Generalised quasilinear approximations for homogeneous shear turbulence

The spectral energetics of a generalised quasilinear (GQL) model is studied in homogeneous shear turbulence. The GQL approximation decomposes the velocity into low and high streamwise wavenumber groups, the former of which is solved by considering the full nonlinear equations whereas the latter is obtained from the linearised equations around the former. In contrast to the QL model, the GQL model exhibits a healthy energy cascade in the streamwise direction, and the energy cascade is mediated by slightly anisotropic small-scale motions, which are close to the nearly isotropic small-scale motions in direct numerical simulation. We examine the spectral energetics of the GQL model in the streamwise and spanwise wavenumber space, and a series of supporting numerical experiments with different combinations of modes are carried out. The turbulence statistics and spectra of velocity, component-wise energy transport and pressure strain are investigated.