Quadrupoles in potential flow: two model problems

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Abstract
The Ffowcs Williams-Hawkings formulation of the sources of the acoustic analogy is examined by reference to two compressible inviscid flows whose density and velocity fields are known exactly. The purpose of this exercise is to generate some feel for the importance of the various source terms in determining the sound of moving surfaces with shocks, and to help quantify the errors involved in approximating those sources. Practical applications of the theory involve flows that cannot be known exactly and the errors of approximation cannot be checked directly. Progress must start with simple cases, and these model problems represent a first move. The flows considered are the one-dimensional flow caused by a plane boundary impulsively accelerated into a fluid, and the two-dimensional flow due to a wedge moving supersonically and supporting a plane attached shock.For each of these flows, a system of analogous acoustic sources is developed, the fields of which, when superposed, produce a density field (an acoustic field) identical with that of the original flow. The acoustic fields generated by the component source terms are calculated and compared. This suggests that the volume quadrupoles of potential flow play only a minor role as sound generators. When properly viewed the field is generated entirely on the bounding surfaces of the flow. A general argument shows that volume quadrupoles in steady rectilinear motion only influence the sound field through propagation effects.

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