Proper orthogonal decomposition and low-dimensional modelling of thermally driven two-dimensional flow in a horizontal rotating cylinder

Abstract

A proper orthogonal decomposition (POD) analysis and low-dimensional modelling of thermally driven two-dimensional flow of air in a horizontal rotating cylinder, subject to the Boussinesq approximation, is considered. The problem is unsteady due to the harmonic nature of the gravitational buoyancy force with respect to the rotating observer and is characterized by four dimensionless numbers: gravitational Rayleigh number (Rag), the rotational Rayleigh number (RaΩ), the Taylor number (Ta) and Prandtl number (Pr). The data for the POD analysis are obtained by numerical integration of the governing equations of mass, momentum and energy. The POD is applied to the computational data for RaΩ varying in the range 102–106 while Rag and Pr are fixed at 105 and 0.71 respectively. The ratio of Ta to RaΩ is fixed at 100 so that the results apply to physically realistic situations. A new criterion, in the form of appropriately defined error norms, for assessing the truncation error of the POD expansion is proposed. It is shown that these error norms reflect the accuracy of the POD-based reconstructions of a given data ensemble better than the widely employed average energy criterion. The translational symmetry in both space and time of the pair of modes having degenerate (equal) eigenvalues confirms the presence of travelling waves in the flow for several different RaΩ values. The shifts in space and time of the structure of the degenerate modes are utilized to estimate the wave speeds in a given direction. The governing equations for the fluctuations are derived and low-dimensional models are constructed by employing a Galerkin procedure. For each of the five values of RaΩ, the low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Sufficient modes are included in the low-dimensional models so that the modelling of the unresolved scales of motion is not needed to stabilize their solution. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for a set of parameters different from those from which they were constructed is also examined.