Absorption of Sound near Abrupt Area Expansions

Abstract

Sound incident onto an abrupt area expansion in a channel is investigated both numerically and analytically. In the presence of a mean e ow, the incident sound leads to unsteady vortex shedding from the lip of the expansion, thereby converting acoustic into vortical energy. We use an acoustic analogy and Green' s functions to determine the sound ree ected and transmitted across the area change. We compare predictions obtained from threedifferent Green' sfunctionswithsourcetermsderivedeitherusinga simpleanalyticalmodelorfrom anumericalcalculation. Thecompact Green' s function, with zero normalderivative on theductwalls, givesthe bestresultsfora low-Mach- number e ow. This Green' s function contains a singularity at the lip of the expansion (and hence acoustic sources neartheliphavethegreatesteffect ).Thismeansthatourestimateoftheoverallvorticitye eldcanberelativelycrude, when using the compact Green' s function, provided it is accurate near the lip. Therefore, although predictions for the radiated sound e eld made using all three Green' s functions are formally correct, the solution made using the compact Green' s function is less susceptible to errors in the source terms and gives more accurate results. In addition, we e nd that thereisa Strouhal number at which sound absorption is maximized and that this absorption can be enhanced by multiple ree ections from the duct ends. Our predictions are compared with an experiment. In this paper we investigate this energy exchange for a simple two-dimensional e ow past a backward-facing step with e ow using anacousticanalogyfromHowe 14 torelatetheacousticsourcestothe shed vorticity. Initially, we consider two semi-ine nite channels of heighth and H(h < H) joinedat y1 = 0.There is a mean e ow along the channelthatseparatesatthe rearward-facing step.Wedetermine the transmission and ree ection of an incident sound wave atthe step including the effects of the mean e ow. This problem is investigated both numerically and analytically. Vortex-sound interaction is of fundamental interest in aeroacous- tics, and this simple geometry enables us to investigate solution techniques and compare theoretical results with experiment. To re- late the sources from the acoustic analogy to the sound that radiates away from the junction, we must introduce a suitable Green' s func- tion for the pipework system. The choice of Green' s function is crucial to the solution, and three different Green' s functions are cal- culated.Two ofthesearederivedfromexpansionsintermsofmodes of straight-walled semi-ine nite pipes, whereas the remaining one, the compact Green' s function, satise es boundary conditions of zero normal velocity on all rigid surfaces. The compact Green' s function has a singularity at the lip of the expansion, and so sources near the lip have a greater effect on the sound that radiates to the far e eld than those farther away. This is important because it implies that, when using the compact Green' s function, our prediction for the vorticity e eld, whether from an analytical model or from a numer- ical calculation, can be relatively crude provided that the vorticity near the lip is accurately predicted. As a result, we postulate that the most accurate results will be obtained using the compact Green' s function. The calculation of these Green' s functions is discussed in Sec. III. The vorticity e eld is obtained in two ways: using an analytical model (Sec. IV.A) and using computational e uid dynamics (CFD; Sec. IV.B) with comparison between the two. The analytical model that we use for the vorticity is that of Howe 17 and assumes that the strength of the shed vorticity is unchanged as it convects down- stream and it forms an ine nitely thin vortex sheet whose strength is determinedbytheapplicationoftheKuttacondition.Incontrast,our numerical calculation uses a crude zero-equation turbulence model and the strength of the vorticity diffuses and decays as it convects downstream as a result of numerical diffusion. Despite the apparent differences between the two models, we show in Sec. V that the radiated sound e elds, which they predict, are very similar because of the importance of the region near the lip, provided the compact Green' s function is used in the calculation of the far-e eld sound. Our initial calculations are for two semi-ine nite pipes, but pre- dictions for e nite length pipes are obtained in Sec. VI by the in- troduction of a ree ection coefe cient at the pipe exit. Solutions for e nite length channels are easily derived from the ine nite channel solutions, which isimportant because it implies that, fora particular expansion ratio and Mach number, the ine nite channel solution is all that is required; the solution for any length channel can then be deduced from this. We justify our results by comparison with an experiment and discuss the implication for the absorption of incident sound waves. In particular, we e nd, for ine nite channels, there is an optimum Strouhal number at which maximum absorption is achieved. When wave ree ection from the ends of e nite length pipes is included, channel resonances can amplify this absorption.