Effect of added wall-transpiration and pressure gradient on the plane Couette flow using resolvent analysis*

The resolvent analysis is applied on the plane Couette flow with constant wall-transpiration velocity V₀ and wall velocity U_w with emphasis on the effect of  V₀ on the coherent structures and the amplification rate σ₁. Various invariant scaling laws for the most amplified first singular value σ₁ and the flow parameters are found. For a constant ratio of the wall-transpiration and streamwise Reynolds numbers γ=Re_V0/Re=V₀ /U_w, σ₁ is the largest for an invariant relationship Re·γ^a = C reaching a peak value for a specific Re for each γ. It is shown that the streamwise structures not only move closer to the upper wall, but also become more confined in the wall-normal and spanwise direction for an increasing Re_V0 while keeping the streamwise Reynolds number Re constant. Direct numerical simulations (DNS) showed that added wall-transpiration could not lead to the destruction of the coherent structures of the plane Couette flow, whereas it was only possible by imposing a pressure-gradient, i.e. for the plane Couette-Poiseuille flow.

The resolvent analysis and the structured singular value analysis as an extension to the resolvent analysis are applied on the Couette-Pouiseuille flow, where the structures indeed get destroyed for large enough pressure gradients agreeing with the results of the DNS.