Galerkin reduced order model for two-dimensional Rayleigh-B\'enard convection

In this work, Galerkin projection is used to build Reduced Order Models (ROM) for two-dimensional Rayleigh-B\'enard (RB) convection with no-slip walls. We compare an uncoupled projection approach that uses separate orthonormal bases for velocity and temperature with a coupled formalism where the equations are projected onto a single basis combining velocity and temperature components. Orthonormal bases for modal projection are obtained as the eigenvalues of the controllability Gramian of the linearized RB equations. Various coupled and uncoupled ROMs with different number of modes are generated and validated against Direct Numerical Simulations (DNS) over a wide range of Rayleigh numbers, $Ra$. DNS and ROM results are compared in terms of mean vertical profiles, heat flux, flow structures, bifurcation diagrams and energy spectra. Coupled ROMs are found to be unstable at high $Ra$ numbers with a stability limit that depends on the basis $Ra$. Uncoupled models show an increasing agreement with DNS as a function of the system dimension. It is found that for the system truncations investigated here, a quantitative agreement with DNS can be obtained up to $Ra\simeq 4\times 10^5$. ROMs are used to perform a bifurcation analysis for $Pr=10$ and the results compared to DNS. They qualitatively predict the transitions between periodic, quasiperiodic and chaotic states as well as the spectral characteristics over a wide range of $Ra$ numbers. Overall, these results show that these ROMs reproduce the main flow features of RB convection and could be used as DNS-surrogates for the development of active control strategies and state estimation applications.