 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 uids can have. This may strongly afiect 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 di–culty of model validation against fleld data. Most of the time, the uid 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 fleld 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 u- ids. (iv) The majority of industry literature concerns the annular displacement ow, but this will be impractical if the uids 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 172.2. Associated fundamental problems rule states that heavy uids displace better in the upward direction in the annulus. However, ensuring a stable density difierence 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 ow 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 deflne some sub- set of typical parameters to be the focus. In particular we will focus mainly on laminar ows. (vi) The laminar ows of importance contain all of: (a) signiflcant buoyancy; (b) signiflcant inertia; (c) difierent inclinations; (d) viscous efiects; (e) interesting rheologies on some of the uids. 2.2 Associated fundamental problems In this section we review the scientiflc in a number of areas that are closely related to our problem. This helps to frame the fundamental mechanisms present in our ow problem. With some generality, although our ow 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 flrst review those previous studies that appear particularly important for our problem. We will then comment on where there are signiflcant deflciencies in the litera- ture and our knowledge. More speciflcally we consider the following: † Laminar ows in the processes that we study typically have high P¶eclet numbers (Pe 1) and long aspect ratios (– ¿ 1), but commonly –Pe & 1. In the absence of instability and dispersive mixing, these ows exhibit sharp interfaces, qualitatively similar to immiscible dis- placements. We present an overview of high Pe regimes in x2.2.1. 182.2. Associated fundamental problems † Our displacement ows are naturally vulnerable to interfacial instabili- ties. We review the most relevant mechanisms in x2.2.2. On increasing the imposed ow, for very large V^0, we logically expect the ow to ex- perience a transition and flnally to fall into a turbulent regime. We comment on this at the end of x2.2.2. † An alternative way of viewing the background to our problem is as a variant of a conflned gravity currents. When V^0 ! 0, we inevitably expect to recover the results of a conflned gravity current. A detailed review of the most signiflcant experimental, analytical and computa- tional studies of gravity currents is given in x2.2.3. † A limit arising in high Pe ows 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 x2.2.4. † As previously stated, studies on the efiect of rheology on displacement ows are somewhat scattered and less deep. In x2.2.5, we will pro- vide short descriptions of most related works considering rheological parameters. 2.2.1 High Pe miscible displacements Displacement of one uid with another can be regarded as an archetypical ow, occurring in many industrial settings, which is made more complex to understand when there are density difierences between the uids. Many practical processing situations involving aqueous liquids in laminar duct ows with diameters D^ » 10¡2 (m) and mean velocities V^0 . 0:1 (m/s) necessarily fall in to the category of high Pe ows, conservatively in the range 103¡107. For such ows the laminar Taylor-dispersion regime [6, 144] (explained in x2.2.4) is strictly found only for duct lengths L^ D^Pe, which are arguably less common in processing geometries, even though D^=L^ ¿ 1 is usual. Thus, in an industrial setting probably the most relevant laminar regime is the non-dispersive high P¶eclet number regime, where the ducts have long aspect ratio, but still lie well below the Taylor dispersion regime. This high P¶eclet 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 ow remains stable, sharp interfaces persist over wide ranges of parameters for dimensionless times (hence lengths) smaller than 192.2. Associated fundamental problems the P¶eclet number. At longer times (lengths) the dispersive limit is at- tained. For flxed lengths and increasing P¶eclet number (while remaining laminar) the ow 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 uids. 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 uid 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 efiects (F ). They also found the asymptotic value of M for large Pe when the viscosity ratio tends to inflnity. They pointed out an interesting argument that displacement ows at infl- nite capillary number can be in fact interpreted as immiscible ows with zero surface tension. Similarly inflnite Pe can be seen as a miscible ow with zero difiusion. Therefore they stated that it is possible to identify the interface between two immiscible uids with that between two miscible uids without molecular difiusion. Thus, the asymptotic values of M should have the same value for both the immiscible and miscible displacement ows. 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 of M 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 difierent 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 difierence. 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 flnd any trustworthy value for the difiusion coef- flcient between glycerine and a known glycerine-water mixture, they chose to measure the average difiusion coe–cient 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 u- ids in a density stable conflguration (i.e. the lighter uid above the heavier one) for large Pe. They observed a well-deflned interface between the two 202.2. Associated fundamental problems uids for which the transverse average concentration proflle has features of a kinematic wave. The important variables in their symmetric displacement were the viscosity ratio and a normalized ow 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 difierent domains in a map of the ow rate number versus the viscosity-ratio. They also analysed the stability of the ow and found critical values for the ow parameters at which the 2D ow 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 ows at increasingly large ow rates but also we consider density and viscosity difierences between the uids. Each of these efiects 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 ows, 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 uids. 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 flrst 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 ow to turbulence with some certainty. The Reynolds number (Re = ‰^U^ D^=„^) describes the relative importance of inertial to viscous forces. For a Poiseuille ow in a pipe, when the Reynolds number is su–ciently small, both large and small perturbations eventually decay (i.e. roughly for Re • 2000). Above this, the ow is believed to be unstable to perturbations of su–ciently large flnite amplitude. Practically, these perturbations are usually introduced into the ow 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 ow becomes random, strongly three-dimensional, very non-axisymmetric and strongly nonlinear everywhere in the ow [45]. 212.2. Associated fundamental problems Kelvin-Helmholtz instability The Kelvin-Helmholtz instability theory is a way to predict the onset of linear instability in stratifled layers of uids with difierent densities which are in relative motion. Let us consider the basic ow of two incompressible inviscid uids in horizontal parallel inflnite streams (two-dimensional in the (x^, z^)-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 uids, it is not too di–cult to obtain the necessary and su–cient condition for the linear instability [45]: q k^2 + l^2g^(‰^21 ¡ ‰^22) < k^2‰^1‰^2(U^1 ¡ U^2)2; (2.1) where k^, l^ are the wavenumbers in x^ and y^ directions respectively (note that y^ axis is perpendicular to the paper). Therefore the ow is always unstable (to modes with su–ciently large k^, that is, to short waves) provided that U^1 6= U^2. When the heavier uid is placed below, condition 2.1 for Kelvin- Helmholtz instability signifles an imbalance between the destabilizing efiect of inertia and the stabilizing efiect of buoyancy. It should be noted that this simple model of analysing Kelvin-Helmholtz instability is only a flrst attempt at understanding the mechanism behind this instability. This model does not include important features of the instability, such as the efiects of viscosity and nonlinear efiects of inertia. x z U2 U1 g ^ ^ ^ ^ ^ Figure 2.2: Principle of the development of Kelvin-Helmholtz instability for uid layers moving with difierent velocities. A small deformation of the interface is magnifled if condition 2.1 is satisfled. 222.2. 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 identifled the Kelvin-Helmholtz instability. He proposed a technique to produce stratifled shear ows 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 flnite 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 uids begin to accelerate. At this moment the uids 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 flnally unfurl inducing transverse mixing between the two uids (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 ow 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 efiects on the stability of the interface. It flrst 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 stratifled ow and stabilizes the growth of waves. Multi-layer ow 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 ows, since axial variations are very slow, the ow on any particular cross-section is not distinguishable from a miscible multi-layer ow. There are only a limited number of studies associated with instability of such ows, e.g. [63, 64] and [125, 126]. There is also extensive literature on the instability of immiscible parallel multi-layer ows, dating from the classical study of [166]. Explanations of the physical mechanisms that govern this type of instability for Newtonian uids have been ofiered by Hinch [70] and 232.2. Associated fundamental problems Figure 2.3: The growth of instabilities at the interface of layer of water and salt water (colored). The density difierence is 7:95 £ 10¡2 (g/cc). Molecu- lar difiusion has acted for 30 (min) on the tube horizontally before it was inclined at 4:4 –. The flrst 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- uid ows there is less work on shear instabilities; note also that the term multi-layer is ill-deflned if the uids can mix. Linear stability studies generally assume a quasi-steady parallel base state. Ranganathan & Govindarajan [117] and Govindarajan [62] analysed the stability of miscible uids of difierent viscosities owing through a chan- nel in a three-layer Poiseuille conflguration. They obtained instabilities at high Schmidt numbers and low Reynolds numbers, resembling those of [166]. In Couette ow it appears that the stability characteristics of the miscible ow are predicted by those of the immiscible ow with zero surface tension; see [49]. However, for core annular ow this is not the case; see [129]. Recent studies have considered convective instabilities in miscible multi- layer ows, 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 ows. Amaouche et al. [5] have recently proposed a weighted-residual-based ap- proach for two-layer weakly inertial ows in channel geometries (see also Mehidi & Amatousse [99]). In their study they extensively compared sta- 242.2. Associated fundamental problems bility predictions of their simplifled thin-fllm 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 ows, we should mention an important turbulent efiect which con- cerns us, namely turbulent entrainment in stratifled ows. 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 stratifled ows. 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 ow of the non-turbulent surrounding uid into the turbulent layer, and therefore a relatively small mean velocity perpendicular to main ow 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 coe–cient 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 coe–cient quickly decreases when Ri increases. 2.2.3 Gravity currents A more general motivation for our work arises since buoyancy-driven ows of miscible Newtonian uids over near-horizontal surfaces occur frequently in the oceanographic, meteorological and geophysical contexts (see [12, 139]) i.e. gravity currents. Such ows are driven by buoyancy, but the physical mechanisms that limit the ow may be inertia or viscosity depending on the geometric conflguration, the mean ow and the type of uids. Most frequently these ows have been studied in unconflned geometries (e.g. [13, 14, 39, 75, 138]). Slightly closer to our study are those of lock-exchange ows in tanks (open channels). Such ows are typically studied in a regime where vis- 252.2. Associated fundamental problems cous efiects are unimportant and buoyancy forces are balanced by inertia. The velocity is essentially constant in each interpenetrating stream. The mathematical approach for studying these ows 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 ows, vulnerable to interfacial instabilities, (loosely of Kelvin-Helmholz type), and local mixing. Typically the edges of gravity currents are not well-deflned, due to local instability and mixing. More recently, due to the importance of these ows in the industrial world, conflned 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 uids of difierent viscosity as well as density. Most of these studies in conflned geometries (ducts) focus on the exchange ow conflguration where there is zero net ow along the duct. Figure 2.4: Flow of cold air in warm air; shadow pictures showing the proflle of a front of a gravity current. The temperature difierence between the uid 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 ow of the interpenetrating front usually has an important role in the development of the ow. For example this front can in a sense limit the gravity current and change the dynamics in the ow. Figure 2.4 shows an example of changes in the front proflle in a gravity current induced by a ow 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 difierence of 1%. In this context, increasing the density contrast can be interpreted as the efiect of increasing the Reynolds 262.2. Associated fundamental problems number of the ow, which can signiflcantly 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 uid. Gravity currents produced by lock-exchange Shin et al. [138] presented a theory supported by experiments on gravity cur- rents in a lock-exchange ow conflguration. Their geometry was a horizontal rectangular channel. The geometry they considered is shown in Fig. 2.5 with current of density ‰^2, propagating with constant velocity U^ into uid of den- sity ‰^1. The depth of the current far behind the front where the interface is at is denoted by h^. 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 ux 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 uids. For such a ow the Froude number is: FH = U^q 2Atg^H^ : (2.2) ^ ^ ^ ^ ^ ^ Figure 2.5: A schematic diagram of a gravity current in a frame moving with current [138]. 272.2. Associated fundamental problems From the above equation, one can obtain the front velocity as U^ = 0:7 q Atg^H^ according to the theory of Benjamin (which gives FH = 1=2). Interestingly as we will see later despite the strong difierences that a conflg- 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 verifled this Benjamin’s theory. In Fig. 2.6 we observe good agreement be- tween the experiment and theoretical potential ow solution (dashed line) despite the mixing between the two uids 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 ow 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 ow 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 ux vs the normalized interface hight, respectively. It can be clearly seen that for maximum dissipation ( _E) we have FH = 0:527, while we can also have FH = 0:5 when energy is conserved. The difierence in speeds between a current with maximum dissipation (h^=H^ = 0:347 and FH = 0:572) and the energy conserving current with (h^=H^ = 0:5 and FH = 0:5) is di–cult 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 p2. They concluded that the dissipation efiect of turbulence and mixing is negligible when the Reynolds 282.2. 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 ˆh/ ˆH 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 ˆh/ ˆH ∆ ˙ E b) Figure 2.7: a) The Froude number FH plotted against the dimensionless height of the current h^=H^. b) The dimensionless net energy ux _E plotted against the dimensionless height of the current h^=H^ [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 ows in sloping channels Birman et al. [13] studied a lock-exchange channel ow problem of two uids of difierent densities. They carried out high resolution simulations accom- panied by complementary experiments. Their simulations reveal that the ow 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 – < fl < 50 – (with a maximum at fl = 40 –). This flnding was also supported by experiments performed in a difierent geometry (i.e. a circu- lar pipe in the experiments of Seon et al. [132]) which surprisingly provided similar qualitative results. The ow 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 ow. 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 difierent. They used this argument to justify the observed transition. In an inclined channel, the bulk ow behind the front in fact experiences a continuous acceleration caused by the longitudinal gravity vector component. This acceleration helps the ow in the stratifled 292.2. Associated fundamental problems layers behind the front move faster than the front. At early times, the uid 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 quantifled the transition time for difierent 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 stratifled 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 conflned geometry In the absence of an imposed mean ow, a detailed experimental study of buoyancy driven miscible ows 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 ow results. Seon et al. [132] experimentally characterised the velocity of the interpenetrating fronts of light and heavy uids, as a function of viscosity, density ratio and inclination angle. For difierent inclinations of the pipe from horizontal to vertical they observed three ow regimes: in- creasing front velocity, constant front velocity and decreasing front velocity. These three regimes are shown in Fig. 2.8. In the flrst regime segregation and mixing efiects control the front velocity. In the second regime, the front velocity is independent of inclination angle and uid viscosity, controlled by the balance between inertia and buoyancy. For the flrst 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 uids 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 efiects start to control the front velocity, it gradually de- creases towards a steady-state value, which is proportional to the sine of the 302.2. 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 fl for At = 4 £ 10¡3; „^ = 10¡3 (Pa.s). The insets are pictures of a 30 (cm) long section of tube just above the gate valve in the corresponding ow domains. The dashed lines qualitatively represent the boundaries between the domains and the dotted lines are only guides for the eye. This flgure is extracted from [132]. inclination angle, from horizontal. This flnal velocity thus tends to zero for a horizontal tube. They also showed that the uid concentration/interface proflles depend on the reduced variable x^=t^, i.e. spreading difiusively. 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 uids 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 ow. In the absence of an imposed mean ow 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 efiect of buoyancy diminishes but the former efiect remains present. We should note that for our study there is a third driving force, that of the imposed ow, which does not diminish over 312.2. 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 V^f=V^t as a function of V^”^ cos fl=V^t for the set of data points related to difierent experiments in the range [At, „^] 2 [4£ 10¡4 ¡ 3:5£ 10¡2, 10¡3 ¡ 4£ 10¡3]. This flgure is extracted from [132]. In the exchange ow context, the driving force is the buoyancy and the physical mechanisms that limit the ow are either inertia or viscosity, depending on the geometric conflguration and the type of uids ([132]). Two characteristic velocities can be deflned. Firstly, a viscous velocity scale V^” : V^” = Atg^D^”^ (2.3) when buoyancy and viscous term are balanced. Secondly, an inertial velocity 322.2. Associated fundamental problems scale V^t: V^t = q Atg^D^ (2.4) when buoyancy and inertia terms are balanced. Here At is the Atwood number, deflned as the ratio of the difierence of densities of the two uids by their sum, g^ is the acceleration due to gravity, D^ is the diameter of the pipe and ”^ is the common kinematic viscosity of the uids, deflned with the mean density. Exchange ows have been classifled as either inertial or viscous according to which efiect is dominant in balancing buoyancy forces. Seon et al. [132] showed that inertial exchange ows are found in pipes if V^” cos fl V^t = Ret cos fl ’ 50; (2.5) and viscous exchange ows otherwise. Here fl measures the inclination of the pipe from vertical. Figure 2.9 depicts the relevance of the characteristic velocities V^t and V^” through plotting the normalized stationary (long time) front velocity V^f=V^t as a function of the ratio V^” cos fl=V^t. It is clearly observed that for V^” cos fl=V^t < 50, all data points corresponding to difierent values of all control parameters collapse onto a single linear variation (i.e. V^f ’ 0:0145V^” cos fl), implying that the front velocity is controlled by viscous dissipation in the uid bulk. For V^” cos fl=V^t > 50 the points are close to a horizontal line corresponding to V^f ’ 0:7V^t, implying the the ow is inertial in this domain (although mixing is weak). Efiects of channel geometry Hallez and Magnaudet [67] used a Direct Numerical Simulation (DNS) tech- nique to observe the evolution of concentration and ow flelds in buoyant mixing of miscible uids in tilted channels, for the pure exchange ow. They were mostly interested in estimating the efiect of the channel geometry and considered difierent geometries including a two-dimensional (2D) channel, a square channel and flnally a three-dimensional (3D) pipe. They reported key difierences in the ow structure among these geometries when in the inertial regime. They claimed that the striking difierences between the ow 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. 332.2. Associated fundamental problems Buoyant currents in the conflned geometry experience difierent phases. While in the initial acceleration phase and also the flrst slumping phase, 2D simulations might provide a good estimation of the ow evolution. However, they fail to predict the ow behaviour in the long-time whenever the efiects of geometry are signiflcant. 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 (deflned by them as Fd = V^f= q g^D^(‰^1 ¡ ‰^2)=‰^1) 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 fleld (top image in each set) for difierent geometries. These photos show the ow 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 deflned as an imaginary part of conjugate eigenvalues of the vorticity gradient tensor. This flgure clearly shows that for the intermediate tilt angles of 60¡80 – (Figs. 2.10a and b) for the 3D ow, vortices caused by the Kelvin-Helmholtz instabilities (produced by shear between the layers) are not su–ciently strong to cut the channel of light (heavy) uid feeding the front with pure uid. Hence, the concentration difierence at the front has its maximum value and the front velocity is independent of the inclina- tion angle (V^f = 0:7V^t). In contrast, for lower tilt angles (Figs. 2.10d and e), the shear overcomes the segregation efiect (created by the transverse grav- ity component) and Kelvin-Helmholtz rolls are this time su–ciently strong to only temporary cut the thin channels of pure uid. Therefore, the local concentration at the tip of the light (or heavy) uid is formed a by a balance between the mixing at the front and feeding of pure uid 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 ow to periodically cut the mechanism of pure 342.2. 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 difierent ge- ometries at two difierent inclination angles with respect to vertical. The flrst 3 sets are at fl = 60 – and the second 3 sets are at fl = 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. uid 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 uid. Interestingly, provided that the vortex chain in the ow is su–ciently strong, it can even tear ofi the head of 352.2. Associated fundamental problems the gravity current and isolate it from the main body of the ow. This head then acts as an isolated drop of light (heavy) uid and slowly diminishes by difiusion efiects or twisting in its own wake. This phenomenon never occurs in the 3D pipe ow simulations since reconnection of the front to the mixing region behind it is always the case in this conflguration. 2.2.4 Taylor dispersion The detailed study of miscible displacement ows and mixing in pipes and channels (x2.2.1) 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 ow the Taylor dispersion is a regime in which although there is no e–cient mixing structures in the ow, dispersion can occur only because of the contribution of molecular difiusion and the velocity gradient. What is known as the Taylor dispersion is a shearing process of a passive tracer in- jected into a ow driven by a pressure gradient along a duct, i.e. a Poiseuille ow. We brie y review this efiect, analyzed for the flrst time in [144] (for a laminar ow) and [145] (for a turbulent ow). If at initial time t^ = 0 a line of constant concentration dye is placed transverse to the Poiseuille ow, in the absence of molecular difiusion it would be quickly stretched into a parabola under the efiect of the velocity proflle. After a time t^, the tracer would be distributed over a distance ¢x^ that increases linearly with time, implying ¢x^ » V^0t^ with V^0 the mean speed. For a su–ciently long time, molecular difiusion perpendicular to the axis of the pipe limits this efiect of stretching and transversely homogenizes the distribution of the tracer. The characteristic time T^ needed for the tracer to difiuse across the pipe is T^ » D^2=D^m. Assuming that we are in frame of reference that moves with mean speed of the ow, the longitudinal dispersion (for the mean concentration C^m) can be expressed by: @C^m @t^ = D^T @2C^m @x^2 (2.6) where D^T is the Taylor dispersion coe–cient. According to (2.6) D^T » ¢x^2=t^. 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 T^ » t^ we deduce: D^T » V^ 2 0 D^2 D^m (2.7) 362.2. Associated fundamental problems The pre-factor above was found by Taylor [144]. Aris [6] also found a flrst order correction and derived the relationship by a difierent method. It may seem counter-intuitive that D^T reversely varies with the difiusion coe–cient. In fact, the efiect of transverse difiusion homogenizes the dis- tribution of the tracer and limits the efiects of the velocity gradient which is the dispersive mechanism. Using (2.7) it can be shown that the Taylor dispersion coe–cient is proportional to D^mPe2 (with Pe = V^0D^=D^m). We should highlight here that even though in our investigation we are in the high Pe regime (which would consequently have a large dispersion coe–- 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 L^, su–ciently long for the tracer to spread in the transverse direction during the process timescale. For a developed turbulent ow, the spreading of the concentration proflle remains linear with time which is similar to the efiect of velocity gradient in the laminar case. However, the mechanism of transverse mixing is not by molecular difiusion but is by uctuations of the turbulent velocity fleld. For a given mean speed, the difiusion coe–cient is lower for a turbulent ow than for a laminar ow. This re ects the fact that transport by transverse velocity uctuations in a turbulent ow is much more efiective than by molecular difiusion in a laminar ow. The transverse homogenization is faster and the resulting longitudinal difiusion coe–cient is much lower. Finally, we should mention that in ows of the type we study, although in a laminar regime (according to the imposed ow Reynolds number) the dominant dispersive efiect is not always related to molecular difiusion. For example, in the experiments of Seon et al. [131, 135] transverse mixing is very e–cient at inclinations away from horizontal and is due to turbulent ow driven by buoyancy. In these inclinations dispersion (i.e. macroscopic difiusion coe–cient) in the axial direction is larger than closer to horizon- tal angles for which viscous efiects become important and mixing in the transverse direction is less e–cient. 2.2.5 Efiects of Rheology The literature for non-Newtonian uid 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 flngering with non- Newtonian uids; see [29, 90, 124, 162] and [95] as examples. Gas-liquid dis- 372.2. Associated fundamental problems placements in tubes have been studied for visco-plastic uids 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 ow around the displacement front is multi-dimensional. Other multi-dimensional dis- placement ows with generalised Newtonian uids 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 uids in the high Pe regime. Gabard and Hulin [58] investigated iso-density miscible displacements in which a more viscous uid is displaced by a Newtonian uid. In their experimental investigation the geometry used was a vertical tube. They observed the efiect of rheology of the displaced uid and the ow velocity on the transient residual fllm thickness during the displacement process. They showed that in displacements of shear-thinning uids 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 uids (38%). For yield stress uids the residual thickness is even further decreased (24¡25%). They suggested that the 3D ow fleld close to the displacement front can play an important role in forming the residual fllm thickness. Their numerical simulations conflrmed 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 ow rates are larger. In [19] a novel reactive miscible displacement technique was studied. Instead of viscosifying the entire displacing uid in order to improve dis- placement e–ciency, the authors engineered a reaction to take place at the front when the two uids mixed. The efiect of the reaction was to locally viscosify the uid mixture, with the idea of using this high viscous plug to improve displacement. The method in [19] did indeed produce enhanced displacement e–ciencies, but not by the anticipated mechanism: instead via locally destabilizing the ow. Moving slightly further, many authors have considered displacement ows of non-Newtonian uids from capillar- ies, driven by a gas ow, 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 Safiman- 382.2. Associated fundamental problems Taylor (viscous flngering) instability in a Hele-Shaw cell while including yield stress uids. 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 flnger does not really feel the presence of walls or other flngers due to the uid’s yield stress. In the viscous dominated regime, yield stress does not play an important role and the flnger can flnd the Hele-Shaw cell. Their observations were conflrmed by a linear stability analysis. They also conducted experi- ments with foams presenting very difierent results due the wall slip. Other investigations of viscous flngering (with stability analyses) include [29] and the earlier Darcy- ow analogues of [108{110]. Slightly less related to our study are studies of viscous spreading of thin layers fed with an imposed ow at a source. These arise in particular in the context of lava dome formation and spreading (i.e. non-Newtonian uid); see [66]. Frequently, the models and experiments used to understand these phenomena are complicated with thermal efiects, 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 uids of the type considered here. Although the lubrication/thin-fllm modeling is simi- lar, these ows are unconstrained single uid ows in which the ux 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 uids by other uids. Experimental studies involving two uid ows of yield stress uids in the pipe geometry include Crawshaw & Frigaard [32] and Malekmohammadi et al. [97] who have studied the exchange ow problem (i.e. buoyancy driven ow in a closed ended pipe). The focus of these studies is stopping the motion using the yield stress of one of the uids. Huen et al. [78] and Hormozi et al. [73] have studied core-annular ows, using a yield stress uid for the outer lubricating layer and a range of difierent Newtonian and non- Newtonian uids for the core. The start-up phase of these experiments is displacement-like, although the flnal steady state is a multi-layer ow. Finally, a number of authors have considered the displacement of yield stress uids by a gas. De Souza Mendes et al. [36] investigated the displace- ment of viscoplastic ows in capillary tubes experimentally through gas in- jection. They showed that below a certain critical ow rate, the visco-plastic liquid is completely displaced by the displacing uid. However above this critical ow rate small lumps of unyielded liquid will remain on the walls. For increased values of imposed ow rate a smooth liquid layer of uniform 392.2. Associated fundamental problems thickness forms. They reported that the thickness of this layer increases with the dimensionless ow rate. There have also been extensive compu- tational studies of these ows [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 uids 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], biofllms [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 uid is that the uid does not deform until a critical shear stress is exceeded locally. Therefore, when these uids flll ducts and are displaced by other uids, there is a tendency for the yield stress uid 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 flrst recognised in the context of oil well cementing by Mclean et al. [98], who identifled 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 oilfleld cementing [30, 82], and latterly also simulation based design models, [15, 111]. Further features of oilfleld 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 flrst con- sidered by Allouche et al. [4] who studied symmetric displacement of two visco-plastic uid owing 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 uid exceeds that of the displacing uid. They were mainly interested in ows for which a (flxed) 402.2. Associated fundamental problems ow rate is imposed and the displacing uid is heavier than the displaced uid. They argued that under such conditions and for a combination yield stresses of the displacing and displaced uids, four qualitatively difierent velocity proflles could exist as shown in Fig. 2.11. The focus of their work was on the unique velocity proflle of Fig. 2.11a in which the uid 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 difierent possible characteristic axial velocity proflles when displacing uid is heavier than the displaced uid; U is the velocity and 1 and 2 denote the displacing and displaced uid respectively; Yi is the interface position [4]. They found a critical ratio of the yield stresses (’y) versus the Bingham number of displacing uid (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 ’y, B1 and a third parameter ’B, giving the ratio between the buoyancy stress and the yield stress of the displaced uid. This concept can be explained by considering a displacement at flxed ow rate when the front has evolved into a quasi-parallel multi-layer ow. In the case of static residual wall layers, the whole imposed ow rate has to pass through the mobile layer of the displacing uid. Assuming now that 412.2. Associated fundamental problems for the flxed ow rate the thickness of the static residual layer is increased, the shear stresses in the displacing uid 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 flnally the shear stress at the wall exceeds the yield stress of the displaced uid and consequently the uid 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 signifl- 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 deflned 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 ow in downstream. They anal- ysed the streamline conflguration 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 ow tends to avoid this situation to minimise dissipation. They flnally introduced the the following relation for the thickness of static residual layer: hstatic = min fhmax; hcircg (2.8) 422.2. Associated fundamental problems Figure 2.13: Schematic illustration of the two types of streamline behavior in displaced uid: (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 ow 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 ow. 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 uid displacements (Newtonian uid displacing Bingham uid) was performed, including the efiect of ow rate oscillations. This has shed further light on the efiects of the main 3 di- mensionless parameters (Reynolds number, Bingham number and viscosity ratio), in the absence of density difierences. 2.2.6 Summary of fundamental literature Through reviewing the scientiflc literature, we understand that buoyant mis- cible displacement ows are associated with various fundamental problems. These displacement ows are naturally complex due to the presence of many 432.2. Associated fundamental problems parameters in the ow, corresponding to a number of competing physical efiects. In this perspective, we have seen that relatively few studies directly address the speciflc problem that concerns us. However difierent behaviors in our ows 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- flned geometries and the Taylor dispersion. We also flnd that the literature on non-Newtonian displacement ows is relatively poor in comparison to its Newtonian counterpart. However, the concept of static wall layers in these displacement ows 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 ows: (a) Newtonian ows: Buoyant exchange ows have been previously stud- ied in depth for difierent inclination angles (fl) and density ratios (At). However, there are no extensive studies examining the efiects of adding a mean imposed ow (V^0) to the buoyant exchange ow. Understat- ing the combined in uence of these three important parameters i.e. the mean imposed ow velocity, V^0, the density difierence, At, and near- horizontal inclination angles, fl, has not been in depth before. There- fore, proper regime classiflcations even for Newtonian ows predicating the behavior of the ow do not currently exist. In this research area, producing reliable data to shed light on ow characteristics by employ- ing experimental, analytical and computational approaches is of major importance. (b) Efiects of viscosity ratio and shear-thinning: Including a viscos- ity ratio in iso-density displacement ows has a long history in liter- ature. However, buoyant displacement ows when a viscosity ratio is present have not been deeply studied. Quantifying the efiects of in- creasing/decreasing the viscosity of the displacing/displaced uid is ex- tremely important especially because, practically speaking, the indus- trial buoyant displacement ows often involve a viscosity ratio. Shear- thinning uids are in fact viscous uids with variation of viscosity with shear stress. Therefore, they can present the efiects of displacements of variable viscosity ratios. There are not many research works which thoroughly investigate the efiects of presence of shear-thinning efiects in buoyant displacement ows in close to horizontal geometries. (c) Efiects of yield stress: Considering the small number of studies relat- ing to Newtonian buoyant displacement ows in slightly inclined ducts, 442.3. Conclusions it is expected that the efiects of a yield stress for buoyant ows is even less investigated. In this context, the case where the displaced uid has a yield stress is of more interest and practical concern. Static residual wall layers are common for these ows. Nonetheless, it is felt that flrstly appearance or non-appearance of these static residual layers should be studied in more depth. Second, the thickness of these layers should be quantifled versus the ow parameters. 2.3 Conclusions In this chapter we have reviewed both engineering and scientiflc backgrounds of our displacement ow problem. The engineering background overview clearly demonstrates the complexity of the problem that industry is faced with. Although the main factors accountable for low e–ciency displacements during primary cementing operations have been long identifled, 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 ow subject. On the other hand, although previous fundamental scientiflc studies can be of enormous help in understanding of the basic mechanisms of misci- ble buoyant displacement ows, there are still many aspects to explore. In particular the combined efiects of many parameters involved are to be de- termined. Below in x2.4 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 efiective, so there is less motivation to improve them. From the scientiflc perspective we should also regard laminar displacements as the more likely to lead to long regions of \mixed" uid. As mentioned before, by \mixed" we mean a part of the pipe where more than one uid is present (i.e. even if the uids are largely in 2 separate layers). The reason for this is that in a fully turbulent ow the uid concentration is typically fairly uniform on each cross-section and spreads axially relative to the mean displacement front via turbulent dispersion. This is a difiusive 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 difierence in speeds between the fastest and 452.4. Research objectives slowest propagating fronts. The fronts often propagate at constant speed (after initial transients) and the difierence 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 uid layers. Therefore, our study of laminar displacement ows, and at inclinations where they are most likely to be found, is efiectively a consideration of the worst case. 2.4 Research objectives The scientiflc and technical aims of the current thesis are to provide reliable knowledge to better understand buoyant miscible displacement ows, 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 ows. The key objective is to quantify the efiects of each parameter. This essentially starts from examining the efiects of the density difierence, mean imposed ow 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 ows where the basic mechanisms are known. However, the efiects of the difierent controlling parameters espe- cially the imposed ow velocity (V^0) is not well investigated. The related literature reveals that depending on difierent parameters, the uids can be mixed across the duct or stay separated. Based on the available literature, we a priori expect that uids segregation is usually the case in a close to horizontal conflguration 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 ows through three activities: (i) Scaled laboratory displacement ow experiments are sought in a cir- cular pipe with orientations close to horizontal. A realistic range of uid properties and ow parameters are considered. It is of interest to quantify the evolution of the interface between the uids when they are separated. It is interesting for us to measure the velocity of interpen- etrating uids and also measure 1D local velocity proflles in a center 462.4. Research objectives line across the pipe. Comments are made on mixing between the uids when it appears. Efiorts are devoted to produce experimental corre- lation predicting the behaviors of the ow when possible. The efiects of each parameter are determined on the displacement ow and ow 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 fllled with two difierent uids. The typical experiment consists of a displacement at flxed ow rate and a flxed inclination angle; this forms the bulk of the exper- imental work. There is a possibility to change the tilt angle to investigate the efiects of fl. † There is possibility to accurately vary the value of the mean im- posed ow added to the control volume to study the efiects of V^0. † Sets of experiments are carried out in which the properties of dis- placing and displaced uids are modifled. It is natural to start with Newtonian uids and flrstly change the density ratio to ob- serve the in uence of At. † Viscosities of displacing and displaced uids are changed to pro- vide an understanding of the efiects a viscosity ratio between the two uids (i.e. m). † Experiments with shear-thinning uids (i.e. uids whose viscosi- ties decrease as shear rates increase) are conducted. The be- haviour of these uids is simply described by a consistency, •^, and a power-law index, n. † Finally, our experimental work ends with including yield stress (¿^Y ) efiects usually in the displaced uids. (ii) In a simplifled 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 efiects are neglected. These models are developed for channel (generalised Newtonian) and pipe (Newtonian) ow displacements. Through this, it is our aim to investigate the ef- fects of the ow parameters on the limits of the buoyant displacement ows (e.g. viscous limit). In this analytical approach the following items are considered: 472.4. Research objectives † The study is proceeded non-dimensionally to produce predictive models for channel and pipe ows. For Newtonian iso-viscous ows, hence, our main focus is on a combination of 3 dimension- less parameters which are Re, Fr and fl (while neglecting the efiects of Pe and At). A viscosity ratio adds another dimension- less parameter (m) to the problem. For non-Newtonian ows, 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 e–ciencies, 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 ow model for non-Newtonian uids. For iso-viscous Newtonian ows both 2-layer and 3-layer channel ows and also 2-layer pipe ows are studied. † Finally, the efiects of including weak inertial terms into the lu- brication model of the 2-layer channel ows are considered. In addition, the instabilities involved in the model are studied. (iii) 2D ow 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 ows and provide ow 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 flrstly benchmarked and validated at sensible mesh resolutions for our usage. † Naturally, the range of dimensional and non-dimensional parame- ters considered is flrstly similar to those of the experiments. Then the parameter coverage is extended to include a wider range. † Similarities and difierences between the computational study and its corresponding experimental investigation are reported. When 482.4. 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 difierent buoyant displacement ows, the attempt is to provide physical arguments and also formulate appropriate balance equa- tions. As much as the details of these ows are exciting for us, it is of inter- est to generate ow regime maps to give predictions of the ow behaviours for a wide range of non-dimensional parameters. In these ow regime dia- grams, compact experimental and computational results are included. Also, simplifled and predictive analytical results are superimposed on the same diagrams. Thus, the key ow features that may occur are described; this is accompanied with established supporting analyses. 49Chapter 3 Research methodology In this thesis we employ experimental, analytical and numerical approaches to better understand miscible displacement ows, in the presence of buoy- ancy and in ducts that are inclined close to horizontal. Although we mostly consider Newtonian uids (with or without a viscosity ratio), we also investi- gate generalized Newtonian uids with both shear thinning and yield stress rheological features. A large part of the efiort 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 uids of difierent densities are initially placed in an unstable conflguration in a long pipe, inclined close to horizontal. A flxed ow rate is applied at the upper end of the pipe. 2. The computational techniques: two miscible uids of difierent densities are initially placed in an unstable conflguration in a long plane channel, inclined close to horizontal. A flxed ow 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 fllled with a lighter uid coloured with a small 503.1. 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 ow rate was controlled by a valve and measured by both a rotameter (Omega, variable-area type) and a magnetic owmeter (Omega, low- ow type), located downstream of the pipe. The role of the gate valve was to initially separate the uids. 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 ow at the valve, the movement (and the resulting shear stress) could slightly disturb the ow (and afiect 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 ow 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 proflles along the tube, averaged concentration proflles on the cross section, and also the spatiotemporal diagrams of changes in these proflles. 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 uorescent light tubes flltered through a difiusive paper giving a homogeneous light. Light absorbtion calibration was carried out for both 513.1. 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. 523.1. 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) = “expfiC , where “ and fi are physical constants, till a maximum value CMAX which depends on the uid 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 uid 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 uid 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 “ and fi. 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 uid and 1 for the transparent heavy uid). This measurement method enabled us to obtain images of the normalized concentration, averaged over the depth of pipe. 533.1. Experimental technique 0 200 400 600 800 1000 10 100 1000 Concentration of ink (mg/l) Average light intensity of the pip e 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 flnd the best possible region of optical measurements, we took an image of the pipe fllled with the dyed uid 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 efiects are signiflcant. 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 543.1. Experimental technique Longitudinal direction of the pipe (pixel) Transverse direction of the pipe (pixel ) 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 Transverse direction of the pipe (pixel ) b) Figure 3.4: a) An image taken by camera #1 of a section of the pipe fllled with dyed uid 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 efiect of the pipe curvature is negligible. diagrams of the concentration proflles along the length of the pipe. The displacement fronts were marked on these diagrams by a sharp boundary between the difierent relative concentrations of the uids (the boundary was identifled 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 uids, the normalized concen- tration across the pipe can be interpreted as the normalized height h(x^; t^) of the interface at each time, an example of which is shown in Fig. 3.5a. This flgure shows a sequence of interface evolution in time obtained for fl = 87 –, At = 3:6 £ 10¡3, ”^ = 1 (mm2.s¡1) (i.e. the kinematic viscosity) and V^0 = 19 (mm.s¡1). The distances are measured with respect to the position of the gate vale. The time interval between interface proflles is 2 (s) while the flrst 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 (x^) and time (t^). Using this diagram, it is easy to observe the movement of the front and obtain the front velocity. The front velocity V^f is equal to the slope of the boundary marked on this diagram. For this case the front velocity is found to be V^f = 29 (mm.s¡1). In the experiments 553.1. Experimental technique a) 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 0.8 1 xˆ (mm) h(xˆ , ˆ t) 0 20 40 60 0 0.2 0.4 0.6 0.8 1 xˆ/ˆt (mm/s) h(xˆ /ˆ t) xˆ (mm) ˆ t (s) 1600140012001000800 600 400 200 0 10 20 30 40 50 b) Figure 3.5: a) Experimental proflles of normalized interface height, h(x^; t^), for t^ = 6; 8; ::; 48; 50 (s), with fl = 87 –, At = 3:6£ 10¡3, ”^ = 1 (mm2.s¡1) and V^0 = 19 (mm.s¡1). The inset shows experimental proflles of normalized h(x^=t^) 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 (> D^) 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 proflles, which can help to understand the ow dynamics from a difierent view. We measured the velocity proflle somewhere downstream of the ow (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 („.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 uids 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 uids 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 ow 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 563.1. Experimental technique of 4-cycles of short bursts and records the echoes back scattered from the particles suspended in uids. 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. Re ection efiects at the lower wall of the pipe afiect the velocity measurement locally, making it hard to measure a zero velocity at the lower wall. For a typical acquisition time of the velocity proflles, 120 (m.s) per pro- flle was set while no real time flltration of signals was applied during the recording process. For the tracer, we used polyamid seeding particles with a mean particle diameter of 50 („m) with volumetric concentration equal to 0:2 (g.l¡1) in the both uids. Considering the trade ofi between a good signal to noise ratio and also small ultrasonic signal re ections [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 difierence of the uids used in our experiment is su–ciently small to neglect the difierences in the speed of sound in the uids. 3.1.4 Fluids characterisation Most of the experiments were conducted using water as the common uid, with salt (NaCl) as a weighting agent to densify the displacing uid. The uid 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 uids, 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 efiects, low percentage Xanthan-water solutions were used. To make a uid with a yield stress, we used Carbopol solutions. All exper- iments reported in this thesis were density unstable, i.e. heavy uid in the upper part of the pipe displacing a less dense uid 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 modifler. Xanthan solutions were mixed at concentrations of 0:3% and less, where the solu- tions are relatively inelastic. In preparation we flrst weighted the Xanthan 573.1. 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 uids 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 flrst 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 efiect 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. 583.1. 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 signiflcant 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 uid 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 ofi 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 flts well the shear behaviour of Carbopol solu- tion is the Herschel-Bulkley model: ¿^ = ¿^Y + •^ _^ n: (3.1) This includes the simpler Bingham, power law and Newtonian models and is deflned by three parameters: a uid consistency index •^, a yield stress ¿^Y , 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 („m) gap at the cone tip, was used for rheometry. Fluid samples were flrst loaded on the bottom plate. The top plate was then lowered to the desired gap height of 150 („m) 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. 593.1. 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 uids (¿^ = •^ _^ n). We used the strain rate range 10 ¡ 100 (s¡1) to flt the uid consistency index, •^, and power-law index, n, from a log-log plot of the efiective 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 difierent 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 difierence 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 ˆγ˙ (s−1) ηˆ (P a . s ) Figure 3.6: Variation of the efiective viscosity · with shear rate _^ for Xanthan-water solutions of various concentrations. The data points cor- respond to 0:3 % (†), 0:2 % (N) and 0:15 % (H); fllled data points are our measurement while hollow data points refer to rheometry of Xanthan-water solutions (+ 70 % glycerol) reported by Gabard & Hulin [58]. 603.1. Experimental technique For yield stress uids, 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 flt to a power law curve. The practical range of shear rate used to obtain repeatable results for determining •^ and n to flt 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 (•^) and the power law index (n) was always below 7 % and 12 % respectively. An example owcurve from the rheometer measured data compared with the curve fltted 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 ˆγ˙ (1/s) τˆ (P a ) Figure 3.7: Example owcurve 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, ¿^ = ¿^Y + •^ _^ n: the solid line shows the curve flt with parameters ¿^Y = 3:05 (Pa), n = 0:60 and •^ = 8:24 (Pa.sn). 3.1.5 Experimental results validation We flrst calibrated our apparatus against exchange ow results of Seon et al. [132, 135] for difierent Atwood numbers at fl = 85 – and fl = 87 –. The 613.2. 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 uids of equal viscosity is given as: [1 + `At] [ut + u ¢ ru] = ¡rp+ 1Rer 2u+ `Fr2eg; (3.2) r ¢ u = 0; (3.3) ct + u ¢ rc = 1Per 2c: (3.4) Here eg = (cos fl;¡ sin fl) and the function `(c) = 1¡2c interpolates linearly between 1 and ¡1 for c 2 [0; 1]. The 4 dimensionless parameters appearing in (3.2) are the angle of inclination from vertical, fl, the Atwood number, At, the Reynolds number, Re, and the (densimetric) Froude number, Fr. These are deflned as follows: At · ‰^1 ¡ ‰^2‰^1 + ‰^2 ; Re · V^0D^ ”^ ; F r · V^0q Atg^D^ : (3.5) Here ”^ is deflned using the mean density ‰^ = (‰^1 + ‰^2)=2 and the common viscosity „^ of the uids. In (3.4) appears a 5th dimensionless group, the P¶eclet number, Pe, deflned by: Pe · V^0D^D^m ; (3.6) with D^m the molecular difiusivity (generally assumed constant for simplic- ity in our work). In our computations, the efiect of molecular difiusion is neglected. This neglect is due to the large P¶eclet number that correspond to our experimental ows, for which we typically have a well deflned interface. The equations (3.2)-(3.4) have been discretised using a mixed flnite ele- ment/flnite volume method. The Navier-Stokes equations are solved using 623.2. Computational technique Galerkin flnite element method, where the divergence-free condition is en- forced by an augmented Lagrangian technique [60]. We use a flxed 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 flnite 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. xeu 0ˆˆ V= Dˆ Lˆ xˆ yˆ on walls ,0ˆ =u gˆ 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 (D^ = 19:05 (mm)). For the channel length, we typically have L^ = 100£ D^. The present numerical algorithm is implemented in C++ as an appli- cation of PELICANS. PELICANS is an object oriented platform developed 633.2. Computational technique a) xˆ (mm) ˆ t (s) 500 1000 1500 2000 10 20 30 40 50 60 70 80 90 100 b) Figure 3.9: Computational concentration fleld evolution obtained for fl = 85 –, At = 3:5 £ 10¡3, ”^ = 1 (mm2.s¡1), V^0 = 15:8 (mm.s¡1), (Re = 300). a) Sequence of images from top to bottom are shown for t^ = 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) uids respectively) along the channel. The heavy broken line shows the temporal evolution of the leading front and its slope is the leading front velocity (V^f = 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 difierential 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 ux of c at the solid walls. A plane Poiseuille ow is imposed at the in ow, along with c = 0. Out ow 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 ow 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 fleld evolution obtained for a typical simulation with parameters fl = 85 –, At = 3:5 £ 10¡3, ”^ = 643.2. Computational technique 1 (mm2.s¡1), V^0 = 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 reflnement was carried out until successively calculated front ve- locities on meshes difiered 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, reflned 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) flner mesh resolution producing only a little difierence 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 reflnement, which would be advisable if e.g. ow 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 fl = 60 –, Re = 200, V^0= q g^D^ = 0:316, „^2=„^1 = 2, ‰^2=‰^1 = 1:5); see Fig. 3.10. We flnd close agreement with the computed front velocity and also observe similar qualitative behavior in the displacement ow 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 ows (”^ = 1 (mm2.s¡1), At = 10¡3), at fl = 83 – and fl = 87 –. In each case we have studied a sequence of in- creasing imposed ow (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 ows than with their code. We have also compared our results with those of Hallez & Magnaudet [67] for exchange ow 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 £ D^) and shorter computational times than we have. By com- 653.2. 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) uids respectively) along the channel for fl = 60 –, Re = 200, V^0= q g^D^ = 0:316, „^2=„^1 = 2, ‰^2=‰^1 = 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 D^ and D^=V^0 respectively. parison, we are concerned with displacement ows, long time ow behaviour and estimating global features such as the front velocity. Our typical com- putational channel length exceeds 100£ D^ and we have signiflcantly coarser meshes. We have however performed a number of simulations for channel ex- change ow conflgurations to compare with [67] over the range fl = 60¡90 –, and captured all the main trends and qualitative behaviors reported in their work. For example, we observe the strong in uence of vortices periodically cutting the channels of pure uid which feed the advancing fronts and help to maintain constant front velocity (see x2.2.3). In near-horizontal channels we have observed an initial inertial phase during which the front velocity re- mains approximately constant. Afterwards, viscous efiects come to play and front velocity decreases and attains a flnal velocity, depending on balance between viscous and permanent/logitudinal buoyancy forces. Also similar to [67], for a wide range of inclinations (fl = 60 ¡ 90 –) for At = 4 £ 10¡3, 663.2. Computational technique ”^ = 1 (mm2.s¡1) and D^ = 20 (mm), we have compared the densimetric Froude number during the initial slumping phase in a channel ow. On increasing the angle from horizontal, we have observed a slight increase in the front velocity and found a constant plateau of modifled Froude number versus tilt angle between fl = 70 ¡ 80 –; see e.g. Fig. 5 in [67]. Although we have good qualitative agreement with [67], some quantitative difierences 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 difierence 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 ows of difiering viscosi- ties develop pearl and mushroom shaped instabilities. Good quantitative comparisons were made. There is numerical difiusion present in solution of (3.2)-(3.4). Imple- menting molecular difiusion within (3.4) was also tested, i.e. by adding (1=Pe)r2c to the right hand side. However, for the mesh sizes we have used it was found that for Pe ‚ 105 there was no discernible difierence in results, i.e. numerical difiusion 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 signifles that the cause of the mixing is primarily dispersion via secondary ows and instability To summarize, our code has produced similar results to the available computational and experimental studies. These are complex ows 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 ow features (such as front velocity) over a range of parameters for which our code is adequate. Usage of our particular code is also partly in uenced by its exibility to be extended to non-Newtonian multi- uid ows, which is the eventual aim of the study of these ows (al- though we do not present any results from non-Newtonian uid simulations in this thesis). Here other researcher in our laboratory has also made some 673.2. Computational technique progress, e.g. [73, 74, 161]. 68Chapter 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 flxed and experiments were performed at successively increasing mean imposed ow velocity V^0. 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 ows. Speciflcally, we identify several commonly observed ow regimes. We demonstrate that the superposition of a pressure-driven ow on an exchange ow strongly in uences the front velocity and the physical mecha- nisms that dissipate energy. The front velocity V^f is presented as a function of the mean ow velocity, V^0, in three difierent ow regimes. An interesting flnding of this work is that a transition of the ow from inertia-dominated behaviour to viscous-dominated behaviour, was observed with increased en- ergy introduced into the system (via V^0). 4.1 Observation of 3 difierent regimes We present the results of a typical experimental sequence, as V^0 is increased from zero, in Fig. 4.1 for fl = 83 –, At = 10¡2 and „^ = 10¡3 (Pa.s). Phenomenologically, we observe 3 distinct behaviours as V^0 is increased from zero. (i) As V^0 ! 0, we observe an exchange- ow dominated regime: the imposed ow has only a slight in uence on the dynamics of the exchange ow. For the case depicted in this flgure, we are in the inertial regime [132], since Ret cos fl = 101 > 50, and the ow develops some shear instabilities at the interface (see x2.2.3 for details and deflnitions). (ii) In the second 4A version of this chapter has been published: S.M. Taghavi, T. Seon, D.M. Martinez and I.A. Frigaard. In uence of an imposed ow on the stability of a gravity current in a near horizontal duct. Phys. Fluids 22, 031702 (2010). 694.1. Observation of 3 difierent 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 1610 3 Re 3 ^ ^ ^ ^ ^ × Figure 4.1: Variation of the front velocity V^f as a function of mean ow velocity V^0 for fl = 83 –, At = 10¡2, „^ = 10¡3 (Pa.s). The dashed line is a linear flt of data points in the mean ow dominated regime whereas the dotted line shows the slope of the flnal mean ow regime (V^f » V^0). The inset displays the data but for higher mean ow values, and as a function of Reynolds (number based on V^0), 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 ow domains. regime, the balance between pressure gradient and dissipative forces still exists but the mean ow becomes stronger than the buoyancy driven ow, and so controls its dynamic. The main feature here is a linear relationship between V^f and V^0 (here V^f=V^0 … 1:3). We have conducted a large number of experiments at various At and ”^ in this regime. We have observed that the slope V^f=V^0 does not vary signiflcantly with At and ”^. In particular we emphasize that this linear relationship is found for cases for which the flrst regime may be either inertial or viscous. These observations will be discussed quantitatively and in more depth in Chapter 7. (iii) For V^0 0, we observe a second linear regime, with V^f … V^0. This third regime is displayed partially on the main curve and more completely in the inset of Fig. 4.1. It is deflned by the buoyancy forces becoming negligible compared 704.2. Stabilizing efiect of the imposed ow (a) (b) (c) Figure 4.2: Three snapshots of video images taken for difierent mean ow and showing the ow stability induced by the Poiseuille ow. These images are obtained for fl = 83 –, At = 10¡2, „^ = 10¡3 (Pa.s) and mean ow veloci- ties: (a) V^0 = 9 (mm.s¡1), (b) V^0 = 71 (mm.s¡1) and (c) V^0 =343 (mm.s¡1) (the corresponding buoyant velocity is V^ V^0=0f = 31 (mm.s¡1)). The fleld 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 ow is turbulent (Re ‚ 3000, see inset). As a result, the two uids mix (see inset) and are completely displaced (V^f … V^0). We also observe a transitional zone between the second and third regimes in Fig. 4.1. 4.2 Stabilizing efiect of the imposed ow We now focus on an interesting flnding of our work, i.e. the in uence of the imposed ow on the stability of the system. To illustrate this we show in Fig. 4.2 images from the ows of Fig. 4.1, for three difierent representative mean imposed ow velocities. Fig. 4.2 displays images of the 70 (cm) long section of the tube, tilted at fl = 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 uid 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 uids transversally across the section. This low mean ow case (V^0 = 9 (mm.s¡1)) is in the flrst regime (see Fig. 4.1) where the ow is driven by a balance between buoyancy and inertia, (since here Ret cos fl > 50, see x2.2.3 for details). In Fig. 4.2b with an increased imposed ow we observe a stable ow in which there are no Kelvin-Helmholtz instabilities at the interface. Consequently there is no mixing between the two uids. Moreover, the front height is small and the slope of the interface with respect to the pipe axis is constant and weak. 714.2. Stabilizing efiect of the imposed ow We infer that the velocity fleld is quasi-1D and is therefore under conditions where the lubrication approximation becomes valid; the ow 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 V^0 is greater than in the previous case. As the mean ow approaches a Poiseuille ow, the ow 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 ow stabilizes the initial inertial exchange ow 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 ow velocity (see Fig. 4.2c) is further increased, i.e. much higher than the buoyant velocity, the ow 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 ow results in a complete displacement. The two pure uids are separated by a mixing zone. If we consider the pure exchange ow in this conflguration, Seon et al [135] showed that this exchange ow 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 q Atg^h^f , where h^f (height of the front) adapts itself to maintain a front velocity equal to the viscous bulk velocity. Such a viscous exchange ow 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 ¢t^ = 0.5 (s), and this sequence corresponds to an experiment conducted at fl = 87 –, where the mean ow (V^0 = 77 (mm.s¡1)) was imposed after the flrst image. We observe in this sequence that the inertial bump disappears under the efiect of the mean ow. 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 ow conflguration, is now valid everywhere due to the mean ow (except perhaps very close to the front). Indeed, the only way for the inertial bump to disappear is to be subjected to a laminar ow in this region and this can only be achieved when the streamlines in this region are parallel. Therefore, the ow is now dominated by the Poiseuille ow and 724.3. Summary Figure 4.3: Sequence of images showing the initial bump shape spread out by the Poiseuille velocity gradient. This sequence is obtained for fl = 87 –, At = 10¡2, „^ = 10¡3 (Pa.s). and the mean ow (V^0 = 77 (mm.s¡1)) is imposed after the flrst image (top one). The fleld 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 ¢t^ = 0.5 (s). the buoyancy driven ow becomes a correction. In order to have a better image of the stabilizing efiect of the imposed ow, Fig. 4.4 illustrates snapshots of an experimental sequence of increasing the imposed ow for fl = 85 –, At = 10¡2 and „^ = 10¡3 (Pa.s). In this flgure 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 in uence of an imposed ow on the well-studied buoyant exchange ow conflguration. We have observed 3 distinct regimes as a function of V^0. † In the flrst regime, deflned for a low mean ow, the dynamics is gov- erned by the balance between buoyancy forces and dissipative forces, 734.3. 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 ˆt (s) Figure 4.4: Illustration of stabilizing efiect of the imposed ow on the waves observed at the interface for fl = 85 –, At = 10¡2 and „^ = 10¡3 (Pa.s). Y-axis is h(t^) ¡P300 (s)t^=100 (s) h(t^), where h(t^) 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 V^0 = 38; 42; 44; 49; 61 (mm.s¡1). which depends on the uid properties and can be either viscous or inertial. † In the second regime, deflned for higher values of the mean ow, the front velocity varies linearly with the imposed ow 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 ow between parallel plates and in Chapter 7 for a pipe geometry. † In the imposed ow dominated regime (i.e. the second regime) the im- posed ow stabilizes the unstable buoyant ow by making the stream- lines more parallel. In other words, it tends to decrease the inertial 744.3. 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 ow (e.g. presence of the inertial bump), is removed by a su–- ciently strong imposed ow. A difierent way of viewing this is to note that when V^0 ! 0, the instabilities at the surface of the current are due to the shear created by the exchange ow (due to buoyancy). If a mean ow is imposed, the relative in uence of buoyancy decreases compared with that of the pressure gradient: the velocity gradient at the surface will decrease whereas the stratiflcation remains unchanged. Thus, the local gradient Richardson number (loosely speaking Ri = StratificationShear ) increases and the ow 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 ow. Indeed in this case, the mean ow required to stabilize the buoyant ow may itself be unstable, and so the ow would transition from an unstable buoyancy dominated regime to a turbulent pressure-driven regime. † Finally, in the third regime, deflned when the buoyancy forces are negligible, the mean ow is turbulent. The two uids are displaced at the mean ow velocity and a mixing zone separates the two pure uids. In this turbulent regime, we can expect that for a suitably strong mean ow and over long enough time-scale, the mixing zone will spread difiusively governed by turbulent Taylor dispersion [145]. Thus, V^f » V^0 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 ows frames much of the work presented in this thesis. 75Chapter 5 Lubrication model approach for channel displacements5 As we have seen in Chapter 4, as the displacement ow rate is increased from zero we enter a regime that is dominated by the imposed ow, where the front velocity increases approximately linearly with the imposed mean velocity. Frequently these ows are viscous dominated and the interface elongates progressively as the front proceeds. This is a classical conflguration where it is common to adopt a thin-fllm or lubrication approach to modelling the ow. This type of model is easiest to develop for a 2D plane channel displacement, rather than the 3D pipe ow. 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 uids of difierent density. † We simplify the Navier-Stokes equations and derive a lubrication/thin fllm approximation. † A semi-analytical solution is found for the ux function that drives the interface propagation problem. † We analyse the ux function and show that there are no steady trav- eling wave solutions to the interface propagation equation. † At short times, difiusive efiects of the interface slope are dominant and there is an exchange ow, relative to the mean ow. We flnd a short-time similarity solution governing this initial counter-current ow. 5A version of this chapter has been published: S.M. Taghavi, T. Seon, D.M. Martinez and I.A. Frigaard. Buoyancy-dominated displacement ows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1-35 (2009). 765.1. Two- uid displacement ows in a nearly horizontal slot † At longer times we analyse the hyperbolic part of the thin fllm model, which allows us to predict the propagation speeds of the displacement (at long times). † We explore the efiects of viscosity ratio, inclinations and rheological properties on the front height and front velocity, which also deflne the displacement e–ciency. † For displaced uids with a yield stress it is possible for the displaced uid to remain static on the wall of the channel. We analyse the maximal static layer thickness. The simpliflcation of the plane channel allows us to develop the lubrica- tion model for a very wide range of uid types. Later in this thesis (Chap- ters 6 and 7) we develop a similar model for iso-viscous Newtonian uids in a pipe. In Chapter 8 we analyse the plane channel model further, for Newtonian and power law uids of difierent viscosities. Thus, the methods of this chapter have much wider application. β Dˆ 0 ˆV Light Heavy yˆ xˆ ˆ ˆˆ ˆ( , )y h x t= Figure 5.1: Schematic of displacement geometry. 5.1 Two- uid displacement ows in a nearly horizontal slot We consider a two-dimensional region between two parallel plates, separated by a distance D^, that are oriented at an angle fl … …=2 to the vertical. The 775.1. Two- uid displacement ows in a nearly horizontal slot slot is initially fllled with uid 2, which is displaced by uid 1, injected at x^ = ¡1 with a mean velocity V^0. Cartesian coordinates (x^; y^) are as shown in Fig. 5.1. Both uids are assumed to be generalised Newtonian uids, with rhe- ologies described below, and although the uids are miscible we consider the large P¶eclet number limit in which no signiflcant mixing occurs over the timescales of interest. The dimensionless equations of motion, valid within each uid region ›k; k = 1; 2, are: `kRe •@u @t + u @u @x + v @u @y ‚ = ¡@p@x + @ @x¿k;xx + @ @y ¿k;xy + `k cos fl St ; (5.1) `kRe •@v @t + u @v @x + v @v @y ‚ = ¡@p@y + @ @x¿k;yx + @ @y ¿k;yy ¡ `k sin fl St ; (5.2) @u @x + @v @y = 0: (5.3) Here u = (u; v) denotes the velocity, p the pressure, and ¿k;ij is the ij-th component of the deviatoric stress in uid k. The parameter `1 · 1, and the 3 dimensionless parameters appearing above are the density ratio `2, the Reynolds number, Re, and the Stokes number, St, deflned as follows. `2 = ` · ‰^2‰^1 ; Re · ‰^1V^0D^ „^1 ; St · „^1V^0 ‰^1g^D^2 : (5.4) Here ‰^k is the density of uid k, „^1 is a viscosity scale for uid 1 and g^ is the gravitational acceleration. Further dimensionless parameters will appear in constitutive laws, deflning the deviatoric stresses. In order to derive (5.1)- (5.3) we have scaled distances using D^, velocities with V^0, time with D^=V^0, pressure and stresses with „^1V^0=D^. On the walls of the slot the no-slip condition is satisfled. Due to the scaling adopted, we have Z 1 0 u dy = 1: (5.5) in each cross-section. The slot is assumed inflnite in x, with the interface between uids initially localised close to x = 0. We shall consider ows that are buoyancy dominated, in which the heavier uid lies at the bottom of the slot, separated from the lighter upper uid by an interface that we denote 785.1. Two- uid displacement ows 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 ow, satisfying a kinematic condition. 5.1.1 Constitutive laws The uids are assumed to be generalised Newtonian uids. In particular we are interested to understand shear thinning and yield stress efiects. A suitable model that incorporates these efiects is the Herschel-Bulkley model, which incorporates also the simpler Bingham, power law and Newtonian models. Constitutive laws for the Herschel-Bulkley uids are: _ (u) = 0 , ¿k(u) • Bk; x 2 ›k; (5.6) ¿k;ij(u) = • •k _ nk¡1(u) + Bk_ (u) ‚ _ ij(u) , ¿k(u) > Bk; x 2 ›k: (5.7) where the strain rate tensor has components: _ ij(u) = @ui@xj + @uj @xi ; (5.8) and the second invariants, _ (u) and ¿k(u), are deflned by: _ (u) = 2 41 2 2X i;j=1 [ _ ij(u)]2 3 5 1=2 ; ¿k(u) = 2 41 2 2X i;j=1 [¿k;ij(u)]2 3 5 1=2 : (5.9) Herschel-Bulkley uids are described by 3 dimensional parameters: a uid consistency •^, a yield stress ¿^Y and a power law index, n. The parameter •1 = 1 and •2 is the viscosity ratio m: m · „^2„^1 = •^2[V^0=D^]n2¡1 •^1[V^0=D^]n1¡1 ; (5.10) where „^2 is a viscosity scale for uid 2. Note that in the case of 2 Newtonian uids, „^k = •^k. The Bingham numbers Bk are deflned as: Bk · ¿^k;Y •^1[V^0=D^]n1 : (5.11) 795.1. Two- uid displacement ows in a nearly horizontal slot 5.1.2 Buoyancy dominated ows 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, fl … …=2 and ` » O(1). The ratio of buoyancy to viscous forces is given by the parameter j` ¡ 1j=St. We suppose that j` ¡ 1j=St 1 so that the interface elongates over some (dimensionless) length-scale –¡1 1. To deflne this length-scale we assume that the dynamics of spreading of the interface, relative to the mean ow, will be driven by buoyant stresses which have size: j‰^1 ¡ ‰^2jg^ sin flD^ in the y-direction. These stresses, which act across the interface where there is a density difierence, translate into axial stresses according to the slope of the interface. If the slope of the interface has size D^=L^, the stress that acts to spread the ow axially has size j` ¡ 1j‰^1g^ sin flD^2=L^. This tendency to spread is resisted by viscous stresses within the uids, of size „^1V^0=D^, which dissipate the energy injected by buoyancy. By matching these two terms, we can obtain the characteristic spreading length in this regime : j`¡ 1j‰^1g^ sin flD^2=L^ = „^1V^0=D^ ) L^ = j`¡ 1j‰^1g^ sin flD^ 3 „^1V^0 (5.12) Thus, the ratio between the axial length-scale and channel width is: –¡1 = L^D^ = j`¡ 1j‰^1g^ sin flD^2 „^1V^0 = j`¡ 1j sin flSt (5.13) Following standard methods, see e.g. [93], we re-scale as follows –x = »; –t = T; –p = P; v = –V; and arrive at the following reduced system of equations, in each uid region ›k; k = 1; 2: –`kRe • @u @T + u @u @» + V @u @y ‚ = ¡@P@» + @ @y ¿k;»y + `k cos fl St +O(– 2); –3`kRe •@V @T + u @V @» + V @V @y ‚ = ¡@P@y ¡ –`k sin fl St +O(– 2); @u @» + @V @y = 0: 805.1. Two- uid displacement ows in a nearly horizontal slot To aid interpretation of our model results, note that the time and length variables, (T; »), are related to the dimensional time and length by: j‰^1 ¡ ‰^2jg^ sin flD^3 „^1V^0 » = x; j‰^1 ¡ ‰^2jg^ sin flD^ 3 „^1V^ 20 T = t (5.14) Note that we have used D^=V^0 to scale t, which is the usual convective timescale based on the mean velocity and D^. Therefore, the scale related to the slow time variable, T , corresponds to the time taken to travel the characteristic spreading length L^ at mean velocity V^0. We now consider the limit – ! 0 with Re flxed: 0 = ¡@P@» + @ @y ¿k;»y + ´ `k j1¡ `j ; (5.15) 0 = ¡@P@y ¡ `k j1¡ `j ; (5.16) where ´ = cot fl=–. The parameter ´ 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 ´ is an order 1 parameter, i.e. we consider inclinations fl = …=2+O(–). For ´ > 0 the slope of the channel is \downhill", in the direction of the ow, and for ´ < 0 the ow is uphill. Note that for larger ´ the model does not necessarily break down, but efiectively we have chosen the wrong scaling as the efiect of the channel slope is dominant. Before proceeding, we observe that there are 2 qualitatively difierent types of displacement ows. (i) HL (heavy-light) displacement: uid 1 is heavier than uid 2, and the lower layer of uid is consequently uid 1. Parameters are: (nH ; •H ; BH ; nL; •L; BL) = (n1; 1; B1; n2;m;B2). (ii) LH (light-heavy) displacement: uid 1 is lighter than uid 2, and the lower layer of uid is consequently uid 2. Parameters are: (nH ; •H ; BH ; nL; •L; BL) = (n2;m;B2; n1; 1; B1). These are illustrated schematically in Fig. 5.2. We do not consider mechan- ically unstable conflgurations, i.e. heavy uid over light uid. 815.1. Two- uid displacement ows in a nearly horizontal slot Heavy0ˆV 0 ˆV Light Light Heavy a) b) Figure 5.2: Schematic of displacement types considered: a) Heavy uid displaces Light uid, (HL displacement); b) Light uid displaces Heavy uid, (LH displacement). We integrate (5.16) across both uid layers to give the pressure: P (»; y; T ) = 8 >>>>>>< >>>>>>: P0(»; T ) + ´ `Hj1¡ `j» ¡ `H j1¡ `jy y 2 [0; h]; P0(»; T ) + ´ `Hj1¡ `j» ¡ `H ¡ `L j1¡ `j h¡ `L j1¡ `jy y 2 [h; 1]; (5.17) where P0(»; T ) is deflned by: P0(»; T ) = P (»; 0; T )¡ ´ `Hj1¡ `j»; with `H = ‰^H=‰^1 for the heavier uid, `L = ‰^L=‰^1 for the lighter uid. On substituting into (5.15), we arrive at: 0 = ¡@P0@» + @ @y ¿H;»y; y 2 (0; h); (5.18) 0 = ¡@P0@» + @ @y ¿L;»y ¡ ´+ @h @» ; y 2 (h; 1): (5.19) 825.1. Two- uid displacement ows in a nearly horizontal slot In the lubrication approximation, the leading order strain rate compo- nent is _ »y = @u@y , and the leading order shear stress ¿k;»y is deflned in terms of _ »y via the following leading order constitutive laws: @u @y = 0 , j¿k;»yj • Bk; x 2 ›k; (5.20) ¿k;»y = 2 664•k flflflfl @u @y flflflfl nk¡1 + Bkflflflfl @u @y flflflfl 3 775 @u @y , j¿k;»yj > Bk; x 2 ›k: (5.21) Thus, for given h and @h@» , (5.18) & (5.19) deflne 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 ¿H;»y = ¿L;»y, representing stress continuity. These 4 conditions are su–cient to determine u for given @P0@» . The pressure gradient is determined by the additional constraint that (5.5) is satisfled. For now we assume that the solution of this problem may be computed and we note that the dependence of u on (»; T ) enters only via h(»; T ), which satisfles @h @T + u @h @» = V: (5.22) Combining the kinematic equation with the divergence free constraint leads, in the usual manner, to the equation: @h @T + @ @» q(h; h») = 0; (5.23) where q(h; h») is deflned as: q(h; h») = Z h 0 u(y; h; h») 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(»; T ) ! 1; as » ! ¡1; h(»; T ) ! 0; as » !1; (5.25) as the channel is assumed full of pure uid 1 and uid 2 at the two ends of the channel. As initial conditions we note that an initial proflle in the unscaled variables h(x; t = 0) = h0(x) is transformed to h(»; T = 0) = h0(»=–). Since h0 should be compatible with the far-fleld conditions we have that as – ! 0, h(»; 0) ! 1¡H(»); (5.26) 835.1. Two- uid displacement ows in a nearly horizontal slot where H(») is the usual Heaviside function. In other words, in terms of », the initial change in h is localised to » = 0. For a LH displacement this is reversed, i.e. h(»; T ) ! 0; as » ! ¡1; h(»; T ) ! 1; as » !1; (5.27) h(»; 0) = H(»); (5.28) since the far-fleld pure uids are reversed. 5.1.3 The ux function q(h; h») In the general case, flnding the ux function q(h; h») requires computation, and this is addressed in Appendix A. For the particular case of a Newtonian uid the analytical solution may be found trivially. Denoting b = ´ ¡ h», for a HL displacement we flnd: 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 ux q(h; b;m): qA(h;m) = 3mh 2(mh2 + (h+ 3)(1¡ h)) 3[(1¡ h)4 + 2mh(1¡ h)(h2 ¡ h+ 2) +m2h4] (5.30) qB(h;m) = [h 3(1¡ h)3(mh+ (1¡ h))] 3[(1¡ h)4 + 2mh(1¡ h)(h2 ¡ h+ 2) +m2h4] : (5.31) For a LH displacement, the ux 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 uids, 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 re ective symmetry, as do those for b = §10 in Fig. 5.3b, (with m = 1). Note also that in Figs. 5.3a & b, the ux 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 uxes are mathematically identical for the same b, in fact b = ´ ¡ h» will not be the 845.1. Two- uid displacement ows 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 uids: a) b = 0 and difierent 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 difierent b; HL or LH displacements with b = ¡10 (–), b = 0 (O), b = 10 (⁄). Examples of q for 2 non-Newtonian uids 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 h» will have difierent sign between the two displacement types. In addition, m is the ratio of displaced to displacing uid 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 ´ = 0, (see later Figs. 5.4c & d, and will be the same shape for small inclinations if we retain the same m and replace ´ with ¡´. This does not therefore contradict observations from lubrication-type 855.1. Two- uid displacement ows in a nearly horizontal slot models of iso-density displacements with central flnger-like interfaces, where the cases m and 1=m also produce markedly difierent results. Figs. 5.3c & d illustrate non-Newtonian efiects on q in HL displacements. In Fig. 5.3c we observe that as the heavy uid yield stress, B1, is increased q = 0 in some interval of small h. For these thin layers the yield stress uid remains static. In Fig. 5.3d we see that the efiects of viscosity ratio m is broadly similar for non-Newtonian and Newtonian uids. For the examples shown q increases monotonically with little apparent efiect 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 di–cult to flnd 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 difierences are signiflcant. 5.1.4 The existence of steady traveling wave displacements One of the most important practical questions in considering this displace- ment ow is whether or not (5.23) & (5.24) admit steady traveling wave so- lutions. This determines whether or not the displacement can be efiective. In this section we demonstrate that, regardless of uid type and of rheo- logical difierences between uids, 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 difierent uid types. First let us note that the slope of the interface h» acts always to spread the interface. To see this note that following the construction of the previous section, we may write q(h; h») = q(h; b) where b = ´¡h». Formally we may write (5.23) as @h @T + @q @h @h @» = ¡ @q @b @b @» = @q @b @2h @»2 ; (5.33) from which we see that the interface spreads difiusively 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 uid 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 = » ¡ T , we see that if the solution is steady in this frame, h = h(z), 865.1. Two- uid displacement ows in a nearly horizontal slot we must have that d dz • h¡ q(h; ´¡ dhdz ) ‚ = 0; and since q = 0 at h = 0, this implies that: h = q(h; ´¡ dhdz ); (5.34) must be satisfled for all h 2 [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 = ´¡ 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; ´) • h for all h 2 [0; 1]. For a LH displacement, a necessary condition for there to be steady traveling wave solution is that q(h; ´) ‚ h for all h 2 [0; 1]. This follows directly since for a HL displacement we require that dhdz • 0 so that q(h; b) ‚ q(h; ´). If this condition is not satisfled we would therefore be unable to flnd 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 su–cient as well as necessary. Finally, we shall show that the conditions of lemma 5.1.2 are in fact never satisfled. We focus only on the HL displacement, the LH displacement being treated similarly. We consider solutions u(y) to the system @ @y ¿H;»y = ¡f; y 2 (0; h); @ @y ¿L;»y = ´¡ f; y 2 (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 ow rate constraint (5.5), which determines f . We flx ´ and consider h = 1 ¡ †, noting flrst that both the velocity solution and f(h) will vary smoothly with h. For any h 2 [0; 1] we note that the shear stress throughout the light uid layer is given by: ¿L;»y(y;h) = ¿L;»y(1;h) + (1¡ y)(f(h)¡ ´); 875.1. Two- uid displacement ows in a nearly horizontal slot and as h ! 1, we have ¿L;»y(y;h) » ¿L;»y(1; 1)¡ † @¿L;»y @h (1; 1) + †(f(1)¡ ´) +O(† 2): Thus, the velocity gradient within the light uid layer is given by: @u @y = @u @y (¿L;»y(1; 1)) +O(†); where the algebraic relation for the velocity gradient comes directly from the constitutive laws. Hence we may straightforwardly compute the ux in the lighter uid layer: qL(†) = Z 1 h u(y) dy » ¡† 2 2 @u @y (¿L;»y(1; 1)) +O(† 3): Now when h = 1 the channel is full with the heavy uid, and the pressure gradient corresponds to the Poiseuille ow 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: ¿L;»y(1; 1) = ¡0:5fH(1) < 0 ) qL(†) » ¡† 2 2 @u @y (¡0:5fH(1)) > 0: Since via the ow rate constraint we have that the total ux is equal to unity, we have that q(h; ´) » 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 su–ciently close to h = 1, regardless of uid type and rheological difierences. Similarly, we can show that for a LH displace- ment the necessary conditions of lemma 5.1.2 are always violated su–ciently close to h = 0, regardless of uid type and rheological difierences. This leads to the following result. Lemma 5.1.3 There are no steady traveling wave solutions to (5.23). Remarks: 885.2. Newtonian uids † This is the key theoretical result of the chapter. It is perhaps surprising that for no combination of rheology or density difierences 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 e–ciency, close to 100%. Secondly, if we wish to improve the e–ciency 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 uid displacement, we might flnd this result rather more directly as the solution may be computed. For example, in [135] the simpler problem of 2 Newtonian uids of identical viscosity in an inclined pipe is considered, in the absence of a mean imposed ow. No traveling wave solutions are found. Here however, the mean ow re- sults in a difierent structure to the ux functions q, i.e. for Newtonian uids the advective and buoyant components, qA and qB, are present whereas only qB is present in [135], (also with an algebraically difierent form). For non-Newtonian uids the division of the ux into qA and qB is not possible, due to nonlinearity. Thus, we have to work with qualitative properties of the uxes for such uids. 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 ux (i.e. a displacement) makes this a natural and legitimate question. † Although we have focused on Herschel-Bulkley uids for deflniteness, the same results could be demonstrated for any of the popular gen- eralised Newtonian models, e.g. Carreau uids, Cross model, Casson model, etc.. . 5.2 Newtonian uids We commence with an analysis of Newtonian uid displacements. Although the industrial applications discussed in Chapter 1 and Chapter 2 typically involve non-Newtonian uids, many of the qualitative behaviours are ex- hibited in a Newtonian uid displacement. Analysis of the Newtonian uid case not only provides simpliflcation in terms of the number of dimensionless parameters, i.e. (m;´), but also since q is given by the analytical expression 895.2. Newtonian uids (5.29) numerical solution is considerably faster. For non-Newtonian uids, each evaluation of q requires numerical solution of the nested iteration de- scribed in Appendix A. The convection-difiusion equation (5.23) was discretized in the conser- vative form, second order in space and flrst order in time; and afterward, integrated straightforwardly by using a Lax-Wendrofi scheme in which an artiflcial dissipation was added to the equation to compensate for the desta- bilizing efiects of the known anti-difiusion due to the flrst 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 ux 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 »: typically h(»; T = 0) = currency1»§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 proflle, e.g. in Fig. 5.4a the initial condition is h(»; T = 0) = ¡»+0:05. Figs. 5.4a & b plot the solution for m = 1, ´ = 0, (i.e. equal viscosities in a perfectly horizontal channel). In the early times, T 2 [0; 1] we observe that the interface develops quickly into a slumping proflle; 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 proflles of h may be conveniently plotted against »=T , in which variable the interface proflles collapse to a single similarity proflle as T ! 1; see Fig. 5.4b. To clarify interpretation of flgures such as Fig. 5.4b, the x-axis of the flnal similarity proflle gives the speed of the interface at difierent heights: vertical lines correspond to segments of the interface that advance at steady speed. Note that the flrst interface proflle in Fig. 5.4b, for T = 1, efiectively shows h(»; 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, » = ¡0:5. The convergence at the upper wall as T ! 1 simply follows »=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 proflle of Fig. 5.4a and the flrst interface proflle of Fig. 5.4b is simply due to the difierent initial conditions. 905.2. Newtonian uids 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(»; T ) for T = 0; 0:1; ::; 0:9; 1, parameters ´ = 0, m = 1; b) h(»=T ) for T = 1; ::; 9; 10, parameters ´ = 0, m = 1. Examples of HL displacements: c) h(»=T ) for ´ = 0: m = 0:1 (–), m = 1 (O), m = 10 (⁄); d) h(»=T ) for m = 1: ´ = ¡10 (–), ´ = 0 (O), ´ = 10 (⁄). The inset flgures 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 \flnal" similarity proflle 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 flnal similarity proflle. Fig. 5.4c shows the flnal shape for 3 difierent values of viscosity ratio m and Fig. 5.4d shows the flnal shape for 3 difierent values of the inclination parameter ´. 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 uid displacing a more viscous uid. The interface above the steadily moving front also transitions from convex to concave curvature as m is increased, further emphasizing the extending flnger. Similarly, for ´ > 0 915.2. Newtonian uids the heavy uid ows downhill through the lighter uid and hf is accordingly smaller in this conflguration. The inset flgures in Fig. 5.4c & d show the analogous LH displacements for the same parameters. The efiects of m are identical with those for the HL displacement, (since m is the ratio of in- situ uid viscosity to displacing uid viscosity). The efiect of varying ´ is however reversed: ´ > 0 retards unsteady spreading for an LH displacement and ´ < 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 ! 1. 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 @» » ¡ 1¡ hf VfT ! 0; as T !1: Therefore, as T ! 1, we have that b = ´ ¡ @h@» ! ´, and the interface motion in the stretched region is governed approximately by: @h @T + @ @» q(h; ´) = 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; ´): Thus, the total area of uid owing behind the interface in the interval [hf ; 1] at long times is T Z 1 hf Vi(h) dh = T [1¡ q(hf ; ´)]: 925.2. Newtonian uids Furthermore, at the front height hf the interface speed should equal the front velocity Vf , i.e. @q @h(hf ; ´) = Vi(hf ) = Vf : (5.37) The total area of uid behind the interface is T and since the area of uid owing behind the interface in the interval [0; hf ] is approximately TVfhf , we have the following relationship: Tq(hf ; ´) = T ¡ T [1¡ q(hf ; ´)] = TVfhf = Thf @q@h(hf ; ´); from which: q(hf ; ´) = hf @q@h(hf ; ´): (5.38) Equation (5.38) is an equation for the front height hf . This is instantly recognisable as the same condition that must be satisfled 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 »=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 uids this appears to arise at more extreme parameter values when physical efiects are somehow opposing one another. An example is shown in Figs. 5.5c & d. In this illustration, the competing efiects 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 difierent values of ´ 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 e–cient displacement, the displacing uid should be more viscous in comparison to the displaced uid. Increasing ´ tends to reduce e–ciency for the HL displacement but increase e–ciency for the LH displacement. Via repeated computations of q for difierent (m;´) we are able to delineate the 6To interpret this flgure for the LH displacement The front height hf for LH displace- ment is deflned as the distance from the top wall to the stretched part of the interface. 935.2. Newtonian uids 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, ´ = 10; m = 8; b) h plotted against »=T for T = 1; ::; 9; 10, parameters ´ = 10; m = 8, broken horizontal line indicates the front height determined from (5.38); c) two front heights, ´ = 10; m = 0:08; d) h plotted against »=T for T = 1; ::; 9; 10, parameters ´ = 10; m = 0:08, broken horizontal lines indicate the front heights. regime in the (m;´)-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 back ow 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 difiusive efiects are always present for h 2 (0; 1). Having determined hf from (5.38) and then Vf from (5.37), we may shift to a moving frame of reference z = »¡VfT and seek a steadily traveling solution to (5.23), which satisfles: d dz • hVf ¡ q(h; ´¡ dhdz ) ‚ = 0; ) hVf ¡ q(h; ´¡ dhdz ) = 0: (5.39) 945.2. Newtonian uids 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 uid HL displacement with ´ = ¡10 (O), ´ = ¡5 (⁄), ´ = 0 (.), ´ = 5 (–), ´ = 10 (/). This flgure also gives the front heights for a Newtonian uid LH displacement with ´ = 10 (O), ´ = 5 (⁄), ´ = 0 (.), ´ = ¡5 (–), ´ = ¡10 (/). For the LH displacement the front height is measured down from the top wall; b) Parameter regime in the (m;´)-plane in which multiple fronts (shaded area). Elsewhere there is only a single front. Equation (5.39) must be solved numerically for h 2 (0; hf ). Example shapes are shown in Fig. 5.7. Fig. 5.7a shows HL displacement front shapes for 2 Newtonian uids for difierent values of viscosity ratio at ´ = 0. Fig. 5.7b shows HL displacement front shapes for 2 Newtonian uids for difierent values of ´ 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; ´¡ dhdz ) ! q(hf ; ´) as h ! h ¡ f ; which implies that dhdz ! 0 as h ! h¡f , as can be seen in Fig. 5.7. Evidently, as T ! 1 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 efiects, compatible with the assumed stratiflcation. Our study of the long- time behaviour has revealed only forward propagating fronts, which of course 955.2. Newtonian uids 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) ´ = 0, m = 0:1 (⁄);m = 1 (–);m = 10 (.); b) ´ = ¡10 (O), ´ = ¡5 (–), ´ = 0 (.), ´ = 5 (⁄), ´ = 10 (/). Compare with transient computations in Figs. 5.4c & d. are more common since a positive ow 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 ow also diminishes. Thus, we cannot expect ow 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 ow. For example, with a HL displacement at flxed positive inclination, ´ > 0, buoyancy acts to push the lighter uid against the mean ow direction. For su–ciently large ´ and small viscosity ratio, we observe that the lighter uid may be driven backwards against the ow, resulting in a sustained ow 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 ow. Since our model is anyway an asymptotic reduction of the full equations in which T efiectively 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-deflned and of physical interest. To study this limit, we shift to the steadily moving frame of reference z = »¡T , recall that b = ´¡h», and consider (5.23) for an HL displacement, 965.2. Newtonian uids −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: Proflles of h(»; T ) for T = 0; 1; ::; 9; 10, with parameters ´ = 50, m = 0:1, illustrating ow reversal. which becomes: @h @T + @ @z • qA(h;m) + (´¡ @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 ux q(h; b;m), which is deflned by (5.29) for 2 Newtonian uids, i.e. Introducing · = z=pT this becomes: 1 2· dh d· ¡ p T dd· (qA ¡ h+ ´qB) + qB d2h d·2 + @qB @h dh d· ¶2 = 0; (5.41) Therefore, provided that T » 0 we may seek a similarity solution satisfying: 1 2· dh d· + qB d2h d·2 + @qB @h dh d· ¶2 = 0; (5.42) or in conservative form: 1 2· dh d· = d d· ¡qB dhd· ¶ : (5.43) 975.2. Newtonian uids Since qB(h;m) vanishes at both h = 0 and h = 1, it is clear that there is some singular behaviour in h(·) at these points. Thus, it is more comfortable to work with the function ·(h). The boundary conditions are, ·(0) = ·0 and ·(1) = ·1, where ·0 & ·1 are unknown at this stage. Physically we expect that ·0 > 0 & ·1 < 0 as the spreading of the interface is caused by gravitational slumping. A Taylor expansion reveals that ·(h) » ·0 +O(h3) as h ! 0, with similar asymptotic behaviour as h ! 1, i.e. ·0(h) ! 0 quadratically at both ends of the interval. We integrate equation (5.43) as follows: 1 2·dh = d ¡qB dhd· ¶ (5.44) Z h 0 1 2· dh = Z h 0 d ¡qB dhd· ¶ = ¡qB(h)dhd· + qB(0) dh d· = ¡qB(h) dh d· ; (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 Z 1 0 · dh = 0: (5.46) Let us now deflne g(h) such that · = g0. Therefore, g(h)¡ g(0) = Z h 0 ·dh; (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: g00g = ¡2qB: (5.48) We use the initial condition g(0) = 0 and g0(0) = ·0. We then integrate forward, with respect to h and iterate on ·0 via a shooting method to satisfy g(1) = 0. This numerical procedure appears to work well. Figure 5.9a plots the similarity solutions ·(h) for various m. Note that the solution is not sym- metric with respect to m. For the heavy-light displacement the heavy uid viscosity is 1 and the light uid viscosity is m. Buoyancy efiects have no bias between the uids, but the more viscous uids evidently resist motion. Thus, we see that for large m the axial extension ·0 ¡ ·1 is smaller than for small m. This efiect might have been removed had we scaled viscosity 985.2. Newtonian uids 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(·) 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 ow should occur, relative to the mean displacement. For smaller mean velocities the parameter – ! 0 and the dimensional time period over which buoyancy dominates extends to inflnity, ensuring compatibility with exchange ow studies, for which there is zero net ow rate and hence a ow reversal in each layer. To explore this analogy further, let us flx fl = …=2, in which case we may note that the similarity variable · is deflned in terms of dimensional variables by: · = zT 1=2 = x^¡ V^0t^ t^1=2 s „^1 j‰^1 ¡ ‰^2jg^D^3 : We may compare this with the analysis in [135] for exchange ows in hor- izontal pipes, wherein difiusive similarity proflles are found for Newtonian uids of the same viscosity. We may note that the scaling j‰^1¡ ‰^2jg^D^3=„^1 is the same as the (V^”D^)1=2 that scales the similarity variable x^=t^1=2 in [135], (see equation (27) and xVII.B in this paper). However, although this is the same viscous-buoyancy balance driving the difiusive spreading in both cases, here we have the additional criterion that T 1=2 ¿ 1, and we have seen 995.3. Non-Newtonian uids numerically that the difiusive regime does not last for longer times. This criterion can be written dimensionally as: t^V^0 D^ ¿ 1 V^0 j‰^1 ¡ ‰^2jg^D^2 „^1 = L^ D^ : The most simplistic interpretation therefore is that t^V^0 ¿ L^ , 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 difiusion is measured in a frame of reference moving with the mean velocity. The criterion t^V^0 ¿ L^ also means that the moving frame has not moved very far relative to the stationary frame in which the usual exchange ow analysis takes place. 5.3 Non-Newtonian uids We turn now to results for non-Newtonian uids. 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 ow of a power law uid. The strain rate in the uid is proportional to the pressure gradient to the 1=n-th power, and hence the areal ow rate also. In a two-layer ow of the type we have, the short-time behaviour is dominated by that part of q(h; h») driven by the pressure gradient due to the slope of the interface. However, the ux in uid layer k is proportional to jh»j1=nk and the two uxes are coupled via the ow rate constraint. Thus, it is immediately obvious that there can be no single similarity variable unless the two uids happen to have the same shear-thinning index. In this case the similarity variable is · = z=tn=(n+1). Although of mathematical interest, the practical interest is limited. 5.3.1 Shear-thinning efiects We commence by considering only shear-thinning efiects, Bk = 0, and shall also focus only on HL displacements. Figs. 5.10a & b show the flnal similarity proflles of the interface for m = 1 and ´ = 0, i.e. the only efiects are the relative values of the two power law indices. We observe that for flxed nH 1005.3. Non-Newtonian uids the front height increases as nL decreases. Conversely, for flxed nL the front height decreases as nH decreases. Both efiects are essentially predictable, in that with all other parameters flxed (or neutralised in the case of inclination, ´ = 0), varying the power law indices makes one uid 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 uids, Bk = 0, ´ = 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 efiects are found when the \bulk" viscosity of one uid is for example large but has smaller power law index than the other uid. 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 uids. These displacements are shown in Fig. 5.10c and we see that in fact the latter case displaces 1015.3. Non-Newtonian uids 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 uids, 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 ´ = ¡10 (O), ´ = ¡5 (⁄), ´ = 0 (.), ´ = 5 (–), ´ = 10 (/), and the heavy broken line indicates multiple fronts. 1025.3. Non-Newtonian uids 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 efiects can result in displacements that are close to 100% e–cient. An example of this are shown in Fig. 5.10d, where for m = 0:1, nH = 1 we show the efiects of decreasing nL. Note that as nL ! 0, the light uid efiectively 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 uid displacements of the previous section. The long-time behaviour can be analyzed over a wide range of parameters by direct treat- ment of the ux 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 difiuse. The front speed is calculated straightforwardly for Newtonian displacements, but for non- Newtonian uids this is more laborious. A slightly difierent interpretation of the front speed is as an indicator of displacement e–ciency. No single measure or deflnition is universal, e.g. for flnite length ducts it is common to present quantities such as the volume fraction displaced after 1 volume of displacing uid has been pumped, or alternatively after an inflnite volume has been pumped. Here we deflne: Displacement E–ciency = 1Vf : (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 uid is flrst seen at unit length down- stream. Examples of variations in front height and speed, for difierent ´ and m, as either nH or nL is reduced, are shown in Fig. 5.11. Essentially the displacement e–ciency increases as the displacing uid 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 difierent transitions occur. For 2 Newtonian uids the occurrence of multiple fronts is relatively easy to identify, as there are essentially only 2 1035.3. Non-Newtonian uids efiects that compete: viscosity and buoyancy, see Fig. 5.6. However, for power law uids we may have uid combinations that are either more or less viscous than each other, for difierent shear rates, and these efiects are then complemented with efiects of difierent channel inclinations. Thus, the possible combinations of efiects 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 ´ > 0 and for suitable viscosity ratios. For example, in Fig. 5.11b at small m for ´ = 10, the heavy broken line indicates 2 moving fronts, but one front has negative speed, (hence the decrease in e–ciency). The jump in Fig. 5.11b, (at small m for ´ = 10), in fact indicates a transition from 2 fronts to 3 fronts: 2 moving forward and 1 moving backwards! 5.3.2 Yield stress efiects We turn now to yield stress uids and for simplicity we set nk = 1, i.e. these are Bingham uids. Such uids are in any case shear-thinning, due to the yield stress, but no additional power law behaviour is considered. We start by examining the efiects of a single yield stress on a Newtonian displace- ment, (for m = 1; ´ = 0), by increasing either BH or BL. Again only HL displacements are considered. Fig. 5.12 shows the interfaces at T = 10, plotted against »=T for each of these cases. It can be observed that increas- ing BH improves the displacement, due to the enhanced efiective viscosity, Fig. 5.12a. Similarly, increasing BL makes the displacement less e–cient, 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 uid 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 ow and as a static situation, see e.g. [4, 54]. We discuss static wall layer solutions further in x5.3.2. The long-time analysis of solutions is qualitatively similar to that dis- cussed earlier. Examples showing the efiects of ´ and m on the front height and speed are shown in Fig. 5.13. General efiects of varying m, ´ and Bk are mostly in line with our physical intuition, i.e. efiects that make the displac- ing uid more viscous usually (but not always) improve the displacement. However, for parameter ranges where some ambiguity exists, this type of computation determines which efiects dominate. We also observe the same 1045.3. Non-Newtonian uids 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: Proflles of h plotted against »=T at T = 10: a) ´ = 0; nk = 1; BL = 0;m = 1, BH = 1(–), BH = 5(O), BH = 20(⁄); b) ´ = 0; nk = 1; BH = 0;m = 1, BL = 1(–), BL = 5(O), BL = 20(⁄). range of difierent 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 flgures previously), Fig. 5.14 illustrates the two difierent type of transition, by showing @q@h(h; h» = 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; h» = 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; h» = 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 e–ciency. The static wall layer The deflning novel feature of a yield stress uid displacement is the pos- sibility for residual uid to remain permanently in the channel, i.e. even asymptotically as T ! 1 a fraction of uid 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 1055.3. Non-Newtonian uids 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: ´ = ¡10(O), ´ = ¡5(⁄), ´ = 0(.), ´ = 5(–), ´ = 10(/) for all plots. Broken heavy line indicates multiple fronts. parallel ow of 2 uids. If the wall stress created by the displacing uid, owing at unit ow rate through the channel, does not exceed the yield stress of the displaced uid, it follows that there could be a static residual layer on the wall. It can also be argued that there exists a uniquely deflned 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 ; ~B1 = BH•H ; ’Y = BH BL ; ’b = ´ BL (5.50) The parameter ~B1 is a rescaled Bingham number, relevant to the displac- 1065.3. Non-Newtonian uids 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, ´ = 10, m = 0:1, multiple fronts; b) m = 0:2, single front; c) ´ = 0, BH = 5, BL = 0, m = 2:3, multiple fronts; d) ´ = 0, BH = 5, BL = 0, m = 2:4, single front. ing uid; ’Y is simply the yield stress ratio and ’b measures the ratio of buoyancy stress due to the slope of the channel and the yield stress of the displaced uid. The critical condition for the existence of any static wall layer is independent of the buoyancy ratio, ’Y . Fig. 5.15 shows the variation in maximum static wall layer Ystatic with the parameters ’Y and 1~B1 for 3 flxed values of the ratio ’b. The shaded areamarks 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 ~B1 = BH=•H . As ’b increases from negative to positive the static layer thickness is increasing. The limit BH ! 0 must be treated separately. Straightforwardly, we flnd that Ystatic depends on nH , ~´ = ´=•H and ~B2 = BL=•H . Fig. 5.16 shows the variation in maximum static wall layer with the parameters ~´ and ~B for 4 1075.3. Non-Newtonian uids ϕY 1 ˜B1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 a) ϕY 1 ˜B1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 b) ϕY 1 ˜B1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 c) ϕY 1 ˜B1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 d) ϕY 1 ˜B1 0 0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1 e) ϕY 1 ˜B1 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 ; ~B1; ’Y ; ’b), with contours spaced at intervals ¢Ystatic = 0:1: a) ’b = ¡2; nH = 1; b) ’b = ¡2; nH = 0:2; c) ’b = 0; nH = 1; d) ’b = 0; nH = 0:2; e) ’b = 2; nH = 1; f) ’b = 2; nH = 0:2. difierent flxed 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 1085.4. Summary we may transition from having no static layer to having a flnite 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 uid that move do so very slowly as the static layer criterion is violated. 1 ˜B2 χ˜ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 a) 1 ˜B2 χ˜ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 b) 1 ˜B2 χ˜ 0 0.1 0.2 0.3 0.4 0.5 0.6 −10 −5 0 5 10 c) 1 ˜B2 χ˜ 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 uid displaces a Herschel-Bulkley uid, 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 fllm model and developed a semi-analytical solution method to flnd the ux function. † We have shown that there are no steady traveling wave solutions to the displacement problem, in the lubrication/thin fllm limit. 1095.4. 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(»; T ) for T = 0; 1; ::; 9; 10, parameters ´ = 10; m = 1; nH = 1=4; nL = 1; B1 = 0; B2 = 2; b) h(»; T ) for T = 0; 1; ::; 9; 10, parameters ´ = 10; m = 1; nH = 1=4; nL = 1; B1 = 0; B2 = 1. † At short times, difiusive efiects of the interface slope are dominant and there is an exchange ow relative to the mean ow. We have found a short-time similarity solution governing this initial counter-current ow. † 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 efiects of viscosity ratio, inclinations, and other rheological properties. (i) More e–cient displacements are generally obtained with a more viscous displacing uid. (ii) Modest improvements in displacement e–ciency 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 e–ciencies by more viscous uids. (iv) Generally, a yield stress in the displacing uid increases the dis- placement e–ciency and yield stress in the displaced uid de- 1105.4. Summary creases the displacement e–ciency, eventually leading to com- pletely static residual wall layers of displaced uid. (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 uid, or with a mild buoyancy difierence opposing the ow. It decreases with buoyancy difierence in the direction of the imposed ow and with increases in the efiective viscosity of the displacing uid. 111Chapter 6 Stationary residual layers in Newtonian displacements7 In Chapter 4 we identifled 3 regimes in our displacement ows, as a function of the mean ow velocity (V^0) increasing from zero: (i) an exchange ow 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 flrst two of these regimes. At the outset it is not obvious how this transition should be deflned. 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 V^0 is increased from zero. We could identify the transition with respect to the curve V^f vs V^0. b) An interesting phenomenon was observed during many of our exper- iments in this range. The layer of displaced uid remained at the top of the pipe (diameter D^) during the entire duration of the experiment, appar- ently stationary for very long times (t^ & 103D^=V^0). 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 ow we have zero net ux and there is an equal ow of heavy uid downstream as there is of light uid upstream. For the displacement ow, denoting the ow rates of heavy and light uids through a given cross-section by Q^ and Q^l, respectively, we always have …D^2V^0 4 = Q^+ Q^l: (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 ows. Phys. Fluids 23 044105 (2011). 1126.1. Pipe displacements At V^0 = 0 we have Q^l = ¡Q^ < 0 in the exchange ow. As V^0 is increased, the net buoyancy force available to resist motion in the imposed ow direction remains constant, but the imposed ow creates viscous stresses which act on the lighter uid layer at the interface and drag the lighter uid along the duct. The viscous drag increases with V^0 and eventually we expect to attain a transition where Q^l = 0, and thereafter Q^l > 0. Could we target the upstream ow and represent the transition between (i) and (ii) by where Q^l = 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 fllm 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 ows in this regime. An excellent agreement is found. In x6.3 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 ow velocity V^0 from zero we came across ows in which the downstream layer of in-situ uid remained apparently stationary and uniform at the top of the pipe, while the displacing uid traveled underneath. Fig. 6.1 displays an example of such a ow in the conflguration where a heavy uid (transparent) is injected to displace the lighter uid (black), which is initially fllling the inclined pipe. The displacement is from left to right. The leading front of the heavy uid slumps underneath the light uid at the start of the displacement (flrst image). We observe that 25 (s) after 1136.1. 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 fleld 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 „^ = 10¡3 (Pa.s) and the mean ow velocity is V^0 = 38 (mm.s¡1). The flgure 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 ¢t^ = 0:5 (s). It shows the variation of the layer height with time. the beginning of the process the two uids are stratifled along the length of the pipe. Since only the transparent uid is injected, it is obvious that the two layers have difierent mean velocities and intuitively we would not expect this conflguration 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 t^ = 0 the image is all black because the pipe is full of black uid. After around 15 (s) the heavy uid arrives in this part of the pipe and we observe on the spatiotemporal diagram the two layers with the transparent uid below the black uid. 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, flve times the volume of the pipe have owed through the pipe. Alternatively, the layers are constant for » 103 times the advec- tive timescale D^=V^0 … 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 ow 1146.1. Pipe displacements becoming near-parallel. (b) (a) (d) (c) 63 cm Gate valve 22 cm xV0^ ^ y^ Figure 6.2: Four snapshots of video images taken at difierent mean ow rates and illustrating the difierent regimes. The heavy transparent uid ows downward under the combined efiects of buoyancy (¢‰^) and pressure gradient (V^0). The light black uid has difierent behaviors ( ows upward or downward) depending on the control parameters values. These images were obtained at fl = 85 –, At = 10¡2, „^ = 10¡3 Pa.s. The mean ow velocities were: (a) V^0 = 29 (mm.s¡1), (b) V^0 = 38 (mm.s¡1), (c) V^0 = 42 (mm.s¡1), and (d) V^0 = 61 (mm.s¡1). The fleld 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 V^0 was increased from zero, the most obvious changes in the ow 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 ow) exhibited 4 difierent characteristic behaviours. Fig. 6.2 illustrates these 4 behaviours in a 1015 (mm) long section of the pipe, tilted at fl = 85 –, for a sequence of displacements at the same density difierence (At = 10¡2) but at difierent V^0. In each image the heavier transparent uid 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 uid is moving upward against the imposed ow 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 (V^0 = 29 (mm.s¡1)) allows a counter-current ow similar to the exchange ow, except that the back ow moves slower. We describe such ows as sustained back ows, i.e. there is a sustained upstream ow which advects the trailing front con- 1156.1. Pipe displacements tinually upstream against the mean ow. In Fig. 6.2b with an increased imposed ow (V^0 = 38 (mm.s¡1)) we observe that the trailing front moves initially upstream against the ow, 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 ows as stationary interface ows. In the next image (Fig. 6.2c), with a slightly higher mean velocity (V^0 = 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 ow, i.e. there is a ow backwards against the mean ow which initially advects the trailing front upstream but the back ow is not sustained over long times. Finally, if the mean velocity is further increased (Fig. 6.2d), the trailing front between clear and dark uid is simply displaced downstream. We call this high mean ow case an instantaneous displacement, (V^0 = 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 ows 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 ow rises. The vertical scale depicts time (500 (s) in each flgure) 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 ow zones) from the gray regions (transparent uid). We observe in Fig. 6.3a that the back ow starts with a constant velocity and then slows down until it stops. It does not move signiflcantly 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 ow and the buoyant ow. For a slightly increased imposed ow we observe in Fig. 6.3b the temporary back ow regime. The back ow 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 flgure but the back ow 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 1166.1. 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 (¢t^ = 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 x^ axis is the same as in Fig. 6.2: downward. These diagrams correspond to the experiments: (a) Fig. 6.2b (V^0 = 38 (mm.s¡1)) and (b) Fig. 6.2c (V^0 = 42 (mm.s¡1)). the slow onset of temporary back ow. Fig. 6.4 displays transverse proflles of the longitudinal velocity (paral- lel to the pipe axis) averaged over time for 3 regimes: sustained back ow (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 ow 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 artiflcially 1176.1. Pipe displacements −40 −20 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 uˆ(yˆ) (mm/s) ˆ D − yˆ (m m ) a) −40 −20 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 uˆ(yˆ) (mm/s) ˆ D − yˆ (m m ) b) −40 −20 0 20 40 60 80 100 120 0 2 4 6 8 10 12 14 16 18 uˆ(yˆ) (mm/s) ˆ D − yˆ (m m ) c) Figure 6.4: Ultrasonic Doppler Velocimeters proflles for the same series of experiment as Fig. 6.2: (a) V^0 = 29 (mm.s¡1) (see Fig. 6.2a) sustained back ow regime, proflles averaged between 60 and 120 (s), (b) V^0 = 38 (mm.s¡1) (see Fig. 6.2b) stationary interface regime, proflles averaged between 240 and 300 (s), (c) V^0 = 74 (mm.s¡1) instantaneous displacement regime, proflles averaged between 120 and 240 (s). The vertical scale represents the distance from the upper wall (y^ measuring distance from the lower one) and the horizontal scale the corresponding value of the longitudinal ow 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 proflles are distorted by instrumental error. complete the proflle to the wall, where the velocity is zero. First of all, we observe that the 3 flgures show a downward global net ow, due to the mean ow. By looking speciflcally at each regime, we 1186.1. Pipe displacements observe that in the sustained back ow 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 uid 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 uid in the upper layer is not motionless but moves in a counter-current recirculatory motion. The displacing uid is observed to pass underneath the upper layer and so we expect that the net ow 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 ow 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 uid has been mostly displaced leaving only a very thin residual layer. The above constitutes a description of the distinct ow regimes observed in our experiments, as V^0 is varied. One could say that we have essentially 3 regimes, with the stationary layer apparently representing a transition state between ows with sustained back ow and those that displace. Below in x6.1.2 we derive a simple model that predicts similar ow regimes and transitions. In x6.1.3 we present the comparison between the predictions of this model and the classiflcation of our experiments. 6.1.2 Lubrication model • β g y x D ^ ^ ^ Xbf ^ f ^ ΩH (heavy) ΩL (light) y=h(x,t)^ ^^^ Cross-section h(x,t)^ ^^ z^ 0V ^ Figure 6.5: Schematic views of the distribution of the two uids in two perpendicular vertical planes of the pipe (diametrical and transversal). The notation is that used in the models. 1196.1. 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 ow occurs within regions where the uids are separated by interfaces that are aligned approximately with the pipe axis. It is there- fore very natural to develop a thin-fllm/lubrication style model for the pipe displacement ow. The procedure is more or less standard and we follow largely that of our previous chapter (x5). At each axial position x^ the ow is assumed stratifled with interface denoted y^ = h^(x^; t^); see the geometry illustrated schematically in Fig. 6.5. The leading order equations are the momentum balances: 0 = ¡@p^@z^ ; (6.2) 0 = ¡@p^@y^ ¡ ‰^kg^ sin fl; (6.3) 0 = ¡@p^@x^ + „^ •@2w^ @z^2 + @2w^ @y^2 ‚ + ‰^kg^ cos fl; (6.4) (z^; y^) 2 ›k k = H;L and the incompressibility condition, r¢ u^ = 0. At the walls u^ = 0, and both velocity and traction vectors are continuous at the interface. For the ows considered a mean ow V^0 is imposed by pumping in the positive x^-direction. Thus, the additional constraint …D^2V^0 4 = Z ›H S›L w dz^dy^; (6.5) is satisfled by the solution. We eliminate p^ and derive the evolution equation for h^: @ @t^A^(h^) + @ @x^Q^ = 0; (6.6) where A^(h^) is the area occupied by the heavier uid, A^(h^) = j›H j, and Q^ = Z ›H w^ dz^dy^: (6.7) The ux consists of a superposition of Poiseuille and exchange ow compo- nents: Q^ = Q^(h^; h^x^) = 2V^0 Z A^(h^) 1¡ 4 z^ 2 + y^2 D^2 ¶ dz^dy^ +…D^ 2 8 F0V^” ˆ 1¡ 4(D^=2¡ h^) 2 D^2 !7=2ˆ cos fl ¡ sin fl @h^@x^ ! 1206.1. Pipe displacements where V^” = At:g^:D^2=”^ and F0 is given in [135] as F0 = 0:0118. The exchange ow component has been estimated (see [135]) by extrapolating from the value at h^ = D^=2 and from asymptotic expressions for h^ » 0 and h^ » D^. In x5 we deflned dimensionless parameters, – and ´ via – = „^V^0[‰^H ¡ ‰^L]g^ sin flD^2 = V^02V^” sin fl ; ´ = cot fl– = 2V^” cos fl V^0 ; (6.8) and scaled the system using a length-scale L^ = D^=– in the x^-direction, with L^=V^0 as timescale. Here we adopt the same scalings and also scale A^(h^) with …D^2=4, Q^ with …D^2V^0=4 and (h^; y^; z^) with D^. The resulting dimensionless equations are @ @T fi(h) + @ @» q(h; h») = 0; (6.9) where h 2 [0; 1] is now dimensionless, fi(h) = 4A^(h^)=…D^2 is the area fraction occupied by the heavy uid: fi(h) = 1… cos ¡1(1¡ 2h)¡ 2… (1¡ 2h) p h¡ h2 q(h; h») = 32… Z fi(h) (14 ¡ x 2 ¡ y2) dxdy + F0[´¡ h»] 4 ¡1¡ (1¡ 2h)2¢7=2 ; T and » are the dimensionless time and length variables, respectively. Although the algebraic form of (6.9) difiers from that analysed for the plane channel, we expect similar behaviour. Let us flrst consider the down- stream behaviour. At long times the interface is expected to elongate (as shown in x5), which negates the efiect 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 ow rate of displacing heavy uid. In the context of (6.9), considered at long times, this implies that the interfacial speed is zero and the ux, q(h; 0) = 1. The interfacial speed Vi is simply the characteristic speed: Vi = @q@fi(h; 0) = @q @h(h; 0) •dfi dh (h) ‚¡1 ; 1216.1. Pipe displacements and since fi 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 ´. In Fig. 6.6 we plot contours of q(h; 0) and the contour @q@h(h; 0) = 0, against (h; ´). The intercept of q(h; 0) = 1 and @q@h(h; 0) = 0 occurs at a critical ´ = ´c = 116:32:: and for h = 0:72::, indicating that there is a unique interface height and value of ´ 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 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 ´ = 116:32:: and h = 0:72::. Considering now the trailing front, for the plane channel displacement large values of ´ resulted in a second front propagating upstream against the ow, i.e. a back ow. Large ´ corresponds to a weak imposed ow relative to the buoyancy driven exchange ow component. For the pipe ow, assuming again that the long time behaviour is dominated by the hyperbolic part of (6.9), the equations determining the back ow front speed Vf < 0 and front height hf , are simply: [1¡ fi(hf )]Vf = [1¡ q(hf ; 0)]; (6.10) Vf = @q@h(hf ; 0) •dfi dh (hf ) ‚¡1 (6.11) For su–ciently large ´ 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 1226.1. Pipe displacements the conditions for the stationary interface are identical with those determin- ing whether or not equation (6.9) has a sustained back ow: for ´ > ´c we have sustained back ow and for ´ < ´c there is no sustained back ow. 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: Proflles of h(»; T ) for T = 0; 1; ::; 9; 10, with ´ = ´c. The broken line shows the theoretical stationary h = 0:72:: at ´ = ´c. The inset shows the extension of the stationary frontal region. In Fig. 6.7 we plot h(»; T ) for the critical ´ = ´c, 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 difiusive efiects are always present). By solving for each h 2 (hf ; 1) the nonlinear equation q(h; h») = 0; we may flnd the steady interface slope, h»(h). This can be integrated to flnd the shape of the steady proflle for h > hf . In this frontal region, buoyancy driven by the slope of the interface (acting to smooth the interface) is in 1236.1. Pipe displacements balance with the buoyancy force driving uid back up the inclined pipe. The frontal proflle 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 fl only at a critical balance V^” cos fl … 58:16V^0; (6.12) (recall ´ = 2V^” cos fl=V^0). In the lubrication model context, sustained back ows are only found upstream for smaller values of V^0 than in (6.12), whereas downstream the interface speed becomes positive for larger V^0 than in (6.12) and instantaneous displacement ensues. Temporary back ows are not strictly covered by the long time analysis of the lubrication model. Our experiments have been performed over the ranges: V^0 2 0¡ 80 (mm.s¡1); At 2 [10¡3 ¡ 10¡2]; fl 2 [83 – ¡ 87 –]: To give an overall perspective of the difierent ow regimes and where they occur, Fig. 6.8 presents the classiflcation of our ows for the full range of experiments. We observe that the sustained back ow regime is clearly separated from the instantaneous displacement regime. Between these two regimes we flnd stationary layers and temporary back ows. We must ac- knowledge potential errors in making the classiflcations depicted in Fig. 6.8. For example, sustained back ow experiments are terminated when the back ow 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 ows. With a flnite duration experiment (with other restrictions and errors) it is di–cult to deflnitively 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 ow component of q, the prediction ofiered by this linear relation (6.12) is surprisingly good. We also note that in those experiments that we have classifled as stationary interface ows the stationary layer occupies approxi- mately 30% of the pipe at the top, which corresponds well to the theoretical stationary h = 0:72:: at ´ = ´c. It is worth commenting that we have plotted our results in dimensionless velocity coordinates, with both V^0 and V^” cos fl scaled with the inertial scale 1246.1. Pipe displacements 0 1 2 3 4 5 6 0 20 40 60 80 100 120 ˆV0 ˆVt ˆVν cos β ˆVt 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 (V^0 is in the range 0¡80 (mm.s¡1), At is in the range 10¡3¡10¡2, fl is in the range 83 –¡87 –). The heavy line represents the prediction of the lubrication model for the stationary interface: 58:16V^0 = V^” cos fl. V^t. This of course does not afiect the relation between V^0 and V^” cos fl which is exemplifled 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 ow. As discussed in x2 pure exchange ow studies [135] have suggested that for V^” cos fl ’ 50V^t the exchange ow 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 ow results in the streamlines becoming progressively aligned with the pipe axis, even in this inertial regime. The consequent stabilization as the ow rate (Reynolds number) is increased is somewhat counterintuitive. We can view (6.12) as being derived from the instantaneous displacement 1256.2. Plane channel geometry (2D) conflguration in the regime V^” cos fl ’ 50V^t, and provided that V^0 is large enough the ows are su–ciently 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 conflrm 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 conflrm our understanding of the stationary layer. This geometry allows for faster computations and more precise asymptotic approximations. The channel has height D^ and is oriented similarly to the pipe, close to horizontal. Again a heavy uid displaces a lighter uid in the downwards direction. 6.2.1 Lubrication model The lubrication/thin-fllm approach is analogous to that developed for the pipe, leading to a dimensionless evolution equation for the interface height, y = h(»; T ): @h @T + @ @» q(h; h») = 0: (6.13) This has been derived and extensively studied in Chapter 5 for a wide range of uid 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 ´ and h for which the entire ux 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; ´)-space in Fig. 6.9, from which we flnd ´c = 69:94 for the plane channel at an interface height h = 0:707. Note here that h 2 [0; 1] as we have scaled with the height D^ of the channel. The relation ´ = ´c again provides a predictor of the stationary interface, which we now test against 2D computational solutions. 1266.2. 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 ´ = ´c = 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 ow, i.e. molecular difiusion is neglected. This neglect is due to the large P¶eclet numbers that correspond to our experimental ows, for which we typically have a well deflned interface. The Navier Stokes equations are made dimensionless using the channel height D^ as lengthscale and V^0 as velocity scale. The model equations are: [1 + `At] [ut + u ¢ ru] = ¡rp+ 1Rer 2u+ `Fr2eg; (6.14) r ¢ u = 0; (6.15) ct + u ¢ rc = 0: (6.16) Here eg = (cos fl;¡ sin fl) and the function ` = `(c) interpolates linearly between ¡1 and +1 for c 2 [0; 1]. The 2 additional dimensionless parameters appearing above are the Reynolds number, Re, and the (densimetric) Froude 1276.2. Plane channel geometry (2D) number, Fr, deflned as follows. Re · V^0D^”^ ; Fr · V^0q Atg^D^ : (6.17) Here ”^ is deflned using the mean density ‰^ = (‰^H + ‰^L)=2, and the mean static pressure gradient has been subtracted from the pressure before scaling. We see that for small At the ow is essentially governed by the 3 parameters fl, Re & Fr. For t > 0, no slip boundary conditions are satisfled at the solid walls (zero ux for c) and out ow conditions imposed at the channel exit. At the in ow the heavy uid concentration is imposed (c = 0), and the velocity u is represented by a fully established Poiseuille proflle. The initial interface position is some way down the channel and our initial velocity fleld is stationary: u = 0 at t = 0. We have selected a range of parameters that resembles that of our pipe ow experiments. Thus, we will describe the simulations in the following section with reference to V^0, V^” and V^t, as these are more natural from the experimental perspective. The mapping between parameters is simply: Re · V^0V^”V^ 2t ; F r · V^0V^t : (6.18) When considering the lubrication model predictions: ´ = 2cos flV^”V^0 = 2Re cos flF r2 : (6.19) Unlike the pipe ow, we have limited our computational study to param- eters for which the pure exchange ow (V^0 = 0) is in the viscous regime. The reason for this restriction is that in general the stabilizing efiect of the imposed ow (as shown in x4) does not afiect the channel exchange ow in the same way as it afiects a pipe exchange ow. For the pure exchange ow Hallez and Magnaudet [67] have reported key difierences in the ow 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 fl = 87 –; ”^ = 2£ 10¡6 (m2.s¡1), At = 3:5 £ 10¡3; V^0 = 9:5 (mm.s¡1), [Re = 90, Fr = 0:37]). The 1286.2. Plane channel geometry (2D) upper image in Fig. 6.10 depicts the initial condition for the concentration fleld at t^ = 0 (s). The subsequent images (from top to down) show the evolution of the concentration fleld at t^ = 25, 50, 100, 200, 300 (s). Although we observe that for t^ > 0 the trailing front initially moves backwards against the mean ow, 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 difiusion is well limited by the MUSCL scheme, dispersion due to (physical) secondary ows 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 xˆ/ ˆD Figure 6.10: Sequence of concentration fleld evolution obtained for fl = 87 –; ”^ = 2£ 10¡6 (m2.s¡1), At = 3:5£ 10¡3; V^0 = 9:5 (mm.s¡1), [Re = 90, Fr = 0:37]. The images are shown for t^ = 0, 25, 50, 100, 200, 300 (s) (from top to bottom). To have a better understanding of the difierent regimes in typical station- 1296.2. Plane channel geometry (2D) xˆ (mm) ˆt (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 uids respectively) along the channel for fl = 87 –; ”^ = 2 £ 10¡6 (m2.s¡1), At = 3:5 £ 10¡3; V^0 = 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 ow velocity (of the lighter uid) is zero (near stationary). ary ows 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) 1306.2. 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 xˆ/ˆD Figure 6.12: The velocity proflles corresponding to Fig. 6.10 for a channel ow at t^ = 0, 25, 50, 100, 200, 300 (s) (from top to bottom). is not constant with time. Initially the trailing front ows 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 efiects control the initial back ow velocity; the corresponding initial viscous velocity, which is proportional to the slope of the interface, would be too large (inflnite at t^ » 0). During the flrst acceleration when the interface between the two motionless uids starts to move, the back ow is accelerated by buoyancy up until it attains approximately the inertial velocity. At this point inertia prevents the uid 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 ow 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/flnal 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 ow is in the stationary regime. Note that this is essentially 1316.2. Plane channel geometry (2D) the same picture that we have observed experimentally. Fig. 6.12 shows the velocity proflles corresponding to Fig. 6.10 at t^ = 0, 25, 50, 100, 200, 300 (s). The flgure shows the region between 20 < x^=D^ < 38. The initial interface is located at x^=D^ = 25. As expected, for t^ < 100 (s) we see a counter-current ow in the longitudinal direction, with net ow equal to the imposed ow rate. In this time frame we transition from an initially inertially limited ow to a viscously limited ow. 0 1 2 0 0.2 0.4 0.6 0.8 1 uˆ/ˆV0 yˆ/ˆD Figure 6.13: The velocity proflle close to the pinned point (with the axial position x^=D^ = 26:25) corresponding to Figs. 6.10 & 6.12 for a channel ow at t^ = 300 (s): illustrating the counter-current inside the stationary (lighter/black) uid. Dashed line represents the local height of the interface. Fig. 6.13 illustrates a single velocity proflle at t^ = 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 ow inside the stationary upper layer. Fig. 6.14 displays the four archetypical regimes for difierent imposed ow velocities. In Fig. 6.14a, we see that for a low imposed ow the velocity of the 1326.2. 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 flnally reaches a constant buoyant velocity, allowing the lighter uid to keep rising (sustained back ow). At a larger mean imposed ow velocity, Fig. 6.14b is the stationary interface regime. A further increase in the imposed ow (Fig. 6.14c) leads to an upstream front which advances, stops and then recedes down the pipe. This corresponds to the temporary back ow regime. Finally, for a su–ciently strong imposed ow (Fig. 6.14d), there is no back ow from the beginning of the displacement process. An instantaneous displacement is achieved. 0 1000 2000 0 50 100 150 xˆ (mm) ˆ t (s) 0 1000 2000 0 50 100 150 xˆ (mm) ˆ t (s) 0 1000 2000 0 50 100 150 xˆ (mm) ˆ t (s) 0 1000 2000 0 50 100 150 xˆ (mm) ˆ t (s) b) c) ) d) Figure 6.14: Four possible conditions for a viscous buoyant channel ow when an imposed ow is present: the parameters are fl = 89 –; ”^ = 10¡6 (m2.s¡1), At = 10¡2; a) V^0 = 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: V^0 is in the 1336.2. Plane channel geometry (2D) range 2 ¡ 30 (mm.s¡1), At is in the range 10¡3 ¡ 10¡2, ”^ is in the range 10¡6 ¡ 2£ 10¡6 (m2.s¡1), fl belongs to the range 85 – ¡ 89 –. Each simula- tion has been classifled 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 V^” cos fl … 34:97V^0, (i.e. ´ = ´c = 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 ows are clearly separated from those showing the instantaneous displacement ows. The transition between temporary back ows and the instantaneous displacement ows seems to be governed by a balance between the imposed pressure gradient (roughly speaking V^0) and the characteristic inertial velocity (V^t). For viscous ow in a channel, this transition (V^0=V^t = ) lies somewhere in the range = 0:6 ¡ 0:8, probably with minor dependency on the inclination angle fl. 0 0.5 1 1.5 2 0 5 10 15 20 25 ˆV0 ˆVt ˆVν cos β ˆVt Sustained Back Flow Stationary Interface Temporary Back Flow Instantaneous Displacement Figure 6.15: Classiflcation of our simulation results in a channel. The heavy line represents the prediction of the lubrication model for the stationary interface: 34:97V^0 = V^” cos fl. 1346.3. Simple physical model In a more inclined channel, where the ows become inertial, we anticipate that there could be an increase in the value of . Experimental observations for an exchange ow in a pipe reveal that this increase can be up to 40% with respect to the horizontal, see [135]. Although the precise value of is of interest, it should be noted that this does not afiect the long time behaviour of the ow/interface. Indeed whether or not the back ow is temporary or the displacement is instantaneous, the displacing uid eventually washes out the displaced uid as long as V^0=V^” cos fl 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 simplifled conceptual model, along the lines of that presented by Seon et al. [135] for pure exchange ows. The objective is to describe the speed of propagation of the trailing front (V^ bff ) and its position (X^bff ) as they move backwards against the imposed ow, see Fig. 6.5. First of all, it is clear that the only driving mechanism to push the lighter uid up the channel is buoyancy. Except at early times, the ows 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 re ects this balance is V^” . Buoyancy acts both axially along the pipe (/ cos fl) and perpendicular to the pipe axis (/ sin fl). The latter trans- verse component acts only when the interface between the uids is tilted with respect to the pipe axis (i.e. if @h^=@x^ 6= 0) and is then proportional to ¡ sin fl@h^=@x^. The second force afiecting the back ow comes from the imposed ow that determines a net pressure gradient pushing uids down- wards, along the pipe. Since the uids are Newtonian we may assume that this force scales approximately linearly with V^0. Therefore, on summing the difierent driving forces we might postulate that: V^ bff = V^” cos fl ˆ Ka ¡Kt@h^@x^ tan fl ! ¡KmV^0; (6.20) where the coe–cients Ka, Kt & Km re ect the relative in uences of axial buoyancy, transverse buoyancy and the mean ow, respectively. For the case V^0 = 0 (exchange ow) 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, V^ bff 1356.3. Simple physical model Inertial Viscous γVt 0 ^ Xbff ^ KaVν cosβ^ KmV0^ = 0 : stopping lengthV bff ^ 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 Xbff ^ ^ (a) (b) Figure 6.16: (a) Schematic variation of the velocity V^ bff +KmV^0 as a function of distance X^bff from the gate valve (continuous line) in a viscous regime and for fl 6= 90 –. The short dashed line represents the flnal viscous velocity V^ bff + KmV^0 = KaV^” cos fl. The dotted line marks the boundary between the transient inertial regime and the viscous regime. We also represent the case V^t > KmV^0 > KaV^” cos fl, 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) V^0=(V^” cos fl) is plotted versus (D^=X^bff ) tan fl for 2 series of experiments at difierent angles fl: 83 – (⁄) and 85 – (–), and same density contrast and viscosity (At = 10¡2, „^ = 10¡3 (Pa.s). The experiments plotted here are either in the temporary back ow regime or in the stationary interface regime, and X^bff represents the position where the front stops (maximal X^bff ). The dashed line is a guide for the eye to show the common linear curve. 1366.3. 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 difierential equation for X^bff by approximating the interface slope with @h^@x^ … ¡D^=X^bff , which leads to: dX^bff dt^ = V^ bf f = V^” cos fl ˆ Ka +Kt D^X^bff tan fl ! ¡KmV^0; (6.21) At short times (and distances) the model (6.21) would predict an inflnite front velocity, which is not physically possible. In practice, in this early period of the ow V^ bff will be limited by inertial efiects rather than viscous efiects. 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, V^t. We see for example that in Fig. 6.3 the front velocity is indeed initially constant before it decreases. This cut-ofi behaviour is illustrated schematically in Fig. 6.16a. Consequently, we may modify (6.21) as follows: dX^bff dt^ = min ( V^t; V^” cos fl ˆ Ka +Kt D^X^bff tan fl !) ¡KmV^0; (6.22) where is a further coe–cient to be determined. Although simplistic we believe (6.22) contains the essential elements of the trailing front dynamics. Evidently V^ bff decreases with time as the front propagates, in all cases. Let us consider some difierent possible be- haviours. First, let us suppose that the imposed ow is weak, so that KmV^0 < KaV^” cos fl. The ow 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 ow: V^ bff ! KaV^” cos fl ¡KmV^0 and the front advances steadily up the pipe. Secondly, suppose that the imposed ow is stronger, so that KmV^0 > KaV^” cos fl, but that KmV^0 < V^t (case represented in Fig. 6.16a). The transient phase of the back ow elongates the interface so that the slope decreases until there is a perfect balance: V^” cos fl ˆ Ka +Kt D^X^bff tan fl ! = KmV^0; (6.23) corresponding to the stationary regime. Rearranging this shows that the 1376.4. Discussion stopping lengths Xf satisfy: V^0 V^” cos fl = 1Km ˆ Ka +Kt D^X^bff tan fl ! : (6.24) Note that at larger V^0 the transient phase of the back ow will be reduced and stopping length too. We might also expect that this delicate balance be afiected over longer times by changes in the interface proflle 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 ow regime. In this simple conceptual model, the stationary interface regime and the temporary back ow 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 V^0, say KmV^0 > V^t, we expect no back ow and the instantaneous displacement regime is entered. In Fig. 6.16b we have plotted V^0=(V” cos fl) against (D^=X^bff ) tan fl for 2 series of experiments at difierent angles. Only those experiments are plotted that were characterised as a temporary back ow or stationary interface and X^bff 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 ow in the presence of a mean ow. In principle, this also allows us to determine directly the fl- independent coe–cients Ka=Km & Kt=Km, via linear regression, and to use the model in equation (6.24) predictively. However, to be more confldent in determining Ka=Km & Kt=Km we would need to conduct more experiments for a wider range of At, ”^ and D^. 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 ow phenomenon in which a buoyant displacement ow retains a stationary upper layer of displaced uid for the duration of our experiment. The same feature was observed in our plane channel displacement simulations. Some aspects of this ow are obvious. For example, as we increase the imposed ow rate from zero we do expect to reach a ow rate for which the upper layer has zero ow. Less 1386.4. Discussion obvious is that the ow structure should remain stationary, i.e. the layer thickness of the lighter uid that is found at the transition state is one at which the interface speed is zero. The ow apparently evolves to select this interfacial position, so that the ow structures observed for V^0 close to the transition persist over very long timescales (as described in detail in x6.1). We have found 2 parallels to this phenomenon in the literature. Hup- pert & Woods [81] have considered a range of porous media ows driven by density difierence, using a lubrication approximation. Part of their study considers two-layer exchange ows between reservoirs and amongst the so- lutions investigated there exist those for which the ow in one layer is zero. There are many difierences between porous media ows and those governed by the Navier-Stokes equations. In the present context we note that the main difierences are that in porous media ows of Huppert & Woods [81] zero ow in one uid layer means the velocity is everywhere zero in that layer and the modifled pressure gradient is also zero. In the Navier-Stokes context (current work) there is a positive pressure gradient driving the light uid layer backwards against the ow and the velocities are non-zero within the stationary layer. In looking simply at lubrication-type models with an imposed ow 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 ow 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 difierent states: (a) sustained back ow, (b) stationary interface, (c) temporary back ow, (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 ow regimes can also be found at long times in thin-fllm/lubrication style models of these ows. 1396.4. Discussion The transition between states is the stationary layer, which is predicted by the lubrication model, at critical conditions: 58:16V^0 = V^” cos fl (6.25) for the pipe geometry and 34:97V^0 = V^” cos fl (6.26) for a plane channel geometry. In the context of Chapter 4 where we have studied ow rate efiects on the downstream front velocity, the stationary layer ows studied mark the boundary between the exchange ow dominated regime and the regime where the downstream front velocity (V^f ) increases linearly with V^0, for which the imposed ow becomes increasingly dominant. The transition between temporary back ows and instantaneous dis- placements appears to be characterised by a condition V^0 = V^t; with = 0:6¡ 0:8; for the plane channel geometry. This estimate has been made using only ow parameters for which the pure exchange ow would be viscous in the plane channel. It is interesting to re ect that although we have classifled 4 difierent states, in our experiments and in each of the models we have used we are only able to identify 3 states deflnitively. For the experimental results we simply classify observed ows within the practical limits of our experiments. Thus, if the back ow exceeds the end of the pipe (above the gate valve) we classify the ow as a sustained back ow (although given a longer pipe some of these might be temporary); the stationary back ow is identifled 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 ow, stationary back ow and instantaneous displacement. Although at short times (and distances) the model presented in Chapter 5 always has a fast initial phase where temporary back ows 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 difierent techniques to understand the dynamics of complex ows: each technique gives difierent insights. 1406.5. Summary Amongst the 4 difierent states classifled, the stationary interface is a transition state, only marking the ow that exists at the boundary between sustained and temporary back ow regimes. This means that it would be near impossible to flnd exactly the correct parameters to capture this state exactly. In all likelihood, any such state would anyway flnally evolve into a temporary back ow via downstream processes such as uid entrainment (see e.g. turbulent entrainment in x2.2.2), thinning the layer below the critical thickness. Thus, it is relevant that in our study we have observed (and classifled 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 D^=V^0, 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. † Identiflcation and physical explanation of the stationary residual layer ow. † Classiflcation of the ow transitions occurring upstream of the initial uid positions. † Usage of the lubrication/thin fllm approach to make predictions of the critical imposed ow for which stationary residual layers occur; see (6.25) and (6.26), together with validation of these approximations with experimental and 2D simulation data. 141Chapter 7 Iso-viscous miscible displacement ows8 The aim of this chapter is to bring together the studies of Chapters 4 ¡ 6 with a more complete investigation of these displacement ows in the iso- viscous setting. We aim to give a complete classiflcation of the types of ow 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 P¶eclet number regime at low At. Three dimensionless groups largely de- scribe these ows: Fr (densimetric Froude number), Re (Reynolds number) and fl (duct inclination). Our results will show that the ow regimes in fact collapse into regions in a two-dimensional (Fr;Re cos fl=Fr)-plane. These regions are qualitatively similar between pipes and plane channels, although viscous efiects 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 efiectively a measure of displacement e–ciency. 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) uids have the same viscosity „^, are miscible and have difiering densities. In general we study laminar ows. This ow may be studied from a number of difierent perspectives. First of all, from a modeling perspective a natural formulation involves a concentration-difiusion equation coupled to the Navier-Stokes equations. The phase change between pure uids 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 ows in near-horizontal ducts at low Atwood number. 1427.1. Problem Setting Stokes equations dimensionless using D^ as length-scale, V^0 as velocity scale, and subtracting a mean static pressure gradient before scaling the reduced pressure, we arrive at: [1 + `At] [ut + u ¢ ru] = ¡rp+ 1Rer 2u+ `Fr2eg; (7.1) r ¢ u = 0; (7.2) ct + u ¢ rc = 1Per 2c: (7.3) Here eg = (cos fl;¡ sin fl) and the function `(c) = 1¡2c interpolates linearly between 1 and ¡1 for c 2 [0; 1]. The 4 dimensionless parameters appearing in (7.1) are the angle of inclination from vertical, fl, the Atwood number, At, the Reynolds number, Re, and the (densimetric) Froude number, Fr. These are deflned as follows: At · ‰^1 ¡ ‰^2‰^1 + ‰^2 ; Re · V^0D^ ”^ ; F r · V^0q Atg^D^ : (7.4) Here ”^ is deflned using the mean density ‰^ = (‰^1 + ‰^2)=2 and the common viscosity „^ of the uids. In (7.3) appears a 5th dimensionless group, the P¶eclet number, Pe, deflned by: Pe · V^0D^D^m ; (7.5) with D^m the molecular difiusivity (here assumed constant for simplicity). It appears that 5 dimensionless parameters are required to fully describe this ow. However, commonly the P¶eclet number is very large as we consider lab/industrial scale ows rather than micro- uidic devices, e.g. Pe > 106 is common. If the uids are initially separated we expect difiusive efiects 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 ows remain structured and partially stratifled we commonly observe interfaces that are sharp over experimental timescales. Such ows are close to their immiscible uid analogues (at inflnite capillary number, i.e. vanishing surface tension), which are modeled by setting Pe = 1 and ignoring the right-hand-side of (7.3). Secondly, we see that the direct efiect of the density difierence on inertia is captured by At. Supposing for example that we restrict our attention to density difierences of the order of 10% (as in our experiments) we see that 1437.1. Problem Setting At • 0:05. We expect therefore that for moderate density difierences 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; fl) in this large Pe, small At limit that is representative of many practical displacement ows. Moreover, we consider only fl such that the duct inclination is close to horizontal since this range of inclinations is where viscous efiects are mostly found. Thus, the overall aim of our study is to build a quantitative description of the difierent ow regimes found, in terms of Re and Fr, for fl close to …=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 uids (cost, ease of cleaning and mixing, rheological and optical properties, etc). To preserve consistency of the uids used it is natural to mix a pair of uids and then to conduct experimental sequences in which we vary the mean ow V^0 at flxed inclination. We observe that both Re and Fr increase linearly with V^0 in such an experimental sequence. The results of pure exchange ow studies are governed by the relative sizes of V^” (denoting the velocity at which buoyancy and viscous stresses balance) and V^t (denoting the velocity at which inertial and buoyancy stresses balance). Thus, if we wish to measure the departure from the exchange ow setting as the ow rate is increased, the natural experimental description revolves around V^0, V^” and V^t, at flxed fl. The relationships between these parameters and Re & Fr are: Re · V^0V^”V^ 2t ; F r · V^0V^t : (7.6) 7.1.1 Viscous and inertial ows Frequently in discussing our results below we shall refer to ows 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 ows are inertial. Secondly, it is obvious that as the imposed ow V^0 is increased, viscosity plays an increasing role in balancing the mean pressure drop, and the amount of inertia injected into the ow increases. Therefore, our usage of viscous and inertial is primarily phenomenological, in describing observed results. Where the ow 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 uids can be considered incompressible. 1447.2. Displacement in pipes fl – ”^ (mm2.s¡1) At (£10¡3) V^0 (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) 2 [1; 10], V^0 2 0¡ 110 (mm.s¡1). Table 7.1: Experimental plan. refer to the ow as viscous. Where we observe two and three-dimensional regions of ow, typically associated with instability and (at least localised) mixing close to the interface, we refer to the ows as inertial. 7.1.2 Outline The main content of this chapter proceeds in 3 sections. The flrst section (x7.2) concerns pipe ow displacements. The main methods are experimen- tal and semi-analytical, using a lubrication/thin-fllm modeling approach. The second section (x7.3) 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 efiects, which we study in x7.4. We also study the ow stability in x7.4. The chapter ends with a brief summary. 7.2 Displacement in pipes The flrst 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 ow behaviour based on the lubrication approximation from x5 and extrapolation from the exchange ow studies of [132] and [135]. This culminates (x7.2.6) 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 ow regimes observed In a typical experiment we observe a short inertial phase following the open- ing of the gate valve. The uids are initially at rest. When the gate valve 1457.2. Displacement in pipes is opened the static head accelerates both uids from rest and at the same time the density difierence between uids accelerates the uids in oppos- ing directions. This flrst 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 ow. 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 ow (buoyancy forces dominate imposed ow) or downstream (imposed ow 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 flrst part of the experiment, when the interface is transverse to the pipe axis, but as the ow elongates it appears that viscous forces dominate, over a wide range of ow 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 ow 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 proflle becomes progressively positive and the ow 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 efiectively an area averaged concentration of dark uid, 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 ow in both interface position and underlying velocity fleld. The total ow rate is flxed, so the initial period of back ow 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 1467.2. Displacement in pipes UDV probe Figure 7.1: Sequence of images showing propagation of waves along the interface for V^0 = 40 (mm.s¡1) along a 1245 (mm) long section of the pipe a few centimeters below the gate valve. Other parameters are fl = 85 –, At = 1:67£ 10¡2 and ”^ = 1 (mm2.s¡1). From top to bottom the images are taken at t^ = 5; 15; 50; 100; 150; 350; 550; 750; 950 (s) after opening the gate valve. progressively steady interface. Considering now longer times, in x4 we reported preliminary experi- mental evidence that the ows transition between 3 distinct stages as the mean imposed ow V^0 is increased from zero. At low V^0, an exchange- ow dominated regime is found, as expected. This exchange ow may either be viscous (low Ret cos fl = V^” cos fl=V^t) or inertial (high Ret cos fl), following [132, 135]. In the latter case the ows are characterized by Kelvin-Helmholtz like instabilities. With increasing V^0 we observed that the ow becomes sta- ble. The speed of the leading front (say V^f ) increases approximately linearly with V^0, with slope larger than 1. We have termed this a viscous regime. At even larger V^0 we flnd that V^f » V^0, as the uids efiectively mix transver- sally. These 3 regimes form the framework for our understanding. Here we 1477.2. Displacement in pipes ˆt (s) 350 400 450 500 550 600 50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 18 ˆ D − yˆ (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 x4. In Fig. 7.3 we plot the variation of the leading front velocity V^f as a function of mean ow velocity V^0 for difierent values of density contrast and viscosity at pipe inclination angles: fl = 87 – and fl = 83 –. Figure 7.4a shows similar data at fl = 85 –. In these flgures we observe mostly the flrst and second regimes of the displacement, i.e. an initial plateau at low V^0 (exchange ow regime) followed by linear increase in V^f at larger V^0. Also shown in Figs. 7.3 & 7.4a is a secondary classiflcation of the front motion, described in detail in x6, that relates to the behaviour of the trailing front. It was found that for low V^0 buoyancy forces were su–ciently strong to produce a sustained upstream motion of the trailing front (a sustained back ow). On increasing V^0 we found a marginal state in which the trailing front advanced upstream against the ow and stopped for the duration of our ex- periment (a stationary interface ow). At the same time the downstream leading front is advected from the pipe leaving behind an apparently station- ary residual layer. At larger V^0 the trailing front moved only upstream for a flnite time, eventually reversing and moving downstream (a temporary back ow). Finally, at large V^0 the trailing front moves directly downstream (an instantaneous displacement). In each flgure we have classifled the displace- ments by examining the spatiotemporal diagram for the trailing front. As well as the transition from exchange ow dominated to viscous displacement regime, we also observe the transition from sustained back ow through to 1487.2. Displacement in pipes 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 ˆV0 (mm/s) ˆ V f (m m /s) a) β = 87 ◦ 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 ˆV0 (mm/s) ˆ V f (m m /s) b) β = 83 ◦ Figure 7.3: Variation of the front velocity V^f as a function of mean ow veloc- ity V^0 for difierent values of density contrast and viscosity at two inclination angles: a) at fl = 87 – data correspond to At = 10¡2 (¥), At = 3:6 £ 10¡3 (†), At = 10¡3 (N) with ” = 1 (mm2.s¡1) and At = 3 £ 10¡3 (H) with ” = 1:8 (mm2.s¡1); b) at fl = 83 – data correspond to At = 4 £ 10¡2 (⁄), At = 10¡2 (¥), At = 3:5 £ 10¡3 (†), At = 10¡3 (N) with ” = 1 (mm2.s¡1) and At = 3:5 £ 10¡3 (H) with ” = 1:7 (mm2.s¡1). In both plots sustained back ows and instantaneous displacements are marked by the superposed squares and circles respectively; data points without marks are either tempo- rary back ows, stationary interfaces or undetermined experiments (i.e. in- su–cient experiment time or short pipe length above the gate valve). instantaneous displacement on each data set, as V^0 is increased. Figure 7.4b examines the second regime more closely for the data at inclination angle fl = 85 –. In the data shown we have excluded those points classifled as sustained back ows and observe that these correspond well to the viscous regime and indeed have an approximately linear variation. The dashed lines give an approximate linear flt to each data set. The inset of Fig. 7.4b shows that by normalizing with V^” cos fl the data in the viscous regime collapses onto a single curve, which we now explain below in x7.2.2. It is this collapse of the data onto a single curve that establishes the essential viscous nature of the ow in this regime. Further explanation is given in x7.2.2 below. Certainly one of the most interesting aspects of the longer-time behaviour in our experiments is the laminarisation as V^0 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 V^0. In this case the pure exchange ow (V^0 = 0) is 1497.2. Displacement in pipes 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 180 200 ˆV0 (mm/s) ˆ V f (m m /s) a) β = 85 ◦ 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 ˆV0 (mm/s) ˆ V f (m m /s) b) 0 0.1 0.2 0 0.1 0.2 0.3 ˆV0/( ˆVν cos β) ˆ V f /( ˆ V ν co s β) β = 85 ◦ Figure 7.4: Variation of the front velocity V^f as a function of mean ow velocity V^0 for difierent values of density contrast and viscosity at fl = 85 –. a) Sustained back ows and instantaneous displacements are marked by the superposed squares and circles respectively; data points without marks are either temporary back ows, stationary interfaces or undetermined experi- ments. b) Illustration of the imposed ow dominated regime where, com- pared to the left plot, only temporary back ows are excluded. The dashed lines are linear flts of data points for each set of increasing V^0 (flxed At and ”^). The inset shows normalized front velocity V^f=V^” cos fl as a function of normalized mean ow velocity V^0=V^” cos fl, for which the data superim- pose. The solid line is a linear flt to all the normalized data points. In both flgures 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 ” = 1 (mm2.s¡1) and At = 3:7£ 10¡3 (H) with ”^ = 1:7 (mm2.s¡1). strongly inertial and in the flrst few snapshots we see a propagating layer of heavy uid at the bottom of the pipe with a signiflcant mixed layer on top. At intermediate imposed velocities we see the clear laminarisation of the ow (e.g. at V^0 = 57 & 72 (mm.s¡1)). Finally at larger V^0 we see progressively more mixing, except now there is su–cient inertia to mix across the whole pipe cross-section. Examples of spatiotemporal diagrams related to ows in the flrst and second regimes are shown in Fig. 7.6a-b, (from a difierent sequence than Fig. 7.5). For the parameters selected the pure exchange ow is inertial. For low V^0 = 30 (mm.s¡1), the ow 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 deflned dark region in 1507.2. Displacement in pipes SB SB SB 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 ˆV0 Figure 7.5: A sequence of snapshots from experiments with increased imposed ow rate; the parameters are fl = 83 –, At = 10¡2 and ”^ = 1 (mm2.s¡1). From top to bottom we show images for V^0 = 9; 19; 31; 44; 56; 57; 72; 108; 257; 474; 841 (mm.s¡1). The flgure shows a 1325 (mm) long section of the pipe a few centimeters below the gate valve. Key: SB = sustained back ow; TB = temporary back ow; ID = instanta- neous displacement. Fig. 7.6a. We observe a range of wave speeds difiering slightly from the front propagation speed. No second front is observed, as for this experiment the trailing front moves backward, upstream against the ow. For an increased V^0 = 75 (mm.s¡1) the ow 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 ow instability. The sustained back ow is evident in the negative veloc- ity values at the top of the pipe. The stable ow is illustrated in Fig. 7.6d. The UDV images are ensemble-averaged over 15 consecutive images, corre- 1517.2. Displacement in pipes xˆ (mm) ˆ t (s) 140012001000800600400200 0 25 50 75 100 125 150 a) Regime 1 xˆ (mm) ˆ t (s) 140012001000800 600400200 0 25 50 75 100 125 150 175 200 225 b) Regime 2 ˆt (s) ˆ D − yˆ (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) ˆt (s) ˆ D − yˆ (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 fl = 85 –, At = 10¡2 and ”^ = 1 (mm2.s¡1): a) & c) V^0 = 30 (mm.s¡1); b) & d) V^0 = 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 proflles, (1:8 (s)). sponding to a time average over a local interval of 1:8 (s). This eliminates small high frequency uctuations, which correspond to the UDV sampling rate. If we look carefully, we can observe the presence of negative values of ow velocity towards the top of the tube. In this experiment there is no back ow of the trailing front, but this does not preclude negative velocities. These regions correspond to a temporary recirculation at this position inside the upper uid, which persists for t^ … 125 (s), by which time the trailing front reaches the UDV probe located at x^ = 80 (cm). After the trailing 1527.2. Displacement in pipes front has passed a more Poiseuille-like ow 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 flxed imposed ow rate. 7.2.2 Lubrication/thin fllm model To explain the similarity scaling evident in our data (e.g. Fig. 7.4b), we re- sort to a lubrication/thin fllm style of model (assuming the immiscible limit Pe ! 1). This type of model has been developed for plane channel dis- placements in x5. Exchange ows have been studied using this type of model in [135] and in x5 we have extended this type of model to the displacement regimes studied here. For brevity, we refer to x5 for the derivation. The interface height evolution is governed by the following dimensionless equation: @ @T fi(h) + @ @» q(h; h») = 0: (7.7) In this model h 2 [0; 1] is the dimensionless interface height (scaled with the diameter), fi(h) 2 [0; 1] is the area fraction occupied by the heavy uid (under the interface) fi(h) = 1… cos ¡1(1¡ 2h)¡ 2… (1¡ 2h) p h¡ h2 (7.8) and the scaled ux of uid in the heavy layer is denoted q(h; h»): q(h; h») = 32… Z fi(h) (14 ¡ x 2 ¡ y2) dxdy + F0[´¡ h»]4 ¡1¡ (1¡ 2h)2¢7=2 : (7.9) The flrst term is the Poiseuille component and the second term is the ex- change ow component; F0 is given by Seon et al. [132] as F0 = 0:0118. The variables T and » are the dimensionless time and length variables, respec- tively: T = t^V^0D^ –; » = x^ D^ –; (7.10) where – = „^V^0[‰^H ¡ ‰^L]g^ sin flD^2 = V^02V^” sin fl : (7.11) 1537.2. Displacement in pipes This type of model contains the balance between viscous, buoyant and imposed ow stresses. Only a single dimensionless parameter ´ remains following the model reduction: ´ = cot fl– = [‰^H ¡ ‰^L]g^ cos flD^2 „^V^0 = 2V^” cos flV^0 = 2Re cos flF r2 ; (7.12) which represents the balance of axial buoyancy stresses and viscous stresses due to the imposed ow. The interface slope h» generates additional axial pressure gradients which contribute to the exchange ow component of ux in (7.9), but as the interface extends progressively longer this efiect 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 ´. Although the algebraic form of (7.7) difiers from that analysed for the plane channel, we flnd qualitatively similar behaviour. Typically we flnd a short initial transient during which the interface elongates from its initial position and during which time difiusive spreading due to the presence of the term h» 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 ´, buoyancy is strong and a second trailing front emerges that moves upstream. As ´ is reduced the trailing front speed decreases until there is no back ow (at a critical ´ = ´c = 116:32:::). The interface displaces only in the positive direction for ´ < ´c. At the upper wall the interface is pinned to the wall. Figure 7.7a shows the result of solving (7.7) numerically, comparing h(»; T ) at T = 10 for difierent ´ = 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: fi(hf )Vf = q(hf ; 0); Vf = @q@h(hf ; 0) •dfi dh (hf ) ‚¡1 ; (7.13) which can be solved numerically. The variation of the front speeds and heights with ´ is plotted in Fig. 7.7b. As ´ ! 0 the imposed ow 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 1547.2. 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 ow displacements based on the lubrication model solution for ´ = 0 (5), ´ = 10 (4), ´ = 50 (–), ´ = 200 (⁄); b) variation of the front speeds (solid line) and heights (broken line) with ´. critical ´ = ´c = 116:32::: with a front height hf = 0:72:::. At this value there is a stationary interface in the downstream part of the ow; see x6. The changes in Vf with ´ are easy to understand mathematically for a Newtonian uid, as the ux function q is composed of two parts, one of which is multiplied by ´. This follows simply from the principle of superposition. As ´ varies we essentially interpolate (nonlinearly) between the ´ = 0 and ´ = 1 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 ´¡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 ´ = ´c = 116:32::: where the stationary interface is found. For values of ´ < ´c, 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- 1557.2. Displacement in pipes 10−4 10−3 10−2 10−1 100 10−3 10−2 10−1 100 ˆV0/( ˆVν cos β) ≡ 2/χ ˆ V f /( ˆ V ν co s β) ≡ 2V f/ χ Displacement Back flow 2 1 Figure 7.8: Normalized front velocity, V^f=V^” cos fl, plotted against normal- ized mean ow velocity, V^0=V^” cos fl, for the full range of experiments in the flrst 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 V^0) for flxed At or viscosity. The heavy solid line indicates the scaled front velocity from the lubrication model. The circle indicates the theoretical transition (´ = ´c = 116:32::). The thin solid line shows V^f = V^0, below which front velocities are not possible (denoted region 1). Region 2 represents ows with increasingly signiflcant inertial efiects. cous forces generated by the imposed ow 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 efiect of inertia. Inertial efiects are di–cult to include in such models for the pipe geometry but we will return to this for the plane channel geometry in x7.4. 7.2.4 The exchange- ow dominated range Region 2 in Fig. 7.8 contains data from ows where inertial efiects are in- creasingly signiflcant in balancing the buoyancy-driven exchange component. Since these efiects are not included in the lubrication approximation, diver- 1567.2. Displacement in pipes gence from the theoretical front velocity curve in Fig. 7.8 is to be expected. Within this exchange- ow dominated regime we have no fully predictive model. However, as V^0 ! 0 we do recover the pure exchange ow results. In inertial exchange ows studied in pipes close to horizontal [135] found that the front velocity can be approximated by V^f … V^t; with … 0:7; where V^t = q Atg^D^ represents the velocity scale at which buoyancy is bal- anced by inertia. In the case that V^t is the relevant scale as V^0 ! 0, we might consider that is simply the leading order term in an expansion with respect to small Fr = V^0=V^t, i.e. for ´ ´c we assume V^f V^t = f(Fr) … f(0) + Frf 0(0) + Fr 2 2 f 00(0) + ::::; (7.14) with f(0) = … 0:7. With this ansatz we rescale V^f with V^t for all our inertial experimental data with ´ > ´c, and flt the coe–cients in (7.14). We flnd f 0(0) = 0:595 and f 00(0) = 0:724, which are in the confldence intervals f 0(0) 2 (0:454; 0:735) and f 00(0) 2 (0:478; 0:970) with confldence level 95%. Figure 7.9a shows a comparison of front velocity data in the exchange ow regime with the prediction: V^f V^t = 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 ´ 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 difierent for difierent experimental sequences. However, the curves appear to converge in this flgure close to the critical value V^0=V^” cos fl = 2=´c which is marked, and diverge thereafter. Note however, that in our experiments we have observed that even inertial exchange ows become viscous on increasing V^0. Thus, above the critical V^0=V^” cos fl = 2=´c our experimental sequences are fltted well by (7.13). 7.2.5 Overall classiflcation of the ow regimes For a more global perspective on our results and in particular to exem- plify the balance with inertia we present the classiflcation of our exper- imental results together with our ow regime predictions from both the 1577.2. Displacement in pipes 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 ˆV0/ ˆVt ≡ Fr ˆ V f /ˆ V t ≡ F rV f a) 10 −4 10−3 10−2 10−1 100 10−3 10−2 10−1 100 ˆV0/( ˆVν cosβ) ≡ 2/χ ˆ V f /( ˆ V ν c o s β) ≡ 2V f/ χ b) Figure 7.9: a) Normalized front velocity, V^f=V^t, as a function of normalized mean ow velocity V^0=V^t = Fr, (equivalently Froude number), plotted for 3 experimental sequences in the inertial regime. Data correspond to At = 9:1 £ 10¡2 at fl = 85 – (†), At = 4 £ 10¡2 at fl = 83 – (currency1), At = 10¡2 at fl = 83 – (¥), all with ” = 1 (mm2.s¡1)s. The broken line shows V^f=V^t = 0:7 + 0:595Fr + 0:362Fr2. b) Normalized front velocity, V^f=V^” cos fl, as a function of normalized mean ow velocity , V^0=V^” cos fl. Data points with the same symbols belong to same experimental sequence: increasing Reynolds number through V^0. The heavy solid line indicates the scaled front velocity from the lubrication model. The thin solid line shows V^f = V^0. The broken lines show our inertial exchange approximation through the simple model. lubrication model and our pure exchange ow curve flt. We plot our exper- imental results in the (V^0=V^t; V^” cos fl=V^t)-plane. Equally, this plane is the (Fr;Re cos fl=Fr)-plane. Note that lines of constant ´ correspond to linear rays through the origin, in the positive quadrant. The quantity Re cos fl=Fr is also related to a Reynolds number based on the inertial velocity scale, denoted Ret by Seon et al. [135], i.e. Ret cos fl · V^tD^ cos fl”^ · V^” cos fl V^t · Re cos flF r : (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 classifled according to the classiflcations of x6. Only data satisfying Re < 2300 has been used. The critical value ´c = 116:32::: corresponds to the line Re cos fl F r = ´c 2 Fr; (7.17) 1587.2. Displacement in pipes which is also marked in Fig. 7.10. Considering pure exchange ows, accord- ing to [135] the dividing line between viscous and inertial exchange ows is at Ret cos fl = V^” cos flV^t … 50; (7.18) (inertial exchange ows for larger Ret than (7.18)), which is marked in Fig. 7.10 with the heavy solid horizontal line. Finally, at su–ciently large imposed velocities we expect to transition to the mixed 3rd regime (e.g. see Fig. 7.5). A su–cient condition for this would be the onset of turbulence. Assuming that at high Re the buoyancy efiects have minimal efiect, we might assume transition at a nominal value Re = 2300. For difierent 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 difierent ow regimes and partly quantify the behaviour within each regime. (a) Inertial exchange ow dominated regime: This regime is found for Ret cos fl & 50 and for Fr = V^0=V^t . 0:9. In Fig. 7.10 this regime is marked by i1. This ow is characterized by development of Kelvin- Helmholtz-like instabilities and partial mixing. Buoyancy forces are su–ciently strong for there to be a sustained back ow. The front velocity in this regime scales with Vt and is approximated reasonably well by the empirical relation (7.15). (b) Inertial temporary back ow regime: In Fig. 7.10 this regime is marked by i2 and is bounded by (7.17) and Fr = V^0=V^t & 0:9. On increasing the imposed ow V^0, the destabilizing in uences of inertia become progressively less e–cient. The bulk ow remains generally in- ertial up until the critical stationary interface ow is encountered, along (7.17), after which the ow 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 ow dominated regime: For V^0 = 0 this regime is observed for Ret cos fl =. 50 [135]. For positive V^0 the same cri- terion appears correct. When the pure exchange ow is viscous, the corresponding displacement ows obtained by adding a small imposed ow V^0, 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 efiects can be observed at the beginning of displacement (i.e. at short time) 1597.2. Displacement in pipes 0 1 2 3 4 5 6 0 20 40 60 80 100 120 ˆV0 ˆVt ≡ Fr ˆ V ν c o s β ˆ V 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: Classiflcation of our results for the full range of experiments in the flrst and second regimes (Re < 2300) in Table 7.1: sustained back ow (¥, ⁄), stationary interface (.), temporary back ow (J, /) and instan- taneous displacement (†). Data point with fllled symbols are viscous and with hollow symbols are inertial. The horizontal bold line shows the flrst order approximation to the inertial-viscous transition (Ret cos fl = 50, from [135]). The dotted line and its continuation (the heavy line) represent the prediction of the lubrication model for the stationary interface, ´ = ´c. The vertical dashed-line is V^0=V^t = 0:9. The thin broken lines are only illustra- tive and show an estimate for the turbulent shear ow 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 efiects are also signiflcant local to the leading displacement front, where they usually appear in the form of an inertial bump. However, in the bulk of the ow energy is dissipated by viscosity. The front velocity can be well predicted by (7.13). (d) Viscous temporary back ow regime: These ows are found in a regime bounded by (7.17) and Fr = V^0=V^t . 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 ow-dominated 1607.2. Displacement in pipes as Fr is increased. The boundary of this regime with the exchange ow dominated regime occurs along (7.17), where stationary residual layers are found; see x6. This is again a viscous regime and the front velocity can be well predicted by (7.13). (e) Imposed ow dominated regime: When the imposed velocity is su–ciently strong, for either the inertial or viscous exchange ow domi- nated regimes, the ow transitions to a laminarised state dominated by viscous efiects. For the inertial exchange ow, the stabilizing efiect is seen on the whole ow while in the viscous exchange ow, the stabiliz- ing efiect 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 fllm model. In Fig. 7.10 this regime is marked by v3. (f) Mixed/turbulent regime: We have not studied in detail this flnal 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 ow regime involves a stratifled stretching of the interface along the pipe, reducing buoyancy efiects, we expect that this transition should be approximately the same as for the transitional ow of a single uid in pipe. At larger At this is less clear. Once in this regime, the front velocity is approximately equal to the imposed ow velocity for our experiments, but at longer times we would expect that dispersion is active. 7.2.6 Engineering predictions and displacement e–ciency Our flndings can be expressed in terms of simple predictive models for the leading front velocity as follows. (a) For Re < 2300, if Ret cos fl & 50 and ´ > ´c the leading front velocity is predicted by (7.15): V^f = V^t[0:7 + 0:595Fr + 0:362Fr2]. (b) For Re < 2300, if Ret cos fl . 50 the leading front velocity is predicted by the lubrication model: V^f = V^0Vf (´), where Fig. 7.7b shows Vf (´). (c) Re ‚ 2300, we assume the ow has mixed across the pipe: V^f = V^0. 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 V^0 is increased from zero, we present two experimental sequences in Figs. 7.11 & 7.12. 1617.2. Displacement in pipes 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 ˆV0 (mm/s) ˆ V f (m m /s) χ = χc Fr = 0.9 2 layer lubrication model Figure 7.11: Variation of the front velocity V^f as a function of mean ow velocity V^0 for fl = 85 –, At = 3:5 £ 10¡3, ” = 1 (mm2.s¡1), (for the pure exchange ow Ret cos fl … 42). Sustained back ows and instantaneous dis- placements are marked by the superposed squares and circles respectively. Data points without marks are either temporary back ows, stationary in- terfaces or undetermined experiments (i.e. insu–cient 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 ow and in- stantaneous displacement, through V^0=V^t = 0:9. The insets are pictures of a 264 (mm) long section of tube a few centimeters below the gate valve in the corresponding ow domains. In Fig. 7.11 we are in the viscous regime initially. The lubrication ap- proximation is very good for ´ > ´c and only begins to diverge as the stationary interface regime is passed for increasing V^0. Looking at the inset pictures below the critical stationary interface, it is clear that the imposed ow changes the shape of the inertial bump at the displacement front. After the critical stationary interface is attained, the inset flgures show that the inertial bump is absent. The divergence of V^f from the lubrication predic- tion as V^0 increases suggests that inertial efiects in the ow are becoming 1627.2. Displacement in pipes important. We return to this in x7.4. 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 ˆV0 (mm/s) ˆ V f (m m /s) χ = χc Fr = 0.9 ˆVf / ˆVt = 0.7 + 0.595Fr + 0.362Fr2 Figure 7.12: Variation of the front velocity V^f as a function of mean ow velocity V^0 for fl = 83 –, At = 10¡2, ” = 1 (mm2.s¡1). Sustained back ows and instantaneous displacements are marked by the superposed squares and circles respectively. Data points without marks are either temporary back ows, stationary interfaces or undetermined experiments (i.e. insu–cient 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 ow and temporary back ow, through V^0=V^t = 0:9. The insets are pictures of a 1325 (mm) long section of tube a few centimeters below the gate valve in the corresponding ow domains. Fig. 7.12 shows data from a sequence in which we are initially in the inertial exchange ow regime. The solid line now shows the approximation with (7.15), which is again very good for ´ > ´c. For ´ • ´c we observe that the imposed ow gradually laminarises the ow. The model (7.15) is no longer a good approximation and the viscous lubrication approximation takes over. The main interest in front velocity V^f is in giving a measure of the displacement e–ciency of the ow. There is no universal deflnition for the 1637.2. Displacement in pipes displacement e–ciency, but one common notion is that it should represent the fraction of the tube that is fllled with displacing uid at a given time t^. Suppose that at t^ the front has advanced to position L^, beyond the gate valve. Then we have that L^ … V^f t^ (assuming any initial efiects are short-lived) and the volume of the pipe in which the displacement has taken place up until that time is L^…D^2=4 … V^f t^…D^2=4. The volume of displacing uid within that length of pipe is simply equal to the volume that has been pumped up until that time: V^0t^…D^2=4. Therefore, the fraction of displacing uid in this volume is simply V^0=V^f . If we consider only ows dominated by the imposed ow, i.e. for which the interface is clean and well deflned in our experimental images, we can make a secondary estimate of the displacement e–ciency by directly inte- grating the volume of the pipe occupied by the displacing uid. This can be compared with V^0=V^f for veriflcation. To compute the displaced volume we wait until the interface has developed for a su–cient time and length below the gate valve. Spatially we typically take L^ > 75D^. Due to the difierent front speeds the (growing) integration length L^ varies between experiments: the largest integration frame we have used had a length of 1540 (mm). To reduce efiects of transients, we average the interface measurements from 3 images separated by time interval 0:5 (s). An example of the comparison between V^0=V^f and the computed dis- placement e–ciency 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 e–ciency increases with V^0 to a plateau, signifled by the linear viscous imposed ow 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 uids the volume integration will clearly be vulnerable, but provided we can estimate V^f the volume- displacement argument still holds and V^0=V^f gives the e–ciency. 7.2.7 Dispersive efiects It has been noted that in the imposed ow dominated regime there is a con- sistent discrepancy between lubrication model prediction and experimental results. A linear relation is observed between V^f and V^0 but the dimension- less Vf (´) of Fig. 7.7b underestimates the slope by a few percent. Since this discrepancy increases with V^0 one possible explanation would be to relate this to inertial efiects, and we consider these later in x7.4. Instead we outline another possible cause of the discrepancy. In deriving the lubrication equation (7.7) and solving numerically we 1647.2. 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 ˆV0 (mm/s) ˆ V 0 /ˆ V f, D isp la ce m en t Effi ci en cy Figure 7.13: Comparison between the ratio V^0=V^f and the value of the displacement e–ciency computed through the interface integral for At = 3:5 £ 10¡3 and ”^ = 1 (mm2.s¡1). The data points computed through the ratio V^0=V^f are shown by fllled symbols: fl = 87 – (†), fl = 85 – (H); com- puted integrals are shown by hollow symbols: fl = 87 – (4), fl = 85 – (⁄). Lubrication model prediction of the imposed velocity for the transition to imposed ow dominated regime is 11 (mm.s¡1) for fl = 87 – and 19 (mm.s¡1) for fl = 85 –. The displacement e–ciency is not deflned for the data points with the imposed ow 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(»; 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 with the velocity at the interface. Since the velocity component u varies across the channel, typically being larger in the centre, the interface will advect further downstream towards the centre of the pipe. The lubrication equation (7.7), which simply represents mass conser- vation, is perfectly valid if h = h(»; z; T ). Where the discrepancy enters 1657.3. Displacement in channels mathematically is that: @ @T fi(h) = Z zR zL @h @T (»; z; T ) dz; where zL and zR denote the intercepts (in z) of the interface with the pipe wall. Thus, equation (7.7) is an equation for the mean interface height, averaged in the transverse, z-direction. When we assume h = h(»; T ) no mathematical inconsistency arises. The chord of the circle representing the interface length simply becomes @fi@h and (7.7) can be reduced to an equation for h(»; T ). As discussed, the interface should generally move faster in the centre of the pipe, away from the walls. What is interesting is that this efiect has not been directly observed in our experiments. Our imaging focuses at the side view of the displacements, but observation from above has not revealed any easily noticeable efiect. One possibility is that as the heavier uid advances in the centre of the pipe, gravity acts to smooth out non- uniformity in the z-direction, i.e. via secondary ows in the cross-sectional plane. Such ows would modify the volumetric ux in (7.7), but it is unclear how. Without more detailed 3D velocimetry it would be hard to determine if this is happening. Nevertheless, we expect this type of interfacial dispersion to be present. The amount of dispersion should increase with V^0 and this efiect could explain the discrepancy between the experimental results and the lubrication model predictions. 7.3 Displacement in channels Similar to the previous chapters, as a second displacement ow geometry we consider a plane channel. Whereas in the pipe ow any detailed com- putations would necessarily be three-dimensional, in the plane channel they are two-dimensional, which has distinct advantages in terms of computa- tional speed. Furthermore the simpler geometry allows room for analysis that would be prohibitively complex in the pipe geometry; see x7.4 later. One can either consider the plane channel as an independent study or as one which allows new perspectives on the pipe displacement ow. In making inferences regarding the pipe ow some caution is needed. For example, as mentioned in x2, Hallez & Magnaudet [67] studied pure ex- change ows in pipes and channels in the inertially dominated regime, when the uids mix, and have shown distinct difierences in the ow structures ob- served. Therefore, direct comparisons are only likely to be valid in regimes 1667.3. Displacement in channels fl – ”^ (mm2.s¡1) At (£10¡3) V^0 (mm.s¡1) Re Fr 81 1 3:5 0¡ 27 0¡ 500 0¡ 1:03 83a 1¡ 2 1¡ 10 0¡ 153 0¡ 2907 0¡ 3:53 85 1¡ 2 1¡ 10 0¡ 27 0¡ 500 0¡ 1:92 87 1¡ 2 1¡ 10 0¡ 27 0¡ 500 0¡ 1:92 88 1 3:5 0¡ 27 0¡ 500 0¡ 1:03 89 1 10 0¡ 27 0¡ 500 0¡ 0:61 90 1¡ 2 0¡ 10 0¡ 27 0¡ 500 0¡1 aMost of the simulations were conducted in the range of [”^ (mm2.s¡1), At (£10¡3), V^0 (mm.s¡1); Re, Fr] 2 [1¡ 2, 1¡ 10, 0¡ 27; 0¡ 500, 0¡ 1:92]. Table 7.2: Numerical simulation plan. where viscous forces dominate in balancing buoyant and imposed pressure drops. Our numerical simulation plan is detailed in Table 7.2. We have selected a range of dimensional parameters that is similar in scope to those of our pipe ow experiments. However, we have not explored very high Re (typically Re < 500 for our simulations). At larger Re we would expect to enter a fully mixed turbulent regime, for which we have not explored the performance of our code. 7.3.1 Exchange ow results We flrst start with an examination of the pure exchange ow in the channel. The objective is to gain an understanding of the transition from inertial to viscous dominated exchange ows (V^0 = 0), parallel to that deduced in the experimental studies of [135]. We have seen the relevance of this transition for pipe ows as a flrst order prediction of the transition from viscous to inertial ows in buoyancy dominated displacement ows (V^0 > 0). The exchange ow results have also given the leading order term in the expansion (7.15). Our results are shown in Fig. 7.14 where we have plotted the normalized front velocity V^f=V^t against the inertial Reynolds number Ret cos fl = V^” cos fl=V^t. The transition between viscous and inertial lock-exchange ows (i.e. with V0 = 0) in plane channels, as determined by our 2D simulations, occurs in the range Ret cos fl = 25§ 5. We see a separation between a linear increase of V^f with the viscous velocity scale V^” cos fl, for Ret cos fl < 25 § 5 (vis- 1677.3. Displacement in channels 0 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 0.6 ˆVν cos(β)/ ˆVt ≡ Ret cos β ˆ V f /ˆ V t ≡ F rV f Figure 7.14: Variation of the normalised stationary front velocity V^f=V^t as a function of the inertial Reynolds number Ret cos fl = V^” cos fl=V^t for lock- exchange ows (V0 = 0). The simulation data correspond to difierent tilt angles, viscosities and density contrasts in the range of [fl –, ”^ (mm2.s¡1), At (£10¡3)] 2 [60¡ 89, 1¡ 4, 1¡ 10]. The transition between viscous and inertial lock-exchange ows for the mentioned simulation range occurs at Ret cos fl = 25 § 5. Guide lines are drawn in this flgure: horizontal dashed line at V^f=V^t = 0:4, vertical dash-dot line at Ret cos fl = 25, and the oblique dashed line showing a more or less linear relation between V^f and V^” cos fl. cous regime) and a constant plateau for which V^f … 0:4V^t for the range of ow parameters studied. This compares with values of Ret cos fl … 50 and V^f … 0:7V^t for the pipe exchange ow transition; see [133]. As Ret cos fl in- creases, the decrease in V^f is due to (geometry dependent) coherent vortices which cut the channel of pure uid feeding the front, which decreases the density contrast at the front and therefore the front velocity. This inter- esting phenomenon has been studied in depth by Hallez & Magnaudet [67]. The extent of the plateau (in Ret cos fl) is not however known. To investi- gate this regime would require a detailed study of mixing regimes occurring at inclinations closer to vertical. This is not the present objective. 1687.3. Displacement in channels 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 ˆV0 (mm/s) ˆ V f (m m /s) a) β = 81 ◦ 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 ˆV0 (mm/s) ˆ V f (m m /s) b) β = 83 ◦ 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 ˆV0 (mm/s) ˆ V f (m m /s) c) β = 85 ◦ 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 ˆV0 (mm/s) ˆ V f (m m /s) d) β = 87 ◦ 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 ˆV0 (mm/s) ˆ V f (m m /s) e) β = 88 ◦ 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 ˆV0 (mm/s) ˆ V f (m m /s) f ) β = 90 ◦ Figure 7.15: Variation of the downstream front velocity V^f as a function of mean ow velocity V^0 for difierent inclination angles: a) fl = 81 –; b) fl = 83 –; c) fl = 85 –; d) fl = 87 –; e) fl = 88 –; f) fl = 90 –. In each plot: At = 10¡2 (¥), At = 3:5 £ 10¡3 (†), At = 10¡3 (N), all with ” = 1 (mm2.s¡1); At = 3:5£10¡3 with ” = 2 (mm2.s¡1) (H). In all plots sustained back ows and instantaneous displacements are squared and circled respectively. The heavy solid line is V^f = 1:5V^0. 1697.3. Displacement in channels 7.3.2 Displacement ow results Turning now to the displacement ow results, an overall comment is that there are many aspects of the ow that are qualitatively similar to the pipe ow, but also signiflcant difierences. Starting with bulk ow parameters, such as the leading displacement front, there is qualitatively similar be- haviour as V^0 is increased from zero. Figure 7.15 shows the variation of V^f with V^0 for a wide range of our data, over various inclination angles fl, Atwood numbers At and kinematic viscosities ”^. Physically this range is similar to that covered in the pipe ow experiments. Low values of V^0 are dominated by the exchange ow characteristics and an approximate plateau in V^f is observed. On increasing V^0 we enter a regime, where the increase in V^f is approximately linear with V^0. We can see this in Fig. 7.15 at dif- ferent inclinations, very similar to the pipe ows. As well as the transition from exchange ow dominated to linear regime, consideration of the trailing front leads to a secondary classiflcation which ranges from sustained back ow through instantaneous displacement, exactly as for the pipe ow. This secondary classiflcation has been explored in detail in x6. Other features of the ow where we flnd similarity with the pipe ow are as follows. † Inertial efiects are more prevalent as At increases and at steeper chan- nel inclinations. † Imposition of the mean ow does have a laminarising efiect on the ow. For example, we can observe an inertial region at the leading front that is strongly afiected by the imposed ow. However, we have not observed strongly inertial exchange ows being fully stabilized, which was the case in the pipe ows. † In the exchange ow dominated regime, whether inertial or viscous, the basic structure is a two-layer ow bounded by one front moving downstream and a second front moving upstream. We now discuss some of the signiflcant difierences with the pipe displace- ment ows, starting with viscous ows (deflned approximately byRet cos fl < 25 § 5). The most obvious difierence is that the front is not observed to displace in a slumping 2-layer pattern, but instead a flnger advances ap- proximately along the centre of the channel leaving behind upper and lower layers of displaced uid. A typical example is shown in Fig. 7.16 where we show snapshots of the concentration proflle at difierent times. The phys- ical parameters correspond to fl = 87 –, At = 10¡3, ”^ = 1 (mm2.s¡1), V^0 = 26:3 (mm.s¡1). 1707.3. Displacement in channels Figure 7.16: Sequence of concentration fleld evolution obtained for fl = 87 –, At = 10¡3, ”^ = 1 (mm2.s¡1), V^0 = 26:3 (mm.s¡1), (Re = 500). The images from top to bottom are shown for t^ = 0; 5; 10; 15; 20; 30; 35 (s). The length shown is the whole channel, L^ = 100D^. This feature is partly expected. In parallel with our experimental study, we have considered low viscosity uids (essentially water) and hence enter the viscous regime by ensuring At is small and fl is close to horizontal. Ob- viously, on taking At ! 0 we have two identical uids and expect to recover a plane Poiseuille ow. This is indeed the case. The smallest values of Atwood number (At = 10¡3) correspond to a 0:2% density difierence be- tween uids and it is hardly surprising to see the front nearly symmetric and advancing close to the channel centreline. For two identical uids in plane Poiseuille ow the leading front speed would be simply V^f = 1:5V^0, which is the heavy solid line marked in Fig. 7.15, (note this is V^f = 4=3V^0 for the pipe, if a stratifled interface is assumed). We can see that as the linear regime is entered the front velocity lies just below this iso-dense limit. Although the front advances towards the centre of the channel, density dif- ferences are expressed through asymmetry of the residual layers above and below. Typically the lower layer is shorter and thinner than the upper layer, except near the tip where it seems that inertial efiects act to point the tip upwards. This is the analogy of the inertial bump that we have observed in the pipe displacement ows. 1717.3. Displacement in channels At = 10−3 At = 3.5 × 10−3 At = 10−2 β = 83 ◦ β = 87 ◦ β = 90 ◦ Figure 7.17: Panorama of concentration colourmaps for displacements with ” = 1 (mm2.s¡1), each taken at t^ = 25 (s). Top panel: fl = 83 –; middle panel fl = 87 –; bottom panel fl = 90 –. In each panel the rows from top to bottom show V^0 = 2:7; 5:3; 10:5; 15:8; 21:0; 26:3 (mm.s¡1) (equivalently Re = 50; 100; 200; 300; 400; 500). The columns from left to right show At = 10¡3; 3:5 £ 10¡3; 10¡2. The length shown is the whole channel, L^ = 100D^. 1727.3. Displacement in channels At = 10−3 At = 3.5 × 10−3 At = 10−2 β = 83 ◦ β = 87 ◦ β = 90 ◦ Figure 7.18: Panorama of velocity proflles corresponding to concentration colourmaps shown in Fig. 7.17. 1737.3. Displacement in channels a) b) Figure 7.19: Sequence of concentration fleld evolution obtained for fl = 87 –, ”^ = 1 (mm2.s¡1), each taken at t^ = 25 (s), close to the front: a) top panel (6 images), At = 10¡3; b) bottom panel (6 images), At = 10¡2. The rows from top to bottom show V^0 = 2:7; 5:3; 10:5; 15:8; 21:0; 26:3 (mm.s¡1) (equivalently Re = 50; 100; 200; 300; 400; 500). The length shown is L^ … 29D^ (so that the flgures are almost 1 : 1). In the context of the plane channel results, it is interesting to review the pipe ow displacement experiments again. For very similar physical parameters we always observed a slumping displacement front. A possible explanation for this would be that the pipe allows for three-dimensional secondary ows, i.e. less dense uid in a layer underneath an advancing flnger can be squeezed azimuthally around the sides of the tube by the heavier flnger. In a strictly 2D geometry this does not happen. In this context it would be interesting to study displacements in a rectangular cross-sectional channel. To give a broader understanding of the ow variations with At, Re and fl, Fig. 7.17 presents a panorama of concentratio
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From displacement to mixing in a slightly inclined duct Taghavi, Seyed Mohammad 2011
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Title | From displacement to mixing in a slightly inclined duct |
Creator |
Taghavi, Seyed Mohammad |
Publisher | University of British Columbia |
Date | 2011 |
Date Issued | 2011-11-24 |
Description | This thesis studies buoyant displacement flows with two miscible fluids in pipes and 2D channels that are inclined at angles (β) close to horizontal. Detailed 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 absence/presence of interfacial instability and mixing. This dual characterisation 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 displacement regime and a slump-type regime for higher density differences. In both regimes, the displaced fluid can remain completely static in residual wall layers. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2011-11-24 |
Provider | Vancouver : University of British Columbia Library |
DOI | 10.14288/1.0059111 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/39272 |
Aggregated Source Repository | DSpace |
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