Nonlinear Vortex Instabilities in Free Convective Boundary Layers in Porous Media
When a thermal boundary layer is formed on a constant temperature heated surface which is embedded in a porous medium, the boundary layer approximation becomes valid only at sufficiently large distances from the leading edge. If the heated surface is upward-facing, then instability of the flow takes the form of longitudinal vortices, and these form at a critical distance from the leading edge which is dependent on the inclination of the surface and the wavelength of the vortices. The review article of Rees  has discussed such instabilities at length and one of the conclusions made is that rigorous mathematical studies of the location where instability first arises cannot be made using approximate methods involving reduction to ordinary differential eigenvalue problems. The primary reason for this is that a contradiction is entertained by asserting simultaneously that x, the non-dimensional streamwise distance, is asymptotically large (so that the boundary layer approximation is valid) and that finite values of x are to be computed as a result of approximating the stability equations. In general this critical value of x is far too small for the boundary layer approximation to be valid.
KeywordsHeated Surface Thermal Boundary Layer Surface Rate Primary Vortex Longitudinal Vortex
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- Chen, C. C, Labhabi, A., Chang, H.-C. and Kelly, R. E. (1991). Spanwise pairing of finite-amplitude longitudinal vortex rolls in inclined free-convection boundary layers. J. Fluid Mech., 231, 72–111.Google Scholar
- Rees, D. A. S. (1998). Thermal boundary layer instabilities in porous media: a critical review. In Transport phenomena in porous media (ed. D. B. Ingham and I. Pop), pp. 233-59. Pergamon, Oxford.Google Scholar