Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-09T09:28:09.694Z Has data issue: false hasContentIssue false
  • English
  • Français

Plume formation and resonant bifurcations in porous-media convection

Published online by Cambridge University Press:  26 April 2006

Michael D. Graham  and
Paul H. Steen
Michael D. Graham
Affiliation:
Department of Chemical Engineering, University of Wisconsin, Madison, WI53706-1691, USA
Paul H. Steen
Affiliation:
School of Chemical Engineering, and Center for Applied Mathematics, Cornell University, Ithaca, NY14853-5204, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The classical boundary-layer scaling laws proposed by Howard for Rayleigh–Bénard convection at high Rayleigh number extend to the analogous case of convection in saturated porous media. We computationally study two-dimensional porous-media convection near the onset of this scaling behaviour. The main result of the paper is the observation and study of instabilities that lead to deviations from the scaling relations.

At Rayleigh numbers below the scaling regime, boundary-layer fluctuations born at a Hopf bifurcation strengthen and eventually develop into thermal plumes. The appearance of plumes corresponds to the onset of the boundary-layer scaling behaviour of the oscillation frequency and mean Nusselt number, in agreement with the classical theory. As the Rayleigh number increases further, the flow undergoes instabilities that lead to ‘bubbles’ in parameter space of quasi-periodic flow, and eventually to weakly chaotic flow. The instabilities disturb the plume formation process, effectively leading to a phase modulation of the process and to deviations from the scaling laws. We argue that these instabilities correspond to parametric resonances between the timescale for plume formation and the characteristic convection timescale of the flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 272 , 10 August 1994 , pp. 67 - 90
Copyright
Copyright © Cambridge University Press 1994

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

Aidun, C. K. & Steen, P. H. 1986 Transition to unsteady convective heat transfer in a fluid-saturated porous medium. AIAA/ASME 4th joint Thermophysics and Heat Transfer Conf., AIAA Paper 86–1264.Google Scholar
Aidun, C. K. & Steen, P. H. 1987 Transition to oscillatory convective heat transfer in a fluid-saturated porous medium. J. Thermophys. H.T. 1, 268–273.Google Scholar
Beck, J. L. 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377–1383.Google Scholar
Boyd, J. P. 1989 Chebyshev and Fourier Spectral Methods. Springer.CrossRefGoogle Scholar
Buretta, R. J. 1972 Thermal convection in a fluid filled porous layer with uniform internal heat sources. PhD thesis, University of Minnesota.Google Scholar
Caltagirone, J.-P. & Fabrie, P. 1989 Natural convection in a porous medium at high Rayleigh numbers Part I - Darcy's model. Eur. J. Mech. B 8, 207–227.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh-Bénard convection. J. Fluid Mech. 204, 1–30.Google Scholar
Doedel, E. J. 1981 AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congress Numerantum 30, 265–284, 1981 (Proc. of the 10th Manitoba Conf. on Numerical Mathematics and Computation, University of Manitoba, Winnepeg, Canada, 1980).Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Elder, J. W. 1967 Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 29–48.Google Scholar
Slder, J. W. 1968 The unstable thermal interface. J. Fluid Mech. 32, 69–96.Google Scholar
Sornberg, B. 1990 An improved pseudospectral method for initial-boundary value problems. J. Comput. Phys. 91, 381–397.Google Scholar
Soster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 1249–1257.Google Scholar
Graham, M. D. 1992 Dynamics and mechanisms of time-dependent natural convection in porous media. PhD thesis, Cornell University.Google Scholar
Graham, M. D. & Steen, P. H. 1992 Strongly interacting traveling waves and quasiperiodic dynamics in porous medium convection. Physica D 54, 331–350.Google Scholar
Saidvogel, D. B. & Zang, T. 1979 The accurate solution of Poisson's equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 167–180.Google Scholar
Sorne, R. N. & O’Sullivan, M. J. 1978 Origin of oscillatory convection in a porous medium heated from below. Phys. Fluids 21, 1260–1264.Google Scholar
Soward, L. N. 1964 Convection at high Rayleigh number. Applied Mechanics, Proc. 11th Cong. Appl. Mech. (ed. Görtler, H.), pp 1109–1115.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions II. Springer.CrossRefGoogle Scholar
Kimura, S., Schubert, G. & Straus, J. M. 1986 Route to chaos in porous-medium thermal convection. J. Fluid Mech. 116, 305–324.Google Scholar
Kimura, S., Schubert, G. & Straus, J. M. 1989 Time-dependent convection in a fluid-saturated porous cube heated from below. J. Fluid Mech. 207, 153–189.Google Scholar
Koster, J. N. & Müller, U. 1982 Free convection in vertical gaps. J. Fluid Mech. 125, 429–451.Google Scholar
Krishnamurti, R. 1979 On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309–320.Google Scholar
Lennie, T. B., McKenzie, D. P., Moore, D. R. & Weiss, N. O. 1988 The breakdown of study convection. J. Fluid Mech. 188, 47–85.Google Scholar
Nayfeh, A. H. & Mook, D. T. 1979 Nonlinear Oscillations. Wiley.Google Scholar
Rand, D. A. 1982 Dynamics and symmetry: predictions for modulated waves in rotating fluids. Arch. Rat. Mech. Anal. 79, 1–37.Google Scholar
Robinson, J. L. & O’Sullivan, M. J. 1976 A boundary-layer model of flow in a porous medium at high Rayleigh number. J. Fluid Mech. 75, 459–467.Google Scholar
Steeen, P. H. 1986 Container geometry and the transition to unsteady Bénard convection in porous media. Phys. Fluids 29, 925–933.Google Scholar
Steen, P. H. & Aidun, C. K. 1988 Time-periodic convection in porous media: transition mechanism. J. Fluid Mech. 196, 263–290.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar