USING STATISTICAL METHODS TO PREDICT TURBULENCE INDUCED SOUND AND VIBRATION IN AEROSPACE AND AUTOMOTIVE APPLICATIONS

ICSV4 Cairns Australia 9-2 July, 2007 USING STATISTICAL METHODS TO PREDICT TURBULENCE INDUCED SOUND AND VIBRATION IN AEROSPACE AND AUTOMOTIVE APPLICATIONS Patrik Ragnarsson, Christophe Schra Michel Tournour, David Freed 2, Sivapalan Senthooran 2 and Fongloon Pan 2 LMS International, Interleuvenlaan 68, 300 Leuven, Belgiu 2 Exa Corporation, 50 North Hill Drive, Suite 2, Brisbane, CA, USA Patrik.ragnarsson@ls.be Abstract Wind induced sound and vibrations are areas where a lot of research efforts are spent today. Due to the rando nature of turbulent flow fields, a deterinistic approach to the proble will just describe one of any possible solutions. To reduce statistical uncertainty and achieve a robust solution of a proble a large nuber of calculations can be perfored (i.e. Monte Carlo siulations). The focus of this paper is on a ethod to siplify the calculations involved in using such stochastic approaches. The experiental and nuerical test case consists of a plate-backed cavity where the rectangular plate is excited by a reattaching turbulent boundary layer, and is supported by a visco-elastic seal. The approach is based on a Power Spectral Density (PSD) loading using a Corcos odel. The statistical paraeters are calculated by processing deterinistic flow data. These data consist of tie-series wall pressure fluctuations (WPF) obtained by CFD siulations. The WPF tie series were coputed using Exa s PowerFLOW CFD software, whereas the Vibro-acoustic responses were coputed using LMS Virtual.Lab and Sysnoise. The agreeent between the rando acoustic ethod using the Corcos odel and a noral deterinistic ethod is quite good. The over all levels are siilar, and the results fro the deterinistic calculation show, as expected, ore fluctuations than the rando acoustic results. The rando acoustic ethod is in a sense a superposition of a large nuber of deterinistic load cases, hence the ore sooth and averaged curve.. INTRODUCTION Turbulent loading of structures is a big issue in any engineering applications. The turbulent flow on the wings of an airplane and the turbulence initiated by the A-pillar of a car are just two exaples of areas where it is iportant to have a good understanding of the turbulent flow field and its effects on the structures and the radiated noise. This paper is focusing on ways to use siulated flow data for vibro-acoustic response calculations. Using noral deterinistic calculation ethods on this type of stochastic proble is very tie consuing. To achieve a robust solution a large nuber of flow siulations are needed. One way of doing this is to perfor a so called Monte Carlo siulation. The ethod studied in this paper is a rando acoustic ethod. The idea is to use one single flow siulation and to use a

statistical approach when solving the vibro-acoustical proble. In this paper the Corcos odel is used to describe the flow field in a statistical anner []. The average PSD of the surface pressure, the streawise and spanwise decay rates and the flow convection speed are the paraeters used to express the flow field in ters of surface pressure fluctuations. Fro these paraeters a set of orthogonal (i.e. uncorrelated) excitations are calculated using truncated singular value decoposition. For each of these orthogonal excitations the response is calculated in a deterinistic anner. Finally the results fro all the different orthogonal excitations are cobined together into a coplete, statistically robust solution for the plate vibration and the radiated sound pressure. 2.. The Corcos odel 2. THEORY A plate is excited by a turbulent boundary layer because the turbulence generates a fluctuating pressure field on the plate. Assuing stationary statistics of the wall pressure field, the resulting vibration response of the plate can be written as [2]: G uu y x µ x, () υ * y j i ( yi, y j, ω) H ω = u F, Φ pp ( xµ, xυ, ω) H u, F, ω daµ daυ where G uu is the response of the plate, H u,f are the frequency response functions relating displaceent at y i and y j with the excitation at x µ and x υ and Ф pp is the turbulent boundary layer cross power spectral density function of the pressure fluctuations between all the loaded points. The superscript * denotes the coplex conjugate transpose. The frequency response function H u,f of the plate can be calculated in any different ways. In this paper a finite eleent odel is used. If a teporally stationary process and spatial hoogeneity of the flow field are assued the cross spectral density function of the pressure loading Ф pp can be separated into an averaged auto power spectru of the pressure loading Φ pp and a coherence function Γ between the excitation points x µ and x υ. The cross power function can then be re-written as: Φ pp ( x, ω) = Φ pp( ω) Γ( ζ, ζ ω) x, (2) µ, υ 3, where ζ and ζ 3 are the streawise and spanwise distance between the excitation points x µ and x υ. Corcos [] hypothesized that the coherence function could be separated into two parts: a streawise coherence function and a spanwise coherence function: ωζ ωζ Γ ζ (3) 3 ( ), ζ 3, ω = A B U c U c where the functions A and B could be siple analytical expressions depending only on the convection speed (U c ), the angular frequency (ω), the distance between the excitation points (ζ and ζ 3 ), and decay constants (α and α 3 ): ωζ ωζ α U c U A U c = e e iωζ c and B ωζ 3 α3 ωζ 3 U c U c = e. (4) (5)

In this work the Corcos odel is used by identifying the auto power spectru and the coherence function fro the CFD siulations. As can be seen in the equations above, the paraeters needed to get the estiate of the coherence function are the convection speed and the spanwise and streawise decay rates. 2.2 Singular value decoposition Given the results fro the Corcos odel above, the cross power spectral density function of the surface pressure can be calculated. To calculate the resulting response of the syste, this rando pressure load case is decoposed into a set of orthogonal (i.e. uncorrelated) deterinistic load cases. The way to obtain this decoposition of the cross power spectral density function (or atrix) is to use the truncated singular value decoposition ethod. The truncated singular value decoposition (SVD) is defined by [3]. Φ U σ V (6) pp * Where Ф pp is the n by n cross power spectral density atrix, U and V are by orthogonal atrices, superscript * denotes the coplex conjugate transpose and σ is an by diagonal atrix where the diagonal eleents are the singular values. The truncation error satisfies: Φ pp U H σ V = σ (7) + where σ + is the (+) largest singular value of Ф pp. The singular values are autoatically sorted by descending order, which allows easy and precise error estiation. This eans that after choosing a truncation error liit for the calculation it is straightforward to decide on the nuber of load cases needed to achieve this liit. Since the load cases are orthogonal, the results fro solving all of these load cases can be recobined to the response of the syste corresponding to the average response of a large nuber of deterinistic calculations. 3. TEST SETUP The test setup is shown in Figure below. A tilted fence is located upstrea of a plate supported by visco-elastic seals. A coplex frequency independent stiffness has been used to odel the seal. This stiffness was odelled as a transverse spring with the stiffness set to 2.2 ( + 0.25i) MPa [4]. The plate diensions are (466x375x3.38) and it is ade of aluiniu. The fence is 52.5 high, 490 wide, tilted 55 o and located 490 upstrea of the plate. h a Air Flow Direction w θ d Plate b Figure. Left: The test setup, showing the plate downstrea of the fence. Right: odel of the cavity.

The plate ean-square velocity and the sound pressure level at a point 70 below the plate in a sound absorbing cavity were easured in the experient. 4. CFD calculations 4. CALCULATION PROCEDURE The exterior flow field and the pressure fluctuations on the plate surface were obtained fro the siulation using the coercial software PowerFLOW 4.0a, fro Exa Corporation. 4.. Nuerical Schee The underlying physics in PowerFLOW is based on the Lattice-Boltzann Method (LBM) in which the fundaental equation of otion for fluid particles, the Boltzann equation, is solved on a cubic lattice using a discrete set of fluid velocity states. The discrete Boltzann equation is coupled with the ideal gas equation of state to represent realistic transfer of ass and oentu sufficient to describe sall-scale fluid otions with very low nuerical dissipation [5,6]. The geoetry of the surface enters the LBM discretization as boundary facets iersed in the lattice, and they provide the boundary conditions for the discrete velocity states by enforcing zero-flux through the wall [7]. For high-reynolds nuber turbulent flow, a turbulence odel is used both for wall boundary layers and for dissipation of energy in the fluid to the scales of turbulence that can not be represented on the grid. The present approach can be thought of as very large-eddy siulation or VLES. The wall odel uses a ulti-layer boundary layer odel with law-ofthe-wall in the inner layer and the RNG k-epsilon odel in the outer layer. The effect of surface pressure gradients is included through a local pressure gradient based rescaling of the usual law-of-the-wall; the resulting wall shear stress leads to correct prediction of boundary layer separation including when driven by a local adverse pressure gradient [8]. The VLES approach is achieved in the fluid doain by solving the RNG k-epsilon equations in the fluid, and coupling the to the Lattice-Boltzann equations through the eddy-viscosity [9,0]. This VLES approach is capable of producing high-fidelity WPF up to very high frequency, where the desired frequency is liited ostly by the choice of grid resolution for the coputational lattice. See, for exaple, Vaillant and Maillard [], Senthooran et al.[2], and Belanger et al. [3] for further details. 4..2 Coputational Methodology The test configuration shown in Figure was replicated in the PowerFLOW digital wind tunnel as shown in Figure 2. The siulation was done at 43.6/s inlet velocity atching the experiental wind tunnel speed. Variable resolution regions were used with fine resolution in critical regions to predict the boundary layer developent and flow separation accurately and coarse resolution in non-critical regions to optiize the coputational effort. The finest cell size used in the siulation is 0.75. The tie step in this nuerical schee is inherently deterined based on the resolution and Mach nuber. The siulation ran for 368,868 tie steps, corresponding to second in physical tie. The transient surface pressure data on the plate was collected in such a way as to eulate.37 diaeter icrophones at the locations used in the experient to validate the CFD calculations. These icrophone locations are shown in Figure 2. The transient pressure data was recorded on the entire plate surface on a esh with the grid size 5 for flow visualizations and to provide the data

required for the vibro-acoustic calculation. These easureents were sapled at 36890 Hz and were recorded for 0.7 seconds. 4.2 Acoustic response calculations Figure 2: CFD configuration with icrophone locations The calculations have been perfored in LMS Virtual.Lab and Sysnoise. The first step was to calculate the odal response of the structure in LMS Virtual.Lab. Thereafter the plate was loaded with a pressure field and the radiated sound could be calculated. The pressure loading were perfored in two different ways: a deterinistic approach and a rando approach. 4.2. Deterinistic approach The procedure was to iport the wall pressure fro the CFD calculation and to transfor it in to point forces distributed over the plate. These point forces represents the turbulent pressure fluctuation that drives the plate. Since the CFD calculations norally are perfored with a very dense esh, the acoustical esh can be coarsened to save calculation tie and eory use. The CFD data are then interpolated to fit the coarsened acoustical esh. The interpolation procedure starts fro a node on the coarser esh; the eight closest nodes within a user defined axiu distance on the corresponding denser esh are taken under consideration. These eight nodes are weighted based on the inverse of the distance fro the new node according to (8) x Ac x ( d ) + x2 ( d 2 ) + x3 ( d 3 ) + K + x8 ( d8 ) ( d ) + ( d ) + ( d ) K( d ) =. (8) 2 3 8 Based on the new acoustical esh with the corresponding point forces, the sound radiation proble could then be calculated by solving the coupled structure-acoustic odel in LMS Virtual.Lab. The structure is solved with FE techniques and the acoustic proble is solved using BEM technique. 4.2.2 Rando approach The pressure field was represented by a cross power spectral density atrix which is derived fro the well known Corcos odel []. The Corcos paraeters have been extracted fro the CFD siulations (see Figure 6). The cross PSD atrix were then decoposed into a nuber of orthogonal load cases which cobined represents the desired pressure loading. Each of these load cases can then be solved in a deterinistic anner in a siilar was as for the deterinistic approach described above. The coplete solution is obtained when the result fro all the load cases are recobined to a final result. In LMS Virtual.Lab and Sysnoise the

process of decoposing the data in to load cases, solving the load cases and recobining the is autoatically perfored in the rando acoustic odule. 5. RESULTS AND DISCUSSION Figure 3 shows the tie averaged velocity agnitude and strealines on a plane on the centreline of the plate. This shows that the tie-average flow reattachent is just past the leading edge of the plate, upstrea to the icrophone locations. Figure 3. Tie averaged velocity agnitude and 3D strealines Figure 4 shows the coparisons between the experiental and predicted power spectra density (PSD) for all four icrophones located on the plate. For both the predictions and the easureents, the spectra are seen to depend only weakly on the streawise location, indicating a fairly hoogeneous turbulent field over that region. This iplies that the assuption in Equation 2 about spatial hoogeneity is at least valid over that region of the flow field. Both the experiental and siulations results show high pressure fluctuation levels below 00 Hz and continuous reduction in pressure fluctuation levels fro 00 Hz to the high frequency region. The cascading slope predicted in the siulation agrees well with the experient up to a very high frequency region. The aplitude of the siulation spectra are slightly over-predicted copared to the experiental spectra. This could be due to slight isatch in the reattachent location between siulation and experient. Another source of error could be a isatch of the upstrea boundary layer and turbulence levels; these were not characterized for experient so they could not be atched in siulation; PSD surface pressure (db/hz rel. 4e-0Pa^2) Figure 4: Power Spectra Density (PSD) of the plate surface pressure. Coparisons between PowerFLOW and experient. Pressure fluctuations on the surface of the plate due ipingeent of the transient flow structures can be characterized by db-aps of pressure fluctuations on the surface. Figure 5 shows the db-aps for the plate surface in octave bands. It can be seen that the coplex pressure fluctuation topology on the plate surface is well captured and represented by these db-aps copared to the icrophone easureents taken at experiental locations.

Figure 5: Octave band db aps of the plate surface pressure These pressure fluctuations predicted fro the PowerFLOW siulation on the entire plate using a fine esh were transferred to LMS Virtual.Lab in tie doain for acoustic response calculations for the deterinistic ethod and have been used to estiate the Corcos paraeters for the rando acoustic ethod. The PSD of the average surface pressure can be seen in Figure 6 left. Figure 6 right shows the estiation of the convection speed of the turbulent flow field. The decay rates are estiated by fitting Equations (4) and (5) to the coherence function in spanwise and streawise direction respectively. The decay rates were estiated to 0.3 in the strea wise direction and 0.7 in the span wise direction. This corresponds well to the decay rates fro the easureents [4]. Figure 6: Left: PSD of the average surface pressure. Right: estiation of the convection speed of the turbulent flow field. In Figure 7 left a coparison of the plate surface velocity can be found. Experiental results (blue) are copared with two siulation results; a deterinistic approach (pink) and a Corcos approach (black). In Figure 7 right, the results fro the sound pressure inside the cavity are presented. The calculations have been perfored in the frequency range 75-500 Hz. 80 80 PSD surface velocity (db/hz rel. e-2 (/s)^2) 70 60 50 40 30 20 0 0 75 25 75 225 275 325 375 425 475 Frequency (Hz) Experient (Purdue University) Deterinistic ethod (Virtual.Lab/Sysnoise) Corcos (Virtual.Lab/Sysnoise) PSD sound pressure (db/hz rel. 4e-0Pa^2) 70 60 50 40 30 20 0 0 75 25 75 225 275 325 375 425 475 Frequency (Hz) Experient (Purdue university) Deterinistic ethod (Virtual.Lab/Sysnoise) Corcos (Virtual.Lab/Sysnoise) Figure 7: Coparison of calculation with a standard deterinistic approach, The Corcos approach and experients. Left: plate surface velocity Right: Sound pressure level in the cavity 70 fro the plate

As it can be seen in Figure 7 both the deterinistic calculation and the rando acoustic calculation over predicts the results fro the easureents. One possible reasons for this is the over prediction of the surface pressure in the CFD calculations. In Figure 4 the calculated plate surface pressure at four icrophone positions are copared with the easureents at these points. The CFD results are 5-0 db-units higher than the easureents for the entire frequency range of interest. Keeping this in ind while looking at the plate velocity and the sound pressure in the cavity, the results atch better. 6. CONCLUSIONS The agreeent between the deterinistic ethod and the rando acoustic ethod is quite good. As expected, the deterinistic ethod shows ore fluctuations than the rando acoustic ethod. The rando acoustic ethod is in a sense a superposition of a large nuber of deterinistic load cases, hence the ore sooth and averaged curve. The soothness of the experiental curve is probably due to a large nuber of averages. Future experients and calculations will be perfored in order to address discrepancy between the experiental results and the siulations. ACKNOWLEDGEMENTS The authors gratefully acknowledge the EC support for the present work, perfored within the Marie Curie EST project 'SIMVIA2' (www.sivia2.eu, contract nr MEST-CT-2005-020263). REFERENCES [] G.M. Corcos, The structure of the turbulent pressure field in boundary-layer flows, Fluid Mechanics 8, 353-378 (963). [2] S.A. Habric, Y.F. Hwang, W.K. Bonness, Vibrations of plates with claped and free edges excited by low speed turbulent boundary layer flow Journal of Fluids and Structures 9, 93-00, 2003. [3] Bendat J S & Piersol A C, Rando data: analysis and easureent procedures New York: Wiley, 97. [4] J. Park, L. Mongeau, T. Siegund, "An Investigation of the Flow-induced Sound and Vibration of Viscoelastically Supported Rectangular Plates: Experients and Model Verification", Journal of Sound and Vibration 2004, 275, 249-265 [5] Y. Li, R. Shock, R. Zhang, and H. Chen, Nuerical Study of Flow Past an Ipulsively Started Cylinder by Lattice Boltzann Method, J. Fluid Mech., Vol. 59, 2004, pp. 273-300. [6] H. Chen, O. Filippova, J. Hoch, K. Molvig, R. Shock, C. Teixera, and R. Zhang, Grid Refineent in Lattice Boltzann Methods Based on Voluetric Forulation, Physica A 362, pp. 57-67, 2006. [7] H. Chen, C. Teixeira, and K. Molvig, Realization of Fluid Boundary Conditions via Discrete Boltzann Dynaics, Intl. J. Mod. Phys. C, 9 (8), 998, p. 28. [8] C. Teixeira, Incorporating Turbulence Models into the Lattice-Boltzann Method, Intl. J. Mod. Phys. C, Vol. 9 No. 8, 998, pp. 59-75. [9] H. Chen, S. Kandasay, S. Orszag, R. Shock, S. Succi, and V. Yakhot, Extended Boltzann Kinetic Equation for Turbulent Flows, Science, Vol. 30, pp. 633-636, 2003. [0] H. Chen, S. Orszag, I. Staroselsky, and S. Succi, Expanded Analogy Between Boltzann Kinetic Theory of Fluid and Turbulence, J. Fluid Mech., Vol. 59, 2004, pp. 307-34. [] O. Vaillant, and V. Maillard, Nuerical Siulation of Wall Pressure Fluctuation on a Siplified Vehicle Shape, AIAA Paper 2003-327, 2003. [2] S. Senthooran, B. Crouse, G. Balasubraanian, D. Freed, and S. Noelting, Siulation Of Wall Pressure Fluctuations On Siplified Autoobile Shapes Using A Lattice Based Method, Proc. 2005 ASME IMECE, Nov. 5-, 2005, Orlando, Florida. [3] A. Belanger, M. Meskine, B. Caruelle, and K. Debatin, Aero-acoustic Siulation of a Double Diaphrag Using Lattice Boltzann Method, AIAA Paper 2005-297, 2005.