0
Research Papers: Fundamental Issues and Canonical Flows

A Lattice Boltzmann Simulation of Three-Dimensional Displacement Flow of Two Immiscible Liquids in a Square Duct

[+] Author and Article Information
Kirti Chandra Sahu

e-mail: ksahu@iith.ac.in
Department of Chemical Engineering,
Indian Institute of Technology Hyderabad,
Ordinance Factory Estate, Yeddumailaram,
Andhra Pradesh, Hyderabad 502205, India

S. P. Vanka

Department of Chemical Engineering,
Indian Institute of Technology Hyderabad,
Ordinance Factory Estate, Yeddumailaram,
Andhra Pradesh, Hyderabad 502205, India;
Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61822

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 5, 2013; final manuscript received July 10, 2013; published online September 19, 2013. Assoc. Editor: Samuel Paolucci.

J. Fluids Eng 135(12), 121202 (Sep 19, 2013) (8 pages) Paper No: FE-13-1132; doi: 10.1115/1.4024998 History: Received March 05, 2013; Revised July 10, 2013

A three-dimensional (3D), multiphase lattice Boltzmann approach is used to study a pressure-driven displacement flow of two immiscible liquids of different densities and viscosities in a square duct. A three-dimensional, 15-velocity (D3Q15) lattice model is used. The effects of channel inclination, viscosity, and density contrasts are investigated. The contours of the density and the average viscosity profiles in different planes are plotted and compared with those obtained in a two-dimensional (2D) channel. We demonstrate that the flow dynamics in a 3D channel is quite different as compared to that of a 2D channel. We found that the flow is relatively more coherent in a 3D channel than that in a 2D channel. A new screw-type instability is seen in the 3D channel that cannot be observed in the 2D channel.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Redapangu, P. R., Sahu, K. C., and Vanka, S. P., 2012, “A Study of Pressure-Driven Displacement Flow of Two Immiscible Liquids Using a Multiphase Lattice Boltzmann Approach,” Phys. Fluids, 24, p. 102110. [CrossRef]
Joseph, D. D., Bai, R., Chen, K. P., and Renardy, Y. Y., 1997, “Core-Annular Flows,” Ann. Rev. Fluid Mech., 29, pp. 65–90. [CrossRef]
Whitehead, J. A., and Helfrich, K. R., 1991, “Instability of the Flow With Temperature-Dependent Viscosity: A Model of Magma Dynamics,” J. Geophysical Res., 96, pp. 4145–4155. [CrossRef]
Homsy, G. M., 1987, “Viscous Fingering in Porous Media,” Ann. Rev. Fluid Mech., 19, pp. 271–311. [CrossRef]
Chen, C.-Y., and Meiburg, E., 1996, “Miscible Displacement in Capillary Tubes. Part 2. Numerical Simulations,” J. Fluid Mech., 326, pp. 57–90. [CrossRef]
Rakotomalala, N., Salin, D., and Watzky, P., 1997, “Miscible Displacement Between Two Parallel Plates: BGK Lattice Gas Simulations,” J. Fluid Mech., 338, pp. 277–297. [CrossRef]
Goyal, N., and Meiburg, E., 2006, “Miscible Displacements in Hele-Shaw Cells: Two-Dimensional Base States and Their Linear Stability,” J. Fluid Mech., 558, pp. 329–355. [CrossRef]
Petitjeans, P., and Maxworthy, P., 1996, “Miscible Displacements in Capillary Tubes. Part 1. Experiments,” J. Fluid Mech., 326, pp. 37–56. [CrossRef]
Sahu, K. C., Ding, H., Valluri, P., and Matar, O. K., 2009, “Linear Stability Analysis and Numerical Simulation of Miscible Channel Flows,” Phys. Fluids, 21, p. 042104. [CrossRef]
Sahu, K. C., Ding, H., Valluri, P., and Matar, O. K., 2009, “Pressure-Driven Miscible Two-Fluid Channel Flow With Density Gradients,” Phys. Fluids, 21, p. 043603. [CrossRef]
Taghavi, S. M.Séon, T.Martinez, D. M. and Frigaard, I. A., 2009, “Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit”. J. Fluid Mech., 639, pp. 1–35. [CrossRef]
Taghavi, S. M., Séon, T., Martinez, D. M., and Frigaard, I. A., 2011, “Stationary Residual Layers in Buoyant Newtonian Displacement Flows,” Phys. Fluids, 23, p. 044105. [CrossRef]
Mishra, M., Wit, A. D., and Sahu, K. C., 2012, “Double Diffusive Effects on Pressure-Driven Miscible Displacement Flow in a Channel,” J. Fluid Mech., 712, pp. 579–597. [CrossRef]
Talon, L., Goyal, N., and Meiburg, E., 2013, “Variable Density and Viscosity, Miscible Displacements in Horizontal Hele-Shaw Cells. Part 1. Linear Stability Analysis,” J. Fluid Mech., 721, pp. 268–294. [CrossRef]
Talon, L., Goyal, N., and Meiburg, E., 2013, “Variable Density and Viscosity, Miscible Displacements in Horizontal Hele-Shaw Cells. Part 1. Nonlinear Simulations,” J. Fluid Mech., 721, pp. 295–323. [CrossRef]
Joseph, D. D., Renardy, M., and Renardy, Y. Y., 1984, “Instability of the Flow of Two Immiscible Liquids With Different Viscosities in a Pipe,” J. Fluid Mech., 141, pp. 309–317. [CrossRef]
Kang, Q., Zhang, D., and Chen, S., 2004, “Immiscible Displacement in a Channel: Simulations of Fingering in Two Dimensions,” Adv. Water Resour., 27, pp. 13–22. [CrossRef]
Chin, J., Boek, E. S., and Coveney, P. V., 2002, “Lattice Boltzmann Simulation of the Flow of Binary Immiscible Fluids With Different Viscosities Using the Shan-Chen Microscopic Interaction Model,” Phil. Trans. R. Soc. London, 360, pp. 547–558. [CrossRef]
Grosfils, P., Boon, J. P., and Chin, J., 2004, “Structural and Dynamical Characterization of Hele-Shaw Viscous Fingering,” Phil. Trans. R. Soc. London, 362, pp. 1723–1734. [CrossRef]
Dong, B., Yan, Y. Y., Li, W., and Song, Y., 2010, “Lattice Boltzmann Simulation of Viscous Fingering Phenomenon of Immiscible Fluids Displacement in a Channel,” Comput. Fluids, 39, pp. 768–779. [CrossRef]
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D., and Yortsos, Y. C., 1999, “Miscible Displacement in a Hele-Shaw Cell at High Rates,” J. Fluid Mech., 398, pp. 299–319. [CrossRef]
Sahu, K. C., Ding, H., and Matar, O. K., 2010, “Numerical Simulation of Non-isothermal Pressure-Driven Miscible Channel Flow With Viscous Heating,” Chem. Eng. Sci., 65, pp. 3260–3267. [CrossRef]
Govindarajan, R., and Sahu, K. C., 2014, “Instabilities in Viscosity Stratified Flow,” Ann. Rev. Fluid Mech. (to be published).
Yih, C. S., 1967, “Instability Due to Viscous Stratification,” J. Fluid Mech., 27, pp. 337–352. [CrossRef]
Yiantsios, S. G., and Higgins, B. G., 1988, “Linear Stability of Plane Poiseuille Flow of Two Superposed Fluids,” Phys. Fluids, 31, pp. 3225–3238. [CrossRef]
Hooper, A. P., and Boyd, W. G. C., 1983, “Shear Flow Instability at the Interface Between Two Fluids,” J. Fluid Mech., 128, pp. 507–528. [CrossRef]
South, M. J., and Hooper, A. P., 2001, “Linear Growth in Two-Fluid Plane Poiseuille Flow,” J. Fluid Mech., 381, pp. 121–139. [CrossRef]
Chen, S., and Doolen, G. D., 1998, “Lattice Boltzmann Method for Fluid Flows,” Ann. Rev. Fluid Mech., 30, pp. 329–364. [CrossRef]
Gunstensen, A. K., Rothman, D. H., Zaleski, S., and Zanetti, G., 1991, “Lattice Boltzmann Model for Immiscible Fluids,” Phys. Rev. A, 43, pp. 4320–4327. [CrossRef] [PubMed]
Shan, X., and Chen, H., 1993, “Lattice Boltzmann Model for Simulating Flows With Multiple Phases and Components,” Phys. Rev. E, 47(3), pp. 1815–1819. [CrossRef]
Swift, M. R., Osborn, W. R., and Yeomans, J. M., 1995, “Lattice-Boltzmann Simulation of Nonideal Fluids,” Phys. Rev. Lett., 75, pp. 830–833. [CrossRef] [PubMed]
Zhang, R., He, X., and Chen, S., 2000, “Interface and Surface Tension in Incompressible Lattice Boltzmann Multiphase Model,” Comput. Phys. Commun., 129, pp. 121–130. [CrossRef]
He, X., Zhang, R., Chen, S., and Doolen, G. D., 1999, “On the Three-Dimensional Rayleigh-Taylor Instability,” Phys. Fluids, 11(5), pp. 1143–1152. [CrossRef]
He, X., Chen, S., and Zhang, R., 1999, “A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability,” J. Comput. Phys., 152, pp. 642–663. [CrossRef]
Langaas, K., and Yeomans, J. M., 2000, “Lattice Boltzmann Simulation of a Binary Fluid With Different Phase Viscosities and Its Application to Fingering in Two-Dimensions,” Eur. Phys. J. B, 15, pp. 133–141. [CrossRef]
Riaz, A., and Meiburg, E., 2003, “Three-Dimensional Miscible Displacement Simulations in Homogeneous Porous Media With Gravity Override,” J. Fluid Mech., 494, pp. 95–117. [CrossRef]
Oliveira, R. M., and Meiburg, E., 2011, “Miscible Displacements in Hele-Shaw Cells: Three-Dimensional Navier-Stokes Simulations,” J. Fluid Mech., 687, pp. 431–460. [CrossRef]
Hallez, Y., and Magnaudet, J., 2008, “Effects of Channel Geometry on Buoyancy-Driven Mixing,” Phys. Fluids, 20, p. 053306. [CrossRef]
Vanka, S. P., Shinn, A. F., and Sahu, K. C., 2011, “Computational Fluid Dynamics Using Graphics Processing Units: Challenges and Opportunities,” Proceedings of the ASME 2011 International Mechanical Engineering Congress and Exposition, Denver, CO, Nov. 11–17.
Redapangu, P. R., Vanka, S. P., and Sahu, K. C., 2012, “Multiphase Lattice Boltzmann Simulations of Buoyancy-Induced Flow of Two Immiscible Fluids With Different Viscosities,” Eur. J. Mech. B/Fluids, 34, pp. 105–114. [CrossRef]
Sahu, K. C., and Vanka, S. P., 2011, “A Multiphase Lattice Boltzmann Study of Buoyancy-Induced Mixing in a Tilted Channel,” Comput. Fluids, 50, pp. 199–215. [CrossRef]
Lee, T., and Lin, C.-L., 2005, “A Stable Discretization of the Lattice Boltzmann Equation for Simulation of Incompressible Two-Phase Flows at High Density Ratio,” J. Comput. Phys, 206, pp. 16–47. [CrossRef]
Carnahan, N. F., and Starling, K. E., 1969, “Equation of State for Non-attracting Rigid Spheres,” J. Chem. Phys., 51, pp. 635–636. [CrossRef]
Premnath, K. N., and Abraham, J., 2005, “Lattice Boltzmann Model for Axisymmetric Multiphase Flows,” Phys. Rev. E, 71, p. 056706. [CrossRef]
Evans, R., 1979, “The Nature of the Liquid-Vapor Interface and Other Topics in the Statistical Mechanics of Non-uniform Classical Fluids,” Adv. Phys., 28, pp. 143–200. [CrossRef]
Selvam, B., Merk, S., Govindarajan, R., and Meiburg, E., 2007, “Stability of Miscible Core-Annular Flows With Viscosity Stratification,” J. Fluid Mech., 592, pp. 23–49. [CrossRef]
Scoffoni, J., Lajeunesse, E., and Homsy, G. M., 2001, “Interface Instabilities During Displacement of Two Miscible Fluids in a Vertical Pipe,” Phys. Fluids, 13, pp. 553–556. [CrossRef]
Hooper, A. P., and Grimshaw, R., 1985, “Nonlinear Instability at the Interface Between Two Viscous Fluids,” Phys. Fluids, 28(1), pp. 37–45. [CrossRef]

Figures

Grahic Jump Location
Fig. 2

Schematic diagram of D3Q15 lattice

Grahic Jump Location
Fig. 1

Schematic describing the geometry and the initial configuration

Grahic Jump Location
Fig. 3

Variation of mass fraction of the displaced liquid (Mt/M0) with time for different grids. The rest of the parameters are Re = 100, m = 10, At = 0.2, Fr = 5, Ca = 29.9, and θ = 45 deg. The dotted line represents the analytical solution of the plug-flow displacement, given by Mt/M0 = 1-tH/L.

Grahic Jump Location
Fig. 4

Contours of the index function, ϕ, at (a) t = 10 and (b) t = 30 in the x-y plane at z = W/2 for different grids. The parameters are the same as those of Fig. 3.

Grahic Jump Location
Fig. 5

Evolution of the isosurface of ϕ at the interface at different times (from left to right: t = 12, 18, 30, and 50) for the simulation domain of 2112 × 66 × 66 grid. The parameters are Re = 100, m = 10, At = 0.2, Fr = 5, Ca = 29.9, and θ = 45 deg. The flow is in the positive x direction.

Grahic Jump Location
Fig. 13

Spatiotemporal evolution of the contours of the index function, ϕ, obtained from the three-dimensional channel for m = 5 with the rest of the parameter values the same as those used to generate Fig. 12. The results shown are in x-y plane at z = W/2 of the channel.

Grahic Jump Location
Fig. 14

The contours of the index function, ϕ, at t = 20 in the x-y plane at z = W/2 for different values of Atwood number. The rest of the parameter values are Re = 100, m = 10, Fr = 1, θ = 30 deg, and Ca = 29.9.

Grahic Jump Location
Fig. 6

Contours of the index function, ϕ, at t = 20 in (a) x-y plane at z = W/2 and (b) x-z plane at y = H/2. The parameters are Re = 100, At = 0.2, m = 10, Fr = 1, and Ca = 29.9.

Grahic Jump Location
Fig. 7

Contours of the index function, ϕ, at t = 20 in the y-z plane at x = L/2. The parameters are Re = 100, At = 0.2, m = 10, Fr = 1, and Ca = 29.9.

Grahic Jump Location
Fig. 8

Axial variation of normalized average viscosity, μ¯yz(≡μyz/μyz0) for (a) θ = 30 deg and (b) θ = 85 deg. The rest of the parameters are Re = 100, At = 0.2, m = 10, Fr = 1, and Ca = 29.9. Here, μyz=(1/HW)∫0W∫0Hμdydz and μyz0 is the value of μyz at x = 0.

Grahic Jump Location
Fig. 9

Vertical variation of normalized average viscosity, μ¯xz(≡μxz/μxz0) for (a) θ = 30 deg and (b) θ = 85 deg. The rest of the parameters are the same as those used in Fig. 6. Here, μxz=1/LW∫0W∫0Lμdxdz and μxz0 is the value of μxz at y = 0.

Grahic Jump Location
Fig. 10

Axial variation of normalized average viscosity, μ¯y(≡μy/μy0), for (a) θ = 5 deg and (b) θ = 85 deg in a two-dimensional channel. The rest of the parameters are the same as those used to generate Fig. 6. Here, μy=1/H∫0Hμdy and μy0 is the value of μy at x = 0.

Grahic Jump Location
Fig. 11

Vertical variation of normalized average viscosity, μ¯x(≡μx/μx0), for (a) θ = 5 deg and (b) θ = 85 deg in a two-dimensional channel. The rest of the parameters are the same as those used to generate Fig. 6. Here, μx=1/L∫0Lμdx and μx0 is the value of μx at y = 0.

Grahic Jump Location
Fig. 12

Spatiotemporal evolution of the contours of the index function, ϕ. (a) x-y plane at z = W/2 (three-dimensional simulation) and (b) the corresponding two-dimensional results. The parameters are Re = 100, m = 20, At = 0.2, Fr = 5, θ = 30 deg, and Ca = 29.9.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In