A Primitive-Variable Boundary Integral Formulation Unifying Aeroacoustics and Aerodynamics, and a Natural Velocity Decomposition for Vortical Fields

Abstract

The paper presents an integrated approach to the study of aeroacoustics and unsteady aerodynamics, which is based upon the use of the boundary integral equation approach. Accordingly, the boundary integral formulation in primitive variables is thoroughly developed, for Euler as well as Navier–Stokes equations. As a result, one obtains explicit time–domain expressions for velocity and pressure in terms of the Cauchy data of the problem (boundary integral), plus the contribution of the nonlinear terms (field integral). It is shown that the representation for the pressure is very closely related to the classical boundary integral formulations in aeroacoustics (such as the Ffowcs Williams and Hawkings equation and the Kirchhoff method). Similarly, the representation for the velocity is closely related to a formulation introduced by the author for unsteady viscous compressible aerodynamics. In the process, however, a new velocity decomposition of the type v = grad φ + w is uncovered. The novelty is that the vortical velocity w is not defined in terms of the vorticity ω; on the contrary, w is defined through its own governing equation. Such a decomposition (and the governing equation for w, which does not involve the pressure) arises in a very natural way from the mathematical approach used, and hence is here named the natural velocity decomposition. In addition, two innovative results are obtained. First, for all the cases addressed (incompressible and compressible, inviscid and viscous fields) there exist generalizations of the Bernoulli theorem (formally very similar to the classical Bernoulli theorems, with the potential φ of the irrotational portion of the velocity replacing the velocity potential φ). Second, it is shown that the velocity field may be obtained independently of the pressure, which may be evaluated a–posteriori in terms of the potential φ (akin to the evaluation of the pressure in a Venturi tube), through the appropriate generalized Bernoulli theorem. The paper is of theoretical nature – no numerical results are included; however, the schemes for computational implementations are presented and the advantages discussed.