Low-dimensional models for coupled momentum and energy transport problems

Low-dimensional models for transitional flow and heat transfer in a periodically grooved channel are developed. The full governing partial differential equations with appropriate boundary conditions are solved by a spectral element method. Spontaneously oscillatory solutions are computed for supercritical values of Reynolds number and proper orthogonal decomposition is used to extract the empirical eigenfunctions at Re = 435. This decomposition enables us to represent the velocity and temperature fields in an optimal way. Using the empirical eigenfunctions as basis functions in a truncated series expansion and applying Galerkin's method, a low-dimensional system of non-linear ordinary differential equations is obtained. Expansion coefficients computed based on the low-order model and by direct projection of the full model data are found to be in very good agreement. Four modes for each field is the smallest set capable of predicting stable, self-sustained oscillations with correct amplitude. Retaining fewer modes in the truncated series expansion either does not produce stable oscillations in time or fails to predict the correct amplitude of the oscillations. It is found that keeping more modes than necessary in the expansion may reduce the accuracy and restrict the validity of the low-order models due to the noise introduced by the low energy level higher modes.