Finite element study of the vibroacoustic response of a structure, excited by a turbulent boundary layer

Finite element study of the vibroacoustic response of a structure, excited by a turbulent boundary layer A. Clement 1, C. Leblond 2, C. Audoly 3 and J.-A. Astolfi 4 1,4 Institut de Recherche de l Ecole Navale (IRENav), Brest 2 DCNS Research, Département Dynamique des Structures (DDS), Nantes 3 DCNS Research, CEMIS, Toulon ABSTRACT The vibration and the acoustic field generated by a structure submerged in fluid, under a turbulent boundary layer flow is investigated. This investigation aims at improving numerical models predicting the radiated noise. Experimental results are compared to a finite element modelization. Modal behaviour of the FSI problem is described through a modal analysis based on the (U,p,φ) formulation with CODE_ASTER. The pressure field is computed as a sum of plane waves and harmonical analysis are performed for each term of the sum. Spectrum models are used to include the spatial correlation in the sum. The studied structure is a plate over a cavity, clamped on the leading edge and with other edges free. The apparatus is immerged in an hydrodynamic tunnel, a turbulent flow is generated on it s upper side and the cavity is filled with fluid at rest. The vibroacoustic behaviour is mesured with a laser doppler scanning vibrometer and an acoustic transducer in the cavity. The validity of the model is assed by comparing numerical and experimental data for different spectrum models. A good reproduction of the behaviour of the plate is obtained in terms of defflection shapes and spectral levels. 1. INTRODUCTION As the behaviour of structures immersed in turbulent flows and fluid structure interaction under turbulent loadings has been studied, mainly for aeroacoustic applications, in the past century, the behaviour of the turbulent boundary layers are well assessed, however there is still few available theoretical models which fully describes the physical phenomena occurring within the boundary layer, particularly in the case of a heavy fluid loading. Thus no model can predict the fluid structure interaction of systems under a turbulent boundary layer loading with the degree of precision needed for industrial applications [1]. The recent evolutions in the field of underwater detection, as well as considerations on the environmental impact of ships and concerns related to the well-being of passengers in ships now cruising at high speeds have brought studies on the hydroacoustic behaviour of structures under turbulent boundary layer loadings [2, 3]. Other studies tackling the same physics can also be found, in the description of the behaviour of turbulent flows in or over pipes and vessels. The structures described in the latter can be heat exchangers found in electric power plants and other types of vessels of varying sizes and shape [4 6]. In the following paper the goal is to describe the vibro-acoustic behaviour of a structure under turbulent flow loading through a numerical model using a finite elements approach, taking fully into account the fluidstructure coupling. Numerical results obtained on a test case are compared to experiment performed in a water tunnel facility. 2. GENERAL STATEMENT OF THE PROBLEM The general statement of the fluid-structure interaction problem considered in this studied can be fully described following the developments made in [7], concerning the displacement of a plate. This statement is the transcription in the wavenumber-frequency space of the problem in the physical space described in [8]. 1 email: adrien.clement@ecole-navale.fr 5560 1

Figure 1. Elastic plate under a turbulent flow 2.1 Displacement of the plate The power cross-spectrum, in the spatio-temporal space S ww (x, x, ω) of the response of a plate (Fig.1) in the (x, y) plane, can be found through the use of the of the power auto-spectrum of the load, described in the wavenumber-frequency space [7]: S ww (x, x, ω) = 1 Γ (2π) 2 ω (x, k )S pb p b (k, ω)γ ω (x, k )dk (1) In this formulation, Γ ω (x, k ) is the space varying Green function of the system and it can be interpreted as a sensibility function of the system for the response at point x when an harmonic load of pulsation ω is applied. S pb p b (k, ω) is the turbulent boundary layer pressure autospectrum in the wave-number space. Here k and k are the variables in the dual space of the Fourier transform, corresponding to the spatial variables x et x, x = (x, y, 0) for a plate in the plane (x, y). 2.2 Plane-wave decomposition of the displacement As the integral problem in (Eq.1) can only be computed for problems with simple boundary conditions, such as fully supported plates, the problem must be transformed into a discrete formulation. This is done, following the work found in [9 11] describing the synthesization of a boundary layer pressure field through the use of speakers. In this work the pressure field is not synthesized through speakers but it is synthesized numerically in order to obtain loadings cases for the finite elements computation. The boundary layer pressure spectrum must be reformulated in order to obtain the discrete formulation of the plate displacement. Thus the pressure field is described as a sum of independent pressure plane waves. These plane waves have an amplitude A rs (t) and are described by the wave-vector k = (k r, k s ): P rs (x, y, t) = A rs (t)e jkrx e jksy (2) Thus the pressure field can be written as a sum of these plane wave, in the following form: p(x, y, t) = r,s P rs (x, y, t) (3) Given the discrete formulation of the pressure field of (Eq.3), the cross-spectral density of the field can be expressed as: S pp (ξ x, ξ y, ω) = S ArsA rs (ω)e jkrξx e jksξy (4) r,s Furthermore, the cross-spectral density of pressure spectrum generated by a turbulent boundary layer can be expressed in the following way: S pp (ξ x, ξ y, ω) = + S pb p b (k x, k y, ω) 4π 2 e jkxξx e jkyξy dk x dk y (5) 5561 2

This equation can then be written in the wavenumber space, in a discrete form: S pp (ξ x, ξ y, ω) = r,s S pb p b (k r, k s, ω) 4π 2 e jkrξx e jksξy k r k s (6) Through equations (Eq.4) and (Eq.6), the plane wave formulation can then be linked to the boundary layer pressure spectrum: S ArsA rs (ω) = S p b p b (k r, k s, ω) 4π 2 k r k s (7) Finally the displacement of the structure under a turbulent boundary layer pressure, as found in [7] and defined by (Eq.1), can be expressed in a discrete manner, for a plane wave model of the load, through (Eq.7). Thus the displacement auto-spectrum of the plate, at point Q is given by: S vv (Q, ω) = r,s S ArsA rs (ω) H v (Q, k r, k s, ω) 2 (8) Here H v (Q, k r, k s, ω) is the displacement of the structure at point Q when loaded by a unit plane wave of wave-vector k = (k r, k s ). This sensitivity function is the counterpart of the Green sensitivity function Γ ω (Q, k) that appears in the integral formulation (Eq.1). 2.3 Radiated acoustic pressure field 2.3.1 General expression of the pressure field The radiated acoustic pressure auto-spectrum can be found using the same approach as the one used to describe the displacement auto-spectrum of the plate. As found in [7], the general statement of this autospectrum can be expressed in a similar manner as in (Eq.1): S p±p ± (z, z, ω) = 1 (2π) 2 T ± ω(z, k )S pb p b (k, ω)t ± ω (z, k )dk (9) The pressure auto-spectrum can thus be written in discrete form in the same manner as the displacement auto-spectrum described by equation (Eq.8): S p±p ± (z, ω) = r,s S ArsA rs (ω) T v (z, k r, k s, ω) 2 (10) Here T v (z, k r, k s, ω) is the acoustic response at point z of the fluid domain to a unit surface plane wave load of wave-vector k = (k r, k s ) on the structure. This sensitivity function is the counterpart of the Green sensitivity function T ± ω (z, k) found in equation (Eq.9) which takes into account the boundary conditions of the studied problem. 2.3.2 Pressure field radiated by a plate As the auto-spectrum displacement of the plate is obtained through equation (Eq.1), the pressure field in the fluid domain can also be obtained through a direct computation of the pressure field radiated by a vibrating structure immersed in fluid. With this approach, a second problem is solved, taking only into account the fluid domain, with the computed speed of the structure as an input field. The sound pressure is then obtained by solving this problem with a finite element method or a boundary elements method. 3. FINITE ELEMENT COMPUTATION The displacement and the radiated pressure of a fluid structure interaction problem, under a turbulent boundary layer loading is obtained through the discrete equation (Eq.1) defined in section (Sec.2), and described by [9 11]. In order to compute a solution to these equations a finite element approach has been chosen. 5562 3

3.1 Finite element formulation INTER-NOISE 2016 The computation of the solution to equation (Eq.1), needs the computation of several transfer functions H v (Q, k r, k s, ω) for a large number of loading cases F (k r, k s ), in order to obtain a converged solution to the sum over (k r, k s ) featured in this expression. For that reason, the computation can not be done, in practice, through direct computation of the solution to a finite element matrix formulation: [ [ ] [ ] [ ){ } ( iω 3 I ω ]Nddl 2 M + iω C + K q = F (k r, k s ) (11) N ddl N ddl ]Nddl N ddl One way to allow the for the computation of the transfer function is through the use of a modal basis. The transfer functions are thus computed with the finite element software Code_Aster, which implements a (u, p, ϕ) formulation of the coupled fluid structure interaction problem as described by [12]. The modal basis obtained using this formulation is orthogonal in regards of the mass matrix of the problem, and the expression is symmetric, thus allowing better computation efficiency than the unsymmetric (u, p) formulation. Boundary conditions Fluid Fluid-structure interface Wall boundary conditions Cavity Clamp boundary condition Elastic solid Figure 2. Modeling of the problem under concern 3.2 Modal analysis A modal analysis is performed in order to obtain the modal basis necessary for the computation of the several harmonic analysis that must be performed. This analysis is done considering the problem described in (Fig.2) which is a finite element representation of the experimental setup, in the hydrodynamic tunnel, used to assess our results. The system under study is a plate mounted clamped-free in the cavitation tunnel of the IRENav. The plate is clamped at the leading edge and all other edges are left free, this allows for greater motion of the system than a fully clamped structure. This apparatus is modeled by an homogeneous material (linear elasticity) clamped at the upstream position, the fluid is considered at rest and it is modeled with acoustic finite elements. The boundary condition for the acoustic problem are considered to be rigid walls. Thus the finite element modal analysis can be expressed with the matrix formulation: [ [ ] [ ] [ ){ } ( iω 3 I ω ]Nddl 2 M + iω C + K q = 0 (12) N ddl N ddl ]Nddl N ddl (I, M, C, K) are respectively the impedance, mass, damping and stiffness matrix of the studied problem. 3.2.1 Acoustic boundary conditions The acoustic boundary conditions are given under the assumption of rigid walls in order to simplify the modal analysis, both the tunnel walls and the baffle supporting the plate are modelised with these boundary conditions. This is consistent with the experimental configuration, given the size of the acoustic domain studied. Therefore, there is no impedance matrix in the studied system and the problem can be reformulated as a quadratic eigenvalue problem: [ ] [ ] [ ){ } ( ω 2 M + iω C + K q = 0 (13) N ddl N ddl ]Nddl N ddl 5563 4

If the hypothesis is made that the modal response of the structure is mostly influenced by the material properties and the mass and inertial coupling due to the fluid loading, then the contribution of the radiated acoustic power to the displacement of the structure can be neglected. Thus, in the case of impedance boundary conditions for the acoustic problem, the modal analysis can be performed without taking into account the impedance matrix. The acoustic radiated pressure can then be obtained through a direct acoustic finite element computation taking into account the impedance boundary conditions with the computed plate speed value as a load case. 3.2.2 Damping In order to account for the structural damping of the experimental apparatus measurements of the response of the plate to a localized Dirac loading case were done in vacuum. The damping matrix of the structure is computed with a Rayleigh model, by elements. The Rayleigh coefficients are obtained through fitting the experimental test case in vacuum with a numerical test case with the same load in vacuum. For each element the damping is given by: c e i = αk e i + βm e i (14) The damping matrix C is constructed by assembling the elementary contributions and is thus not a linear combination of the mass M and stiffness K matrix, unlike computations done considering a global Rayleigh damping. The Rayleigh model of the damping matrix is only valid on a small frequency band, thus the fitting of the Rayleigh coefficients with the experimental data was only conducted for the frequency band considered in this study, from 0Hz to 4kHz. If the analysis was to be conducted over a larger frequency band, the use of modal damping would be a more appropriate approach. 3.3 Harmonic analysis Given the complex modal basis obtained through the analysis presented in the previous sections, several harmonic analysis are performed for the different load cases, in order to obtain all of several transfer functions H v (Q, k r, k s, ω), for each mode, appearing in the equation (Eq.1). The harmonic problem considered is the same as the problem presented for the modal analysis, thus an impedance matrix should be added to the harmonic problem. However adding an impedance matrix to the problem is of little use to the resolution as we neglect the contribution of the acoustic radiated pressure to the displacement of the structure. As there is no impedance matrix considered in the resolution of the harmonic problem, the pressure in the fluid will have to be computed through a fully acoustic computation with the displacement of the plate used as an input. The following matrix finite element problem must be solved for each load case: ( iω 3 [ I ]Nddl ω 2 [ M ] [ ] + iω C N ddl N ddl + [ K with F (k r, k s ) a unitary plane wave of wave-vector k = (k r, k s ). Expressed without the impedance matrix, the problem is thus: [ ] [ ] [ ){ } ( ω 2 M + iω C + K q N ddl N ddl ]Nddl 3.3.1 Projection of the harmonic problem ]Nddl ){ q N ddl = } N ddl = F (k r, k s ) (15) { } F (k r, k s ) In order to solve the finite element problem described by equation (Eq.16), this problem is projected on the modal basis. As the modal basis is complex, due to the damping matrix, the projections have to be computed out of the finite element software, as there is currently no possibility of performing this projection within the software used in this study. On the one hand, the matrices are projected on the modal basis with the following approach: }t [ ] } A red ij = {Φ i A {Φ j (17) Here, A is a complex matrix accounting for the matrix M, C or K and {Φ} n is a complex modal basis containing n modes. (i, j) are the row and column indexes of the matrix and [A red ] n is the reduced matrix obtained through projection of the matrix A 5564 5 (16)

On the second hand the reduced loading vector is projected on the modal basis following the same approach: }t { } F red i = {Φ i F (18) with, {F } the load vector,{φ} n the modal basis, (i) the row index and {F red } n the projected load vector 3.3.2 Computation of the transfer functions The last step in the finite element analysis is to compute the solutions of the projected harmonic problem for different load cases: ] ] ]){ } { } ( ω [M 2 red + iω [C red + [K red q = F red (k r, k s ) (19) N modes This matrix system is solved for a number N freq of frequencies and for a number N k of wave-vectors. This is done through the use of the Python function numpy.linalg.solve which computes the solution of matrix linear systems through the use of the library LAPACK s gesv resolution function. The choice of solving the linear system out of the finite element software was driven by the need of performing a large number of such computation, for all the load cases. Each computation is independent thus solving these system out of the software allows for a greater flexibility in the processing and optimization of the computations. Through the resolution of the different load cases, gives us the N freq N k reduced transfer functions. These reduced transfer functions will be used to compute the solution to the equation (Eq.8), as this equation can be expressed with these functions, thus reducing the size of the initial problem: S vv (Q, ω) = i,j b i,j (ω)φ i (Q)Φ j (Q) (20) H v (Q, k r, k s, ω) 2 = i,j a i,j (k r, k s, ω)φ i ((Q)Φ j ((Q) (21) b i,j = r,s a i,j (k r, k s, ω)s ArsA rs (ω) (22) 3.4 Displacement spectrum As shown in the previous sections the solution of equation (Eq.8) is computed without reconstructing the transfer functions in the physical space through the use of the equation (Eq.22). Only the resulting displacement auto-spectrum is reconstructed in the physical space through equation (Eq.20). However computing the N 2 modes coefficients b i,j can be time consuming, thus a different approach is used. On the first hand the displacement auto-spectrum is sought as a linear combination of the modal basis: S vv (Q, ω) = i c i (ω)φ i (Q)Φ i (Q) (23) This is a weaker formulation than the formulation given by equation (Eq.20), but it is valid as long as the modes are well separated one from another. On the second hand, the solution to the equation (Eq.8) is computed for a reduced number N r of points Q, with N r > N modes. Finally the coefficients c i (ω) are sought by solving the over constrained linear system defined by equation (Eq.23) and the N r chosen points. This involves solving the problem approximatively N modes time rather than Nmodes 2 times as it was the case with (Eq.22). 4. RESULTS OBTAINED WITH THE NUMERICAL MODEL 4.1 Modal analysis Three different modal analysis formulations are studied in the following section. These formulations are obtained from the formulations described previously, for different mesh fineness, each case is approximatively 5565 6

9000 7000 Frequency (Hz) 5000 3000 1000 23660 elements 27976 degrees of freedom 66332 elements 76411 degrees of freedom 130708 elements 148772 degrees of freedom 0 0 20 40 60 80 100 120 Numbering of the mode Figure 3. Numerical frequencies as a function of mode order twice as fine as the precedent case. The studied structure is a clamped-free plate modeling the experimental apparatus, of length 160mm, of width 80mm and of thickness 1mm. The cavity under the plate has a height of 20mm and the tunnel has a height of 90mm and a width of 180mm. A modal analysis is realized in the frequency band 0-14kHz with the finite element software and the results obtained by the three cases are compared. As shown by the analysis, the first meshing case is not fine enough to achieve convergence over the frequency band (Fig.3). The two other cases seem to converge rather well over the frequency band 0-8kHz, as the two curves are almost coincident. In the following study, the two last meshing cases will be used. Figure 4. Error between experimental and numerical frequencies As the numerical results are obtained from a model for which experimental data is given, the different results obtained are compared with the experimental data. As far as the modal analysis is concerned, the only comparison made is between the numerical and experimental modal frequencies and the error between these frequencies (Fig.4). The deformed shaped obtained numerically and experimentally are also compared and the two systems show the same behaviour. Thus the numerical model proves to be a good approximation of the experimental set-up, as far as the modal analysis is concerned. 4.2 Harmonic analysis The solution of the problem described in section (Sec.2) by the equation (Eq.8) or equation (Eq.22) need the computation of a sum over several wavevectors. The conditions under which this sum converges to the 5566 7

INTER-NOISE 2016 integral solution defined by equation (Eq.1) must be studied. The (kx, ky) space over which the wavevectors must vary to take into account the characteristics of the turbulent boundary layer wavenumber-frequency spectrum, as well as some of the characteristics of the structure such as the bending wavenumber, as the bending wavenumber for each mode gives the position of the maximum of the transfert reponse function. This study is realized through the use of the Fourier transforms of the modal deformed shapes of the structure 4.2.1 Length of the wavenumber space 10 4 k c = /U c /U 10 3 0,5k c k a = /c K mx 10 2 10 10 1 1 10 2 10 3 10 4 k Figure 5: Longitudinal bending wavenumber for each mode and convective and acoustic wavenumbers The position of the bending wavenumber of the structure for the different mode shapes is shown in (Fig.5). It is the position in the (k x, k y ) space of the maximum of the transfer function defined by the Fourier transform of the mode shapes. The position of the bending wavenumber towards the convective ridge and towards the acoustic ridge can be observed. Small order modes are mainly influenced by the convective ridge whereas high order modes are mainly in the low wavenumber region of the spectrum. The mode of order 3 is coincident with the convective ridge and the modes of order 4 and 5 are close to the ridge. 4.2.2 Transfer functions and wavenumber spectrum In order to assess the wavenumber range and discretisation chosen with the help of the spatial Fourier transform of the mode-shapes, the parameters leading to a converged numerical integration of the equation describing the problem are studied. For example, the mode of order 35 and frequency 3080Hz is investigated. The Fourier transform of the mode shape, as well as the turbulent boundary layer wavenumber spectrum obtained with the Corcos model for this frequency and the product of these two values can be plotted (Fig.6). On the plotted data, curves for 9dB and 18dB have been added, as the integration will mostly be influenced by the discretisation within these limits. Out of the 18dB limit, the integration steps can be much more loose as only a few points are necessary to perform a good integration. The choice of the 18dB limit was influenced by the behaviour of the convective ridge given by the Corcos model, it is mostly the limit of the convective ridge and gives information on the width of this ridge. For other wavenumber-frequency spectrum models, the limit should be modified, as these cases could present a different behaviour concerning the low wavenumber average level. 4.3 Displacement of the plate The displacement of the plate is computed with the intermediate meshing that was defined previously and on a modal basis containing 44 modes, in the frequency band 0-4kHz. Similar results were obtained with a modal basis containing 100 and 200 modes over the same frequency range. Therefore the modal basis chosen 5567 8

Figure 6. Mode 35 (3080 Hz) Transfer function, Corcos spectrum and product for the computation can be limited to 44 modes as the truncation of the basis doesn t impact the results over the studied frequency range. This is consistent with the fact that the the modal frequencies are well separated one from the others in the studied system and that the response of the structure is mainly influenced by the turbulent component of the boundary layer spectrum and not the acoustic component, thus reducing the modal coupling with high order modes. The mean displacement of the plate can bee seen (Fig.7), for different formulations of the wavenumberfrequency excitation spectrum. Here the spectrum chosen are the Corcos formulation, for its ease of use and the Smol yakov spectrum which is known for having a better behaviour in the low wavenumber region than the Corcos spectrum. The mean displacement of the experimental setup has been acquired in the hydrodynamic tunnel with a scanning laser Doppler vibrometer and approximatively 300 points over the structure. For the lower frequencies, results are almost the same as far as the different spectrum are concerned. This is consistent with the fact that most spectrums found in litterature show the same behaviour over the convective ridge, and that the low frequencies are mainly under the influence of the convective part of the excitation, for our studied system, as shown in section (Sec.4.2.2). For higher frequencies the two studied spectrums show different behaviours but they still keep levels close to the level obtained with the experimental setup. As far as the Corcos spectrum is concerned, levels are higher in the high frequency region and the experimental response is over estimated by the spectrum, whereas the Smol yakov spectrum underestimates the levels for higher frequencies. This behaviour is consistent with data shown in [4], taken from a genuine study and from literature, concerning the low wavenumber behaviour of the turbulent boundary layer wavenumber-frequency spectrum. As found in the data, experimental spectrum tend to be higher than the Smol yakov prediction of an average of 10dB and lower than the Corcos one, of an average of 20dB. 5. CONCLUSION The computation method presented in this paper gives a good representation of the displacement behaviour of a structure immersed in fluid, excited by a turbulent boundary layer. The method has been developed to be more exhaustive than what was previously found in the literature and takes into account the acoustics response of the plate. Thus the acoustic response can be obtained in the fluid domain, as far as the modal basis has been computed for a problem including the acoustic propagation of the fluid. If the modal basis doesn t take into account the the acoustic propagation, the hypotheses is made that there is little influence of the acoustic field on the displacement, thus the displacement of the plate can be computed, then the acoustic pressure is obtained through direct computation of the sound radiated by a vibrating plate. 5568 9

Figure 7. Mean displacement Finally, as the method has been implemented with a finite element software, it is not limited to simple geometries, where exact solution of the deformed shapes are known. Thus more complex industrials structures can be studied, as far as a description of the boundary layer turbulent spectrum is known. ACKNOWLEDGEMENTS This work has been conducted as part of a Ph.D. thesis, with financial support from the French DGA, in partnership with the DCNS company and the IRENav. The authors would like to thank the members of these different institutions who helped in the realization of this study. REFERENCES [1] J. L. Lumley and A. M. Yaglom. A century of turbulence. Flow, Turbulence and Combustion, 66(3):241 286, 2001. [2] E. Ciappi and F. Magionesi. Full scale analysis of the response of an elastic ship panel excited by turbulent boundary layers. Noise Control Engineering Journal, 57(3):179 192, 2009. [3] E. Ciappi, F. Magionesi, S. De Rosa, and F. Franco. Analysis of the scaling laws for the turbulence driven panel responses. Journal of Fluids and Structures, 32(0):90 103, july 2011. [4] W. K. Bonness, D. E. Capone, and S. A. Hambric. Low-wavenumber turbulent boundary layer wallpressure measurements from vibration data on a cylinder in pipe flow. Journal of Sound and Vibration, 329(20):4166 4180, 2010. [5] M. Esmailzadeh, A. A. Lakis, M. Thomas, and L. Marcouiller. Prediction of the response of a thin structure subjected to a turbulent boundary-layer-induced random pressure field. Journal of Sound and Vibration, 328(1-2):109 128, 2009. [6] M. Esmailzadeh and A. A. Lakis. Response of an open curved thin shell to a random pressure field arising from a turbulent boundary layer. Journal of Sound and Vibration, 331(2):345 364, 1/12 2012. [7] C. Maury, P. Gardonio, and S. J. Elliott. A wavenumber approach to modelling the response of a randomly excited panel, part i: General theory. Journal of Sound and Vibration, 252(1):83 113, 2002. [8] D. Habault and P. J. T. Filippi. Light fluid approximation for sound radiation and diffraction by thin elastic plates. Journal of Sound and Vibration, 213(2):333 374, 1998. [9] M. Aucejo, L. Maxit, and J. L Guyader. Experimental simulation of turbulent boundary layer induced vibrations by using a synthetic array. Journal of Sound and Vibration, 331(16):3824 3843, 7/30 2012. 5569 10

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