Sensitivity of the Orr–Sommerfeld equation to base flow perturbations with application to airfoils

Abstract

The sensitivity of the Orr–Sommerfeld equation due to base flow deviations is investigated by means of a Monte Carlo-type perturbation strategy and a Chebyshev collocation method. In particular, the sensitivities of the peak growth rate and frequency, both of which are of importance in airfoil applications, are investigated in the low Reynolds number regime. A separated boundary layer with a nominal shape factor of H=5.9 was perturbed for both velocity and wall-normal position deviations. Wide bands of eigenvalue spectra are obtained due to both types of perturbations. The standard deviation of the peak growth rate and frequency due to both perturbations does not exhibit a pronounced Reynolds number dependence. To broaden the results, six boundary layers were investigated with shape factors ranging from H=5.9−22. Perturbations resulting in a standard deviation of 1% of the nominal shape factor were applied. It is found that sensitivities of both the peak growth rate and frequency are more pronounced at lower shape factors, with a decrease in sensitivity with increasing shape factor. This result suggests that, at low Reynolds numbers, boundary layers with larger separated regions are less sensitive to base flow perturbations.