Numerical determination of bifurcation points and catastrophe phenomena in thermal convection problems

The nodal integral approach has been developed for the accurate numerical solution of linear and nonlinear problems. Recently, the basic ideas have been extended to develop a nodal integral method (NIM) for nonlinear natural thermal convection problems treated in the Boussinesq approximation, and used to obtain numerical solutions for natural convection in a closed cavity with insulated top and bottom and one hot and one cold side. Comparison of the solutions with reference solutions extrapolated from fine-mesh, finite difference solutions were used to establish the very high accuracy of the NIM, even on coarse meshes. The authors have now extended the NIM for nonisothermal problems to include a Euler-Newton continuation procedure in both the cavity tilt angle, theta, and the Rayleigh number, Ra. This has made it possible to study the very interesting two-parameter family of problems, which has as a special case the classic Benard convection problem a closed square cavity with insulated sides, a hot bottom, and cold top. In particular, it has enabled study of numerically the unfolding of the first bifurcation in the Benard problem into a structurally stable cusp catastrophe and the preservation of the second as a structurally unstable pitchfork bifurcation as the cavity tilt angle is varied from its Benard problem value theta = 0. Bifurcation of solutions in natural thermal convection problems has become increasingly important in reactor technology with the advent of inherently safe, innovative, advanced liquid-metal reactor designs, such as SAFR and PRISM