The generation of a Tollmien-Schlichting wave by a sound wave

Abstract

A spectral numerical method is used to study the two-dimensional unsteady flow over a flat plate in the presence of a plane sound wave propagating parallel to the flow. For s = ωx / U ∞ > O (1), no observable interaction or energy interchange between the sound wave and the Tollmien-Schlichting wave is present. In the region s ≤ O (1), near the leading edge, an interaction occurs; the magnitude of this interaction is computed. A plane sound wave with dimensionless frequency ωv / U ∞ 2 = 56 x 10 -6 generates a Tollmien-Schlichting wave of the same frequency, and of amplitude, at the first neutral point, 10 -4 of that of the sound wave. Computations at a second, lower frequency result in a smaller neutral point amplitude. The response of the boundary layer to large unsteady disturbances is discussed, and compared with published behaviour based on the boundary layer equations. Solutions of the Orr-Sommerfeld equation are examined and found to have a different character for s ≶ O (1), consistent with the more complete equations.