Transitional flow analysis in a grooved channel

Proper Orthogonal Decomposition (POD) is applied in order to identify the coherent structures and derive a low-order model of transitional flow in a periodically grooved channel. The direct numerical simulation of the governing partial differential equations with the related boundary conditions is obtained for a Reynolds number greater than the critical value. The resulting spontaneously oscillatory solutions of these equations supply the input data for POD analysis. Empirical eigenfunctions obtained through POD methodology are used to extract the coherent structures. These eigenfunctions are also used as basis functions in a truncated series expansion to optimally represent the flow field. The eigenfunctions are phase-shifted and occur in pairs of similar magnitude. Using Galerkin's method, a low-order system of ordinary differential equations is derived. This low-order system is solved using a fourth-order Runge-Kutta method to yield temporal series expansion coefficients. The effect of the number of retained eigenfunctions in the series expansion on the performance of the low-order model is discussed.