Unsteady lift and sound produced by an airfoil in a turbulent boundary layer

An analysis is made of the unsteady lift exerted on a stationary rigid body immersed in an incompressible, plane-wall turbulent boundary layer. The li ft is expressed as a surface integral over the body involving the upwash ve locity induced by the "free" vorticity Omega (found by taking explicit acco unt of the interaction of the body with the flow and excluding the bound vo rticity) and a harmonic function X-2 that depends only on the shape of the body. The upwash velocity is the free-field velocity given in terms of Omeg a by the Biot-Savart formula, augmented by the velocity field of a conventi onal distribution of image vortices in the wall. The function X-2 can be in terpreted as the velocity potential of flow past the body, produced by moti on of the wall at unit speed towards the body. Detailed predictions are mad e of the lift on a slender airfoil placed in the outer region of the bounda ry-layer. When the airfoil chord is large compared to the boundary-layer th ickness, vortex shedding into the wake causes the magnitude of the net upwa sh velocity near the trailing edge to be small. The main contributions to t he surface integral are then from the nose region, where the upwash velocit y may be estimated independently of the fluctuations near the trailing edge . Analytical results for a thin plate airfoil of chord 2a at distance h fro m the wall show that the lift increases as alh increases; it is ultimately independent of a and scales with the ratio of h to the hydrodynamic wavelen gth. Application is made to determine the sound generated by the airfoil in a weakly compressible boundary layer flow over a finite elastic plate. (C) 2001 Academic Press.