On Sound Generation by Moving Surfaces and Convected Sources in a Flow

Abstract

The theory of sound generation by surfaces in arbitrary motion is presented, with two generalizations relative to the Ffowcs-William Hawkins (FWH) equation: (i) it allows for the presence of a steady, non-uniform potential flow of low Mach number; (ii) it includes the effects on the radiation field of reflections from solid surfaces, e.g. these which cause non-uniformity of the flow. The final result is a generalization of the Kirchhoff integral with: (i) a retarded time modified by sound convection by the mean flow; (ii) position coordinates of observer and source modified to account for the presence of the obstacles which reflect sound waves and cause the mean flow to be non-uniform. An alternative generalization of the Kirchhoff integral is presented for sources in arbitrary motion in an uniform mean flow with unrestricted Mach number. The two generalized forms of the Kirchhoff integral also apply to convected sources of sound, such as the turbulence noise associated with vorticity and the entropy noise due to convected fluid inhomegeneities. The two noise sources are shown to be dominant at low Mach number, relative to other noise sources present in an inhomogeneous potential flow with unrestricted Mach number. In the latter case applies the (i) inhomogeneous high-speed wave equation, that includes as particular cases the (ii) convected and (iii) classical wave equations. The latter two (ii–iii) are obtained from (α) the Laplace equation using the retarded time. All three (i–iii) are obtained by two more distinct methods: (β) elimination among the equations of fluid mechanics for the stagnation enthalpy as acoustic variable; (γ) an acoustic variational principle for the acoustic potential in a steady homentropic non-homogeneous potential mean flow with unrestricted Mach number.