Galerkin Reduced Order Models for Rayleigh-Bénard Convection Without Closure Models
Researchers have developed Galerkin reduced-order models (ROMs) for two-dimensional Rayleigh-Bénard convection that remain numerically stable without requiring closure models or DNS snapshot databases. The models use eigenfunctions of the controllability Gramian of linearized equations as orthonormal bases, bypassing the need for traditional POD-based approaches. This advance significantly reduces computational cost while enabling detailed bifurcation analysis across periodic, quasiperiodic, and chaotic flow regimes.
A new study posted to arXiv presents Galerkin projection-based reduced-order models (ROMs) for two-dimensional Rayleigh-Bénard convection with no-slip boundary conditions. The work compares two projection strategies: an uncoupled approach using separate bases for velocity and temperature, and a coupled formalism projecting onto a single combined basis. Orthonormal bases are derived from eigenfunctions of the controllability Gramian of the linearized Rayleigh-Bénard equations, eliminating the reliance on direct numerical simulation (DNS) snapshot databases that conventional POD-based methods require. The ROMs are validated against DNS across a wide range of Rayleigh numbers, with comparisons covering mean vertical profiles, heat flux, flow structures, dynamical regimes, and energy spectra. The coupled approach demonstrated superior agreement with DNS in mean vertical profiles and Nusselt number scaling. Notably, the models remain numerically stable without closure models, a persistent challenge in prior POD-based thermal convection ROMs. The framework was further exploited to perform detailed bifurcation analysis at Prandtl number 10, using Poincaré sections and Lyapunov exponents to precisely identify transitions between periodic, quasiperiodic, and chaotic states at a fraction of the computational cost of full DNS.
What's missing
The study is a preprint and has not yet undergone formal peer review. The models are validated only in two dimensions; their extensibility to three-dimensional Rayleigh-Bénard convection, which is more physically realistic, is not addressed.
What different sources said
- arXiv physicsCenter
Galerkin reduced order model for two-dimensional Rayleigh-Benard convection
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