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 by a posteriori indications 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.

Concave Surface Boundary Layer Flows in the Presence of Streamwise Vortices

“…Computations [34][35][36] showed increasing wall shear stress τ w in concave surface boundary layer flow as the effect of nonlinear development of Gortler instability. The present study reveals this phenomenon, as shown in Fig.…”

Visualizing shear stress in Görtler vortex flow

“…The wall shear stress subsequently increases well beyond the turbulent curve for flat plate boundary layer. Computational studies of Sabry and Liu (1991) showed that the further enhancement of the wall shear stress was due to the effect of the nonlinear steady longitudinal Görtler vortices. In the absence of the secondary instability, the spanwise-averaged wall shear stress appeared to bridge the Blasius curve and the flat plate turbulent boundary layer curve.…”

Longitudinal Vortices in Turbulent Boundary Layers Subjected to Wall Curvature and Strong Adverse Pressure Gradients: Numerical Investigations Using LES and DNS

“…Sabry and Liu (1991) [28] have shown that boundary layer flows over concave surfaces lend themselves to qualitative comparison with computations obtained using the temporal model. They obtained good agreement with experiments by Swearingen and Blackwelder [29] well into the nonlinear regime.…”