WALL-BOUNDED BLADE-TIP VORTEX INTERACTION NOISE

Abstract An analytical treatment is given of the canonical problem of the generation of sound when a locally rectilinear vortex is drawn through the clearance gap between a wall and the square tip of a rigid blade at low Mach number. The mean flow in the neighborhood of the blade tip is assumed to be locally two-dimensional, and is modelled by free streamline potential flow theory, according to which the flow into the gap on the pressure side emerges as a wall jet on the suction side. The additional vorticity shed from the tip (in accordance with the unsteady Kutta condition) during the passage of the vortex is assumed to convect at constant speed along the free streamline boundary of the jet. Sound is generated by a dipole source, the magnitude and orientation of which are determined by the unsteady lift experienced by the blade. We calculate the separate contributions to the radiation (i.e., to the dipole strength) from the vortex and the shed vorticity. The sound is generated while the vortex is within a distance from the blade tip comparable to the clearance, and has wavelength which is order 1/Mlarger, whereMis the characteristic Mach number of the flow. The conclusions of this analysis are supported by an alternative, simplified treatment in which the vortex motion through the gap is assumed to be steady and the shed vorticity convects at a fixed distance from the duct wall rather than along the free streamline.