A modeling approach to transitional channel flow

Low-dimensional dynamical models of transitional flow in a periodically grooved channel are numerically obtained. The governing partial differential equations (continuity and Navier-Stokes equations) with appropriate boundary conditions are solved by a spectral element method for Reynolds number Re = 430. The method of empirical eigenfunctions (proper orthogonal decomposition) is then used to extract the most energetic velocity eigenmodes, enabling us to represent the velocity field in an optimal way. The eigenfunctions enable us to identify the spatio-temporal (coherent) structures of the flow as travelling waves, and to explain the related flow dynamics. Using the computed eigenfunctions as basis functions in a truncated series representation of the velocity field, low-dimensional models are obtained by Galerkin projection. The reduced systems, consisting of few non-linear ordinary differential equations, are solved using a fourth-order Runge-Kutta method. It is found that the temporal evolution of the most energetic modes calculated using the reduced models are in good agreement with the full model results. For the modes of lesser energy, low-dimensional models predict typically slightly larger amplitude oscillations than the full model. For the slightly supercritical flow at hand, reduced models require at least four modes (capturing about 99% of the total flow energy). This is the smallest set of modes capable of predicting stable, self-sustained oscillations with correct amplitude and frequency. POD-based low-dimensional dynamical models considerably reduce the computational time and power required to simulate transitional open flow systems.