Abstract
A Galerkin/POD reduced-order model from eigenfunctions of non-converged time evolution transitory states in a problem of Rayleigh–Bénard is presented. The problem is modeled in a rectangular box with the incompressible momentum equations coupled with an energy equation depending on the Rayleigh number R as a bifurcation parameter. From the numerical solution and stability analysis of the system for a single value of the bifurcation parameter, the whole bifurcation diagram in an interval of values of R is obtained. Three different bifurcation points and four types of solutions are obtained with small errors. The computing time is drastically reduced with this methodology.
Highlights
A way to avoid this is the use of reduced-order models, i.e., reduced basis [4–10] or proper orthogonal decomposition (POD) [11–20] models
Eigenfunctions of the linear stability analysis of unconverged states of a time evolution scheme for a single value of the bifurcation parameter are used as snapshots of the POD method
The eigenfunctions of the linear stability analysis of the non-converged states are the snapshots of the POD analysis
Summary
The study of bifurcation and instability phenomena is of great interest because it is a mechanism that explains many physical and engineering processes These processes are modeled by partial differential equations which are solved numerically. Bifurcation problems require solving the partial differential equations for many values of the parameters This requires a huge computing time using standard numerical solvers [1–3]. Eigenfunctions of the linear stability analysis of unconverged states of a time evolution scheme for a single value of the bifurcation parameter are used as snapshots of the POD method. Other reducedorder methods calculate the complete bifurcation diagram using solutions in different values of the parameter as snapshots [44]
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