Oblique stripes and local energy flux vectors in wall turbulence model without walls

Globally subcritical flows that jump to nontrivial branches due to nonlinearities in finite amplitude perturbations become turbulent via spatially localized turbulent states such as puffs, spots, and stripes. Waleffe proposed a self-sustaining process and derived a reduced-order model of the process from the Navier-Stokes equations for sinusoidal forcing. This approach was extended by Manneville as a Swift-Hohenberg-like model. Chantry et al. developed a Waleffe model that retains only four Fourier modes in the shear direction and showed that this model has the potential to capture a range of spatially localized turbulent conditions. These results suggest that the driving mechanisms of spatially localized turbulent states may be investigated under periodic boundary conditions, i.e., without walls. Recently, we have revealed the anisotropic structure of cascades in wavenumber space by introducing local flux vectors for invariants. In this talk, we use the local flux vector in the wavenumber space to investigate the driving mechanisms of oblique stripes in the Waleffe flow model. The model also shows that the turbulence consists of anisotropic streamwise structures called streaks. The driving mechanisms of these will be discussed in terms of the local energy flux vectors in the wavenumber space too.