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Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. (English) Zbl 1241.76199

Summary: Excitation of Görtler vortices in a boundary layer over a concave wall by free-stream vortical disturbances is studied theoretically and numerically. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the development of the induced perturbation is formulated for the generic case where the Görtler number \(G_{\Lambda }\) (based on the spanwise wavelength \(\Lambda \) of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by S. J. Leib et al. [J. Fluid Mech. 380, 169–203 (1999; Zbl 0951.76032)] for the flat-plate boundary layer. Numerical solutions show that for a sufficiently small \(G_{\Lambda }\), the induced perturbation exhibits essentially the same characteristics as streaks occurring in the flat-plate case: it undergoes considerable amplification and then decays. However, when \(G_{\Lambda }\) exceeds a critical value, the induced perturbation exhibits (quasi-) exponential growth. The perturbation acquires the modal shape of Görtler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Görtler vortices are excited. The amplitude of the Görtler vortices excited is found to decrease as the frequency increases, with steady vortices being dominant. Comprehensive quantitative comparisons with experiments show that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the entire evolution of the amplitude. An asymptotic analysis is performed for \(G_{\Lambda } \gg 1\) to map out distinct regimes through which a perturbation with a fixed spanwise wavelength evolves. The centrifugal force starts to influence the generation of the pressure when x*   \(\Lambda R_{\Lambda }G^{ - 2/3}_{\Lambda }\), where \(R_{\Lambda }\) denotes the Reynolds number based on \(\Lambda \). The induced pressure leads to full coupling of the momentum equations when x*   \(\Lambda R_{\Lambda }G_{\Lambda }^{ - 2/5}\). This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit growing asymptotic eigensolutions, which develop into fully fledged Görtler vortices of inviscid nature when x*   \(\Lambda R_{\Lambda }\). From this position onwards, local eigenvalue formulations are mathematically justified. Görtler vortices continue to amplify and enter the so-called most unstable regime when x*   \(\Lambda R_{\Lambda }G_{\Lambda }\), and ultimately approach the right-branch regime when x*   \(\Lambda R_{\Lambda }G^{2}_{\Lambda }\).

MSC:

76E09 Stability and instability of nonparallel flows in hydrodynamic stability
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76F06 Transition to turbulence

Citations:

Zbl 0951.76032
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References:

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