Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-19T05:39:28.372Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional Navier–Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements

Published online by Cambridge University Press:  02 July 2014

F. H. C. Heussler ,
R. M. Oliveira ,
M. O. John  and
E. Meiburg
F. H. C. Heussler
Affiliation:
Rheinisch-Westfaelische Technische Hochschule Aachen, D-52056 Aachen, Germany Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
R. M. Oliveira
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
M. O. John
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA ETH, Institute of Fluid Dynamics, CH-8092 Zurich, Switzerland
E. Meiburg*
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Gravitationally and viscously unstable miscible displacements in vertical Hele-Shaw cells are investigated via three-dimensional Navier–Stokes simulations. The velocity of the two-dimensional base-flow displacement fronts generally increases with the unfavourable viscosity contrast and the destabilizing density difference. Displacement fronts moving faster than the maximum velocity of the Poiseuille flow far downstream exhibit a single stagnation point in a moving reference frame, consistent with earlier observations for corresponding capillary tube flows. Gravitationally stable fronts, on the other hand, can move more slowly than the Poiseuille flow, resulting in more complex streamline patterns and the formation of a spike at the tip of the front, in line with earlier findings. A two-dimensional pinch-off governed by dispersion is observed some distance behind the displacement front. Three-dimensional simulations of viscously and gravitationally unstable vertical displacements show a strong vorticity quadrupole along the length of the finger, similar to recent observations for neutrally buoyant flows. This quadrupole results in an inner splitting instability of vertically propagating fingers. Even though the quadrupole’s strength increases for larger destabilizing density differences, the inner splitting is delayed due to the presence of a secondary, outer quadrupole which counteracts the inner one. For large unstable density differences, the formation of a secondary, downward-propagating front is observed, which is also characterized by inner and outer vorticity quadrupoles. This front develops an anchor-like shape as a result of the flow induced by these quadrupoles. Increased spanwise wavelengths of the initial perturbation are seen to result in the formation of the well-known tip-splitting instability. For suitable initial conditions, the inner and tip-splitting instabilities can be seen to develop side by side, affecting different regions of the flow field.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 752 , 10 August 2014 , pp. 157 - 183
Copyright
© 2014 Cambridge University Press 

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.)

Footnotes

‡

Present address: Halliburton Brazil Technology Center, Halliburton, Rio de Janeiro, RJ 21941-907, Brazil.

References

Alba, K., Taghavi, S. M. & Frigaard, I. A. 2012 Miscible density-stable displacement flows in inclined tube. Phys. Fluids 24, 123102.CrossRefGoogle Scholar
Chen, C.-Y. & Meiburg, E. 1996 Miscible displacements in a capillary tube. Part 2. Numerical simulations. J. Fluid Mech. 326, 57–90.Google Scholar
Chen, C.-Y. & Meiburg, E. 1998 Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities. J. Fluid Mech. 371, 269–299.Google Scholar
D’Errico, G., Ortona, O., Capuano, F. & Vitagliano, V. 2004 Diffusion coefficients for the binary system glycerol $\, +\, $ water at $25\, ^\circ \mathrm{C}$ : a velocity correlation study. J. Chem. Engng Data 49, 1665–1670.CrossRefGoogle Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2003 Transient displacement of a Newtonian fluid by air in straight or suddenly constricted tubes. Phys. Fluids 15, 1973–1991.Google Scholar
Fairbrother, F. & Stubbs, A. E. 1935 Studies in electro-endosmosis. Part VI. The ‘bubble-tube’ method of measurement. J. Chem. Soc. 1, 527–529.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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
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.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
John, M. O., Oliveira, R. M., Heussler, F. H. C. & Meiburg, E. 2013 Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 2. Nonlinear simulations. J. Fluid Mech. 721, 295–323.CrossRefGoogle Scholar
Kim, J. & Moin, P. 1985 Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323.Google Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1988 Nonlinear unstable viscous fingers in Hele-Shaw flows. I. Experiments. Phys. Fluids 31, 242–249.Google Scholar
Kuang, J., Petitjeans, P. & Maxworthy, T. 2004 Velocity fields and streamline patterns of miscible displacements in cylindrical tubes. Exp. Fluids 37, 301–308.CrossRefGoogle Scholar
Lajeunesse, E. & Couder, Y. 2000 On the tip-splitting instability of viscous fingers. J. Fluid Mech. 419, 125–149.Google Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 1997 3D instabilities of miscible displacements in a Hele-Shaw cell. Phys. Rev. Lett. 79 (26), 5254–5257.Google Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 2001 The threshold of the instability in miscible displacement in a Hele-Shaw cell at high rates. Phys. Fluids 13 (3), 799–801.Google Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1999 Miscible displacement in a Hele-Shaw cell at high rates. J. Fluid Mech. 398, 229–319.Google Scholar
Martin, J., Rakotomalala, N. & Salin, D. 2002 Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys. Fluids 14 (2), 902–905.Google Scholar
Maxworthy, T. 1987 The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell. J. Fluid Mech. 177, 207–232.Google Scholar
Oliveira, R. M. & Meiburg, E. 2011 Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations. J. Fluid Mech. 687, 431–460.Google Scholar
Park, C. W. & Homsy, G. M. 1984 Two phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37–56.CrossRefGoogle Scholar
Rai, M. M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96 (1), 15–53.Google Scholar
Rakotomalala, N., Salin, D. & Watzky, P. 1997 Miscible displacement between two parallel plates: BGK lattice gas simulations. J. Fluid Mech. 338, 277–297.Google Scholar
Rashidnia, N. & Balasubramaniam, R. 2004 Measurement of the mass diffusivity of miscible liquids as a function of concentration using a common path shearing interferometer. Exp. Fluids 36, 619–626.CrossRefGoogle Scholar
Ruith, M. & Meiburg, E. 2000 Miscible rectilinear displacements with gravity override. Part 1. Homogeneous porous medium. J. Fluid Mech. 420, 225–257.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.Google Scholar
Soares, E. J. & Thompson, R. L. 2009 Flow regimes for the immiscible liquid–liquid displacement in capillary tubes with complete wetting of the displaced liquid. J. Fluid Mech. 641, 63–84.Google Scholar
Taghavi, S. M., Alba, K. & Frigaard, I. A. 2012a Buoyant miscible displacement flows at moderate viscosity ratios and low Atwood numbers in near-horizontal ducts. Chem. Engng Sci. 69, 404–418.Google Scholar
Taghavi, S. M., Alba, K., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2012b Miscible displacement flows in near-horizontal ducts at low Atwood number. J. Fluid Mech. 696, 175–214.Google Scholar
Talon, L., Goyal, N. & Meiburg, E. 2013 Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis. J. Fluid Mech. 721, 268–294.Google Scholar
Tan, C. T. & Homsy, G. M. 1988 Simulation of nonlinear viscous fingering in miscible displacement. Phys. Fluids 31 (6), 1330–1338.CrossRefGoogle Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165.CrossRefGoogle Scholar