Upper bounds on the heat transport in a porous layer

Numerical and asymptotic analytical upper bounds on the convective heat transport in a fluid saturated porous layer of infinite Darcy–Prandtl number are obtained by the Howard–Busse variational method of the optimum theory of turbulence. The nonlinear Euler–Lagrange equations of the variational problem possess different numerical solutions as well as different approximate analytical solutions in the region of high Rayleigh numbers. We use symmetric fields for the numerical investigation and obtain slightly different numerical bounds on the heat transport in comparison to the numerical bounds obtained by Gupta and Joseph (J. Fluid. Mech. 57 (1973) 491–514) as well as different numerical values of the wave numbers connected to the optimum fields. For the analytical investigation we use new approximate solution of the Euler–Lagrange equations for the intermediate layers of the optimum fields. This solution leads to different thickness of the boundary layers of the optimum fields as well as to a lower analytical upper bounds on the heat transport in comparison to the boundary layer thicknesses and analytical upper bounds obtained by Gupta and Joseph.