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)].

Görtler vortices in Falkner–Skan flows with suction and blowing

“…Wadey [15] used asymptotic methods for linear theory to study Görtler vortices in wall jet flows for large spanwise wavenumbers. It was found that curved wall jets on both concave and convex walls were more stable to streamwise vortices than boundary-layer flows.An experimental study of streamwise vortices appeared for wall jet flow on a concave wall, see Matsson [16], followed by nonlinear simulations of the same flow by Le Cunff and Zebib [17]. The simulations were able to capture the primary instability of streamwise vortices in the experiments.…”

“…An experimental study of streamwise vortices appeared for wall jet flow on a concave wall, see Matsson [16], followed by nonlinear simulations of the same flow by Le Cunff and Zebib [17]. The simulations were able to capture the primary instability of streamwise vortices in the experiments.…”

Görtler-crossflow vortices

“…The results are mathematically reliable only for ␣Ͼ0.1 since the matrices become ill-conditioned below this threshold. 18 The neutral curves are presented for decelerated and accelerated boundary layers in Fig. 4 on a convex wall where the curvature has a stabilizing effect.…”

“…4 Le Cunff and Zebib 18 verified the consistency of a u-normalization with a direct integration of the equation where no splitting is introduced.…”

“…In both cases, the Chebyshev expansion is based on 80 polynomials which gives reliable results. 18 In the marching calculations, the freestream velocity U e is a function of x. The scaling is defined at l 1 where xϭ1 and U e ϭ1.…”