Spatial quasilinear theory for slowly-developing free shear flows

Quasilinear (QL) theory has proven to be a useful tool for analyzing shear-driven turbulence. For such flows, the QL reduction generally involves parsing dependent field variables into streamwise-average and fluctuation components and then retaining in the reduced dynamics only those fluctuation--fluctuation nonlinearities that feed back upon the evolution of the mean fields. To date, QL theory for shear flows has been implemented primarily in the context of temporal initial-value problems. Here, a spatial QL theory is derived for free shear flows that evolve slowly in the streamwise direction, e.g., wakes, jets, and free shear layers, for which streamwise averaging is inappropriate. The derivation exploits the spatial anisotropy of the (temporal) mean flow (e.g., the slenderness of the wake). The resulting asymptotically consistent, extended QL system can be marched in the streamwise direction and consistently retains the leading fluctuation--fluctuation nonlinearities in the equations governing the evolution of the fluctuation fields. An application of this spatial QL theory to planar wakes is described, with an eye toward the efficient modeling of the multiscale fluid dynamics of wind farm wakes.