Stability and Resolvent Analysis of plane Couette flow in the distinguished limit Re → ∞, α → 0 with Re · α = O(1) without and with wall-transpiration

For turbulent plane Couette flow at large Reynolds numbers, large streamwise- elongated very persistent channel-wide coherent structures are observed with weak streamwise variation. For a deeper understanding of this phenomenon we investigated this flow at high Reynolds numbers Re → ∞ and small streamwise wavenumbers α → 0 in the distinguished limit Reα = Re · α = O(1) using the temporal linear stability theory as well as the resolvent analysis approach. In this distinguished limit, Reα plays a significant role in the behaviour of the first singular value σ1 over both Re and α and thus the energy of the system. Furthermore, Reα has a significant influence on both the eigenfunctions obtained from the linear stability theory and the response modes obtained from the resolvent analysis. We find that the appearance of the structures in this asymptotic limit is mainly influenced by Reα and not by Re or α alone. The conducted analysis is expanded on the plane Couette flow with constant wall-transpiration V0 and wall velocity Uw, where the first singular value σ1 is the largest for a certain invariant depending on V0 and Uw within the limit of Reα =O(1).