NONLINEAR TOLLMIEN-SCHLICHTING WAVES FOR PLANE POISEUILLE FLOW WITH COMPLIANT WALLS

The influence of a compliant boundary on the two-dimensional wave-like equilibrium states in plane Poiseuille flow is investigated numerically. The underlying physical model is that of Kramer for compliant surfaces. Because of surface displacement, a dependency between the wall-normal and streamwise coordinates exists, a difficulty overcome by the use of a mapping. However, this generates nonlinear terms in the Navier-Stokes equations which are expanded up to third order in the displacement amplitude. The nonlinear system obtained after discretization is solved numerically via Keller's pseudo-arclength continuation method. In the present study we focus on the effects of spring stiffness. It appears that for sufficiently flexible walls the constant wavenumber cuts in the equilibrium surface consist of isolated branches. There is also a substantial increase in the critical Reynolds number for the existence of two-dimensional equilibrium states. The examination of perturbation flow quantities, such as the streamlines and the mean velocity profile, confirms that at finite amplitudes the equilibrium solutions for compliant boundaries are qualitatively different from those in the rigid case. Finally the computation of the skin friction factors for a fixed Reynolds number, which is formed using the constant pressure gradient condition, indicates that the maximal skin friction decreases with increasing flexibility. This reinforces the idea of drag-reducing capabilities of compliant coatings.