Boundary Elements in Primitive Variables and FWH Equation Revisited - Viscosity Effects

Abstract

A boundary element formulation for aeroacoustics in primitive variables is presented. For the Euler equations, the formulation is addressed in details. The result is a self-contained set of boundary integral equations, coupling velocity and pressure. For the sake of simplicity, only the formulation for bodies in uniform translation, such as airplanes is presented. This allows one to obtain some further elaboration of the equations with respect of existing results. This in turn yields a deeper understanding of the formulation. The final result may be considered as an extension of the Kirchho, and Ffowcs Williams and Hawkings formulations, in that the expression for the pressure is fully equivalent those, but an expression for the velocity is also available. The most interesting result however is that, having started from a primitive variable formulation, the final expression may be considered also as an extension of the quasi-potential formulation to vortical flows (indeed, the two formulations coincide if the non-linear terms are neglected). In addition, the extension of the formulation to the NavierStokes equations is outlined (the entropy is used as the additional unknown for the case of viscous flows). Contrary to the traditional formulations (Ffowcs Williams and Hawkings; Kirchho), where the viscous terms are included in the Lighthill tensor thereby making the viscous-flow numerical implementation computationally expensive, here the viscous terms are included in the dierential equation and hence aect the fundamental solution, thereby clarifying the role of viscosity in the propagation of sound.