Receptivity of a laminar boundary layer to the interaction of a three-dimensional roughness element with time-harmonic free-stream disturbances

Abstract

Receptivity of a laminar boundary layer to the interaction of time-harmonic free-stream disturbances with a three-dimensional roughness element is studied. The three-dimensional nonlinear triple–deck equations are solved numerically to provide the basic steady-state motion. At high Reynolds numbers, the governing equations for the unsteady motion are the unsteady linearized three-dimensional triple-deck equations. These equations can only be solved numerically. In the absence of any roughness element, the free-stream disturbances, to the first order, produce the classical Stokes flow, in the thin Stokes layer near the wall (on the order of our lower deck). However, with the introduction of a small three-dimensional roughness element, the interaction between the hump and the Stokes flow introduces a spectrum of all spatial disturbances inside the boundary layer. For supercritical values of the scaled Strouhal number, S0 > 2, these Tollmien–Schlichting waves are amplified in a wedge-shaped region, 15° to 18° to the basic-flow direction, extending downstream of the hump. The amplification rate approaches a value slightly higher than that of two-dimensional Tollmien–Schlichting waves, as calculated by the linearized analysis, far downstream of the roughness element.

Acoustic Receptivity in Boundary Layers with Surface Roughness

“…The scale of the disturbance wedge appears to correlate favourably with the predictions made by Tadjfar & Bodonyi [9).…”

“…Their analysis indicated that, for certain values of the asymptotic frequency/Reynolds number parameter, a disturbance wedge would form with off-centerline peak amplitude developing several wavelengths downstream from the roughness. Later, Tadjfar & Bodonyi [9] tackled the same problem with a similar scheme to that developed by Bodonyi et al Although both analyses agree on the character of the wedge, they appear to disagree on the scale of it. For critical values of their frequency/Reynolds number parameter, they predict the wedge will spread between 15° and 18° to the flow direction, with the region of maximum amplification falling along a ray 7.5° to 9° to the flow.…”

Direct Numerical Simulation of Instabilities in Parallel Flow with Spherical Roughness Elements

“…Several numerical studies investigated the mean flow and disturbance field around isolated three-dimensional roughness elements [Mason & Sykes, 1979, Mason & Morton, 1987, Tadjfar, 1990. Tadjfar employed a three-dimensional triple-deck with time-harmonic free-stream disturbances; and Mason et al solved the full threedimensional, unsteady Navier-Stokes equations.…”

Influência de uma rugosidade tridimensional isolada na transição de uma camada limite sem gradiente de pressão

“…Dentre aqueles trabalhos pode-se citar Tani and Hama (1940), Tani (1961), Sedney (1973), Dryden (1953), Gibbings et al (1986), Goldstein (1985), Tadjfar and Bodonyi (1992), e Klebanoff et al (1992). Naqueles trabalhos houve uma maior preocupação com a geração de perturbações na rugosidade e pouca importância se deuà interação dessa rugosidade com as perturbações já existentes na camada limite.…”

“…Neste regime a principal influência das rugosidades na transição da camada limite se deve aos mecanismos de receptividade. Alguns importantes estudos realizados nesse regime são os trabalhos de Goldstein (1985), Tadjfar and Bodonyi (1992), e Würz et al (2003).…”