Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels

The evolution of two-dimensional Tollmien-Schlichting waves propagating along a wall shear layer as it passes over a compliant panel of finite length is investigated by means of numerical simulation. It is shown that the interaction of such waves with the edges of the panel can lead to complex patterns of behaviour. The behaviour of the Tollmien-Schlichting waves in this situation, particularly the effect on their growth rate, is pertinent to the practical application of compliant walls for the delay of laminar-turbulent transition. If compliant panels could be made sufficiently short whilst retaining the capability to stabilize Tollmien-Schlichting waves, there is a good prospect that multiple-panel compliant walls could be used to maintain laminar flow at indefinitely high Reynolds numbers. We consider a model problem whereby a section of a plane channel is replaced with a compliant panel. A growing Tollmien-Schlichting wave is then introduced into the plane, rigid-walled, channel flow upstream of the compliant panel. The results obtained are very encouraging from the viewpoint of laminar-flow control. They indicate that compliant panels as short as a single Tollmien-Schlichting wavelength can have a strong stabilizing effect. In some cases the passage of the Tollmien-Schlichting wave over the panel edges leads to the excitation of stable flow-induced surface waves. The presence of these additional waves does not appear to be associated with any adverse effect on the stability of the Tollmien-Schlichting waves. Except very near the panel edges the panel response and flow perturbation can be represented by a superposition of the Tollmien-Schlichting wave and two other eigenmodes of the coupled Orr-Sommerfeld/compliant-wall eigensystem. The numerical scheme employed for the simulations is derived from a novel vorticity-velocity formulation of the linearized Navier-Stokes equations and uses a mixed finite-difference/spectral spatial discretization. This approach facilitated the development of a highly efficient solution procedure. Problems with numerical stability were overcome by combining the inertias of the compliant wall and fluid when imposing the boundary conditions. This allowed the interactively coupled fluid and wall motions to be computed without any prior restriction on the form taken by the disturbances.