Vortex Instability of the Asymptotic Dissipation Profile in a Porous Medium
Received: 14 October 2003 Accepted: 26 August 2004 DOI:
Cite this article as: Rees, D.A.S., Magyari, E. & Keller, B. Transp Porous Med (2005) 61: 1. doi:10.1007/s11242-004-2556-2 Abstract
In this note we consider the thermoconvective stability of the recently-discovered asymptotic dissipation profile (ADP). The ADP is a uniform thickness, parallel-flow boundary layer which is induced by a cold surface in a warm saturated porous medium in the presence of viscous dissipation. We have considered destabilisation in the form of stream-wise vortex disturbances. The critical wavenumber and Rayleigh number for the onset of convection have been determined for all angles of the cooled surface between the horizontal and the vertical for which the ADP exists. The paper closes with a presentation of some strongly nonlinear computations of steady vortices.
Keywords free convection porous media boundary-layer viscous dissipation vortex instability 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 1 119 143 Google Scholar Bejan, A. 1995Convection Heat Transfer (2nd Edition) John Wiley & Sons New York Google Scholar
Brandt A. (1984). Multigrid techniques: 1984 guide with applications to fluid dynamics. Lecture notes in C.F.D. at the Von Karman Institute for Fluid Dynamics. Published by G.M.D. mbH, Bonn as GMD–Studien 85.
Cheng, P., Minkowycz, W.J. 1977 Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike J. Geophys. Res. 82 2040 2044 Google Scholar Ene, H.I., Sanchez-Palencia, E. 1982 On thermal equation for flow in porous media Int. J. Eng. Sci. 20 623 630 Google Scholar Gill, A.E. 1969 A proof that convection in a porous vertical slab is stable J. Fluid Mech. 35 545 547 Google Scholar Ingham, D.B., Pop, I., Cheng, P. 1990 Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation Transp. Porous Media 5 381 398 Google Scholar Lewis, S., Bassom, A.P., Rees, D.A.S. 1995 The stability of vertical thermal boundary layer flow in porous medium Eur. J. Mech. B: Fluids 14 395 408 Google Scholar Magyari, E., Keller, B. 2003 The opposing effect of viscous dissipation allows for a parallel free convection boundary-layer along a cold vertical flat plate Trans. Porous Media 51 227 230 Google Scholar Nakayama, A., Pop, I. 1989 Free convection over a non-isothermal body in a porous medium with viscous dissipation Int. Comm. Heat Mass Transfer 16 173 180 Google Scholar Nield, D.A., Bejan, A. 1999Convection in Porous Media2 Springer New York Google Scholar Rees, D.A.S. 1993 Nonlinear wave stability of vertical thermal boundary layer flow in a porous medium J. Appl. Math. Phy. (Z.A.M.P.) 44 306 313 Google Scholar Rees, D.A.S., Bassom, A.P. 1993 The nonlinear nonparallel wave instability of free convection induced by a horizontal heated surface in fluid-saturated porous media J. Fluid Mech. 253 267 296 Google Scholar Rees, D.A.S., Magyari, E., Keller, B. 2003 The development of the asymptotic viscous dissipation profile in a vertical free convective boundary layer in a porous medium Transp. Porous Media 53 347 355 Google Scholar Rees, D. A. S., Magyari, E., Keller, B. 2005 ‘Effect of viscous dissipation on the flow in fluid saturated porous media’ Vafai, K. eds. Handbook of Porous Media II Marcell-Dekker New York Google Scholar