Nonlinear spatially developing Görtler vortices in curved wall jet flow

Abstract

Linear and nonlinear spatial developments of two-dimensional wall jets on curved surfaces are computed using pseudospectral-finite difference methods. Inviscid analysis shows that the instability originates from the inner/outer region on a concave/convex wall; the shear layer is thus always unstable regardless of the curvature. This primary instability is a steady spanwise vortex structure similar to Görtler instability in a Blasius flow. In the present study, a perturbation of prescribed wave number α is assumed. In the limit of high Reynolds number (Re) and small curvature (ε), a parabolic set of nonlinear equations describes the spatial evolution of the disturbance. Direct marching simulation of the perturbation and a parabolic stability approach are employed. Both give the same results with different computational efficiencies. For the concave case at low Görtler numbers (G2=ε√Re), perturbations are unstable for small α. Their energy reaches a maximum and then decays. At high G, the most unstable disturbance occurring at larger α will grow exponentially and reach saturation. The convex case is the most unstable situation. But as for the concave case, the most dangerous disturbance moves from small to larger α as G increases. The numerical results are able to capture the primary instability as observed in the experiment of Matsson [Phys. Fluids 7, 3048 (1995)].