Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-17T19:20:00.977Z Has data issue: false hasContentIssue false
  • English
  • Français

Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations

Published online by Cambridge University Press:  12 October 2011

Rafael M. Oliveira  and
Eckart Meiburg
Rafael M. Oliveira
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Eckart Meiburg*
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: meiburg@engineering.ucsb.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Three-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This streamwise vorticity convects resident fluid from the wall towards the centre of the gap in the cross-gap symmetry plane of the finger, while it transports injected fluid laterally away from the finger centre within the mid-gap plane. In this way, it results in the emergence of a longitudinal, inner splitting phenomenon some distance behind the tip that has not been reported previously. This inner splitting mechanism, which leaves the tip largely intact, is fundamentally different from the familiar tip-splitting mechanism. Since the inner splitting owes its existence to the presence of streamwise vorticity and cross-gap velocity, it cannot be captured by gap-averaged equations. It is furthermore observed that the role of the Péclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Péclet numbers result in wider fingers, while immiscible flows display narrower fingers for larger capillary numbers. Furthermore, while higher capillary numbers are known to promote tip-splitting, inner splitting is delayed for larger Péclet numbers.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 431 - 460
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Aubertin, A., Gauthier, G., Martin, J., Salin, D. & Talon, L. 2009 Miscible viscous fingering in microgravity. Phys. Fluids 21, 054107.CrossRefGoogle Scholar
2. Bear, J. 1972 Dynamics of Fluids in Porous Media. American Elsevier.Google Scholar
3. Chen, C.-Y. & Meiburg, E. 1996 Miscible displacements in a capillary tube. Part 2. Numerical simulations. J. Fluid Mech. 326, 57–90.Google Scholar
4. Chuoke, R. L., van Meurs, P. & van der Poel, C. 1959 The instability of slow, immiscible, viscous liquid–liquid displacements in permeable media. Trans. AIME 216, 188–194.Google Scholar
5. Demuth, R. & Meiburg, E. 2003 Chemical fronts in Hele-Shaw cells: linear stability analysis based on the three-dimensional Stokes equations. Phys. Fluids 15 (3), 597–602.Google Scholar
6. Fernandez, J., Kurowski, P., Petitjeans, P. & Meiburg, E. 2002 Density-driven unstable flows of miscible fluids in a Hele-Shaw cell. J. Fluid Mech. 451, 239–260.CrossRefGoogle Scholar
7. Ferziger, J. H. & Perić, M. 2002 Computational Methods for Fluid Dynamics. Springer.Google Scholar
8. Goyal, N. & Meiburg, E. 2004 Unstable density stratification of miscible fluids in a vertical Hele-Shaw cell: influence of variable viscosity on the linear stability. J. Fluid Mech. 516, 211–238.Google Scholar
9. Goyal, N. & Meiburg, E. 2006 Miscible displacements in Hele-Shaw cells: two-dimensional base states and their linear stability. J. Fluid Mech. 558, 329–355.Google Scholar
10. Goyal, N., Pichler, H. & Meiburg, E. 2007 Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability. J. Fluid Mech. 584, 357–372.Google Scholar
11. Graf, F., Meiburg, E. & Härtel, C. 2002 Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations. J. Fluid Mech. 451, 261–282.CrossRefGoogle Scholar
12. Gresho, P. & Sani, R. 1987 On pressure boundary conditions for the incompressible Navier–Stokes equations. Intl J. Numer. Methods Fluids 7, 1111–1145.Google Scholar
13. Hill, S. 1952 Channeling in packed columns. Chem. Engng Sci. 1, 247–253.Google Scholar
14. Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.Google Scholar
15. Jiang, G.-S. & Peng, D. 2000 Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21 (6), 2126–2143.CrossRefGoogle Scholar
16. Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
17. Kopf-Sill, A. R. & Homsy, G. M. 1987 Narrow fingers in a Hele-Shaw cell. Phys. Fluids 30, 2607–2609.CrossRefGoogle Scholar
18. Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 1997 3D instabilities of miscible displacements in a Hele-Shaw cell. Phys. Rev. Lett. 79 (26).Google Scholar
19. Martin, J., Rakotomalala, N & Salin, D. 2002 Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys. Fluids 14 (2).Google Scholar
20. McLean, J. W. & Saffman, P. G. 1981 The effect of surface tension of the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 261–282.CrossRefGoogle Scholar
21. Meiburg, E. & Homsy, G. M. 1988 Nonlinear unstable viscous fingers in Hele-Shaw flows. Part II. Numerical simulation. Phys. Fluids 31, 429–439.CrossRefGoogle Scholar
22. Oliveira, Rafael M. 2012 Three-dimensional DNS simulations of confined miscible flows. PhD thesis, UCSB.Google Scholar
23. Osher, S. & Fedkiw, R. 2003 Level Set Methods and Dynamic Implicit Surfaces. Springer.Google Scholar
24. Pacheco, P. 1997 Parallel Programming with MPI. Morgan Kaufmann.Google Scholar
25. Paterson, L. 1985 Fingering with miscible fluid in a Hele-Shaw cell. Phys. Fluids 28 (1).CrossRefGoogle Scholar
26. Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37–56.CrossRefGoogle Scholar
27. Pitts, E. 1980 Penetration of fluid into a Hele-Shaw cell. J. Fluid Mech. 97, 53–64.Google Scholar
28. Qin, J. & Nguyen, D. T. 1998 A tridiagonal solver for massively parallel computers. Adv. Engng Software 29, 395–397.Google Scholar
29. Rai, M. M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96 (1), 15–53.Google Scholar
30. Rakotomalala, N, Salin, D. & Watzky, P. 1997a Fingering in 2D parallel viscous flow. J. Phys. II France 7 (967).Google Scholar
31. Rakotomalala, N, Salin, D. & Watzky, P. 1997b Miscible displacement between two parallel plates: BGK lattice gas simulations. J. Fluid Mech. 338, 277–297.Google Scholar
32. Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329.Google Scholar
33. Schafroth, D., Goyal, N. & Meiburg, E. 2007 Miscible displacements in Hele-Shaw cells: nonmonotonic viscosity profiles. Eur. J. Mech. B/Fluids 26, 444–453.CrossRefGoogle Scholar
34. Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29 (11), 3549–3556.Google Scholar
35. Tan, C. T. & Homsy, G. M. 1988 Simulation of nonlinear viscous fingering in miscible displacement. Phys. Fluids 31 (6), 1330–1338.Google Scholar
36. Vanaparthy, S. H. & Meiburg, E. 2008 Variable density and viscosity, miscible displacements in capillary tubes. Eur. J. Mech. B/Fluids 27, 268–289.Google Scholar
37. Wooding, R. A. 1969 Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech. 39, 477–495.CrossRefGoogle Scholar
38. Zeng, J., Yortsos, Y. C. & Salin, D. 2003 On the Brinkman correction in unidirectional Hele-Shaw flows. Phys. Fluids 15 (12).CrossRefGoogle Scholar