J. Fluid Mech. (1991), vol. 226, pp. 371-382 Printed in Great Britain 37 1 Onset of convection in an anisotropic porous medium with oblique principal axes By PEDER A. TYVAND' AND LEIV STORESLETTEN' Department of Agricultural Engjneering, Agricultural University of Norway, 1432 As-NLH, Norway * Department of Mathematics, Agder College, 4600 Kristiansand, Norway (Received 31 May 1990) We investigate the onset of Rayleigh-Be'nard convection in a horizontal porous layer with anisotropic permeability. The permeability is transversely isotropic, whereas the orientation of the longitudinal principal axes is arbitrary. This is sufficient to achieve qualitatively new flow patterns with a tilted plane of motion or tilted lateral cell walls. The critical Rayleigh number and wavenumber at marginal stability are calculated. There are two different types of convection cells (rolls) : (i) the plane of motion is tilted, whereas the lateral cell walls are vertical; (ii) the plane of motion is vertical, whereas the lateral cell walls are tilted as well as curved. It turns out that type (i) occurs when the transverse permeability is larger than the longitudinal permeability, and for the converse case type (ii) is preferred. 1. Introduction Rayleigh-Be'nard instability in a porous medium was first studied by Horton & Rogers (1945) and later by Lapwood (1948). They identified the important dimensionless group (the Rayleigh number) and determined its critical value at the onset of convection. Palm, Weber & Kvernvold (1972) performed an analytical study of the steady supercritical roll motion and the associated heat transfer. Their results were extended numerically to higher Rayleigh numbers by Straus (1974), who also investigated the stability of finite-amplitude convection rolls. In isotropic porous layers of infinite lateral extent, the preferred motion at the onset of convection is in the form of rolls with square cross-sections. In anisotropic layers this is usually not true. By adopting the physical arguments given by Busse (1981, p. 104) we may predict some of the effects of anisotropy at the onset of convection, provided that the principal axes of the medium are directed along the coordinate axes : let us keep the vertical permeability and the vertical conductivity fixed, and vary the corresponding horizontal quantities. Then an increased horizontal permeability will promote the horizontal motion. This increases the preferred cell width and reduces the critical Rayleigh number. An increased horizontal con- ductivity will speed up the decay of the local buoyancy force. This increases the preferred cell width and increases the critical Rayleigh number. Reducing the horizontal permeability or conductivity will lead to the opposite effects. Castinel & Combarnous (1974) performed the first study of the onset of convection in a horizontal layer with anisotropic permeability, and their work was extended by Epherre (1975) to porous layers with anisotropic thermal conductivity. The results in these papers fully conform to the physical arguments given above.