The nonlinear development of Görtler vortices in growing boundary layers

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Abstract
The development of Görtler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel-flow analysis impossible except in the high-wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using a Fourier series expansion whilst the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact the equilibrium state essentially independent of the initial conditions. In fact the equilibrium state reached is qualitatively similar to the large-amplitude high-wavenumber solution described asymptotically by Hall (1982b). In general it is found that the nonlinear interactions are dominated by a ‘mean field’ type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of our calculations is the finding that the downstream velocity field develops a strongly inflexional character as the flow moves downstream; the latter result suggests that the major effect of Görtler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability.