Exact Coherent States

In the (quick) overview of the SSP model, we discussed how the shearing of x-dependent modes by the mean shear leads to a positive feedback on the mean flow. In the SSP model these are the W 2 term in the M equation and the −MW term in the W equation. Although this interaction is not necessary for the self-sustaining process itself, it is the key effect that leads to the R−1 scaling of the transition threshold, and of the V and W components of the lower branch steady state (while U and 1−M are O(1), see lectures 1 and 5). Advection by a shear flow leads to enhanced dissipation and an R−1/3 scaling characteristic of linear perturbations about shear flows, or evolution of a passive scalar. The R−1/3 enhanced damping, instead of R−1, was included by Chapman for the x-dependent modes in his modification of the WKH model (as discussed in lecture 1). However, that is because Chapman considers the weak nonlinear interaction of eigenmodes of the laminar flow, U(y). In contrast, the basic description of the SSP consists of the weak nonlinear interaction of streaky flow eigenmodes, that is, neutral eigenmodes of the spanwise varying shear flow U(y, z) consisting of the mean shear plus the streaks. An important aspect of the streaky flow U(y, z) is that the mean shear has been reduced precisely to allow that instability, as illustrated by the SSP model where σwU − σmM − σvV > 0 is needed for streak instability and growth of W . So it is unclear a priori whether the R−1/3 should apply to x-dependent modes in the SSP. In section 2 below we review the numerical evidence that the 3D nonlinear lower branch SSP states in plane Couette flow do have R−1/3 critical layers as R→∞ [24]. But why R1/3?