Generalised quasilinear approximations of turbulent channel flow: spanwise triadic scale interactions*

The generalised quasilinear (GQL) approximation is studied in turbulent channel flow at $Re_\tau \approx 1700$, continuing from the work of {(Hernández \emph{et al.}, \emph{J. Fluid Mech.}, vol. 936, A33, 2022)}. The flow is decomposed into two groups, the former of which contains a set of low-wavenumber spanwise Fourier modes and the latter are composed of the rest high-wavenumber modes. The former group is then solved by considering the full nonlinear equations, while the latter group is obtained from the linearised equations about the former. This is in contradistinction to the flow decomposition employed in our previous work, based on streamwise Fourier modes. This decomposition leads to the nonlinear low-wavenumber group that supports the self-sustaining process within the given integral length scales, whereas the linearised high-wavenumber group is not able to do so, unlike the GQL models in our previous work which place a minimal mathematical description for the self-sustaining process across all integral scales. Finally, a set of numerical experiments suppressing certain triadic nonlinear interactions are carried out with the aim of unveiling key roles of those including energy cascade and inverse energy transfer in the near-wall region.