Multiscale spatial quasilinear wake modeling

Wake arrays arise in myriad applications, including wind energy and the direct air capture of carbon dioxide. As a first step toward developing reduced descriptions of the collective dynamics of wake arrays, we use multiple scales asymptotic analysis to derive a spatial quasilinear theory for a spanwise-periodic planar wake. The analysis leverages the anisotropy associated with the slow streamwise development of the time-mean wake flow, leading to a parabolic set of reduced equations that can be marched in the streamwise direction. The mean flow equations retain the Reynolds stress divergence induced by wake instabilities, which are represented using a WKBJ approximation. The transverse spatial structure of the leading-order instability fields is determined by solving a local spatial eigenvalue problem, but a higher-order analysis is required to determine the slow evolution of the modal amplitude. The resulting amplitude equation is novel, with coefficients that are functionals not only of the leading-order mean and fluctuation fields but also of their higher-order corrections and the slow derivatives of the leading fields. We solve the amplitude and mean flow equations numerically to analyze the streamwise evolution of a planar wake with single and multiple temporal frequencies.