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Surface Integral Methods in Computational Aeroacoustics

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Surface Integral Methods in Computational Aeroacoustics Surface Integral Metho ds in Computational Aeroacoustics -From the CFD Near-Field to the Acoustic Far-Field y Anastasios S. Lyrintzis Scho ol of Aeronautics and Astronautics Purdue University W. Lafayette, IN 47907-2023 Abstract A review of recent advances in the use of integral metho ds in Computational AeroAcoustics CAA for the extension of near- eld CFD results to the acoustic far- eld is given. These integral formulations i.e. Kirchho 's metho d, p ermeable p orous surface Ffowcs-Williams Hawkings FW-H equation allow the radiating sound to b e evaluated based on quantities on an arbitrary control surface if the wave equation is assumed outside. Thus only surface integrals are needed for the calculation of the far- eld sound, instead of the volume integrals required by the traditional acoustic analogy metho d i.e. Lighthill, rigid b o dy FW-H equa- tion. A numerical CFD metho d is used for the evaluation of the ow- eld solution in the Presented at the CEAS Workshop \From CFD to CAA" Athens Greece, Nov. 2002. y Professor, e-mail: [email protected]. 1 near eld and thus on the control surface. Di usion and disp ersion errors asso ciated with wave propagation in the far- eld are avoided. The surface integrals and the rst derivatives needed can b e easily evaluated from the near- eld CFD data. Both metho ds can b e ex- tended in order to include refraction e ects outside the control surface. The metho ds have b een applied to helicopter noise, jet noise, prop eller noise, ducted fan noise, etc. A simple set of p ortable Kirchho /FW-H subroutines can b e develop ed to calculate the far- eld noise from inputs supplied byany aero dynamic near/mid- eld CFD co de. 1 Background - Aeroacoustic Metho ds For an airplane or a helicopter, aero dynamic noise generated from uids is usually very im- p ortant. There are many kinds of aero dynamic noise including turbine jet noise, impulsive noise due to unsteady ow around wings and rotors, broadband noise due to in ow turbu- 1 lence and b oundary layer separated ow, etc. e.g. Lighthill . Accurate prediction of noise mechanisms is essential in order to b e able to control or mo dify them to comply with noise regulations, i.e. Federal Aviation Regulations FAR part 36, and achieve noise reductions. Both theoretical and exp erimental studies are b eing conducted to understand the basic noise mechanisms. Flight-test or wind-tunnel test programs can b e used, but in either case diculties are encounted such as high exp ense, safety risks, and atmospheric variability, as well as re ection problems for wind-tunnel tests. As the available computational p ower increases numerical techniques are b ecoming more and more app ealing. Although complete noise mo dels have not yet b een develop ed, numerical simulations with a prop er mo del are increasingly b eing employed for the prediction of aero dynamic noise b ecause they are low- cost and ecient. This research has led to the emergence of a new eld: Computational AeroAcoustics CAA. CAA is concerned with the prediction of the aero dynamic sound source and the transmis- sion of the generated sound starting from the time-dep endentgoverning equations. The full, time-dep endent, compressible Navier-Stokes equations describ e these phenomena. Although recent advances in Computational Fluid Dynamics CFD and in computer technology have 2 made rst-principle CAA plausible, direct extension of current CFD technology to CAA re- quires addressing several technical diculties in the prediction of b oth the sound generation 23 and its transmission. A review of aerospace application of CAA metho ds was given by 4 Long et al. Aero dynamically generated sound is governed by a nonlinear pro cess. One class of problems is turbulence generated noise e.g. jet noise. An accurate turbulence mo del is usually needed in this case. A second class of problems involves impulsive noise due to moving surfaces e.g. helicopter rotor noise, prop eller noise, fan noise etc.. In these cases an Euler/Navier Stokes mo del or even a full p otential mo del is adequate, b ecause turbulence is not imp ortant. Once the sound source is predicted, several approaches can b e used to describ e its propagation. The obvious strategy is to extend the computational domain for the full, nonlinear Navier-Stokes equations far enough to encompass the lo cation where the sound is to b e calculated. However, if the ob jective is to calculate the far- eld sound, this direct approach requires prohibitive computer storage and leads to unrealistic turnaround time. The impracticality of straight CFD calculations for sup ersonic jet aeroacoustics was p ointed 5 out by Mankbadi et al. Furthermore, b ecause the acoustic uctuations are usually quite small ab out three orders of magnitude less than the ow uctuations, the use of nonlinear equations whether Navier-Stokes or Euler could result in errors, as p ointed out by Stoker 6 and Smith. One usually has no choice but to separate the computation into two domains, one describing the nonlinear generation of sound, the other describing the propagation of sound. There are several alternatives to describing the sound propagation once the source has b een identi ed. 1.1 Field solution of Simpler Equations Linearized Euler Equations LEE The rst alternative is to use simpler equations in the acoustic far- eld. The Linearized Euler Equations LEE have b een used in order to 3 7 8 extend the CFD solutions to the far- eld e.g. Lim et al. , Viswanathan and Sankar , Shih 9 et al. The LEE equations employ a division of the ow eld into a time-averaged ow and a time-dep endent disturbance which is assumed to b e small. The hybrid zonal approach consists of the near- eld evaluation using an accurate CFD co de e.g. for jet noise the co de is usually based on Large Eddy Simulations: LES and the extension of the solution to the mid- eld using LEE. Considerable CPU savings can b e realized, since the LEE calculations are muchcheap er than the CFD calculations. This approachisvery promising, b ecause it accounts for a variable sound velo city outside the near- eld where usually an LES mo del is applied. This metho d may also b e appropriate for an intermediate region in some problems, outside from the reactive near- eld where the sp eed of sound is still not constant, b efore moving to another integral metho d for the far- eld. 10 Other Equations Hardin and Pop e hava prop osed a decoupling of the time- dep endent incompressible ow and the compressible asp ects acoustics of the ow. This technique was used successfully to predict the owover a two-dimensional cavity. A eld solution of 11 11 the wave equation can also b e used e.g. Freund . Freund claims that the eld solution of the wave equation is cheap er than the surface integral solutions see section 1.2.2, when the solution everywhere in the eld is sought. However, in most applications only a few lo cations are needed to study directivity and compare with microphone measurements. Also, for anynumerical solution of eld equations dissipation and disp ersion errors still exist and an accurate description of propagating far- eld waves is compromised. 1.2 Integral Metho ds 1.2.1 Volume Integral Metho ds Traditional Acoustic Analogy The rst integral approach for acoustic propagation is 12 the acoustic analogy. In the acoustic analogy, the governing Navier-Stokes equations are rearranged to b e in wave-typ e form. There is some question as to which terms should b e 4 identi ed as part of the sound source and retained in the right-hand side of the equation and 13 which terms should b e in the left-hand side as part of the op erator e.g., Lilley . The far- eld sound pressure is then given in terms of a volume integral over the domain containing the sound source. Several mo di cations to Lighthill's original theory have b een prop osed to account for the sound- owinteraction or other e ects. The ma jor diculty with the acoustic analogy,however, is that the sound source is not compact in sup ersonic ows. Errors could b e encountered in calculating the sound eld, unless the computational domain could b e extended in the downstream direction b eyond the lo cation where the sound source has completely decayed. Furthermore, an accurate account of the retarded time-e ect requires keeping a long record of the time-history of the converged solution of the sound source, which again represents a storage problem. The Ffowcs Williams and Hawkings FW-H 14 equation was intro duced to extend acoustic analogy in the case of solid surfaces. However, when acoustic sources i.e., quadrup oles are present in the ow eld a volume integration is needed. This volume integration of the quadrup ole source term is dicult to compute and 15 is usually neglected in most acoustic analogy co des e.g. WOPWOP . Recently, there 16;17 have b een some successful attempts in evaluating this term e.g. WOPWOP+ . 1.2.2 Surface Integral Metho ds Kirchho Metho d Another alternative is the Kirchho metho d which assumes that the sound transmission is governed by the simple wave equation. Kirchho 's metho d consists of the calculation of the nonlinear near- and mid- eld, usually numerically, with the far- eld solutions found from a linear Kirchho formulation evaluated on a control surface surrounding the nonlinear- eld.
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