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 nonhomogeneous potential mean flow with unrestricted Mach number. §1. INTRODUCTION In most of the aeroacoustics literature [1], the modeling of sound generation by surfaces in arbitrary motion is based on the FWH-equation [2], which has effectively superseded an earlier attempt at the same result [3]. Two of the most important applications of the FWH equation are propeller and rotor noise. A variety of methods have been used in connection with the noise aircraft propellers and the noise of helicopter rotors *luis.campos@ist.utl.pt

On Physical Aeroacoustics with Some Implications for Low-Noise Aircraft Design and Airport Operations

“…The sources of propeller noise are well modeled by the first theory of aerodynamic sound due to Lighthill Proudman [1][2][3][4][5][6] and extended to solid boundaries [7] and surfaces in arbitrary motion [8,9]. The acoustic or classical wave equation, that applies to linear, non-dissipative sound waves in a homogeneous medium at rest [10][11][12][13][14][15][16] …”

“…In the case of BVI the vortices, as the fluid blobs of entropy noise (Section 2.2.3) are convected by a non-uniform flow past obstacles (rotor blades) that act as a reflector modifying the sound field ( Figure 1). In this case further insight can be gained from the second theory of aerodynamic sound [9,[62][63][64][65][66][67][68] concerning sound generation in a non-uniform potential mean flow for which the classical wave Equation (1) is replaced by [6,9,[66][67][68]: of (3) is the convected wave operator. For a non-uniform and/or unsteady mean potential flow the high-speed wave operator [66][67][68] on the .…”

“…h .  of (3) has several other terms [9,68] and can be extended to non-linear sound [69][70][71]. The sound sources on the r.h.s.…”

“…of (3) corresponding to the second theory of aerodynamic sound consist of: (i) a monopole term J associated with compressibility in a high-speed flow [9,68]; (ii) the force dipole G  is the dominant noise source for low Mach number mean flow, and consists of two terms, namely the vortical force (considered next) and the inhomogeneity force (Section 2.2.3). The vortical force [72] corresponds to the Lamb [73] vector, and is equivalent [62] to the quadrupole of Reynolds stresses.…”

“…A consequence of the incomplete combustion process is the presence in the exhaust jet of patches of unburned gas or fluid blobs associated with "entropy noise" [9,[73][74][75][76][77][78]. It is important to distinguish: (i) acoustic-entropy waves and acoustic-vortical-entropy waves associated with propagation in a compressible vortical mean flow with heat exchanges; (ii) "entropy noise" due to inhomogeneities that radiate sound when they are convected in a potential homentropic flow.…”