Turbulent trailing edge noise estimation using a RANSbased statistical noise model

Transcription

1 Pape Numbe 43, Poceedings of ACOUSTICS Novembe 011, Gold Coast, Austalia Tubulent tailing edge noise estimation using a RANSbased statistical noise model C.A. Albaacin, C.J. Doolan, C.H. Hansen and L.A. Books School of Mechanical Engineeing, The Univesity of Adelaide, Adelaide, South Austalia ABSTRACT The pediction capability of a RANS-based Statistical Noise Model (RSNM) is assessed by applying it to a shapedged symmetic flat stut at a Reynolds numbe of Re = 500,000 and zeo angle of attack. The method uses a igidhalf-plane Geen's function to calculate the fa-field specta geneated by the tubulence in the vicinity of the tailing edge. Because the exact fom of the tubulent souces is in geneal not available, a model of the tubulent velocity coss-spectum must be assumed. The coss-spectum model must accuately eflect the fequency and phase distibution in the bounday laye close to the tailing edge, but is othewise abitay. The model used in this study is Gaussian, containing velocity, time and length scales that ae defined in tems of the tubulence statistics obtained fom a steady RANS solution. The pedicted sound spectum shows the coect slope and levels fo fequencies anging fom 300 Hz to khz. INTRODUCTION Tailing edge (TE) noise is an issue in many engineeing applications such as fixed and otay wing aicaft, fans, wind tubines and submaines. Efficient pediction methods fo tailing edge noise ae essential fo enabling the design of silent aifoils fo these applications. Tailing edge noise occus when flow unsteadiness (tubulence) in the bounday laye ove an aifoil inteacts with the tailing edge. The edge acts as an abupt change in acoustic impedance, scatteing the sound waves poduced by the tubulence and enhancing the sound adiation to the fa field. The edge effectively destoys some of the cancelations that would nomally occu between the individual monopole components of the quadupole souces (tubulent eddies) if the edge wee not pesent, tansfoming the nea field pessue fluctuations into popagating acoustic waves. A detailed explanation of the tailing edge noise mechanism can be found in Goldstein (1976, pp ). Thee ae thee main methods geneally used fo calculating TE noise; semi-empiical methods, diect methods and hybid methods. Semi-empiical methods elate the fa field noise to bounday laye popeties such as bounday laye thickness o displacement thickness. The most widely used is the Books, Pope and Macolini (BPM) method, which povides an expession fo the fa field noise in 1/3 octave bands, based on a spectal shape function that depends on the Stouhal numbe and a scaling based on bounday laye displacement thickness. While efficient and accuate, the ange of application of semi-empiical methods is limited, and they cannot take into account the effect of tailing edge modifications such as seations. Diect methods calculate the fluid dynamics and acoustics in a single step. They do so by solving the compessible Navie Stokes equations using eithe diect numeical simulation (DNS) o lage eddy simulation (LES). This appoach has been successfully used by seveal eseaches (Masden et al., 008; Jones and Sandbeg 009; LeGaec et al., 008), but the computational demands of DNS and LES ae too lage fo this appoach to be pactical fo aifoil design applications. Hybid methods de-couple the flow calculation fom the sound calculation, as the latte can be done as a post pocessing step. This sepaation of the sound geneation and popagation pocesses makes the hybid appoach moe efficient, as a fine mesh is only equied in the egion close to the aifoil, whee viscous effects ae impotant. The sound souces ae geneally obtained fom LES o DNS and the fa field noise can be calculated by means of an acoustic analogy. This appoach has been shown to be as accuate as the diect appoach (Khalighi et al., 010) at a faction of the computational cost. Recent examples of TE noise calculations using this appoach include Chistophe et al., (009) and Winkle et al., (009). While moe computationally efficient than diect methods, hybid methods based on LES o DNS calculations ae still too computationally demanding to be of pactical use in aifoil design. Due to its moe modest computational equiements, the solution of the steady Reynolds Aveaged Navie Stokes (RANS) equations pesents itself as a moe pactical appoach to obtain the equied flow data to pefom TE noise calculations. Howeve, sound geneation is inheently time dependent, so the time aveaged infomation available fom a RANS solution is not sufficient by itself to pefom noise calculations. To cope with this limitation, diffeent appoaches based on statistical data povided using RANS equations have been developed; namely stochastic noise geneation and adiation (SNGR) and statistical modelling of the tubulent souces. The SNGR appoach geneates a synthetic tubulent souce field in the time domain, based on pescibed statistical infomation of the flow, which can be obtained fom a RANS solution. It has been succesfully applied to TE noise calculations (Ewet 008; Ewet et al., 009), and has also been used successfully fo a vaiety of aeodynamic noise poblems, such as landing gea noise (Dobzynski et al., 008), and jet and cavity noise (Mesbah 006). The dawback of this appoach is that the computational equiements to geneate and Acoustics 011 1

2 -4 Novembe 011, Gold Coast, Austalia Poceedings of ACOUSTICS 011 stoe the tubulence time data, and to calculate the fa field noise, ae of the same ode o lage than the RANS calculations that povide the statistics to geneate the tubulent souces. Most othe RANS based methods fo TE noise modelling ely on an estimation of the suface pessue spectum, which is then conveted to a fa field acoustic spectum using a diffaction analogy technique (Chandiamani 1974). Recent examples of TE noise calculations using this appoach include the wok of Kamuzzaman et al., (007, 008) and Glegg et al., (010). A limitation of suface pessue models is the assumption of homogeneous tubulence in the spanwise and steamwise diections, a condition that is unlikely to hold in many tailing edge configuations, paticulaly when seations o othe spanwise modifications ae used. In this aticle, the following notation is used: u * is the Fouie tansfom of u, û is the complex conjugate of u. F is a mean flow function defined by θ0 F( y) = ( U fauθ) cos ( U f U ) + a θ0 θ sin Whee θ 0 is the angle between a tubulent souce and the tailing edge plane, Ū is the time aveaged component of velocity and u is the fluctuating component of velocity (u = Ū + u ) and the subscipts and θ ndicate adial and tangential components espectively. The quantity f a is an anisotopy facto. Fo isotopic tubulence, f a = 1. (4) A RANS-based statistical noise model (RSNM) has been poposed by Doolan et al., (010), that does not equie the assumption of homogeneous tubulence. This method uses a semi-infinite had-plane Geen s function to calculate the acoustic fa field diectly, using a statistical model of the tubulent souces in the bounday laye in the vicinity of the tailing edge. The method equies a model of the tubulent velocity coss-spectum, which must accuately epesent the fequency and phase distibution in the bounday laye. This appoach is based on the wok of Tam and Auiault (1999) and Mois and Faassat (00), who developed a simila model fo jet noise applications. In this pape, the pediction capability RSNM is assessed by applying it to a shap-edged symmetic flat stut at a Reynolds numbe based onchod of Re = 500,000 at zeo angle of attack. NOISE PREDICTION METHOD The acoustic spectum S(x,ω) at a point x in the fa field as a function of fequency is given by ' * ' * S( x, ω) = { γ u ( y ˆ 1) u ( y ) whee and V ( y1) V ( y ) } F ( y ) F ( y ) dv ( y ) dv ( y ) 1 1 θ ρ0ω sinφ cos γ = 8 π ( ) ( ) ( ) ( ) sinφ = 3 3/ 3/ c0 y1 0 y R y1 R y ( ) + z z whee ω is the angula fequency, θ is the angle between the obseve and the chod plane with the oigin located at the tailing edge plane (see figue 1), c is the speed of sound, R is the distance between the obseve and the fluid element, 0 is the distance fom the edge to the fluid element,v is the volume of fluid consideed in the noise summation, y is the vecto location of the fluid element (souce) and the subscipts 1 and efe to the fluid elements used in each summation step. 0 (1) () (3) Figue 1. The Coodinate system used in the definition of equations 1,, 3 and 4. With the exception of the velocity coss spectum (the tem in squae backets in Equation 1), all elements in equation 1 can be calculated diectly fom the RANS solution. In ode to pefom a noise calculation, a model fo the velocity coss spectum is equied, ' * ' * u ˆ ( y1) u ( y ) = Φ( y1. y, ω) Coss spectum model The coss spectum model used in this pape is given by (5) ( 1 M ) A π ξ ω 1 ξ ω u s Φ ( y,, ) = exp ω s ls 4ωs whee M c is the convection Mach numbe and A is an empiical constant that detemines the stength of the coelation. The emaining vaiables ae defined as ξ = η M c i + i η= y y 1 c ( cosθ0 θ sin θ0 ) τ, and the tubulent scales ae defined by, c (6) (7) Acoustics 011

3 Poceedings of ACOUSTICS Novembe 011, Gold Coast, Austalia u =, =, k / 3 ω π / τ s s s 3/, l s = c l k. τ s = cτ k / ε / ε whee k (tubulent kinetic enegy) and ε (dissipation ate of tubulent kinetic enegy) ae obtained fom a RANS simulation and c τ, c l and A ae empiical constants detemined by best fit to data. The constants c τ and c l affect the time and length scales of the tubulence, espectively. This has the effect of modifying the fequency content of the esulting noise spectum. RANS SIMULATIONS The mean flow data equied fo the calculation of the tubulent scales (k and ε) is obtained by means of a steady state RANS simulation, using the semi-implicit method fo pessue-linked equations (SIMPLE) algoithm (Fezige and Peic 1999), with a toleance level set to Closue fo the RANS equations was povided by the k-ω SST tubulence model (Mente 199). The OpenFOAM (Welle et al., 1998) softwae package was used fo all flow calculations. (8) The gid used fo this study is shown in Figue 3. It was composed of fou stuctued blocks and the necessay gid spacing was detemined by compaing the esults fom a seies of meshes that wee successively efined. The initial mesh had a esolution of cells, which was inceased by doubling the numbe of cells in both x and y diections fo each efinement step. The esolution was inceased until no change was obseved in the mean velocity and tubulence intensity pofiles at the nea wake (0.6 mm fom the TE). The mean velocity and tubulence intensity pofiles ae plotted against the expeimental data fom Moeau et al. (011) in Figues 4 and 5 espectively. It is clea that thee is no change in the esults between the second and thid efinement steps. Theefoe, a esolution of cells was deemed sufficient to povide a gid independent solution. The RANS model calculates the velocity pofile vey well fo most of the bounday laye, but undepedicts the velocity fo y + < 30. It does not pefom so well in pedicting the tubulence intensity, whee it is unable to esolve two distinct featues of the expeimentally measued pofile; namely the peak at y + = 60 and the boad hump located in the egion 400 < y + < 700. This may be caused by the assumption of isotopic tubulence inheent in the k-ω SST tubulence model. It is expected that the use of a moe complex anisotopic tubulence model like the Reynolds Stess Model (RSM) will povide bette esults in the calculation of the tubulence intensity pofiles. This possibility will be investigated in futue wok. Contou plots of the tubulence intensity and mean velocity in the egion nea the tailing edge ae shown in Figues 6 and 7, espectively. Test case: symmetic stut The test case studied was a 00 mm chod symmetic stut of 5 mm thickness, with a cicula leading edge and a shap tailing edge with a 1 degee apex angle and a span of 450 mm. In this study, two dimensional RANS simuations wee pefomed, so the span of the aifoil was taken into account by means of an empiical spanwise coelation length in the noise calculations (Cocos 1964). The Reynolds numbe was set at Re c = 500,000. A fee steam tubulence intensity of 1.6 % was set fo the simulations. The obseve position was set at m diectly above the tailing edge. Moeau et al. (011) povides the efeence expeimental data fo this pape. A diagam of the stut is shown in Figue. Figue 3. Mesh used fo the RANS calculations. Flow diection fom left to ight. Figue. Diagam of symmetic stut. Gid efinement study Figue 4. Mean velocity pofiles calculated using the k-ω SST model and thee mesh esolutions compaed to expeimental hot wie measuements at chods downsteam fom the tailing edge. ACOUSTIC RESULTS Acoustic gid efinement Once the flow data wee obtained fom a RANS calculation, the data wee sampled onto an acoustic gid in ode to pefom the noise calculations. This gid extended ove the bounday laye thickness δ in both wall nomal and upsteam diections, as shown in Figue 8. A gid efinement study was conducted in ode to detemine a gid independent solution. The esults ae shown in Figue 9. A esolution of was deemed sufficient, since a futhe incease in esolution had no appaent effect on the pedicted sound levels Acoustics 011 3

4 -4 Novembe 011, Gold Coast, Austalia Poceedings of ACOUSTICS 011 fo fequencies between 300 and 000 Hz, but the fequency oll off is too steep and seveely unde-pedicts the sound levels fo highe fequencies. The spectal hump in the expeimental data close to 1.5 khz coesponds to the inteaction of the sound adiated fom the tailing edge and two extension plates, placed at the contaction outlet of the wind tunnel in ode to extend the potential coe egion (Moeau et al., 011). It is theefoe not supising that the model is not able to pedict this featue of the noise spectum. Figue 5. Tubulent intensity pofiles calculated using the k- ω SST model fo thee mesh esolutions compaed to expeimental hot wie measuements at chods downsteam fom the tailing edge. Figue 8. Diagam of the gid used fo the acoustic calculations. The symbol δ epesents the bounday laye thickness at the tailing edge. Figue 6. Tubulence intensity contous nea the tailing edge. The egion in white is pat of the inne volume of the aifoil. The tailing edge is located at y/c = x/c = 0. Figue 9. Spectal density at the obseve position calculated using RSNM with fou diffeent acoustic gid esolutions. Figue 7. Mean velocity contous nea the tailing edge. The egion in white is pat of the inne volume of the aifoil.the tailing edge is located at y/c = x/c = 0. Compaison with expeimental esults The spectal density calculated with RSNM is plotted against the expeimental data obtained in the anechoic wind tunnel at the Univesity of Adelaide (Moeau et al., 011) in Figue 10. The values used fo the empiical constants ae A = 10, c l =1 and c τ =10.5. The model pedicts the coect slope and levels Figue 10. Spectal density in the fa field calculated using a RSNM simulation, plotted against expeimental data of Moeau et al., (011). CONCLUSIONS The RSNM method has been applied to a symmetic flat stut at zeo angle of attack and a Reynolds numbe of 4 Acoustics 011

5 Poceedings of ACOUSTICS Novembe 011, Gold Coast, Austalia Re=500,000. The pedicted spectum shows the coect slope and levels fo fequencies between 300 Hz and khz, but the pediction deteioates significantly fo highe fequencies. The unde pediction of the adiated sound at highe fequencies is most likely due to a poo pediction of the tubulent kinetic enegy by the RANS model, which is used to calculate the tubulent scales in the coss spectum model. It is also possible that the fom of the coss spectum model used in this wok is not the most appopiate fo bounday laye flows, since it is based on a two-point velocity coelation model deived fo jets. This will be investigated in futue wok. The values chosen in this pape fo the empiical constants A, c τ and c l might not be suited fo all of flow conditions, and a study of thei ange of application will be conducted in the futue, when RSNM is applied to othe aifoil shapes. REFERENCES Books, T. F., Pope, D. S. & Macolini, M. A. (1989), Aifoil self-noise and pediction, NASA Refeence publication, 118. Chandiamani, K. L. (1974), Diffaction of evanescent waves, with applications to aeodynamically scatteed sound and adiation fom unbaffled plates, Jounal of the Acoustical. Society of Ameica, 55, 19-9 Chistophe, J., Anthoine, J. & Moeau, S. (009), Tailing edge noise of a contolled-diffusion aifoil at modeate and high angle of attack, Poceedings of the 30th AIAA Aeoacoustics Confeence. Miami, Floida, May. Cocos, G. M. (1964), The stuctue of the tubulent pessue field in bounday laye flows, Jounal of Fluid Mechanics 18, Dobzynski, W., Ewet, R., Pott-Pollenske, M., He, M. & Delfs, J. (008), Reseach at DLR towads aifame noise pediction and eduction, Aeospace Science and Technology, 1, Doolan, C., Albaacin, C., Hansen, C. (010), Statistical estimation of tubulent tailing edge noise, Poceedings of the 0th Intenational Congess on Acoustics, ICA 010, Sydney, Austalia, 3 7 August. Ewet, R. (008), Boadband slat noise pediction based on CAA and stochastic sound souces fom a fast andom paticle-mesh (RPM) method Computes & Fluids, 37, Ewet, R. & Al, E. (009), RANS/CAA based pediction of NACA 001 boadband tailing edge noise and expeimental validation, Poceedings of the 15th AIAA/CEAS Aeoacoustics Confeence. Miami, Floida May. Fezige, J., & Peic, M. 1999, Computational methods fo fluid dynamics. Spinge Belin. Glegg, S., Moin, B., Atassi, O. & Reba, R. (010), Using Reynolds-Aveaged Navie Stokes Calculations to Pedict Tailing-Edge Noise AIAA JOURNAL, 48. Goldstein, M. Aeoacoustics. McGaw Hill, New Yok, Jones, L. & Sandbeg, R. (009), Diect numeical simulations of noise geneated by the flow ove an aifoil with tailing edge seations, Poceedings of the 15th AIAA/CEAS Aeoacoustics Confeence (30th AIAA Aeoacoustics Confeence). Miami, Floida May, 009. Kamuzzaman, M., Lutz, T., Heig, A. & Kame, E. (008), RANS based pediction of aifoil tailing edge fa-field noise: impact of isotopic & anisotopic tubulence, Poceedings of the 14th AIAA/CEAS Aeoacoustics Confeence (9th AIAA Aeoacoustics Confeence). Vancouve, Bitish Columbia Canada. 5-7 May. Kamuzzaman, M., Lutz, T. & Kame, E. (007), An appoach to RANS based pediction of aifoil tailing edge fa-field noise, Poceedings of the Second Intenational Meeting on Wind Tubine Noise. Lyon Fance. 0-1 Septembe. Khalighi, Y., Mani, A., Ham, F. & Moin, P. (010), Pediction of sound geneated by complex flows at low Mach numbes AIAA JOURNAL, 48. Le Gaec, T., Gloefelt, X. & Coe, C. (008), Diect noise computation of tailing edge noise at high Reynolds numbes, Poceedings of the 14th AIAA/CEAS Aeoacoustics Confeence (9th AIAA Aeoacoustics Confeence). Vancouve, Bitish Columbia Canada. 5-7 May. Masden, O., Bogey, C. & Bailly, C. (008), Diect Noise computation of the tubulent flow aound a zeoincidence aifoil AIAA JOURNAL, 46. Mente, F. (199), Impoved two-equation k w tubulence models fo aeodynamic flows, NASA Technical Memoandum. NASA Ames, CA. Mesbah, M. 006 Flow noise pediction using the stochastic noise geneation and adiation appoach, PhD Thesis, Katholieke Univesiteit Leuven, Belgium, 006. Moeau, D., Books, L. & Doolan, C. (011), Expeimental investigation of boadband tailing edge noise fom shap-edged stuts, Poceedings of the 17th AIAA/CEAS Aeoacoustics Confeence. Potland Oegon. 5-8 Jun. Mois, P. & Faassat, F. (00), Acoustic analogy and altenative theoies fo jet noise pediction, AIAA JOURNAL, 40. Tam, C. & Auiault, L. (1999), Jet mixing noise fom finescale tubulence. AIAA JOURNAL, 37. Welle, H., Tabo, G., Jasak, H. & Fueby, C. (1998), A tensoial appoach to CFD using object oientated techniques, Computes in Physics, 1(6): Winkle, J., Moeau, S. & Caolus, T. (009), Lage-eddy simulation and tailing-edge noise pediction of an aifoil with bounday-laye tipping, Poceedings of the 30th AIAA Aeoacoustics Confeence. Miami, Floida May. Acoustics 011 5