Excitation of steady and unsteady Gortler vortices by free-stream vortical disturbances

Excitation of Gortler 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 Gortler 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 Leib, Wundrow & Goldstein (J. Fluid Mech., vol. 380, 1999, p. 169) 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 Gortler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Gortler vortices are excited. The amplitude of the Gortler 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) >> 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* similar to Lambda R(Lambda)G(Lambda)(-2/3), where R-Lambda denotes the Reynolds number based on Lambda. The induced pressure leads to full coupling of the momentum equations when x* similar to 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 Gortler vortices of inviscid nature when x* similar to Lambda R-Lambda. From this position onwards, local eigenvalue formulations are mathematically justified. Gortler vortices continue to amplify and enter the so-called most unstable regime when x* similar to Lambda R(Lambda)G(Lambda), and ultimately approach the right-branch regime when x* similar to Lambda R(Lambda)G(Lambda)(2).