Boundary-layer receptivity to unsteady pressure gradients: experiments and overview

The experimental evidence on the mechanisms of forcing of unstable vorticity waves (the Tollmien–Schlichting–Schubauer or TS waves) of circular frequency ω and wavelength λTS in wall layers by unsteady pressure gradients of amplitude A and frequency ω is reviewed and found to be confused and contradictory. It is proposed that a likely effective receptivity mechanism rests on the fact that under realistic conditions A varies with distance x along any body of finite thickness, A(x), and introduces thereby additional characteristic lengths which can match λTS. Heuristic arguments suggest that through A(x) the pressure gradient infuses vorticity at the wall and forces spatial growth of the TS mean-square vorticity $\overline{\zeta^2}$, at a rate proportional to the real part of ΔAF(κTS), the contribution of A(x) between x−½λTS and x + ½λTS to the Fourier transform AF(κ) at κ = κTS. A second input into $\overline{\zeta^2_{\rm TS}}$ growth corresponds to the conversion of the steady boundary-layer vorticity into unsteady ζTS, through the action of vf, the normal velocity of the forcing field. The rate is given by $\int_0^{\delta}\overline{v_{\rm f}\zeta_{\rm TS}}U^{\prime\prime}(y)\,{\rm d}y $ and is proportional to the imaginary part of κTSΔAF(KTS).

The proposition is consistent in all currently verifiable respects with one numerical and a series of laboratory experiments. In the laboratory experiments, various configurations of a pulsating pressure source and shielding plates located in the free stream supplied the variable-amplitude pressure gradients over the nearby flat-wall boundary layer. Three of the cases presented here demonstrate that stationary unsteady pressure fields induce Stokes-like sublayers when the boundary layer is stable and self-excited vorticity waves when it is unstable. The results of a fourth experiment suggest that unsteady pressure sources in wakes near the boundary layer can force the growth of unstable wall waves at the wake frequencies even though their propagation speeds differ. Material is also presented on key Soviet experiments and views on receptivity. Finally, these experiments and ours are examined for consistency and complementarity.