Interaction of a vortex with a contraction in a 2-dimensional channel: Incompressible flow prediction of sound pulse

In large scale Solid Rocket Motors (SRM) flow pulsations can be driven by coupling between vortex shedding and acoustic standing waves in the engine. Sound generation by a vortex ingested by the nozzle is described by a model considering the interaction of a line vortex by a contraction in a channel. The flow is described by 2-Dimensional incompressible potential flow theory. Using conformal mapping, an analytical solution is found for the potential describing the flow and the convection velocity of the vortex. The vortex path is obtained by using a 4th order Runge-Kutta integration method. Given the flow, the low frequency and low Mach number approximation of Vortex Sound Theory is used to deduce the sound wave generated by this flow for a contraction in an infinitely long channel. Results are compared to results obtained in the literature using either a compressible frictionless flow model or an incompressible model in which a discrete panel describes the interaction between the vortex and the wall. The sound appears to be mainly produced in the approach phase, where the flow is essentially subsonic. This explains that a low Mach number incompressible flow approximation is meaningful. A conformal mapping is found to map a channel with a contraction with rounded edge onto a plane wall. This allows to explore the effect of the edge shape on sound production. While the amplitude of the pulse is reduced by rounding off the edge, the time integral of the stagnation enthalpy pulse appears to be only weakly dependent on the edge shape. Furthermore this integral is not sensitive to the contraction ratio and to the height at which the vortex approaches the nozzle. This integral is proportional to the square of the vortex circulation. This model can be used as a test case for more complex numerical models and provides some insight into the influence of parameters on the sound production.