Galerkin reduced order model for two-dimensional Rayleigh-Bénard convection

Abstract

In this work, Galerkin projection is used to build reduced-order models (ROM) for two-dimensional Rayleigh-Bénard (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 eigenfunctions of the controllability Gramian of the linearized RB equations, eliminating the need for DNS snapshot databases required by traditional POD-based approaches. Various coupled and uncoupled ROMs with different numbers of modes are generated and validated against direct numerical simulations (DNS) over a wide range of Rayleigh numbers. One of the objectives is to determine their domain of validity as a function of the system dimension and the Rayleigh number. DNS and ROM results are compared in terms of mean vertical profiles, heat flux, flow structures, dynamical regimes and energy spectra. Crucially, unlike previous POD-based Galerkin models for thermal convection, these ROMs do not require closure models and remain numerically stable. The coupled approach shows better agreement with DNS in terms of mean vertical profiles and Nusselt number scaling. The capabilities of these models are exploited to conduct a detailed bifurcation analysis at $Pr = 10$ using Poincaré sections and Lyapunov exponents, precisely identifying the transitions between periodic, quasiperiodic, and chaotic states with significant reductions of computational cost.