WAVENUMBER SELECTION IN STEADY RAYLEIGH-BENARD CONVECTION (STABILITY, NUMERICAL ANALYSIS, NATURAL)

Steady convection in a layer of fluid heated from below is studied for several patterns. These patterns are such that the difference between them and either straight periodic rolls or three-dimensional bimodal cells is proportional to a small parameter. This leads to an asymptotic expansion of the field variables and their governing equa- tions, and to a solvability condition which determines the wavenum- ber of the periodicity. The first pattern considered is axisymmetric convection at large distances from the origin. This pattern is closely associated with the zigzag instability and, in general, slightly curved rolls. The corresponding solvability condition is in the form of a sca- lar nonlinear algebraic equation. The second pattern is parallel rolls transverse to an imposed gradient of the temperature difference. The solvability condition in this case can be written in the form of an ordi- nary differential equation for the selected wavenumber as a function of the Rayleigh number, which is integrated starting from the value at the critical point. The third and fourth patterns are generalizations of these to bimodal convection in an infinite Prandtl-number fluid. Results are presented as a function of the Rayleigh and Prandtl numbers for roll convection, and as a function of the Rayleigh num- ber, second wavenumber and the initial condition, when necessary, for bimodal convection. Comparisons of calculated wavenumbers with available experimental measurements are all very good.; The three-dimensional governing conservation equations are solved in poloidal-toroidal form by a mixed finite difference/Galerkin method. It is developed by expanding the dependent variables in Fourier series in both horizontal directions. The Galerkin method then yields ordinary differential equations in the vertical direction for the coefficients of the series. These are solved by fourth-order accurate operator compact implicit finite differencing, where new difference formulas for fourth order equations are derived here. This method is shown to yield a great improvement over standard finite difference methods.