On the nonlinear development of the most unstable Görtler vortex mode

Abstract

The nonlinear development of the most unstable Görtler vortex mode in boundary-layer flows over curved walls is investigated. The most unstable Görtler mode is confined to a viscous wall layer of thickness O(G−1/5) and has spanwise wavelength O(G−1/5); it is, of course, most relevant to flow situations where the Görtler number G [Gt ] 1. The nonlinear equations governing the evolution of this mode over an O(G−3/5) streamwise lengthscale are derived and are found to be of a fully non-parallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Görtler equations valid for O(1) Görtler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite-difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flow field as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.