Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T11:29:48.731Z Has data issue: false hasContentIssue false

Free convection in an undulating saturated porous layer: resonant wavelength excitation

Published online by Cambridge University Press:  21 April 2006

D. A. S. Rees
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW
D. S. Riley
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW

Abstract

Thermal convection in a saturated porous medium contained between two undulating fixed boundaries of mean horizontal disposition is considered when the layer is heated from below. In an analytic study, the amplitudes of the two-dimensional undulations are assumed to be small compared with the mean depth, and the wavelength is taken to be close to the critical wavelength for the onset of Lapwood convection. For values of the mean Darcy-Rayleigh number Ra below the Lapwood critical value Rac an analytical formula is found for the mean Nusselt number. As RaRac, convection driven by baroclinic effects induced by boundary variations is greatly amplified by convective instabilities. The natures of the resultant bifurcations are examined when the configuration is varicose and also non-varicose. Consideration is given to both longitudinal and transverse modes and to the effects of detuning. The effects of finite amplitude and larger Rayleigh number are examined, for the varicose configuration, in a numerical study of two-dimensional convection. Periodic solutions are found and the existence of the flows delimited in the parameter space of Ra and the boundary amplitude a.

Type
Research Article
Copyright
© 1986 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.)

References

Arakawa, A. 1966 Computational design of long-term numerical integration of the equations of fluid motion. I. Two-dimensional incompressible flow. J. Comp. Phys. 1, 119143.Google Scholar
Beck, J. L. 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 13771383.Google Scholar
Caltagirone, J. P. 1975 Thermoconvective instabilities in a horizontal porous layer. J. Fluid Mech. 72, 269287.Google Scholar
Caltagirone, J. P., Cloupeau, M. & Combarnous, M. A. 1971 Convection naturelle fluctuante dans une couche poreuse horizontale. C. R. Acad. Sci. Paris B 273, 833836.Google Scholar
Combarnous, M. A. & Bories, S. A. 1975 Hydrothermal convection in a saturated porous medium. Adv. Hydr. 10, 231307.Google Scholar
Combarnous, M. A. & Le Fur, B. 1969 Transfer de chaleur par convection naturelle dans une couche poreuse horizontale. C. R. Acad. Sci. Paris B 269, 10091012.Google Scholar
Eagles, P. M. 1980 A Bénard convection problem with a perturbed lower wall. Proc. R. Soc. Lond. A 371, 359379.Google Scholar
Elder, J. W. 1967 Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 2948.Google Scholar
Frick, H. & Müller, U. 1983 Oscillatory Hele-Shaw convection. J. Fluid Mech. 126, 521532.Google Scholar
Horne, R. N. 1979 Three-dimensional natural convection in a confined porous medium heated from below. J. Fluid Mech. 92, 751766.Google Scholar
Horne, R. N. & O'Sullivan, M. J.1974 Oscillatory convection in a porous medium heated from below. J. Fluid Mech. 66, 339352.Google Scholar
Horne, R. N. & O'Sullivan, M. J.1978 Origin of oscillatory convection in a porous medium heated from below. Phys. Fluids 21, 12601264.Google Scholar
Kelly, R. E. & Pal, D. 1976 Thermal convection between non-uniformly heated horizontal surfaces. In Proc. 1976 Heat Transfer and Fluid Mech. Inst. pp. 117. Stanford University Press.
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86, 433456.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508521.Google Scholar
Pal, D. & Kelly, R. E. 1978 Thermal convection with spatially periodic non-uniform heating: non-resonant wavelength excitation. In Proc. 6th Intl Heat Trans. Conf. Toronto, Vol. 2.
Pal, D. & Kelly, R. E. 1979 Three-dimensional thermal convection produced by two-dimensional thermal forcing. ASME Paper 79-HT-109.
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153161.Google Scholar
Riahi, N. 1983 Nonlinear convection in a porous layer with finite conducting boundaries. J. Fluid Mech. 129, 153171.Google Scholar
Roache, P. J. 1972 Computational Fluid Dynamics. Hermosa.
Schubert, G. & Straus, J. M. 1979 Three-dimensional and multicellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers. J. Fluid Mech. 94, 2548.Google Scholar
Schubert, G. & Straus, J. M. 1982 Transitions in time-dependent thermal convection in fluid-saturated porous media. J. Fluid Mech. 121, 301313.Google Scholar
Straus, J. M. 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 5163.Google Scholar
Tavantzis, J., Reiss, E. L. & Matkowsky, B. J. 1978 On the smooth transition to convection. SIAM J. Appl. Maths 34, 322337.Google Scholar
Vozovoi, L. P. & Nepomnyaschii, A. A. 1974 Convection in a horizontal layer in the presence of spatial modulation of temperature at the boundaries. Gidrodinamika 8, 105117.Google Scholar
Walton, I. C. 1982a The effects of slow spatial variations on Bénard convection. Q. J. Mech. Appl. Maths 35, 3348.Google Scholar
Walton, I. C. 1982b On the onset of Rayleigh—Bénard convection in a fluid layer of slowly increasing depth. Stud. Appl. Maths 67, 199216.Google Scholar
Walton, I. C. 1983 The onset of cellular convection in a shallow two-dimensional container of fluid heated non-uniformly from below. J. Fluid Mech. 131, 455470.Google Scholar
Watson, A. & Poots, G. 1971 The effect of sinusoidal protrusions on laminar free convection between vertical walls. J. Fluid Mech. 49, 3348.Google Scholar