Cross proper orthogonal decomposition

A method is proposed in order to optimally decompose the trace of cross-covariances of flow fluctuations, such as Reynolds stresses. Such method, referred to as cross proper orthogonal decomposition (CPOD), leads to a basis of modes extracted from a flow database that are optimal in representing an inner product related to the cross-covariance of interest. A sample application is shown for the representation of Reynolds shear stress in a turbulent channel flow with friction Reynolds number equal to 179. Leading modes are shown to comprise streamwise vortices and streaks with phase opposition between streamwise (u') and wall-normal (v') velocities, representing ejections and sweeps, and higher-order modes show similar structures, but with u' and v' in phase. A combination of such structures leads to an accurate reconstruction of the Reynolds stress, and consequently of the mean flow, with a reasonable near-wall reconstruction with the leading CPOD mode pair (even and odd modes) for each considered wave number, and a close match of the profiles with the five leading CPOD mode pairs. The present method is thus a valuable modal decomposition technique targeting cross-covariances of flow quantities such as Reynolds stresses.