From Primitive-Variable Boundary-Integral Formulation to FWH Equation

Boundary integral formulations for aeroacoustics in primitive variables (velocity and pressure), forcompressible Euler and Navier-Stokes equations, were developed by the author and his collaborators.Such formulations appear much more complicated and cumbersome to use than the classical (scalar)aeroacoustic formulations, such as the FWH equation, the Kirchho formula, and the Howe expressionfor the vorticity-generated sound. This paper explores the connection between the above scalar formulationsand the proposed primitive-variable formulation, and recasts this into a more manageableform. For the sake of simplicity and clarity, the paper is limited to inviscid isentropic flows. The resultis that, through a series of mathematical manipulations, one obtains that the formulation properlyreduces to the existing ones. The resulting expressions for velocity and pressure are much simplerthan the preceding ones and are surprisingly close to existing ones, in aeroacoustics as well as aerodynamics.Specifically, for incompressible flows, the formulation yields an integral representation for thepressure and one for the velocity. The integral representation for the pressure coincides essentiallywith the Kirchhoff equation for the acoustic pressure, which in turn is equivalent to the FW-H equation.The relationship with the Howe expression for the vorticity-generated sound is also addressed.On the other hand, the resulting rotational-flow expression for the velocity is closely related to thatobtained from the Helmholtz decomposition, as well as that from a non-primitive variable formulationproposed by the author, as a rotational-flow extension of his integral formulation for potential flows.The extension to compressible inviscid isentropic flows is also presented, with some aspects limitedto bodies in uniform subsonic translation. The paper is of theoretical nature; no numerical results areincluded.