An LES investigation of the 2D Tollmien-Schlichting wave instability in channel flow

Classical linear hydrodynamic stability analysis predicts the existence of an unstable 2D ‘Tollmien-Schlichting’ (T-S) wave, which may be excited in parallel boundary layer flows by mechanisms of receptivity. Of particular interest is the evolution of these disturbances beginning with the linear growth phase governed by the Orr-Sommerfeld equation, subsequent non-linearity with the development of 3D flow structures and the breakdown to turbulent flow. Understanding the mechanisms by which these unstable waves derive energy from the mean flow has significance with regard to the aim of maintaining laminar boundary layer flow and reducing the drag generated by aerodynamic surfaces. In this paper we solve the Orr-Sommerfeld eigenvalue problem numerically via a spectral method in MATLAB, approximating the solution via a truncated series of Lagrange polynomials. The stream function of the single unstable mode was found from the spectral solution from which the streamwise and wall-normal components of the perturbation’s velocity were derived.