用大涡模拟方法研究湍流边界层流动噪声_英文_

第13卷第6期船舶力学Vol.13No.6 2009年12月Journal of Ship Mechanics Dec.2009 Article ID：1007-7294（2009）06-0990-12LES Method for Investigation of Noise Generated byTurbulent Boundary LayerPAN Yu-cun1,2,ZHANG Huai-xin1(1State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University,Shanghai200030,China;2Department of Naval Architecture and Ocean Engineering,Naval University of Engineering,Wuhan430033,China)Abstract:Large Eddy Simulation(LES)was used to investigate space-time field of the low Mach number,fully developed turbulent boundary layer on a smooth,rigid flat plate.The flowfield simu-lated by LES was taken as near-field sound sources and radiated sound from turbulent boundary layer(TBL)was studied using Lighthill’s acoustics analogy.The radiated sound of dipole type,of quadrupole type,and of the sum of the two above were paring their power spectral densities,it is concluded that in low Mach number,the wall shear stress(dipole type of source)is the predominant factor responsible for sound radiation from TBL.Key words:turbulent boundary layer;hydroacoustics;large eddy simulation;Lighthill’s acoustic analogyCLC number:O427.4Document code:A1IntroductionThe sound generated by turbulence is an important part of flow-generated noise.The pas-sengers in aircrafts and automobiles always suffer from it.In the past,the jet noise was be-lieved to be dominant in aircraft noise,and the turbulent noise was considered secondary. However,accompanying the progress in jet-engine construction,material and design to con-trol jet-noise over recent years,the airframe turbulence-noise problem has received an in-creasing amount of attention.In classical acoustic problems,the sound wave was induced by the mechanical vibration of particular object.The fluid was only the medium of acoustic wave propagation.However,the objective of this study is to investigate the sound induced by the turbulence of fluid medium itself.The sound radiated from the low Mach number,fully developed turbulent boundary layer on a smooth,rigid flat plate is computed in this paper.The mechanism of hydrodynamic noise generation by turbulent flow has been a very interesting subject for fluid researches since Lighthill[1-2]laid the foundations to a general theory.Initially,the solution of the Lighthill equation was attained in suppose of free space.In Received date：2009-03-03Foundation item:Supported by the National Natural Science Foundation of China(Grant No.10772119)and theResearch Fund of State Key Laboratory of Ocean Engineering,Shanghai Jiao Tong University.Biography：PAN Yu-cun(1980-),male,Ph.D.student of Shanghai Jiao Tong University.第6期PAN Yu-cun et al:LES Method for Investigation of (991)the flow without solid boundary such as jet flow or free shear flow,the quadrupole source was the only sound source and the radiation efficiency was very low.Curle[3]extended Lighthill the-ory to the flow with stationary,solid boundary using Kirchhoff method.The result shown that the rigid boundary was equivalent to distribution of dipole sources,which enhanced the radia-tion efficiency and changed the components of flow noise.In this study,the Lighthill’s acoustic analogy method was used to investigate the hydro-dynamic noise radiation.Lighthill pointed out that,in a low Mach number turbulent flow,the acoustic and turbulent fields are only weakly coupled so that the turbulent fluctuations,which are not influenced by the acoustic disturbances,act as sources of sound.In this study,the hy-brid approach is used,in which the hydrodynamic and acoustic fields are decoupled.In the near-field,the hydrodynamic velocity and pressure are computed,which describe the nonlin-ear process of sound generation.And the near-field was also called source field.The hydro-dynamic terms are used to calculate the far-field acoustic field,which describe the linear propagation of sound wave.Ffowcs Williams[4]pointed out that for sound generated by low Mach number flow,the influence of compressible could be neglected.So,in computation of fluid in near field,we can investigate it based on the equation of incompressible flow.The fully developed turbulent boundary layer on a smooth,rigid flat plate is a typical tur-bulent shear flow.In the vicinity of the plate,the characteristic of wall shear turbulence is ob-vious;however,at the region away the wall,the characteristic of free shear turbulence is dom-inant.There are violent vortices in the wake behind blunt body such as sphere and circular cylinder,but no such vortex in the turbulent boundary layer in this study.So,vortex-shedding is not the main source of noise;this study is concerned with sound radiation from the tiny ed-dies in the vicinity of wall within the turbulent boundary layer.In this study,we should obtain the hydrodynamic terms of turbulent fluctuations,so,the Reynolds-averaged Navier-Stokes equations(RANS)method couldn’t be used here,in which the governing equations are time-averaged.The direct numerical simulations(DNS)can pro-vide the most precise,detailed fluctuating information,but the present computational resource can not support it to simulate high Reynolds-numbers flow.The information obtained by Large Eddy Simulation(LES)is no as exact as that obtained by DNS,but it can also reveal the ele-mentary mechanism of turbulent flow.Moreover,the computational cost of a LES is much low-er than that of a DNS.So,LES method was used in this study.The paper was divided into essentially two sections.In Sec1the unsteady incompressible Navier-stokes equations were solved numerically using LES method to give an approximate description of the space-time field of the low Mach number,fully developed turbulent bound-ary layer on a smooth,rigid flat plate;In Sec2the far-field radiation of sound was calculated based on Curle’s expansion to the Lighthill acoustic analogy.2Numerical simulation of turbulent boundary layerFig.1shows the computational domain of turbulent boundary layer.The Reynolds number,defined as Re=u τδ/ν=800,(u τis the wallshear velocity,δis the thickness of theboundary layer).Based on the theory ofLES,the filtering operation was applied tothe Navier -Stokes equations,and theSmagorinsky ’s [5]model was chosen to pa -rameterize the subgrid scale (SGS)stress -es.In order to account for low-Reynolds-number SGS turbulence near the wall,theVan Driest [6]exponential damping function was applied to the eddy viscosity νt ,so that it could be damped to zero at the wall.The computation result shows the space-time field of fluctuat -ing turbulent fluid motion near the wall.The more detailed computation process and result can be seen in Ref.[7].3Flow-induced noise3.1Lighthill ’s acoustic analogyThe density fluctuation due to acoustic wave propagation from the hydrodynamic source region is governed by the convected wave equation:坠坠t +U ∞坠坠x 1∞∞2-c 02坠2坠x j 坠x j ∞∞ρ′=坠2T i j 坠x i 坠x j (1)where,ρ′=ρ-ρ∞is the density fluctuation,ρ∞is the free-stream density,c 0is the sound speed in water at undisturbed conditions.We assume the free-stream velocity is U ∞,and T i j =ρv i v j +δi j p -c 02ρ-ρ∞∞∞∞∞-τi j is the Lighthill stress tensor defined in terms of the fluctuating velocity relative to the free-stream value,v i =u i -U ∞δi 1.Here,the usual summation convention applies for repeated subscripts,andδi j =0i ≠∞∞j 1i=∞∞j ≠is the Kronecker delta;τi j =μ坠v i j +坠v j i -2δi j 坠v k k ∞∞is the viscous part of Stokes stress tensor.Strictly speaking,it is almost impossible to resolve the equation (1)exactly.The hydro -dynamic and acoustic terms are coupled each other,flow can induce sound and sound can be scattered by flow.To get the source item on the right-hand side of equation (1),the nonlinear equation should be solved strictly,which was a hard work.In order to study complex engineer -ing applications,we should make some simplifying assumptions:according to the practical flow condition,some components of stress tensor are so small that can be neglected;in low Mach number flow,the two-way coupling is neglected,that is,the acoustic terms do not affect the Fig.1Computational domain of the turbulent boundary layer 992船舶力学第13卷第6期hydrodynamic terms.Thus,we can consider the source terms on the right-hand side of equa -tion (1)only contain the influence of flow to noise and the source terms can be obtained from the flow field.If solid body exists in the sound source region T i j ≠≠≠0and the sound is generated by the solid surfaces,Curle derived a simple solution for noise produced by the rigid surface mov -ing through a quiescent medium.ρx 軆,軆軆t -ρ0=-14πc 02坠坠x i S 乙n j p i j y 軆,t-r 軆0軆軆r 軆d 2y 軆+14πc 02坠2坠x i x j V 乙T i j y 軆,t-r 軆0軆軆r 軆d 3y 軆(2)where x軆and y 軆represent the position vectors of the observer and the sound sources,respective -ly,r ＝r 軆=x 軆-y 軆;r 軆0is called the delay time and represents the time interval of sound wave traveling from the sound sources to the observer.In the preceding equation p i j ＝p δi j -τi j and n j is the directional cosine of the outward normal (into the fluid)to the rigid surface S over which the surface integration take place.The volume integral is taken over the entire unsteady flow region V external to the body.In the acoustic far field defined by r 軆>>l e /M ,where l e is the typical eddy size,M=U ∞/c 0is the freestream Mach number,equation (2)is simplified to a form most suitable for numeri -cal evaluation.ρx 軆,軆軆t -ρ∞≈14πc 03坠坠tS 乙r i r 軆2n j p i j y 軆,t-r 軆c 0軆≠d 2y 軆+14πc 04坠2坠t 2V 乙r i r j r軆3T i j y 軆,t-r 軆0軆≠d 3y 軆(3)In the computation,the spatial coordinates x i are non-dimensionalized by a characteristic length L ,and time t is non-dimensionalized by L /U ∞.Also,v i are the components of the fluidvelocity,non-dimensionalized by U ∞,ρthe fluid density,non-dimensionalized by the refer -ence value ρ∞,p the pressure non-dimensionalized by ρ∞U ∞2.Equation (3)can be rewritten asρx 軆,軆≠t -1≈M 34π坠坠tS 乙r i r 軆2n j p i j y 軆,t-M r 軆軆≠d 2y 軆+M 44π坠2坠t 2V 乙r i r j r軆3T i j y 軆,t-M r 軆軆≠d 3y 軆(4)Furthermore,if both the body and the unsteady flow region are small compared with the typical acoustic wavelength l e /M ,the source region is acoustically compact.The far-field den-第6期PAN Yu-cun et al:LES Method for Investigation of (993)sity can be approximated byρx 軆,軆軆t -1=M 3x i x D 觶i t-M x 軆軆+M 4x i x j x Q 咬i j t-M x 軆軆(5)whereD 觶i 軆軆t =坠坠tS 乙n j p i j y 軆,軆軆t d 2y (6)Q 咬i j 軆軆t =坠2坠t V 乙T i j y 軆,軆軆t d 3y (7)In equation (5),there are two kinds of sound source:dipole source D 觶iand quadrupole source Q 咬i j,which represent the compact surface and volume sound radiation,respectively.The pressure fluctuations and viscous shear tensors on the wall boundary generate sound radiation of dipole type;the Lighthill stress tensors behave as quadrupole.3.2Computation conditionTo predict the far-field sound radiation,we assumethere are symmetrical flow field at the both sides of the flat plate.And the center of theplate was chosen to be the origin of coordinates,the vector of the observation point is chosen to be(2m,5m,2m).In this study,the lengths of three di -rections of computational box are chosen to be:8δ×2δ×8乙乙δ＝0.1941264m ×0.0485316m 乙×0.1941264乙m (length of streamwise -L 1×length of spanwise -L 2×length of wall-normal direction -L 3).The time-advanced step is chosen to be 1δτ.To satisfy the sampling precision and the resolu -tion,the computation continued for 50000steps af -ter the time-averaged statistics had converged.Re=u τδ/ν=800.The Mach number M =U ∞/c 0=0.00054947(the sound speed c 0=1500m/s).3.3The dipole and quadrupole sources There are three components of dipole source:D 觶1,D 觶2,D 觶3.According to the assumption of symmetrical flow field on both side of the flat plate,we can deduce that the D 觶2=0.There are nine components of quadrupole source:Q 咬11,Q 咬12,Q 咬21,Q 咬13,Q 咬31,Q 咬22,Q 咬23,Q 咬32,Q 咬33,in which three pairs are symmetrical:Q 咬12and Q 咬21,Q 咬13and Q 咬31,Q 咬23and Q 咬32.So,only six quadrupole terms should be solved respectively.Fig.2The computational domain of the far-field sound radiation994船舶力学第13卷第6期(a)Variation of Q咬11t=2.5～7!"s(b)Variation of Q咬11t=4～4.05!"sFig.3Variation of quadrupole Q咬11(a)Variation of Q咬12t=2.5～7!"s(b)Variation of Q咬12t=4～4.05!"sFig.4Variation of quadrupole Q咬12(a)Variation of Q咬13t=2.5～7!"s(b)Variation of Q咬13t=4～4.05!"sFig.5Variation of quadrupole Q咬13Figs.3~8show the variation of quadrupole sources,(a)show a course of long time(t= 2.5s~7s);(b)show the variation in a short course(t=4s~4.05s).We can find the amplitudes ofthe quadrupole Q咬11,Q咬22,Q咬33are relatively larger,and Q咬11is the largest,which shows the Lighthill第6期PAN Yu-cun et al:LES Method for Investigation of…995stress tensor in streamwise is the largest.In Figs.9and 10,the variation of dipole sources D 觶1,D 觶3was shown.To observe the variation of quadrupole and dipole source in frequency domain,Figs.11～16show the auto-power spectral density of quadrupole sources,Figs.17and 18show the auto-powerspectral density ofdipole sources.(a)Variation of Q 咬22t =2.5～7!"s (b)Variation of Q 咬22t =4～4.05!"s Fig.6Variation of quadrupole Q 咬22(a)Variation of Q 咬23t =2.5～7!"s (b)Variation of Q 咬23t =4～4.05!"s Fig.7Variation of quadrupole Q 咬23(a)Variation of Q 咬33t =2.5～7!"s (b)Variation of Q 咬33t =4～4.05!"s Fig.8Variation of quadrupole Q 咬33996船舶力学第13卷第6期Fig.9Variation ofdipole D 觶1Fig.10Variation of dipole D 觶3Fig.11Power spectral density of quadrupoleFig.12Power spectral density of quadrupole source Q 咬11source Q 咬12Fig.13Power spectral density of quadrupoleFig.14Power spectral density of quadrupole source Q 咬13source Q 咬22Fig.15Power spectral density of quadrupoleFig.16Power spectral density of quadrupole source Q 咬23source Q 咬33第6期PAN Yu-cun et al:LES Method for Investigation of (997)Fig.17Power spectral density of dipoleFig.18Power spectral density of dipole source D 觶1source D 觶3In the above figures,we can find the frequency of quadrupole fluctuations is relativelyhigher.All the quadrupole sources except Q 咬13concentrate their energy in range of 1200～2500Hz.The frequency of dipole fluctuations is relatively lower,and the energy is concentrat -ed in the domain of low frequency (under 500Hz).The reason may be that the dipole fluctua -tions were mainly caused by the fluctuations of viscous shear stress near the wall.As we known,in the vicinity of the walls,the high -speed fluid elements correspond to the sweep event toward wall;the low-speed fluid elements are generally being ejected from the wall re -gions.The viscous shear stress intensively affected by the sweep and ejection events,and these events have low frequency and large amplitude.So,the dipole source fluctuation is mainly in low frequencies.And the quadrupole sources associated with the second-order time derivative of Lighthill stress tensor.The Lighthill stress T i j ′≈ρu i ′u j ′is the product of velocities of differ -ent directions.The frequency velocity of fluctuations is relatively high,thus,the fluctuation of quadrupole source is in the domain of high frequencies.3.4Sound radiationFrom Figs.11to Fig.18,the sum of dipole D 觶i is smaller than the sum of quadrupole Q 咬i j.But according to the equation (5),acoustic densities due to the contribution of quadrupole sources at far-field is proportional to the fourth power of the Mach number,whereas that due to the contribution of dipole sources is proportional to the M 3.So,for the low Mach numbers,the con -tribution of the shear-stress dipole sources to the sound generation is larger than that of the quadrupole sources.In Fig.19,the variation of acoustic density due to the contribution of dipole and quadrupole sources were shown respectively.Fig.20shows the variation of acoustic density due to the contribution by the summation of dipole and quadrupole sources.From these fig -ures,we can conclude that for Mach number M =0.00054947,the turbulent shear-stress dipole sources dominates the whole acoustic field,and the contribution of quadrupole sources is al -most negligible.It is observed that,the characteristic of acoustic field behaves as the charac -teristic of dipole sources.According to the Lighthill analogy,the acoustic wave propagation was governed by a lin -ear,inhomogeneous wave equation of a nearly incompressible fluid.So,we can obtain the re-998船舶力学第13卷第6期lationship between acoustic pressure fluctuation p ′and acoustic density fluctuation ρ′:p ′=p-p ∞=c 02ρ′=c 02ρ-ρ∞p p .Thus,after we obtain the acoustic density fluctuation ρ′by Equation (5),we can research the variation of the acoustic pressure fluctuation p ′.To observe the characteristic of acoustic pressure in frequency domain,we obtain the spectral density of it by Fourier trans -formation:准m p p ω=1π∞-∞乙R pppp τe -i ωτd τwhere,R pp pp τ=〈p ′pp t p ′t+p p τ〉is the auto-correlation function of acoustic pressure fluctua -tions.Fig.21Power spectral density of sound pressureFig.22Power spectral density of sound pressure induced by dipole source induced by quadrupole sourceFig.21shows the power spectral density of sound pressure induced by dipole sources.For frequencies under 500Hz,the power spectral density drops sharply with increasing fre -quency.When the frequency is over 500Hz,the power spectral density tends toward calming and the amplitude of its fluctuation is small.So,the energy is concentrated in the domain of low frequencies.Fig.22shows the power spectral density of sound pressure induced by quadrupole sources.For frequencies under 180Hz,the power spectral density drops with in-Fig.19Variation of acoustic density at theFig.20Variation of acoustic density at the far-field due to the quadrupolefar-field (summation of Quadrupole (———)and dipole (---)and dipole)第6期PAN Yu-cun et al:LES Method for Investigation of (999)creasing frequency.When the frequency isover 180Hz,the power spectral density increases with increasing frequency andreaches its peak at 2300Hz.Over 2300Hz,the power spectral density drops again.So,the energy is concentrated in the domainbetween 1000to 2500Hz.Fig.23showsthe power spectral density of sound pres -sure induced by the summation of dipole and quadrupole sources.This curve is very similar to that of the power spectral densityof sound pressure induced by dipole sources in Fig.21.We can say that the sound pressure in -duced by dipole sources dominates the power spectral density of far-field sound radiation.4ConclusionsLarge Eddy Simulation was used to investigate space-time field of the low Mach number,fully developed turbulent boundary layer on a smooth,rigid flat plate.The radiated sound from turbulent boundary layer was studied using Lighthill ’s acoustics analogy.It is observed that the energy of the computed far-field sound radiation is mainly concentrated in the domain of low frequency (under 500Hz).In addition,the sound level is so small that only precision in -strument could measure it.Thus,It was difficult in laboratory to distinguish between the noise from TBL and the background noise.Numerical method would not introduce background noise and provide us a choice to resolve the problem.There were many different assumptions about mechanism of acoustic sources in the tur -bulent boundary layer.Intensity of quadrupole sources associate with the fluctuation of Lighthill stress tensor,and intensity of dipole source depend on the fluctuation of wall shear stress.There are still debates among scholars about which is the predominant factor responsible for sound radiation from TBL:the dipole sources of wall shear stress fluctuations or the quadrupole source of turbulent Reynolds stress?In this study,the computational results show:for low Mach number,the power spectral of sound radiation induced by dipole source determines the power spectral of whole sound radiation.The dipole source dominates over the sound radiation field.The contribution of quadrupole source to the sound field can be neglected in some degree.References[1]Lighthill M J.On sound generated aerodynamically I.Gerneral theory[J].Proc Royal Soc London,Ser A,1952,211(1107):564-587.[2]Lighthill M J.On sound generated aerodynamically II.Turbulence as a source of sound[J].Proc Royal Soc London,Ser A,1954,222:1-25.Fig.23Power spectral density of sound pressure induced by the summation of dipole andquadrupole source 1000船舶力学第13卷第6期第6期PAN Yu-cun et al:LES Method for Investigation of (1001)[3]Curle N.The influence of solid boundaries upon aerodynamic sound[J].Proc Royal Soc London,Ser A,1955,231(1187):505-514.[4]Ffowcs Williams J E,Hawkings D L.Sound generation by turbulence and surface in arbitrary motion[J].Phil.Trans.Roy.Soc.,1969,264A:321-342.[5]Smagorinsky J.General circulation experiments with the primitive equations[J].Monthly Weather Review,1963,91(3):99-164.[6]Van Driest E R.On turbulent flow near a wall[J].J Aero.Sci.,1956,23:1007-1011.[7]Pan Yucun,Zhang Huaixin.Investigation of wall pressure fluctuations beneath turbulent boundary layer by LES method[J].Journal of ship Mechanics,2010,14(1).(En Chinese)[8]Landahl M T.Wave mechanics of boundary layer turbulent and noise[J].J.A.S.A,1975,57(4):824-831.[9]Lauchle G C.Noise generated by exisymmetric turbulent boundary layer flow[J].J.A.S.A,1977,61(3):694-703.[10]Hardin J C.Acoustic sources in the low Mach number turbulent boundary layer[J].J.A.S.A,1991,90(2):1020-1031.[11]Howe M S.Surface pressures and sound produced by turbulent flow over smooth and rough walls[J].J.A.S.A,1991,95:1041-1047.用大涡模拟方法研究湍流边界层流动噪声潘雨村1，2，张怀新1（1上海交通大学海洋工程国家重点实验室，上海200030；2海军工程大学船舶与海洋工程系，武汉430033）摘要:利用大涡模拟方法，对刚性光滑平板上充分发展的低马赫数湍流边界层流动进行了数值模拟。