Hidden symmetries and pattern formation in Lapwood convection

Abstract

We study Lapwood convection (convection of afluid in aporous medium) on a two-dimensional rectangular domain. The linearized eigenmodes are symmetric p x q cellular patterns, which we call (p, q) modes. Numerical calculations of the branching structure near mode interaction points have derived bifurcation diagrams for the (3, 1)/(1, 1) and (3, 1)/(2, 2) mode interactions which are non-generic, even when the rectangular symmety of the domain is taken into account. This has raised questions about the accuracy of the numerical method used, a finite-element Galerkin approximation implemented using Harwell's E N T W F E code. We show that this apparent lack of genericity is partly a consequence of 'hidden' translational symmetries, which arise when the problem is extended to one with periodic bounda y conditions. This extension procedure has become standard for partial diferential equations (PDEs) with Neumann or Dirichlet bounday conditions, and it reveals restrictions on the Liapunou-Schmidt reduced bifurcation equations and the resulting singulan'ty-theoretic normal forms. Its application to Lapwood convection is unusual in that the PDE involves a mixture of both Neumann and Dirichlet bounday conditions. Specifically, on the vertical sidewalls the stream function satisfies Dirichlet bounda y conditions (is zero), but the temperature satisfies Neumann (no-flux) bounda y conditions. Nevertheless, we show that for abstract group-theoretical reasons the same symmety constraints that occur for purely Neumann bounda y conditions are imposed on the LiapunowSchmidt reduced bzfurcation equations, and therefore the same list of normal forms is valid. The hidden symmetries force certain terms in the reduced bzfurcation equations to be zero and change the generic branching geomety. With the aid of M A C S Y M A , we determine a small number of low-order coeficients of the reduced bzfurcation equations which are needed to find the correct normal form. We show that in some cases the normal form is more degenerate than might be anticipated, but that when these degeneracies are taken into account the resulting branching geomety reproduces that found in the earlier numerical approach. In particular, we obtain an analytic vindication of the numerical method. 0268-1 110/96/030155-38 Q 1996 Journals Oxford Ltd Downloaded by [Virginia Tech Libraries] at 03:40 15 March 2015 156 M. IMPEY ET AL.4

Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

“…Many studies 10 demonstrate the existence of multiple flow patterns and show that the stability of the flow strongly depends on the Rayleigh number and the aspect ratio of the cavity (see, e.g. [5,11,12,13,14,15]). There are also many numerical and analytical studies of the transition of convective patterns from the stationary state to steady flows and then to unsteady chaotic regimes (see, e.g.…”