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Numerical experiments on Hele Shaw flow with a sharp interface

Published online by Cambridge University Press:  20 April 2006

Gretar Tryggvason  and
Hassan Aref
Gretar Tryggvason
Affiliation:
Division of Engineering, Brown University, Providence, Rhode Island 02912, USA
Hassan Aref
Affiliation:
Division of Engineering, Brown University, Providence, Rhode Island 02912, USA
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Abstract

The fingering instability of an interface between two immiscible fluids in a Hele Shaw cell is simulated numerically. The algorithm used is based on a transcription of the equations of motion for the interface in which it formally becomes a generalized vortex sheet. The evolution of this sheet is computed using a variant of the vortex-in-cell method. The resulting scheme and code make it possible to follow the collective behaviour of many competing and interacting fingers well into the nonlinear, large-amplitude regime. It is shown that in this regime the evolution is controlled essentially by just one dimensionless parameter, the ratio of fluid viscosities. The effects of varying this parameter are studied and the results compared with experimental investigations. Scaling properties of the average density profile across the evolving mixed layer between the two homogeneous fluid phases are investigated. Many phenomena are observed that must be characterized as collective interactions and thus cannot be understood in terms of flows with just a single finger.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 136 , November 1983 , pp. 1 - 30
Copyright
© 1983 Cambridge University Press

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References

Aref, H. 1983 Integrable, chaotic, and turbulent vortex motion in two-dimensional flows Ann. Rev. Fluid Mech. 15, 345–389.Google Scholar
Aref, H. & Siggia, E. D. 1980 Vortex dynamics of the two-dimensional tubulent shear layer J. Fluid Mech. 100, 705–737.Google Scholar
Aref, H. & Siggia, E. D. 1981 Evolution and breakdown of a vortex street in two dimensions J. Fluid Mech. 109, 435–463.Google Scholar
Aref, H. & Tryggvason, G. 1982 Numerical experiments on statistical fingering in stratified Hele Shaw flows. Bull. Am. Phys. Soc. 27, 1172 (abstract only).Google Scholar
Aref, H. & Tryggvason, G. 1984 Vortex dynamics of passive and active interfaces. Physica D (to appear).
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulation of the Rayleigh-Taylor instability Phys. Fluids 23, 1485–1490.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems J. Fluid Mech. 123, 477–501.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.
Birkhoff, G. 1954 Taylor instability and laminar mixing. Los Alamos Sci. Lab. Rep. LA-1862; appendices in Rep. LA-1927.Google Scholar
Chuoke, R. L., VAN MEURS, P. & VAN DER POEL, C. 1959 The instability of slow immiscible, viscous liquid-liquid displacements in permeable media J. Petrol. Tech. 11, 64.Google Scholar
Christiansen, J. P. 1973 Numerical simulation of hydrodynamics by the method of point vortices J. Comp. Phys. 13, 363–379.Google Scholar
De Josselin De Jong, G. 1960 Singularity distributions for the analysis of multiple-fluid flow through porous media J. Geophys. Res. 65, 3739–3758.Google Scholar
Dussan, V., E, B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Ann. Rev. Fluid Mech. 11, 371–810.Google Scholar
Glimm, J., Marchesin, D. & Mcbryan, O. 1980 Statistical fluid dynamics: unstable fingers. Commun. Math. Phys. 74, 1–810.Google Scholar
Gupta, S. P., Varnon, J. E. & Greenkorn, R. A. 1973 Viscous finger wavelength degeneration in Hele Shaw models Water Resources Res. 19, 1039–1046.Google Scholar
Hele Shaw, J. H. S. 1898 The flow of water Nature 58, 34–36.Google Scholar
Lamb, H. 1932 Hydrodynamics, art. 330. Dover.
Langer, J. S. 1980 Instabilities and pattern formation in crystal growth Rev. Mod. Phys. 52, 1–28.Google Scholar
Leonard, A. 1980 Vortex methods for flow simulation J. Comp. Phys. 37, 289–335.Google Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. II. Growth of normal-mode instabilities. Proc. R. Soc. Lond A 364, 1–28.Google Scholar
Mclean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele Shaw cell J. Fluid Mech. 102, 455–469.Google Scholar
Meng, J. C. S. & Thomson, J. A. L. 1978 Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods J. Fluid Mech. 84, 433–453.Google Scholar
Meyer, G. H. 1981 Hele-Shaw flow with a cusping free boundary J. Comp. Phys. 44, 262–276.Google Scholar
Moore, D. W. 1981 On the point vortex method SIAM J. Sci. Stat. Comp. 2, 65–84.Google Scholar
Overman, E. A., Zabusky, N. J. & Ossakow, S. L. 1983 Ionospheric plasma cloud dynamics via regularized contour dynamics. I. Stability and nonlinear evolution of one-contour models Phys. Fluids 26, 1139–1153.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell J. Fluid Mech. 113, 513–529.Google Scholar
Perkins, T. K., Johnston, O. C. & Hoffman, R. N. 1965 Mechanics of viscous fingering in miscible systems Soc. Petrol. Engng J. 5, 301–317.Google Scholar
Pitts, E. 1980 Penetration of fluid into a Hele Shaw cell: the Saffman-Taylor experiment. J. Fluid Mech. 97, 53–810.Google Scholar
Pollard, D. D., Muller, O. H. & Dockstader, D. R. 1975 The form and growth of fingered sheet intrusions Geol. Soc. Am. Bull. 86, 351–363.Google Scholar
Prandtl, L. & Tietjens, O. G. 1934 Fundamentals of Hydro- and Aeromechanics. Dover.
Richardson, J. G. 1961 Flow through porous media. In Handbook of Fluid Dynamics (ed. V. L. Streeter). McGraw-Hill.
Rigels, F. 1938 Zur Kritik des Hele-Shaw-Versuchs Z. angew. Math. Mech. 18, 95–106.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond A 134, 170–192.Google Scholar
Rosensweig, R. E. 1982 Magnetic fluids. Sci. Am. 247 (4), 136–810.Google Scholar
Saffman, P. G. 1959 Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell Q. J. Mech. Appl. Maths 12, 146–150.Google Scholar
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 A 245, 312–329.Google Scholar
Shampine, L. F. & Gordon, M. K. 1975 Computer Solution of Ordinary Differential Equations. Freeman.
Slobod, R. L. & Thomas, R. A. 1963 Effect of transverse diffusion on fingering in miscible-phase displacement Soc. Petrol. Engng J. 3, 9–13.Google Scholar
Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium Q. J. Mech. Appl. Maths 12, 265–279.Google Scholar
Thompson, B. W. 1968 Secondary flow in a Hele-Shaw cell J. Fluid Mech. 31, 379–395.Google Scholar
Todd, D. K. 1955 Flow in porous media studied by Hele-Shaw channel Civil Engng slab 25, 85.Google Scholar
Van Meurs, P. 1957 The use of transparent three-dimensional models for studying the mechanism of flow processes in oil reservoirs Trans. AIME 210, 295–301.Google Scholar
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.Google Scholar
Wooding, R. A. & MOREL-SEYTOUX, H. J. 1976 Multiphase fluid flow through porous media Ann. Rev. Fluid Mech. 8, 233–274.Google Scholar