Response of an open curved thin shell to a random pressure field arising from a turbulent boundary layer

Journal of Sound and Vibration 331 (2012) 345–364

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Response of an open curved thin shell to a random pressure field arising from a turbulent boundary layer M. Esmailzadeh, A.A. Lakis n ´ cole Polytechnic of Montre´al, C.P. 6079, Succursale Centre-ville, Montre´al, Que´bec, Canada H3C 3A7 Mechanical Engineering Department, E

a r t i c l e i n f o

abstract

Article history: Received 25 February 2010 Received in revised form 4 September 2011 Accepted 4 September 2011 Handling Editor: L. Huang Available online 4 October 2011

A method is presented to predict the root mean square displacement response of an open curved thin shell structure subjected to a turbulent boundary-layer-induced random pressure field. The basic formulation of the dynamic problem is an efficient approach combining classic thin shell theory and the finite element method, in which the finite elements are flat rectangular shell elements with five degrees of freedom per node. The displacement functions are derived from Sanders’ thin shell theory. A numerical approach is proposed to obtain the total root mean square displacements of an open curved thin structure in terms of the cross spectral density of random pressure fields. The cross spectral density of pressure fluctuations in the turbulent pressure field is described using the Corcos formulation. Exact integrations over surface and frequency lead to an expression for the total root mean square displacement response in terms of the characteristics of the structure and flow. An in-house program based on the presented method was developed. The total root mean square displacements of a curved thin blade subjected to turbulent boundary layers were calculated and illustrated as a function of free stream velocity and damping ratio. A numerical implementation for the vibration of a cylinder excited by fully developed turbulent boundary layer flow was presented. The results compared favorably with those obtained using software developed by Lakis and Paı¨doussis (J. Sound Vib. 25 (1972) 1–27) using cylindrical elements and a hybrid finite element method. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Thin shells are major components in industrial structures such as skins of aircraft fuselage, hulls of ships and blades of turbines. These structures are commonly subjected to excitation forces such as turbulence, which are intrinsically random. Random pressure fluctuations induced by a turbulent boundary layer are a frequent source of excitation and can cause small amplitude vibration and eventual fatigue failure, therefore determination of the response of shell structures to these pressures is of importance. An extensive review of turbulence-induced vibration reveals that most of the studies in the literature are devoted to investigating characteristics of pressure fluctuations beneath a turbulent boundary layer. However, studies on the displacement response of shell structures to these excitations are scarce. Characteristics of a turbulent boundary layer such as wall-pressure power spectral density and cross correlation functions have been extensively studied and well established [1–9]. However, experimental and theoretical investigations of wall pressure characteristics continue to be a subject of

n

Corresponding author. Tel.: þ1 514 340 4711x4906; fax: þ1 514 340 4172. E-mail address: [email protected] (A.A. Lakis).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.09.006

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great interest and there is still much ongoing research in this field. For example, Leclercq and Bohineust [10] conducted an experimental investigation to obtain wall-pressure power spectral density in a turbulent boundary layer. A recent treatment of this problem was given by Lee et al. [11] who developed a methodology to predict the wall-pressure fluctuation spectrum based on time-mean flow characteristics of turbulent flow obtained from solving Reynolds averaged Navier–Stokes equations. Smol’yakav [12] proposed a new model for cross spectrum and wavenumber-frequency spectrum of turbulent pressure fluctuations in a boundary layer. In this paper, the Corcos model for the cross spectral density of the pressure fluctuations is adopted in order to describe the turbulent pressure field due to its simplicity compared to other wall-pressure models. Moreover, using the Corcos model we are able to integrate the root mean square response expression over frequency and space. The Corcos cross correlation model correctly captures the correlation decay rates in turbulent boundary layer flow, but is known to overestimate the wall pressure levels at low wavenumber. Early contributions for determining the structural response to a general random pressure field dealt mainly with beams and plates. For example, the linear response problem of beams and plates under stochastic loading was dealt with by Eringen [13]. He evaluated the cross correlations of displacements and stresses for a white noise random load with a constant power spectral density at all frequencies. Dyer [14] studied analysis of the vibratory response of a plate exposed to a relatively simple random pressure field. Wagner and Baht [15] studied the response of plates to the actual type of loading. They were able to calculate the mean square acceleration of the plate. Stanisic [16] developed a technique to determine the spectral density of transversal displacement of a plate. The first theoretical study of simply supported uniform thin cylindrical shells excited by turbulent boundary layer pressure fluctuations was carried out by Cottis and Jasonides [17] based on Reissner’s simplified shallow shell equations of motion. They derived a general expression for the space-time correlation function of the radial displacement of a thin cylinder with no attempt to evaluate mean square response. The first effort to make a comparison of measured and predicted simply supported thin cylindrical shell vibration when subjected to internal turbulent water flow at high frequencies was made by Clinch [18] using Powell’s joint-acceptance method. Following Clinch’s study, Lakis and Paı¨doussis [19] presented a theory using a cylindrical finite element to predict the mean square displacement response of a uniform or axially non-uniform thin cylindrical shell when subjected to an arbitrary random pressure field, or to a pressure field arising from a turbulent boundary layer caused by internal subsonic flow. Recently, innovative methods for the stochastic response of a plate or cylinder have been carried out by many researchers. Much research effort has been invested in evaluating the auto spectral density of the response due to a turbulent boundary layer pressure field, but no attempt to calculate total root mean square displacement response has been undertaken. For example, the auto spectral density of the velocity of a plate subjected to pressure fluctuations originating from a turbulent boundary layer was studied by Birgersson et al. [20] using spectral finite element and dynamic stiffness methods. Finnveden et al. [21] investigated the vibration response of the velocity of a thin plate excited by a turbulent boundary layer experimentally and numerically. Mazzoni [22] proposed a deterministic model based on an analytical wavenumber integration procedure to predict the power spectrum of the acceleration of a fluid-loaded thin plate subjected to turbulent flow. De Rosa and Franco [23] presented an exact and predictive response to evaluate the power spectral density of acceleration of a simply supported plate wetted on one side subjected to a turbulent boundary layer. Birgersson et al. [24] predicted the power spectral density of velocity of thin cylindrical shells excited by turbulent flow using Arnold–Warburton theory and the spectral finite element method; however, the effect of fluid-structure coupling was not accounted for in their analysis. In the literature survey, it appears that studies on predicting the total root mean square displacement response of thin shells to random pressure fluctuations beneath a turbulent boundary layer are few. For example, an investigation was carried out by Lakis and Paı¨doussis [19] to determine the total root mean square displacement response of cylindrical shells to turbulent flow. Their work, however, was restricted to cylindrical forms, since a cylindrical finite element was employed in order to model dynamic behavior of a cylinder in a vacuum, subjected to potential flow and turbulent flow. It is evident that cylindrical elements are unable to model curved structures such as blades. Therefore, their method is only applicable to cylindrical shells and this is the limitation of their method. A recent work [25] was carried out by the authors in order to predict the total root mean square (rms) displacement response of a thin plate to a turbulent boundary-layerinduced random pressure field, which is solely suitable to thin plates. Therefore, the current models are insufficient to determine the total root mean square displacement response of open curved thin shell structures. In the present work, we are seeking a numerical method to predict rms response of such structures to turbulent boundary layer-induced random pressure fluctuations. Here, the finite element method developed by the authors are employed to model curved thin shell structures such as blades in a vacuum, subjected to potential flow and finally to turbulent flow. However, it should be mentioned that a cylinder can be modeled using the method developed here by employing our flat rectangular shell element. It is worth mentioning that there is no reference in the literature on the total rms response of open curved thin shells; therefore, a thin cylindrical shell was analyzed in order to validate our method. To the authors’ knowledge, no exact treatment of the total root mean square displacement response of an open curved thin shell excited by a turbulent boundary layer has been performed to date. In the foregoing studies conducted on turbulent flow-induced vibration, shell structures were generally plate or cylinder and frequently some simplifying assumptions were considered in order to predict the response, such as considering unit power spectral density of wall pressure, or neglecting the effects of fluid-structure coupling. For more complex shell geometries and more realistic

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random pressure fields, developing a method to predict total root mean square displacement response is desirable. It should be mentioned that the root mean square (rms) displacement values computed for a thin shell subjected to a turbulent boundary layer excitation are more useful in certain engineering applications than the frequency spectrum. In nuclear, aerospace and other engineering applications, engineers need to know the total rms displacement values in order to calculate dynamic stress and other related components. The objective of the present work is to develop a numerical method to predict the total rms displacement response (response over significant modes of the structure) of a curved thin shell structure such as a curved blade to random pressure fluctuations arising from turbulent boundary layer. Many research studies on fluid–structure interaction problems have been devoted to modeling plates and cylinders and consequently, the developed finite elements are solely applicable to these structures. In fact, analyses of three-dimensional thin shell structures in contact with fluid are scarce in the literature. It is therefore necessary to develop an element to correctly model the behavior of curved thin shell structures in contact with fluid. The developed method is capable of predicting the dynamic behavior of thin shells and is generally applicable to three dimensional structures such as turbine blades in a vacuum, in contact with fluid and subjected to turbulent flow. From the finite element point of view, the current models are insufficient to model a curved thin structure (such as a blade) coupled with fluid. As mentioned earlier, most of the studies in the literature are devoted to investigating power spectral density of response of cylinders or plates and studies on total rms response are few such as works done by Lakis on cylindrical shells using cylindrical elements and a previous work carried out by the authors on plates. This study is concerned with the linear vibration of open curved thin shells, either uniform or non-uniform, due to a random pressure field arising from a turbulent boundary layer. The dynamic behavior of such structures in contact with fluid can be determined using a method previously developed by the authors [26] since dynamic characteristics of structures are needed as an input to determine the structural response due to turbulent flow. A general method is developed to predict the total root mean square displacement response of such shells to a turbulent random pressure field arising from a turbulent boundary layer. An expression for the rms displacement response of an open curved shell is obtained in terms of the wetted natural frequencies, mode shapes and other characteristics of the structure and flow. An in-house program was developed to predict the rms displacement response of an open curved thin shell structure such as a turbine blade to a random pressure field arising from a turbulent boundary layer. To validate our method, a thin cylindrical shell subjected to internally fully developed turbulent flow was studied and compared favorably with the results obtained using software developed by Lakis and Paı¨doussis [19]. In this analysis we assume: (1) the random pressure field is stationary, ergodic and homogeneous; (2) excitation only arises from pressure fluctuations in the turbulent boundary layer since acoustic pressure fluctuations induced by flow at moderate speeds are very small compared to turbulent pressure fluctuations; (3) the compressibility effect is negligible; and (4) flow is subsonic and Mach number is below 0.3. 2. Equation of motion The dynamic behavior of a shell subjected to external loads is governed by the following equation: ½½Ms ½Mf fd€ g þ½½Cs ½Cf fd_ g þ ½½Ks ½Kf fdg ¼ fFg,

(1)

where d, d_ and d€ are displacement, velocity and acceleration vectors of the system, respectively, Ms, Cs and Ks are, respectively, the global structural mass, damping and stiffness matrices of the shell structure, F is a vector of external forces, and Mf, Cf and Kf represent the inertial, Coriolis and centrifugal forces induced by potential flow on the structure. In Ref. [26] we have indicated how the inertial effects of a quiescent fluid interacting with a curved thin shell can be taken into account. The inertial force applied on the structure is determined by combining potential flow theory, Bernoulli’s equation and impermeability condition and is expressed as a function of the acceleration of the normal displacement of the shell (@2W/@t2). However, a flowing fluid imparts additional forces such as centrifugal and Coriolis-type forces in connection with the flow velocity on the shell. The former is proportional to U2@2W/@x2 and considered as the stiffness of the fluid and has a diminishing effect on the natural frequencies of the system. The latter is proportional to 2U@2W/@x@t and has a damping effect on vibration. The magnitudes of these forces depend on the dimensionless flow velocity ðU½ð1n2 Þr=E1=2 Þ, where U is the flow velocity, n is Poisson’s ratio, r is the fluid density and E is Young’s modulus. Therefore, when we are dealing with metal shells in the vicinity of fluid with moderate flow velocity, the effects of fluid damping and stiffness are small and can be neglected without significant loss of accuracy on the dynamic behavior of the structure. The acoustic radiation of the structure constitutes a structural dissipation of energy. The effect of this type of damping is less important than the flowing fluid damping on the structure. Such kind of damping is highly nonlinear with vibration amplitude. Accordingly, the acoustic damping has negligible effect on dynamic behavior of the shell and will also not be taken into account in this paper. Therefore, the effect of the flow on the shell can be considered the same as the effect of the stationary fluid in contact with the shell. On the other hand, in the case of high radiation modes, the acoustic damping might result in significant errors. Our theory is based on a method developed for analysis of three-dimensional curved thin structures in a vacuum and in stationary fluid [26]. In this method, global mass and stiffness matrices were determined to obtain the free vibration characteristic of such shells. The approach was developed using a combination of classic thin shell theory and finite element analysis in which the finite elements were four-noded flat rectangular elements with five degrees of freedom per

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Fig. 1. Flat rectangular shell element.

node. These degrees of freedom represent the in-plane and out-of-plane displacement components and spatial derivatives of lateral displacement (i.e. two rotations about the in-plane axes). The displacement functions were derived from Sanders’ thin shell equations. The flat shell element is formulated using previously available theories of membrane and plate bending elements. The solution of membrane displacements is assumed to be a bilinear polynomial and the normal displacement is determined from shell’s equation of motion instead of an arbitrary polynomial function. In standard finite element method, polynomials using Pascal triangle are used to define displacement functions; however, here we use an exponential form for out-of-plane displacement which is a general solution of Sanders’ thin shell equations. This specific form of displacement functions renders a more accurate finite element than the conventional finite element method. It is capable of predicting both high and low natural frequencies with high accuracy which is of great importance for calculating random response due to turbulent flow. Moreover, this method may easily be adapted to take hydrodynamic effects into account. For more details regarding obtaining the structural mass and stiffness matrices see Appendix A. The fluid pressure applied on the structure was determined by combining the velocity potential function, Bernoulli’s equation and the impermeability condition and was expressed as a function of acceleration of the normal displacement of the structure and the inertial force of the quiescent fluid. An analytical integration of the fluid pressure over the element produced the added virtual mass matrix of stationary fluid [26] (see Appendix B for more details on the fluid modeling). The geometry of the shell element used in this analysis is shown in Fig. 1. The nodal displacement at node i is defined by n oT fdi g ¼ U i V i W i @W i =@x @W i =@y , (2) where Ui and Vi are the in-plane displacements of the shell reference surface in the x- and y-directions, respectively, Wi represents the transverse displacement of the shell middle surface, and @Wi/@x and @Wi/@y are the rotations about y- and x-axes, respectively. The displacement vector d, for a shell subdivided into finite elements having n nodes has the form T

fdg ¼ ffd1 gT ,fd2 gT ,. . .,fdn gT g :

(3)

3. Decoupling of the equations of motion The shell response d(x,y,t) at any point x, y and at any time t may be expressed as a normal mode expansion in terms of the generalized coordinates vector q(t) and the modal matrix in a vacuum U(x,y) as follows: (4)

fdðx,y,tÞg ¼ ½Uðx,yÞfqðtÞg: T

Substituting the above relationship into Eq. (1), pre-multiplying by U and then adopting the viscous damping type of structural damping into the equation of motion, the uncoupled set of equations is obtained as follows: 2 q€ r þ 4pzr f r q_ r þ 4p2 f r qr ¼

1 fUgTr fFg, Mr

r ¼ 1,2,. . .,N,

(5)

where fr is the rth natural frequency in Hz, zr is the rth generalized damping ratio, {U}r is the rth mode shape, N is the number of mode shapes to be used in the analysis and M r is the rth element of the generalized mass matrix, ½M ¼ ½UT ½½Ms ½Mf ½U. Upon solving the above equation the instantaneous response at a typical node c, dc in terms of the generalized coordinates is found fdc ðx,y,tÞg ¼

N X

fUcr ðx,yÞgqr ðtÞ,

r¼1

where Ucr(x,y) is the rth mode shape at node c with qr(t) being the associated generalized coordinates.

(6)

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4. Representation of a continuous random pressure field at the nodal points The continuous random pressure field acting on the shell will be approximated by a finite set of discrete forces and moments acting at the nodal points. The shell is divided into finite elements, each of which is a rectangular flat shell element. A pressure field is considered to be acting on an area Sc surrounding the node c of the coordinates lc and dc as 0 00 0 00 shown in Fig. 2. This area Sc is delimited by the positions lc and lc with respect to the origin in the x-direction and dc and dc with respect to the origin in the y-direction. The lateral force acting at an arbitrary point, A, on the area Sc is given by (see Fig. 2) Z d00c Z l00c F A ðtÞ ¼ Pðx,y,tÞdx dy, (7) d0c

l0c

where P(x,y,t) is the instantaneous pressure on the surface. The force FA(t) acting at point A is transformed into one force and two moments acting at node c, as illustrated in Fig. 2. The external load vector in local coordinates acting at a typical point c, {F(t)}Local can be written in the following form: 9 8 0 > > 9 8 > > > > 0 > > > > > > 0 > > > > > > > > 00 R 00 > > > > R di li > > > > 0 > > > >  > > > > 0 0 Pðx , yi ,tÞdxi dyi > > > i di li = = < n 00 00 , (8) ¼  R dp R lp ðx l ÞPðx ,y ,tÞdx dy fFðtÞgLocal ¼ 0 0 > > > p p p p p p> dp lp > > > > > > > My > > > > > > > > > R 00 R 00 > > > > > > > > >  d0j l0j ðy dj ÞPðxj ,yj ,tÞdxj dyj > > Mx > > ; > : j > > dj lj > > 0 Local > > ; : 0 where Fn is the lateral force in the local z-direction, Mx and My are the moments in the local x- and y-directions acting at 00 00 00 00 0 0 0 0 00 00 00 00 0 0 0 0 node c, respectively, and li ¼ lp ¼ lj ¼ lc , li ¼ lp ¼ lj ¼ lc , di ¼ dp ¼ dj ¼ dc , di ¼ dp ¼ dj ¼ dc , lp ¼lc and dj ¼dc. The external load vector in global coordinates acting at each node, {F}Global is obtained using the relation fFgGlobal ¼ ½TfFgLocal ,

(9)

where {F}Local is the local load vector stated in Eq. (8) and [T] is the nodal transformation matrix made up of the direction cosines expressed as follows (see Ref. [27] for more details): 2 3 Lxx Lyx Lzx 0 0 0 6L 0 0 0 7 6 xy Lyy Lzy 7 6 7 6 Lxz Lyz Lzz 0 0 0 7 6 7 ½T ¼ 6 , (10) Lyy Lxy Lzy 7 0 0 7 6 0 7 6 6 0 0 0 Lyx Lxx Lzx 7 4 5 0

0

0

Lyz

Lxz

Lzz

where Lxx ¼ cosðx,xÞ denotes the cosine of the angle between two sets of local and global x coordinates. 5. Mean square response in terms of the cross spectral density of an arbitrary homogenous random pressure A random process can only be described in statistical terms. Here, we assume that the random pressure is stationary in time and homogeneous in space so the response can be expressed in terms of the cross spectral density of the pressure.

Fig. 2. Transformation of a continuous pressure field into a discrete force field acting at node c.

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Assuming that we are dealing with an ergodic process and using the correlation theorem, the mean square displacement response at node c, d2c ðxc ,yc ,tÞ, can be expressed as follows: Z 1 þ1 n d2c ðxc ,yc ,tÞ ¼ lim Dc ðxc ,yc ,f ,TÞ  Dc ðxc ,yc ,f ,TÞdf , T-1 T 0

(11)

where D and D* are the finite Fourier transforms of the nodal displacements and its complex conjugate, respectively, f is the excitation frequency in Hz and T is the period. Taking the Fourier transform of Eq. (5) and then introducing the Fourier transform of the generalized coordinates into the Fourier transform of Eq. (6), the Fourier transform of the nodal displacement, Dc, can be written as follows:   N Hr ðf Þeiyr ðf Þ X Dc ðf ,TÞ ¼ Ucr UTr Fðf ,TÞ, (12) 2 4p2 f r Mr r¼1 where 9Hr(f)9 and pffiffiffiffiffiffiffiyr(f) are the magnification and phase factors, respectively, f is the forced frequency, fr is the rth natural frequency, i ¼ 1, M r is the rth element of the generalized mass matrix, F(f,T) is the Fourier transform of the force vector and Hr(f) is the frequency response function for the rth mode defined as follows: (    )1 2   f f þ 2izr ¼ Hr ðf Þeiyr ðf Þ , (13) Hr ðf Þ ¼ 1 fr fr where zr is the rth generalized damping ratio. Introducing Eq. (12) and its complex conjugate into Eq. (11), the mean square response at node c is obtained Z N X N X   Ucr ðxg ,yg ÞUcs ðxg ,yg Þ 1 1 iðys yr Þ  d2c ðxc ,yc ,tÞ ¼ lim e Hr ðf ÞHs ðf ÞfUgTs Fn ðf ,TÞfUgTr Fðf ,TÞdf , 4 2 2 T-1 T 0 ð2 p Þ f f M M r s r¼1s¼1

(14)

r s

where Ucr and Ucs are the crth and csth elements of the modal matrix, 9Hr(f)9 and 9Hs(f)9 are the magnification factors for the rth and sth modes, yr and ys are the phase lags between the force and the response, and F(f,T) and F*(f,T) are the Fourier transforms of the force vector and its complex conjugate, respectively. For a lightly damped multi-degree-of-freedom system with well separated natural frequencies, the cross-product terms are small in comparison to the self-product terms. If we plot frequency response function against frequency for a lightly damped multi-degree-of-freedom system, we observe that Hr(f) displays pronounced peaks in the neighborhood of the natural frequencies of the system (the products of Hr(f)Hs(f) are small). Moreover, the self-product terms are always positive whereas the cross-product terms may be positive or negative depending on the sign of the product of mode shapes and therefore they can cancel each other. As clearly seen from Eq. (14), the rms response is inversely proportional to the square of the natural frequency which means that the contribution of the higher frequencies to the rms displacement response due to turbulent boundary layer pressure fluctuations decreases significantly. Therefore, the limited numbers of modes suffice to calculate the response, consequently the error due to the neglect of the cross-product terms are small. Hence, the contribution of the cross-product terms to the mean square displacement response can be ignored without significant loss of accuracy. It should be mentioned that the contribution of cross-product terms are neglected in this article in order to reduce the burden of the computation cost. However, they can readily be taken into account in the calculation and in our in-house program if desired by the researcher or if natural frequencies are very close to each other. Eq. (14) can now be rewritten as follows: Z N X 2 U2cr ðxc ,yc Þ 1 1  d2c ðxc ,yc ,tÞ ¼ lim Hr ðf Þ fUgTr Fðf ,TÞfUgTr Fn ðf ,TÞdf : (15) 2 T-1 T 4 4 0 r ¼ 1 16p f r M r Substituting the Fourier transform of the external force vector from Eq. (9) into Eq. (15) and transforming the vector by summation, the mean square response is obtained in terms of the one-sided cross spectral density of pressure. The one-sided cross-spectral density of pressure,Gpp, is defined as follows: Gpp ðxi ,yi ,xj ,yj ,f Þ ¼ lim

1

T-1 T

Pn ðxi ,yi ,f ,TÞPðxj ,yj ,f ,TÞ,

(16)

where P and P* are the Fourier transforms of the pressure and its complex conjugate, respectively. The cross spectral density of a homogenous pressure field can be expressed in terms of distances of separation xx ¼9xi xu9 and xy ¼9yi  yu9 instead of the coordinates themselves. Therefore, the one-sided cross spectral density of pressure can be written as Gpp ðxx , xy ,f Þ ¼ lim

1

T-1 T

Pðxi ,yi ,f ,TÞP n ðxu ,yu ,f ,TÞ:

(17)

The one-sided cross spectral density of the wall pressure could be described by Gpp ðxx , xy ,f Þ ¼ Gpp ðf ÞCðxx , xy ,f Þ,

(18)

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where Gpp(f) is the one-sided power spectral density of the pressure and C(xx,xy,f) is a spatial correlation function between two points at the wall with a spatial separation of xx and xy streamwise and spanwise, respectively. The terms in Eq. (18) should be determined experimentally. 6. Response to a turbulent boundary layer pressure field 6.1. Empirical data for a pressure field arising from a fully developed turbulent boundary layer In the previous section the response of a thin shell subjected to an arbitrary homogeneous random pressure field was obtained. In order to obtain numerical results the cross spectral density of wall pressure must be determined experimentally and then incorporated into the development. Here we consider the particular case where the pressure field arises from a fully developed turbulent boundary layer in subsonic flow. As stated in Eq. (18), the cross spectral density of wall pressure can be defined as the product of the power spectral density of pressure and the cross correlation function. Corcos suggested that a correlation function between two points in an oblique direction can be presented by the product of the lateral and longitudinal correlation functions [28] as follows:

Cðxx , xy ,f Þ ¼ Cðxx ,0,f ÞCð0, xy ,f Þ,

(19)

where C(xx,0,f) and C(0,xy,f) are the streamwise and spanwise spatial correlation functions, respectively. Corcos postulated that the longitudinal and lateral correlations may be presented as an exponentially decaying oscillating function in the flow direction and a simple exponentially decaying function in the cross-flow direction respectively, as follows:     Cðxx ,0,f Þ ¼ eax 2pjSxx j ei2pSxx , Cð0, xy ,f Þ ¼ eay 2p Sxy (20) f xx UC ,

f xy UC

are the Strouhal numbers, ax and ay are empirically determined constants, f is the excitation pffiffiffiffiffiffiffi frequency, xx and xy are the streamwise and spanwise distances of separation, respectively, and i ¼ 1. The convection velocity UC is assumed to be a constant given by UC ¼0.6 UN where UN is the free stream velocity. Blake recommends that ax ¼0.116 and ay ¼0.7 be used for smooth walls [29]. The power spectral density (PSD) of wall pressure beneath a turbulent boundary layer is the most significant characteristic of turbulent flow, which is similar in terms of both feature and magnitude for different structures subjected to different fluids. Fluctuating wall-pressure beneath a turbulent boundary layer for various velocities has been measured for different structures subjected to various fluids. The non-dimensional wall-pressure PSD as a function of Strouhal number has been proposed by several investigators. For example, the power spectral density of the turbulent wall pressure fluctuations on a body of revolution in a water medium was obtained by Bakewell [30] and the result was in good agreement with pressure data obtained on flat plates and in fully developed turbulent pipe flow. The general trends of wall-pressure power spectral density obtained by Bakewell et al. in pipe airflow, by Clinch in pipe water flow, by Willmarth and Wooldridge over a flat plate boundary layer, and in a straight flow channel are similar as reproduced by Au-Yang [31] and are presented for different Strouhal numbers in Fig. 3. Even though there are differences in investigations, namely fluid properties, flow velocity and even the geometry of structures, the general features and magnitude of the wall-pressure PSD are nevertheless similar as observed in Fig. 3. Therefore, it seems that the one-sided power spectral density of wall pressure proposed by Lakis and Paı¨doussis [19] based on Bakewell’s measurements in pipe airflow [1] is applicable to an open curved thin shell such as a blade. Fig. 4 depicts Bakewell’s measurements of the power spectral density versus Strouhal number as well as Lakis’ curve of best fit which is as follows [19]: where Sxx ¼

Sxy ¼

n

Gpp ðf Þ ¼ k2 r2f d U 31 ek1 d n

f =U 1

(21) 6

where Gpp(f) is the one-sided power spectral density of pressure, k1 ¼0.25, k2 ¼2  10 , rf is the density of the fluid, f is the excitation frequency in Hz, d* is the boundary layer displacement thickness and UN is the free stream velocity. While the proposed expression by Lakis and Paı¨doussis has limitations, it is representative of the TBL wall pressure spectrum and is adequate for developing a numerical method to predict the response of a curved thin shell to turbulent flow. We adapt the abovementioned expression for the wall-pressure PSD which is easy to incorporate into the rms response expressions. 6.2. Mean square response Substituting empirical data of the power spectral density of the pressure and the longitudinal and lateral correlation functions into the mean square response and taking integrations over surface and frequency, the mean square displacement response at node c, d2c , is obtained as follows: ! N 6 X X U2cr n 0 00 0 00 2 dc ðxc ,yc ,tÞ ¼ Rb ðFir ,f r , zr ,U C ,U 1 , rf , d ,li ,li ,li ,di ,di ,di Þ (22) 2 4 4 r ¼ 1 16p f r M r b¼1

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1.E-03 Chen & Au-Yang Flat platesor straight flow channels Clinch Pipe water flow

Power Spectra, Φpp(ω) / ρ2U3δ*

1.E-04

1.E-05

Willmarth & Wooldridge Flat plate boundary layer

Skudrzyk & Haddle Bakewell et al. Pipe airflow

1.E-06

1.E-07

1.E-08

1.E-09 0.01

0.1 1 10 Strouhal number, ωδ*/ U

100

Normalized power spectral density of pressure

Fig. 3. Wall-pressure power spectral densities versus Strouhal number [31].

Lakis’ curve fit

Bakewell’s line of best fit

Strouhal number Fig. 4. Bakewell measurements of wall pressure power spectral density versus Strouhal number. Solid line: Bakewell, Dashed line: Lakis’ curve fit. Reynolds number: J 100,000;  150,000; n 200,000; m 250,000; & 300,000 [19].

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where Ucr is the rth mode shape at node c, Fir is the irth element of the modal matrix, fr is the rth natural frequency in Hz, M r is the rth element of the generalized mass matrix, N is the number of mode shapes chosen for evaluation, Rb’s are given in Appendix C, zr is the rth damping ratio, UC and UN are the convection and free stream velocities, respectively, rf is the density of the fluid, d* is the boundary layer displacement thickness, li and di are the coordinates of a typical node i in the curvilinear x- and y-directions, respectively, l0i and li00 are limits of the area surrounding the node i in the curvilinear x-direction, d0i and di00 are limits of the area surrounding the node i in the curvilinear y-direction. 7. Calculations and discussions 7.1. Verification of the present method As mentioned earlier, Lakis and Paı¨doussis introduced [19] a theory to predict the response of a thin cylindrical shell to the pressure field originating from a turbulent boundary layer. They developed an in-house program to predict the rms value of the displacements of a thin cylindrical shell subjected to a fully developed turbulent boundary layer in subsonic flow. This software is a reliable computer code which has been employed for more than three decades to study dynamic behavior of thin cylindrical shells in a vacuum [32], in quiescent fluid [33,34], in flowing fluid [35] and subjected to turbulent boundary-layer-induced random pressure fields [19]. In order to validate our method, a thin cylindrical shell clamped at both ends whose length, radius and thickness are 0.664 m, 0.175 m and 0.001 m, respectively, was studied. The material properties are; Young’s modulus E¼ 2.06  1011 Pa, Poisson’s ratio n ¼ 0.3 and density rs ¼7680 kg m  3. The damping ratio of the shell is z ¼0.01 for all modes. Water is flowing through the cylinder with a density of rf ¼1000 kg m  3 and a centerline velocity (Ucl) of 25 m s  1. The cylinder is discretized into 32 and 22 flat shell elements in circumferential and longitudinal directions, respectively. The maximum rms radial displacement of the cylinder subjected to a fully developed turbulent boundary layer pressure field in subsonic flow was obtained using our method. The effects of the fluid on the shell can be considered as the effects of a stationary fluid in contact with the shell, therefore only the inertial dynamic effects of the mean flow are taken into account. Hence, the natural frequencies of the shell in a stationary fluid were employed in the calculation. To validate the calculation, the maximum rms response of the shell was obtained using the software developed by Lakis and Paı¨doussis [19], in which the cylindrical frustum was used as a finite element. The cylindrical finite element was bounded at its extremity by two circular nodes, with four degrees of freedom per node that represent three translations in axial, circumferential and radial directions, and one spatial derivative of normal displacement with respect to the axial direction. In the theory developed by Lakis, displacements of the middle surface of the shell were expressed in terms of circumferential wavenumber (i.e. n). Accordingly, the rms response can be calculated for each individual circumferential wavenumber. The total response can be obtained by summation over the circumferential wavenumbers. The cylindrical shell was subdivided into 10 identical cylindrical finite elements. The natural frequencies of the shell were calculated first and it was observed that the lowest natural frequency occurs at circumferential mode n¼5. It should be noted that the rms response is inversely proportional to the square of the natural frequency, as can be observed in Eq. (22). Due to this relationship, the contribution of total response diminishes very quickly with increasing frequency. In general, natural frequencies of a thin cylinder that correspond to the breathing mode (n ¼0) and the beam-like mode (n ¼1) are very high. These modes were therefore factored out and the calculations of the response were confined to the circumferential modes from n¼ 2 to 9, from which the approximate total response was obtained by summation. The first few natural frequencies of the fluid-filled clamped cylinder were calculated using the present method and compared with those obtained using the software developed by Lakis and Paı¨doussis [36] in Table 1. Good agreement between the present method and Lakis’ theory was found. The maximum total rms radial displacement of the aforementioned thin cylindrical shell was obtained using the present method and compared with that obtained using the method developed by Lakis and Paı¨doussis as listed in Table 2. In our proposed method, the Corcos lateral and longitudinal correlations were used, but in Lakis’ results Bakewell correlations were employed. Even though different correlations were adapted in the numerical calculations, good agreement was achieved. Table 1 Comparison of the natural frequencies of a fluid-filled cylinder. Coupled natural frequency (Hz)

Discrepancy (%)

Lakis and Paı¨doussis’ method [36] (cylindrical finite element)

Present method (flat shell element)

138.44 152.61 158.24 201.73 205.59

139.49 147.90 167.83 180.24 227.21

0.75 3.18 5.70 11.92 9.52

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Table 2 Comparison of the maximum total rms radial displacement of a thin cylindrical shell subjected to random pressure fluctuations arising from a fully developed turbulent boundary layer for Ucl ¼ 25 m s  1 and z ¼ 10  2. Max. total rms radial displacements (m)

Discrepancy (%)

Lakis and Paı¨doussis’ method [19] (cylindrical finite element)

Present method (flat shell element)

2.12533E  05

2.05761E  05

3.29

Fig. 5. First mode shapes of a blade in a vacuum and submerged in water (bounded by far-off fluid extremities) as compared to an undeformed blade.

Table 3 Comparison of the natural frequencies (Hz) of a clamped blade in a vacuum. ANSYS (SHELL63)

Present method (Flat rectangular element)

Discrepancy (%)

185.10 646.31 1142.90 2135.80 2772.90 2925.30

186.08 648.07 1144.63 2136.32 2770.90 2929.33

0.52 0.27 0.15 0.02 0.07 0.14

In order to model dynamic behavior of a thin cylindrical shell, the method developed by Lakis is more accurate since a cylindrical frustum is employed and the displacement functions of the cylindrical finite element are the exact solutions of the Sanders’ cylindrical shell theory. However, it should be mentioned that the method developed by Lakis and Paı¨doussis is applicable only to cylindrical shells. The finite element used in this study is not limited to a specific geometry; it is versatile to model sophisticated three-dimensional structures such as thin blades. 7.2. Response of a curved thin blade subjected to turbulent boundary-layer-induced random pressure fields Consider a curved thin blade which is clamped at one short side as shown in Fig. 5. The blade has a length of 0.15 m, a width of 0.1 m and a thickness of 0.005 m. The material properties are: Young’s modulus E ¼2  1011 Pa, Poisson’s ratio n ¼0.3 and density rs ¼7970 kg m  3. The boundary layer displacement thickness is assumed equal to 0.00018 m in the calculation of the response. Since the blade is slightly curved and not substantially twisted we approximate d* using Schlichting formula for the plate. In order to get a more exact value, the boundary layer displacement thickness would have to be measured experimentally or estimated with a CFD calculation. The blade is discretized into 30  40 flat rectangular shell elements. The natural frequencies of the blade in a vacuum were calculated and compared favorably with those obtained using ANSYS [37] as listed in Table 3. Natural frequencies of the abovementioned blade completely submerged in water far from fluid extremities (free surface and rigid wall are the

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355

fluid boundary conditions over and under the blade, respectively) were calculated using the flat rectangular element and the method outlined in Appendix B and Ref. [26]. The first mode shapes of the blade in a vacuum at 186.08 Hz and submerged in water far from fluid extremities at 122.10 Hz were visualized using Tecplot and compared with the undeformed blade as shown in Fig. 5. It is worth mentioning that the level of fluid over and under the blade as well as the boundary conditions of the fluid (free surface, rigid wall or elastic object) affect the dynamic behavior of the blade until a certain height of fluid is reached (50 percent of the length of the blade) [26]. Variation of the first three coupled natural frequencies of the blade investigated in the present article but having a thickness of 0.002 m in conjunction with ho1/L (height of fluid over blade-to-blade length ratio) variations where blade is far from rigid wall is demonstrated in Fig. 6. As shown in this figure, as the height of fluid over the blade increases the frequency decreases; however, this reduction ceases when the fluid height to blade length ratio reaches 50 percent. Fig. 7 illustrates the frequency variations versus variations of the ratio of the height of fluid under blade to blade length (ho2/L), while the blade is sufficiently far from the free surface. Note that when the height of fluid under the blade increases the frequency increases as well, but this increase stops when fluid level (ho2) reaches 50 percent of the blade length (L). As we observe, fluid height and its boundary conditions no longer affect the behavior of the blade when the ratio of the height of fluid to the length of the blade exceeds 50 percent. In the present calculations, a blade far from fluid extremities (ho1 and ho2 are large enough) was considered to predict the coupled natural frequencies and rms response. Comprehensive discussion on the dynamic behavior of a blade submerged in fluid can be found in the previous works carried out by the authors (see Refs. [26,27]). Table 4 shows the convergence of the natural frequencies of the system. It is observed that convergence has been reached for a mesh of 30  40 and refining the mesh beyond that does not affect the natural frequencies of the system. Therefore, a mesh of 30  40 has been used for calculation of the random response. On the basis of the proceeding analysis, the total rms displacement responses of the blade were obtained by summation over all significant modes of vibration at all nodes and then the maximum total rms response of the blade was calculated. It can be clearly seen from Eq. (22) that the rms response is inversely proportional to the square of the natural frequency. This therefore indicates that the first four natural frequencies suffice to calculate the total response for the blade studied here.

270

45

Frequency, Hz

43 42 41

250

First frequency 50%

40 39 38

Frequency, Hz

44

37

Second frequency Third frequency

230 210 190

50%

170 150

36

130

35

0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 ho1/L

0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 ho1/L

38 36 34 32 30 28 26 24 22 20 18

230 210 50%

First frequency

190 Frequency, Hz

Frequency, Hz

Fig. 6. Frequency variations of a submerged blade 0.1  0.15  0.002 m3 as a function of ho1/L variations at sufficiently large ho2.

50%

170 150 130 110 90

Second frequency Third frequency

70 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 ho2/L

0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 ho2/L

Fig. 7. Frequency variations of a submerged blade 0.1  0.15  0.002 m3 as a function of ho2/L variations at sufficiently large ho1.

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Table 4 Convergence of the coupled natural frequencies (Hz) of a blade bounded by far-off fluid extremities. Mesh 10  20

Mesh 20  30

Mesh 30  40

Mesh 35  45

123.26 425.94 754.33 1400.07 1881.42 2167.32

122.27 424.95 752.70 1399.82 1880.33 2163.40

122.10 424.64 752.40 1399.74 1880.04 2162.60

122.08 424.57 752.35 1399.72 1879.97 2162.43

Table 5 Comparison of the maximum total rms displacements (m) (U, V and W) of a thin blade subjected to turbulent boundary-layer-induced random pressure fluctuations from both sides when water is flowing along the short side of the blade for UN ¼ 16 m s  1 and z ¼10  3.

U V W

Mesh 10  20

Mesh 20  30

Mesh 30  40

Mesh 35  45

7.2056E  07 4.9851E  07 1.7063E  05

9.0393E  07 6.2037E  07 2.2161E  05

1.0855E  06 7.4023E  07 2.6964E  05

1.1807E  06 8.0318E  07 2.9444E  05

Max. total rms displacements U, V and W [m]

1.E-03

1.E-04

1.E-05

1.E-06

1.E-07

1.E-08

1.E-09 0

4

8

12

16

20

24

28

Free stream velocity [m s-1] Fig. 8. Maximum total rms displacement responses (U, V and W) versus free stream velocity and damping ratio (z) for a blade subjected to a fully developed turbulent flow from both sides, where water is flowing along the short side of the blade: U, z ¼ 10  3, V, z ¼ 10  3, W, z ¼10  3, U, z ¼10  2, V, z ¼ 10  2, W, z ¼10  2. It is observed that the total rms displacements are inversely proportional to damping ratio and directly proportional to free stream velocity.

It should be noted that the Corcos formula characterizes the wall pressure cross spectral density. Once the pressure formula is established, it could be introduced into the numerical model to predict the rms response once the convergence of the finite element method is satisfied. The effect of mesh refinement on the maximum total rms displacements of the blade subjected to a turbulent boundary layer from both sides where fluid is flowing along the short side of the blade for UN ¼16 m s  1 and z ¼10  3 is presented in Table 5. It is observed that mesh refinement beyond 30  40 has almost no considerable effect on the response and the total rms displacements differ by about 8 percent. Therefore, a mesh of 30  40 is used for further calculations. The maximum total rms displacements (U, V and W) versus free stream velocity in the range of 4–24 m s  1 and for damping ratios z ¼10  2 and 10  3 for the clamped blade subjected to turbulent boundary layer from both sides, where water is flowing along the short side of the blade are depicted in Fig. 8. The results reveal that the total rms displacements

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357

are directly proportional to the free stream velocity and inversely proportional to the damping ratio. The same shift in the rms response was observed as the damping ratio changed through all velocities. It is noted that the maximum total rms displacements are small for the set of calculations, as expected. 8. Conclusion In this paper we have presented a method to predict the total rms displacement response of an open curved thin shell structure to a random pressure field originating from a turbulent boundary layer in subsonic flow. This research is based on a method previously developed by the authors to predict the dynamic behavior of three-dimensional curved thin shells in the vicinity of fluid. The formulation of the dynamic problem is a combination of classic thin shell theory and finite element analysis in which the finite elements are four-noded flat shell elements with five degrees of freedom per node, i.e. in-plane and out-of-plane displacements and two rotations about in-plane axes. This method is capable of calculating both high and low frequencies which are of great importance for determining the response of structures subjected to turbulent random pressure fields. Dynamic characteristics of the structure in a vacuum and in contact with fluid such as mode shapes in a vacuum and wetted natural frequencies obtained using a method previously developed by the authors are incorporated into the calculation of random response. A random pressure field is estimated at each node of the finite element. Description of the turbulent pressure field is based on the Corcos formulation for the cross spectral density of pressure fluctuations. Exact integrations over surface and frequency lead to an expression for the rms displacement response in terms of the wetted natural frequency of the system, undamped mode shapes, generalized mass matrix and other characteristics of the structure and flow. An in-house program was developed to calculate the total root mean square displacement response of a thin structure to random vibration due to a turbulent boundary layer of subsonic flow. To the author’s knowledge, there is no reference in the literature for the total rms displacement of an open curved thin shell structure subjected to such random pressure. Therefore, in order to validate our method a cylindrical shell was analyzed and the total rms radial displacement was predicted and compared favorably with that obtained using software developed by Lakis and Paı¨doussis using cylindrical element and a hybrid finite element method. The maximum total rms displacements of a curved thin blade subjected to turbulent boundary-layer-induced random pressure fields were obtained and it was observed that the response is directly proportional to the free stream velocity and inversely proportional to the damping ratio. Appendix A. Derivation of structural mass and stiffness matrices A.1. Displacement functions The displacement functions were derived from Sanders’ thin shell equations. The flat shell element is formulated using previously available theories of membrane and plate bending elements. The solution of membrane displacements (U and V) is assumed to be a bilinear polynomial and the normal displacement (W) will be determined from the shell’s equation of motion instead of an arbitrary polynomial function [26] Uðx,y,tÞ ¼ C 1 þ C 2

x y xy þ C3 þ C4 , A B AB

(A.1)

Vðx,y,tÞ ¼ C 5 þC 6

x y xy þC 7 þ C 8 , A B AB

(A.2)

C j eipððx=AÞ þ ðy=BÞÞ þ iot ,

(A.3)

Wðx,y,tÞ ¼

20 X j¼9

where A and B are the blade length and width in the x- and y-directions, respectively, Cj are unknown constants, i is a complex number i2 ¼  1 and o is the natural frequency of the blade in rad s  1. The displacement functions can be written in matrix form as follows:  T U V W ¼ ½R½A1 fdg ¼ ½Nfdg, (A.4) where matrix [N] is the shape 2 x y xy 1 A B AB 6 6 60 0 0 0 ½R ¼ 6 6 6 4 0 0 0 0

function of the finite element, matrices [R] and [A]  1 are given as follows [26]: 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 7 x y xy 0 0 0 0 0 0 0 0 0 0 0 0 7 1 7 A B AB 7 7 x y x2 xy y2 x3 x2 y xy2 y3 x3 y xy3 5 0 0 0 0 1 A B 2A2 AB 2B2 6A3 2A2 B 2AB2 6B3 6A3 B 6AB3

(A.5)

358

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2

1

6 A 6 xe 6 B 6 y 6 e 6 AB 6 xe y 6 e 6 0 6 6 6 0 6 6 6 0 6 6 0 6 6 6 0 6 6 0 6 6 6 0 6 1 ½A ¼ 6 0 6 6 6 0 6 6 6 0 6 6 6 6 0 6 6 6 0 6 6 6 0 6 6 6 6 0 6 6 6 0 6 4 0

0 0

0 0

0 0

0 0

A xe

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

B ye

0

0

0

0

0

0

0

AB xe ye

0

0

0

0

AB xe ye

0

0

0

0

AB xe ye

0

0

0

1

0 0

0 0

0 0

0 0

0 A xe

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

B ye

0

0

0

0

0

0

AB xe ye

0

0

0

0

AB xe ye

0

0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

A xe B ye

0

AB xe ye

0

0

0

0

AB xe ye

0 0 0

1 0 0

0 A 0

0 0 B

0 0 0

0 0 0

0 0 0

0 0 0

0

6A2 x2e

4A2 xe

0

0

0

6A2 x2e

2A2 xe

0

0

0

0

0

0

0

0

0

0

0

AB xe ye

AB ye

AB xe

0

0

AB xe ye

0

AB xe

0

0

AB xe ye

0

0

0

0

AB xe ye

AB ye

0

6B2 y2e

0

4B2 ye

0

0

0

0

0

0

0

0

0

0

0

0

6B2 y2e

0

6A x2e

3

3

6A x2e

3

3

0

12A x3e

0

0

0

12A x3e

0

0

0

0

0

0

0

0

0

0

0

6A2 B x2e ye

4A2 B xe ye

0

0

0

6A2 B x2e ye

2A2 B xe ye

0

0

0

6A2 B x2e ye

2A2 B xe ye

0

0

0

6A2 B x2e ye

4A2 B xe ye

0

6AB2 xe y2e

0

4AB2 xe ye

0

0

6AB2 xe y2e

0

4AB2 xe ye

0

0

6AB2 xe y2e

0

2AB2 xe ye

0

0

6AB2 xe y2e

0

0

12B3 y3e

0

6B3 y2e

0

12B3 y3e

0

0

12A3 B x3e ye

6A3 B x2e ye

0

6A3 B x2e ye

0

12AB3 xe y3e

0

6AB3 xe y2e

0

0

0

0

0

0

0

12A3 B x3e ye

6A3 B x2e ye

0 0

0

0

12AB3 xe y3e

0

6AB3 xe y2e

0

0

0

0

0

0

12A3 B x3e ye

6A3 B x2e ye

0

0

0

0

0

12A3 B x3e ye

0

0

12AB3 xe y3e

0

6AB3 xe y2e

0

0

12AB3 xe y3e

3

0 0 7 7

7 7 7 0 7 7 0 7 7 7 0 7 7 7 0 7 7 0 7 7 7 0 7 7 7 0 7 7 0 7 7 0 7 7 7 0 7 7 7 2B2 7 ye 7 7 7 0 7 7 7 0 7 7 7 2AB2 7 xe ye 7 7 6B3 7 7 y2e 7 7 0 7 7 5 3 0 7

6AB xe y2e

(A.6) A.2. Kinematics relationship The strain vector in terms of displacements is given by ( @U @V @U @V @2 W þ feg ¼ @x @y @y @x @x2

@2 W @y2

2@2 W @x@y

)T (A.7)

:

Substituting the displacements, the strain vector is obtained in terms of the nodal displacements as follows: feg ¼ ½Q ½A1 fdg ¼ ½Bfdg, where matrix [Q] is as follows [26]: 2 1 y 6 0 A 0 AB 0 6 6 60 0 0 0 0 6 6 6 60 0 1 x 0 6 B AB 6 ½Q  ¼ 6 6 60 0 0 0 0 6 6 6 60 0 0 0 0 6 6 6 4 0 0 0 0 0

0

0

0

0

1 B

1 A

0

x AB y AB

0

0

0 0

(A.8)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

x

y

2

0

0

0

0

0

0

0

0

0

x

y B3

0

0

0

0

0

0

2 AB

AB2 2y

1 A

1 B2 0

3

2

A

A B

0

0

0

2x 2

A B

AB

2

0

xy 3

A B 0 x2 A3 B

3 0 7 7 7 0 7 7 7 7 0 7 7 7 7 7 0 7 7 7 xy 7 7 3 7 AB 7 7 y2 5

(A.9)

AB3

A.3. Structural mass and stiffness matrices The rigidity and mass matrices of a single rectangular finite element in its local coordinates are given by finite element theory Z Z ½BT ½P½BdA, (A.10) ½Ke ¼ A

M. Esmailzadeh, A.A. Lakis / Journal of Sound and Vibration 331 (2012) 345–364

½Me ¼ rs h

Z Z

½NT ½NdA,

359

(A.11)

A

where rs is the density of shell, h is the shell thickness, [P] is the elasticity matrix and dA is the element surface area. Substituting matrices [B] and [N], the structural mass and stiffness matrices can be determined analytically by exact integrations over x and y as follows: Z ye Z xe  ½Ke ¼ ½AT ½Q T ½P½Q dxdy ½A1 , (A.12) 0



½Me ¼ ½AT rs h

0

Z

ye 0

Z

xe

 ½RT ½Rdxdy ½A1 ,

(A.13)

0

where xe and ye are the element dimensions in the x- and y-directions, respectively as shown in Fig. 1. The structural mass and stiffness matrices in global coordinates can be computed using the transformation matrix given in Eq. (10). Appendix B. Effect of the fluid on the dynamic behavior of the structure B.1. Fluid–structure interaction The dynamic behavior of a structure is affected by the presence of fluid. The fluid pressure applied on a structure can be expressed as a function of displacement, velocity and acceleration. These forces are interpreted as centrifugal, Coriolis and inertial effects of the fluid, respectively. The fluid force matrices are superimposed onto the structural matrices to form the dynamic equations of a coupled fluid–structure system. The dynamic behavior of a shell subjected to a potential flow can be represented as follows: ½½Ms ½Mf fd€ g þ ½½Cs ½Cf fd_ g þ½½Ks ½Kf fdg ¼ f0g,

(B.1)

where d, d_ and d€ are displacement, velocity and acceleration vectors of the entire structure, respectively, Ms and Ks are, respectively, the structural mass and stiffness matrices of the shell structure determined in Appendix A, Cs is the structural damping taken to be proportional to the mass and stiffness matrices of the system and Mf, Cf and Kf represent the inertial, Coriolis and centrifugal forces induced by potential flow on the structure. The magnitude of the two latter forces changes directly with flow velocity and fluid density, and inversely with Young’s modulus of the structure. Therefore, when we are dealing with metal shells in the vicinity of fluid with flow velocity in the normal engineering range, the effects of fluid damping and stiffness are small and can be neglected without significant loss of accuracy on the dynamic behavior of the structure. The effects of the flow on the shell can therefore be considered the same as the effects of a stationary fluid in contact with the shell. The global equations of motion of a shell can be reduced to the following system: ½½Ms ½Mf fd€ g þ ½Cs fd_ g þ ½Ks fdg ¼ f0g:

(B.2)

B.2. Derivation of fluid added virtual mass matrix In Ref. [26] we have indicated how the inertial effects of a quiescent fluid on a shell submerged in a stationary fluid can be taken into account. The salient point is given here. The inviscid incompressible flow is modeled using linear potential flow. The velocity potential function f satisfies the Laplace equation throughout the fluid domain:

r2 f ¼

@2 f @2 f @2 f þ 2 þ 2 ¼ 0: @x2 @y @z

(B.3)

The dynamic condition at the interface of a stationary fluid and shell can be determined using Bernoulli’s equation as follows: P9at the wall ¼ rf

@f : @t

(B.4)

The impermeability condition ensures permanent contact at the interface of the fluid and shell, stating that the fluid normal velocity at the interface matches the instantaneous rate of change of the normal displacement of the shell  @ f @W : (B.5) ¼ @t @z at the wall A free surface and rigid wall are considered as fluid boundary conditions for a submerged blade, respectively, as follows (see Fig. B1):   @ f 1 @2 f @ f ¼  , ¼ 0: (B.6) g @t 2 @z z ¼ h1 @z z ¼ hh2

360

M. Esmailzadeh, A.A. Lakis / Journal of Sound and Vibration 331 (2012) 345–364

Fig. B1. A submerged blade.

where h1 and h2 are the heights of fluid over and under an arbitrary element in direction normal to the element, respectively, as shown in Fig. B1, h is the thickness of blade and g is the gravitational acceleration. Putting all of these together, the fluid pressure applied on each element is determined by combining the velocity potential function, Bernoulli’s equation, the impermeability condition and the boundary conditions of the fluid and is expressed as a function of acceleration of the normal displacement of the structure and the inertial force of the quiescent fluid as follows [26]:   1e2mh1 1þ e2mh2 @2 W @2 W P ¼ rf ¼ rf Z f 2 , (B.7)  2 2 m h 2 m h mð1 þe 1 Þ mðe 2 1Þ @t @t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the coefficient Zf depends on the fluid–structure contact model and m ¼ ð1=pÞ ð1=A2 Þ þð1=B2 Þ. The fluid-induced force vector can be obtained by performing the following integration: 8 9 > Z Z <0> = fFge ¼ ½NT fPgdA ¼ ½NT 0 dA, (B.8) > A A : > ; P where [N] is the shape function matrix of the finite element expressed in Appendix A and {P} is a vector expressing the pressure applied by the fluid on the shell element and dAis the fluid–structure interface area of the element. By introducing the displacement functions and pressure, the fluid force vector of a stationary fluid is obtained as follows: Z ye Z xe  ½RT ½Rf dxdy ½A1 fd€ g ¼ ½Mf e fd€ g, (B.9) fFge ¼ rf Z f ½AT 0

0

where [Mf]e is the fluid added virtual mass matrix of the finite element, matrices [R] and [A]  1 are given in Appendix A and [Rf] is as follows [26]: 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6  60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 7 Rf ¼ 6 (B.10) 7 2 2 3 2 2 3 3 x y x xy y x x y xy y x y xy3 5 4 0 0 0 0 0 0 0 0 1 2 2 3 2 2 3 3 3 A B 2A AB 2B 6B 6A 2A B 2AB 6A B 6AB The fluid mass matrix in global coordinates can be computed using the transformation matrix given in Eq. (10). Appendix C. Statements used in the rms response (Eq. (22))

Rb ¼

n X n X g ¼1q¼1

þ

n X n X g ¼1k¼1

þ





Fgr Fqr LLzq xq ðGF n F n Þgq  þ

n X n X g ¼1v¼1

n X n X





Fgr For LLzo yo ðGF n F n Þgo  þ

g ¼1o¼1

n X n X





Fgr Fur LLzu zu ðGF n F n Þgu 

g ¼1u¼1

n X n n X n   X   X   Fgr Fkr LLyk yk ðGF n My Þgk  Fgr Fkr LLxk yk ðGF n Mx Þgk  Fgr Fvr LLyv xv ðGF n My Þgv  g ¼1k¼1

g ¼1v¼1

n X n n X n   X   X   Fgr Fvr LLxv xv ðGF n Mx Þgv  Fgr Fhr LLyh zh ðGF n My Þgh  þ Fgr Fhr LLxh zh ðGF n Mx Þgh , (C.1) g ¼1h¼1

g ¼1h¼1

M. Esmailzadeh, A.A. Lakis / Journal of Sound and Vibration 331 (2012) 345–364

361

where L is a generic form for direction cosines defined as follows: if if if If

b¼1, then g ¼m, L ¼ Lzm xm b¼2, then g ¼w, L ¼ Lzw yw b¼3, then g ¼i, L ¼ Lzi zi b ¼4 to 6 Rb’s are as follows: n X n X

R4 ¼



j¼1q¼1

Fjr For Lxj yj Lzo yo

 ðGF

n X n X

n X n X

j¼1v¼1

j¼1v¼1



n X n X



Fpr Fqr Lyp xp Lzq xq ðGF n My Þpq  þ





p¼1u¼1

p¼1k¼1

p¼1k¼1

p¼1v¼1

p¼1v¼1

n X n n X n   X   X   Fpr Fvr Lyp xp Lxv xv ðGMx My Þpv  þ Fpr Fhr Lxp xp Lxv zv ðGMx Mx Þph  Fpr Fhr Lyp xp Lxv zv ðGMx My Þph  p¼1h¼1



n X n X

Fpr Fhr Lxp xp Lyv zv

 ðGM

n X n X



  þ

e¼1k¼1 n X n X e¼1k¼1 n X n X



n X n X





Fer Fqr Lxe ze Lzq xq ðGF n Mx Þeq 

e¼1q¼1

n X n X





Fer For Lye ze Lzo yo ðGF n My Þeo 

e¼1o¼1

n X n n X n   X   X   Fer For Lxe ze Lzo yo ðGF n Mx Þeo  Fer Fur Lye ze Lzu zu ðGF n My Þeu  þ Fer Fur Lxe ze Lzu zu ðGF n Mx Þeu 

n X n X e¼1o¼1 n X n X

(C.3)

p¼1h¼1

Fer Fqr Lye ze Lzq xq ðGF n My Þeq  þ

e¼1q¼1

p¼1h¼1

n X n  X   Þ þ Fpr Fhr Lyp xp Lyv zv ðGMy My Þph  x M y ph

p¼1h¼1





p¼1o¼1

p¼1u¼1

p¼1v¼1

þ



Fpr For Lyp xp Lzo yo ðGF n My Þpo 

n X n n X n   X   X   Fpr Fkr Lxp xp Lxv yv ðGMx Mx Þpk  þ Fpr Fvr Lyp xp Lyv xv ðGMy My Þpv  þ Fpr Fvr Lxp xp Lxv xv ðGMx Mx Þpv 

n X n X

R6 ¼ 

n X n X



n X n n X n   X   X   Fpr Fkr Lyp xp Lyk yk ðGMy My Þpk  þ Fpr Fkr Lxp xp Lyk yk ðGMx My Þpk  þ Fpr Fkr Lyp xp Lxk yk ðGMx My Þpk 

p¼1k¼1

2



Fpr Fqr Lxp xp Lzq xq ðGF n Mx Þpq 

p¼1q¼1

p¼1k¼1 n X n X

(C.2)

n X n n X n   X   X   Fpr For Lxp xp Lzo yo ðGF n Mx Þpo  Fpr Fur Lyp xp Lzu zu ðGF n My Þpu  þ Fpr Fur Lxp xp Lzu zu ðGF n Mx Þpu 

p¼1o¼1 n X n X

j¼1h¼1

j¼1v¼1

p¼1q¼1

þ

j¼1v¼1

n X n   X   Fjr Fhr Lxj yj Lyh zh ðGMx My Þjh  Fjr Fvr Lyj yj Lyv xv ðGMy My Þjv 

j¼1h¼1

n X n X

j¼1k¼1

n X n n X n   X   X   Fjr Fhr Lyj yj Lyh zh ðGMy My Þjh  Fjr Fvr Lxj yj Lxv xv ðGMx Mx Þjv  þ Fjr Fhr Lyj yj Lxh zh ðGMx My Þjh 

n X n X

n X n X

j¼1u¼1

j¼1k¼1

j¼1h¼1

R5 ¼ 



n X n n X n   X   X   Fjr Fhr Lxj yj Lxh zh ðGMx Mx Þjh  þ Fjr Fvr Lyj yj Lxv xv ðGMx My Þjv  þ Fjr Fvr Lxj yj Lyv xv ðGMx My Þjv 

n X n X

þ



Fjr For Lyj yj Lzo yo ðGF n My Þjo 

j¼1o¼1

j¼1u¼1

j¼1h¼1



n X n X

n X n n X n X   X     Fjr Fkr Lyj yj Lyk yk ðGMy My Þjk  þ Fjr Fkr Lxj yj Lxk yk ðGMx Mx Þjk 2 Fjr Fkr Lyj yj Lxk yk ðGMx My Þjk 

j¼1k¼1





n X n n X n  X   X   Þ þ Fjr Fur Lyj yj Lzu zu ðGF n My Þju  Fjr Fur Lxj yj Lzu zu ðGF n Mx Þju  n M x jo

j¼1o¼1

þ



Fjr Fqr Lxj yj Lzq xq ðGF n Mx Þjq  þ

j¼1q¼1

n X n X



n X n X



Fjr Fqr Lyj yj Lzq xq ðGF n My Þjq 

e¼1u¼1

e¼1u¼1

n X n n X n   X   X   Fer Fkr Lye ze Lyk yk ðGMy My Þek  þ Fer Fkr Lxe ze Lyk yk ðGMx My Þek  þ Fer Fkr Lye ze Lxk yk ðGMx My Þek  e¼1k¼1

e¼1k¼1

n X n n X n   X   X   Fer Fkr Lxe ze Lxv yv ðGMx Mx Þek  þ Fer Fvr Lye ze Lyv xv ðGMy My Þev  Fer Fvr Lxe ze Lyv xv ðGMx My Þev  e¼1v¼1

e¼1v¼1

n X n n X n   X   X   Fer Fvr Lye ze Lxv xv ðGMx My Þev  þ Fer Fvr Lxe ze Lxv xv ðGMx Mx Þev  þ Fer Fhr Lye ze Lyv zv ðGMy My Þeh 

e¼1v¼1 n X n X e¼1h¼1

e¼1v¼1 n X n X     Fer Fhr Lxe ze Lxv zv ðGMx Mx Þeh 2 Fer Fhr Lye ze Lxv zv ðGMx My Þeh 

e¼1h¼1

(C.4)

e¼1h¼1

where Fir is the irth element of the modal matrix, n is the number of nodes, Lxi xi , Lyi yi and Lzi zi denotes the cosines of the angle between two sets of the local and global x, y and z-coordinates of node i, respectively, and GF n F n , GF n Mx , GF n My , GMx Mx , GMy My and GMx My are complicated functions, too long to give here. Here, only GF n Mx is given in detail. Note that the indices

362

M. Esmailzadeh, A.A. Lakis / Journal of Sound and Vibration 331 (2012) 345–364

m and q are associated with forces in the x-direction, w and o with forces in the y-direction, i and u with lateral forces in the z-direction, j and k with moments about x, p and v with moments about y and e and h with moments about z. 9 8  0 0 0 0 00 00 0 0 0 00 0 0 00 0 0 0 > > ðB2 A2 Þ F c5 ðli ,lk ,di ,dk Þ þ F c5 ðli ,lk ,di ,dk ÞF c5 ðli ,lk ,di ,dk ÞF c5 ðli ,lk ,di ,dk Þ > > > > > >  s 0 0 0 0 > > 0 0 s 00 00 0 0 s 0 00 0 0 s 00 0 > > > > ð2ABÞ F 5 ðli ,lk ,di ,dk Þ þ F 5 ðli ,lk ,di ,dk ÞF 5 ðli ,lk ,di ,dk ÞF 5 ðli ,lk ,di ,dk Þ > > > > > >  > 00 00 00 00 > 2 2 c 0 0 c 00 00 00 00 c 0 00 00 00 c 00 0 > > > þðB A Þ F ðl ,l ,d ,d Þ þF ðl ,l ,d ,d ÞF ðl ,l ,d ,d ÞF ðl ,l ,d ,d Þ > > k k k k > 5 i k i 5 i k i 5 i k i 5 i k i >  s 0 0 00 00 > > > 00 00 00 00 0 00 00 00 00 0 00 00 > s s s = < ð2ABÞ F 5 ðli ,lk ,di ,dk Þ þ F 5 ðli ,lk ,di ,dk ÞF 5 ðli ,lk ,di ,dk ÞF 5 ðli ,lk ,di ,dk Þ > K2  ðGF n Mx Þik ¼ 3 2 2 2 > ðB2 A2 Þ F c ðl0 ,l0 ,d00 ,d0 Þ þ F c ðl00 ,l00 ,d00 ,d0 ÞF c ðl0 ,l00 ,d00 ,d0 ÞF c ðl00 ,l0 ,d00 ,d0 Þ > > C ðA þ B Þ > k k k k 5 i k i 5 i k i 5 i k i 5 i k i > > > >  s 0 0 00 0 > 00 0 > s 00 00 00 0 s 0 00 00 0 s 00 0 > > > > ðl ,l ,d ,d Þ þ F ðl ,l ,d ,d ÞF ðl ,l ,d ,d ÞF ðl ,l ,d ,d Þ þ ð2ABÞ F > > k k k k 5 i k i 5 i k i 5 i k i 5 i k i > > >  c 0 0 0 00 > > > 00 00 0 00 0 00 0 00 00 0 0 00 2 2 c c c > > > > A Þ F ðl ,l ,d ,d Þ þF ðl ,l ,d ,d ÞF ðl ,l ,d ,d ÞF ðl ,l ,d ,d Þ ðB i k i k i k i k i k i k i k i k 5 5 5 5 > > > >  > > > ; : þ ð2ABÞ F s5 ðl0i ,l0k ,d0i ,d00k Þ þ F s5 ðl00i ,l00k ,d0i ,d00k ÞF s5 ðl0i ,l00k ,d0i ,d00k ÞF s5 ðl00i ,l0k ,d0i ,d00k Þ > 9 8  c 0 0 0 0 0 0 0 2 2 c 00 00 0 0 c 0 00 0 0 c 00 0 > > > > > > ðB A Þðdk dk Þ F 4 ðli ,lk ,di ,dk Þ þ F 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk Þ > > > > 0 0 0 0 0 s 0 0 s 00 00 0 0 s 0 00 0 0 s 00 0 > > > > d Þ F ðl ,l ,d ,d Þ þ F ðl ,l ,d ,d ÞF ðl ,l ,d ,d ÞF ðl ,l ,d ,d Þ ð2ABÞðd k > > 4 i k i k 4 i k i k 4 i k i k 4 i k i k k > > > >  > 00 00 00 00 00 > 2 2 c 0 0 c 00 00 00 00 c 0 00 00 00 c 00 0 > > > þðB A Þðdk dk Þ F 4 ðli ,lk ,di ,dk Þ þ F 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk Þ > > > > >  > > > > = < ð2ABÞðd00k dk Þ F s4 ðl0i ,l0k ,d00i ,d00k Þ þ F s4 ðl00i ,l00k ,d00i ,d00k ÞF s4 ðl0i ,l00k ,d00i ,d00k ÞF s4 ðl00i ,l0k ,d00i ,d00k Þ K2  c 0 0 00 0 þ 2 2 0 00 00 00 0 0 00 00 0 00 0 00 0 2 2 c c c ðB A Þðdk dk Þ F 4 ðli ,lk ,di ,dk Þ þF 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk Þ > > > C ðA þ B2 Þ2 > > > > >  > > 0 0 0 00 0 00 00 00 0 0 00 00 0 00 0 00 0 > > > > þð2ABÞðdk dk Þ F s4 ðli ,lk ,di ,dk Þ þ F s4 ðli ,lk ,di ,dk ÞF s4 ðli ,lk ,di ,dk ÞF s4 ðli ,lk ,di ,dk Þ > > > > > >  > > 00 0 00 0 00 2 2 c 0 0 c 00 00 0 00 c 0 00 0 00 c 00 0 > > > > ðB A Þðd d Þ F ðl ,l ,d ,d Þ þ F ðl ,l ,d ,d ÞF ðl ,l ,d ,d ÞF ðl ,l ,d ,d Þ k k 4 i k i k 4 i k i k 4 i k i k 4 i k i k > > > >  s 0 0 0 00 > > 00 00 00 0 00 0 00 0 00 00 0 0 00 s s s > > ; : þð2ABÞðdk dk Þ F 4 ðli ,lk ,di ,dk Þ þ F 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk ÞF 4 ðli ,lk ,di ,dk Þ

0

(C.5)

00

where li and di are the coordinates of a typical node i in the curvilinear x- and y-directions, respectively, li and li are limits 0 00 of the area surrounding node i in the curvilinear x-direction, di and di are limits of the area surrounding node i in the curvilinear y-direction (see Fig. 2). Appendix D. Functions used in Appendix C

Z

F c4 ðli ,lu ,di ,du Þ ¼ 0 0

0

0

1 0

0 0 0 0 0 0   e½K 1 þ C jdi du j þ Bjli lu jf cosðAli lu f Þ p 1 g1 g 1 g3 g 3 " # ! df ¼ 5e sin g 5e sin g þ ðe cos g þe cos g Þ , 2 4 2 4 3 4 2 zr 8f r 4zr 2 2 4 f þ þ 1 f f 4 2 fr fr (D.1)

F s4 ðli ,lu ,di ,du Þ ¼ 0 0

0

0

Z

1 0

0 0 0 0 0 0   e½K 1 þ C jdi du j þ Bjli lu jf sinðAli lu f Þ p 1 g1 g 1 g3 g 3 " # ! df ¼ 5e cos g þ5e cos g þ ðe sin g þe sin g Þ , 2 4 2 4 3 2 4 zr 8f r 4zr 2 2 4 f þ þ 1 f f 4 2 fr fr (D.2)

F c5 ðli ,lu ,di ,du Þ ¼ 0 0

0

0

Z

1

0

0 0 0 0 0 0   e½K 1 þ C jdi du j þ Bjli lu jf cosðAli lu f Þ p 1 g1 g1 g 3 g 3 " # ! df ¼ 6e sin g 6e sin g þ ðe cos g þ e cos g Þ , 2 4 2 4 4 4 2 zr 8f r 4zr 2 2 5 f þ þ1 f f 4 2 fr fr (D.3)

F s5 ðli ,lu ,di ,du Þ ¼ 0 0

0

0

Z

1 0

0 0 0 0 0 0    e½K 1 þ C jdi du j þ Bjli lu jf sinðAli lu f Þ p 1 g 1 g 1 g 3 g3 " # df ¼ 6e cos g þ 6e cos g þ e sin g þ e sin g , 2 4 2 4 4 2 4 zr 8f r 4zr 2 2 5 f þ ð Þf þ 1 f 4 2 fr fr (D.4)

and 









g1 ðl0i ,l0u ,d0i ,d0u Þ ¼ K 1 þ C d0i d0u  þ ðAzr þ BÞl0i l0u  f r , 









g2 ðl0i ,l0u ,d0i ,d0u Þ ¼ K 1 zr C zr d0i d0u  þðABzr Þl0i l0u  f r ,

(D.5) (D.6)

M. Esmailzadeh, A.A. Lakis / Journal of Sound and Vibration 331 (2012) 345–364









363



g3 ðl0i ,l0u ,d0i ,d0u Þ ¼ K 1 þ C d0i d0u ðAzr BÞl0i l0u  f r ,

(D.7)



(D.8)









g4 ðl0i ,l0u ,d0i ,d0u Þ ¼ K 1 zr þC zr d0i d0u  þ ðA þBzr Þl0i l0u  f r , where the constants are given as follows: A¼

2p , UC

B¼

ax 2p UC

,

C¼

ay 2p UC

n

,

K1 ¼

k1 d , U1

K 2 ¼ k2 r2f d U 31 , n

0

00

where f is the forced frequency in Hz, fr is the rth natural frequency in Hz, zr is the rth damping ratio, li and li are the limits 0 00 of the area surrounding node i in the curvilinear x-direction, di and di are the limits of the area surrounding node i in the curvilinear y-direction, UC and UNare the convection and free stream velocities, respectively, rf is the density of the fluid, d* is the boundary layer displacement thickness, k1 ¼0.25, k2 ¼2  10  6, ax ¼0.116 and ay ¼0.7. References [1] H.P. Bakewell, G.F. Carey, J.J. Libuka, H.H. Schloemer, W.A. Von Winkle, Wall pressure correlations in turbulent pipe flow U. S. Navy Underwater Sound Laboratory, New London, CT. Report No. 559, 1962. [2] G.M. Corcos, Structure of turbulent pressure field in boundary-layer flows, Journal of Fluid Mechanics 18 (3) (1964) 353–378. [3] W.W. Willmarth, C.E. 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