Direct and adjoint problems for sound propagation in non-uniform flows with lined and vibrating surfaces

Abstract

This paper presents a systematic analysis of direct and adjoint problems for sound propagation with flow. Two scalar propagation operators are considered: the linearised potential equation from Goldstein, and Pierce's equation based on a high-frequency approximation. For both models, the analysis involves compressible base flows, volume sources and surfaces that can be vibrating and/or acoustically lined (using the Myers impedance condition), as well as far-field radiation boundaries. For both models, the direct problems are fully described and adjoint problems are formulated to define tailored Green's functions. These Green's functions are devised to provide an explicit link between the direct problem solutions and the source terms. These adjoint problems and tailored Green's functions are particularly useful and efficient for source localisation problems, or when stochastic distributed sources are involved. The present analysis yields a number of new results, including the adjoint Myers condition for the linearised potential equation, as well as the formulation of the direct and adjoint Myers condition for Pierce's equation. It is also shown how the adjoint problems can be recast in forms that are readily solved using existing simulation tools for the direct problems. Results presented in this paper are obtained using a high-order finite element method. Several test cases serve as validation for the approach using tailored Green's functions. They also illustrate the relative benefits of the two propagation operators.

Low Mach number aeroacoustic computations of airfoils in application with Pierce's wave equation

“…It should be stated that this speed of sound is constructed from the incompressible flow field, for which ρ 0 is kept constant and may arbitrarily be defined. Thus, gradients of a 0 only account for the gradients in the pseudosound p 0 which effects on the propagation of sound is in many applications of lesser important than variations of the mean density ρ 0 , as obtained for example in the case of temperature gradients of the media [25]. This has to be seen as an intrinsic limitation to acoustic analogies based on EIF approaches.…”

“…In this study, an EIF for Pierce's wave equation [22] is presented to generalise Ribner's dilatation theory to include sound propagation effects. This wave operator is selected as it is unconditionally stable due to its energy conservation property [23], fairly accurate [24,25] and numerically less expensive than the complete set of linearised Euler's equations. An acoustic analogy based on Pierce's wave equation, that relies on a Reynolds decomposition of the flow, has recently been used to compute the sound radiated by a mixing layer [26].…”