Invariant scaling laws for plane Couette flow with wall-transpiration

Abstract

The effect of the wall-transpiration Reynolds number ReV0 on the large-scale rolls of the plane Couette flow with wall-transpiration is investigated using resolvent analysis. We observe streamwise-elongated large-scale rolls, moving closer to the suction wall and becoming narrower in the wall-normal and spanwise direction as well as shortened in the streamwise direction for growing ReV0 due to the decreasing characteristic length of the sheared region. Invariant scaling laws describing the growth of the amplification gain over various parameters and determining the optimal parameters yielding most amplification are found enhancing our understanding of the influence on the coherent structures, which can be used to predict their exact size. In a potential second step, this allows to manipulate and efficiently modify these flow characteristics increasing the functionality for various engineering applications as for mass, momentum and heat transfer, which is dominated by turbulent superstructures. For large enough wall-transpiration, the most amplified structures are oblique at optimal streamwise wavenumbers αmax≠0 growing linearly with ReV0. The optimal spanwise wavenumber βmax yielding most amplified structures shows a polynomial dependence of second order on ReV0. For a constant ratio of the wall-transpiration and streamwise Reynolds numbers γ=ReV0Re, the most amplified structures occur at optimal streamwise Reynolds numbers represented by Remax·γa=C. The amplification gain changes its trend from growing with Re2 to decreasing with Re−1.368 after Remax is reached. The resulting streamwise structures are identical for each set of Remax and γ within this invariant scaling law.