Longitudinal vorticity elements in boundary layers: nonlinear development from initial Görtler vortices as a prototype problem

Abstract

The nonlinear effects of longitudinal vorticity elements in boundary layers are studied via a prototype problem: the development of such vorticity elements from initial Görtler vortices in the amplified regime. While a time-dependent, quasi-two-dimensional formulation greatly simplifies the computational framework, full three-dimensionality of the velocity components is obtained. This temporal analogy for spatially developing flows approximates the nonlinear streamwise advection by a constant convection velocity, but the strong cross-sectional, advective nonlinearities are retained. Such an approximation lacks the stretching effect of the streamwise vorticity, since such elements are lifted into regions of higher streamwise velocities. That the temporal analogy is a good theoretical (and experimental) approximation to real developing flows is shown bya posterioriindications that the streamwise vorticity remains weak throughout the nonlinear region (though it has far-reaching nonlinear effects in upwelling in the peak region) and that the region of strong nonlinearities remains in the cross-sectional plane.The aim of this work is to elucidate the nonlinearities producing sites of secondary instabilities and turbulence generation. The present mushroom-like computed iso-streamwise velocity contours surrounding the peak, as well as the streamwise velocity profiles in the peak and valley regions, agree well with experimental measurements up to the region of expected wavy secondary instabilities. Three local intense vorticity and enstrophy areas are found to be significant and these are thoroughly diagnosed. One such intense vorticity region arises from the upwelling of existing spanwise vorticity and subsequent spanwise stretching in the outer layers, leading to intense local high-shear layers of spanwise vorticity in the vicinity of the peak region, as expected. Primarily through the stretching of the vertical vorticity, intense vertical vorticity (and associated enstrophy) develop (i) in the shoulder regions of the mushroom-like iso-streamwise velocity structures in the outer layers of the boundary layer and (ii) in the inner regions of about thirty viscous lengths from the wall, close to the base of the mushroom-like structures. Of the three regions of intense local ‘free’ shear-layer vorticity, the vertical vorticity in the inner regions near the mushroom stem is dominant. This is entirely consistent with experimental observations of sites of high-frequency secondary and fine-scaled wavy instabilities. This theoretically/computationally obtained ‘parent flow’ essentially sets the stage for continued studies of its wavy instabilities. In order to analyse and control the shear stress at the wall and nonlinear flow development, the effect of initial parameters such as Görtler number, initial amplitudes and wavenumbers is fully explored.