Dynamic response of pipes conveying two-phase flow ...

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Abstract The dynamic behavior of pipes subjected to internal gas-liquid two-phase flow has been studied using the Timoshenko beam model and the slip-ratio.

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Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model Tianqi Ma · Jijun Gu · Menglan Duan

Received: date / Accepted: date

Abstract The dynamic behavior of pipes subjected to internal gas-liquid two-phase flow has been studied using the Timoshenko beam model and the slip-ratio factor model. In this paper, the governing equations were carried out using the generalized integral transform technique (GITT) by transforming the governing partial differential equations into a set of secondorder ordinary differential equations. The comparison between Timoshenko beam model and Euler-Bernoulli beam model has been conducted through parametric study on dimensionless frequencies and amplitudes over various aspect ratios, internal fluid flow rates and volumetric gas fractions. The results show that the frequencies of Timoshenko beam model is less and the amplitude is larger than that of Euler-Bernoulli beam model at low aspect ratio. In addition, the amplitude for Timoshenko beam model increases more dramatically than that of Euler-Bernoulli beam model when the pipe is going to lose stability. The high flow rate leads to the divergence of the dynamic system as well as the two phase flow accelerates the instability and has significant influence on the dynamic response when the pipe is long and the internal liquid flows fast. Jijun Gu Tel.: +0086-10-89731669 Fax: +0086-10-89731669 E-mail: [email protected] Tianqi Ma Department of Mechanics and Engineering Science, Fudan University, Shanghai, 200433, China Jijun Gu College of Mechanical and Transportation Engineering, China University of Petroleum-Beijing, Beijing, 102249, China Menglan Duan Offshore Oil/Gas Research Center, China University of Petroleum-Beijing, Beijing, 102249, China

Keywords Pipe-conveying two-phase flow · Fluidstructure interaction · Internal flow · Timoshenko beam model

1 Introduction Pipe subjected to internal flow is an essential part in ocean engineering and many other areas, since the subsea pipelines or risers are used to transport the oil and gas during offshore exploration. The coupling effects between fluid and structure often deflects the pipe, causes pipe vibration and even rupture. The dynamic behavior of pipes conveying fluid should be completely analysed, including natural frequency, mode shape and amplitude, to design structures and guide operation conditions. The first serious study of the dynamics of pipes conveying fluid is due to Bourrires [2], who derived the linearized motion equations of pipe conveying fluid and experimentally examined the flutter instability of a cantilevered pipe. A plenty of studies have been conducted in the past decades to investigate the vibration behavior of pipes excited by internal flowing fluid. The fluidelastic instability behavior of flexible pipes conveying internal flow was well studied by Pa¨ıdoussis [16], who also reviewed the basic dynamics of pipes conveying fluid and established why this system is a model dynamical problem[17]. Ibrahim[10][11] studied mechanics of pipes conveying fluid extensively, involving various aspects of the dynamic characteristics. Sinha et al.[22] used non-linear optimization method involving the limited measured responses together with nite element model to predict the excitation forces acting all along the pipe conveying uid. Huang et al.[9] investigated the natural frequency of fluid conveying pipeline with different boundary conditions by elimi-

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nated element Galerkin method. Gu et al.[6] analyzed dynamic response of pipes conveying fluid by generalized integral transform technique (GITT). Although the dynamics of pipes conveying single-phase flow had been well studied, the dynamic behavior of pipes subjected to multiphase internal flow need further research. The multiphase flow may lead to differences in material properties, phase change process and the excessive turbulence of air-water mixtures. To date, there has been very few research conducted on the pipes in gasliquid two-phase flow to investigate the vibration behavior. Pettigrew and Taylor[20] carried out some surveys on two-phase flow-induced vibration including dynamic parameters and vibration excitation mechanisms, such as fluidelastic instability, phase-change noise, and random excitation. Monette and Pettigrew[15] proposed a modified two-phase mode and conducted excellent theoretical and experimental study of the dynamics of cantilevered pipes conveying air-water mixtures downwards. Cargnelutti et al.[3] studied the two-phase flow induced forces on bends in small scale pipes. Zhang and Xu[26] conducted experiment on wall vibrations of pipe conveying flow with injected uniform bubble cloud and measured the vibrations for different bubble void fractions and averaged bubble sizes. The existence of bubble enhances the wall vibrations which mainly depend on void fraction. Pontaza and Menon [21] presented a flow-induced vibration screening procedure based on the 3-D numerical simulation of unsteady internal multi-phase flow in subsea well jumpers, predicted the flow-induced forces in flow-turning elements, and the structural response. An and Su [1] adopted generalized integral transform technique (GITT) to investigate the dynamic behavior of pipes conveying gasliquid twophase flow and analyzed the effects of the volumetric gas fraction and the volumetric flow rate on the dynamic behavior of pipes conveying air-water two-phase flow. In the foregoing, the pipe conveying internal flow are based on Euler-Bernoulli beam model, which has assumptions that the pipe is sufficiently slender and the shear force on the section doesnt cause any shearing strain, the wavelength of deformation is sufficiently long for the model to be acceptable. This is conditional even may be incorrect actually when Euler-Bernoulli beam theory is applied to short pipes or to the study of high-mode dynamical behaviour of long pipes. The Timoshenko beam model, where transverse shear strain is no longer zero but a constant, modifies the EulerBernoulli beam model. This theory can be applied to study the dynamics of articulated pipes in the limit of a very large number of articulations and also be applicable to continuously flexible short pipes, as well as

Tianqi Ma et al.

for obtaining the dynamical behavior of long pipes in their higher modes. In the literature, Timoshenko beam model have been attracted by some researchers. It was first applied to the study of dynamics of pipes conveying fluid by Pa¨ıdoussis and Laithier [19], who derived the equations of motion by Newtonian method and solved the equation by finite difference and variational techniques. The motion equation was rederived by Laithier and Pa¨ıdoussis[13] via Hamiltons principle. Then the Timoshenko beam model theory has been used for more dynamical analysis of pipe conveying fluid. Li et al.[14] adopted Timoshenko beam theory to model pipes conveying fluid and deduced the dynamic stiffness for the free vibration. The first three natural frequencies of a three span pipe were calculated through using the proposed method. Zhai et al.[25] established the dynamic equation of Timoshenko pipe via the finite element method and determined the dynamic response of fluid-conveying Timoshenko pipes under random excitation via the pseudo excitation method in conjunction with the complex mode superposition method. Yu et al.[24] analyzed vibration band gap in a periodic fluidconveying pipe system based on the Timoshenko beam model. Gu et al.[7] studied the effect of aspect ratio on the dynamic response of a fluid-conveying pipe using the Timoshenko beam model. However, the applicability of Timoshenko beam model theory to pipes conveying gas-liquid two-phase internal flow has not been studied especially for the short pipes. Hence, it is necessary to improve the study of the dynamic behavior of pipes conveying two-phase flow by means of Timoshenko beam model and compare the results with that of Euler-Bernoulli beam model.

In the present study, the Timoshenko beam model with two-phase flow model was adopted to model pipes conveying gas-liquid flow, which is described in the following section with coupled partial differential equations of the transverse vibration formulated. The third section solved the equations by implementing integral transform and obtained the semi-analytical numerical solution, which included the lateral deflection, natural frequencies at different mode. Subsequently, the numerical results based on Euler-Bernoulli beam model and Timoshenko beam model were compared comprehensively with parametric study on natural frequencies and amplitudes for various aspect ratios, volumetric flow rates and volumetric gas fractions. Conclusions and recommendations are outlined in the last section finally.

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model

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2 Model description

(4)

2.1 Timoshenko beam model theory A vertical fluid-conveying pipe as illustrated in Fig. 1, consists of length L, the flexural rigidity EIp , which depends on both the Youngs modulus E and the areamoment of inertia of the empty pipe cross-section Ip , shear rigidity GAp , in which G is the shear modulus and Ap is the cross-sectional area of the pipe, Poissons ratio ν, mass per unit length m, density ρp , conveying fluid with an axial velocity which in the undeformed, straight pipe is equal to U , moment of inertia If , mass per unit length M . Timoshenko beam theory, modifies the Euler-Bernoulli beam theory, takes into account the shear deformation and rotatory inertia. For Timoshenko beam model, the θ denotes the slope of the deflection curve by bending and Υ the angle of shear at the neutral axis in the same cross section, as Fig. 1 showing, and the total slope (dw/dx) is dw =θ+Υ dx with

′ ∂w ∂2θ − θ) (ρp IP + ρf If )θ¨ = EIp 2 + κ GAp ( ∂x ∂x − {[T¯ − p¯A(1 − 2νδ)]− dU ∂w [M − (M + m)g](L − x)}( − θ) dt ∂x

(5)

T¯ denotes the tension at x = L which is always zero unless there is an externally applied tension and the p¯ is the environmental mean pressure at x = L, which equals zero when the pipe discharged to atmosphere or water unless there is a mean pressure there. δ = 0 if there is no axial constraint and δ = 1 if it is prevented. c is the damping coefficient which is due to friction with surrounding fluid. If internal damping, externally imposed tension and pressurization effects are either absent or neglected and U is constant, the equation of the pipe takes the simple form as follows:

(1) ( 2 ) 2 ∂2w ∂ w ∂2w 2∂ w + M + 2U + U ∂t2 ∂t2 ∂x∂t ∂x2 ∂2w ∂θ ∂w − kGAp ( 2 − ) + (M + m)g ∂x ∂x ∂x ∂2w − [(M + m)g(L − x)] 2 = 0 ∂x

m

dθ H Q = ,Υ = ′ dx EIp k GAp

(2)

where w and x are the lateral deflection and the axial coordinate. H and Q indicate the bending moment and ′ transverse shear force respectively. k is the shear coefficient which depends on the cross-section shape of the beam and for the circular cross-section of beam here it is approximately given as

(6)

∂2θ ∂w + κGAp ( − θ)− 2 ∂x ∂x ∂w [(M + m)g(L − x)]( − θ) − (ρp IP + ρf If )θ¨ = 0 ∂x EIp



k =

2

6(1 + ν)(1 + α2 ) 2

(7 + 6ν)(1 + α2 ) + (20 + 12ν)α2

(3)

in which α is the ratio of internal to external radius of the pipe. Following closely the work by Pa¨ıdoussis and Issid[18] , applying Newtons second law, the dynamic equations of Timoshenko beam are derived, see Appendix A. The equations are suitable for pipes which are either clamped at both ends or cantilevered.

( 2 ) 2 ∂w ∂ w ∂2w ∂2w 2∂ w +M + 2U +U m 2 +c ∂t ∂t ∂t2 ∂x∂t ∂x2 2 ′ ∂θ ∂ w ) + {[T¯ − p¯A(1 − 2νδ)]− = k GAp ( 2 − ∂x ∂x dU ∂2w ∂w [M − (M + m)g](L − x)} 2 − (M + m)g dt ∂x ∂x

(7)

2.2 Two phase flow theory in vertical pipes It has been observed that the flow in pipes always occurs in a multiphase flow conditions and there are several flow regimes in two-phase flow, such as bubbly, slug, annular or churn flow. Therefore, in this work we studied the pipe conveying air-water mixtures. The model takes into account that the phases can have different physical properties and velocities. Then M , M U , M U 2 of the two-phase flow can be written as follows:

M=

∑ k

Mk , M U =

∑ k

Mk Uk , M U 2 =

∑ k

Mk Uk2 (8)

4

Tianqi Ma et al.

where k = 1, 2 represents the fluid and gas phase. There are some essential parameters defining the two-phase flow. The volume occupied by the gas in a slice of pipe is Vg and that by the liquid is Vl , the corresponding volumetric flow rates Qg and Ql , and the flow velocities Ug and Ul , then we can defined the void fraction α, the volumetric gas fraction εg , and the slip factor K by: Qg Ug Vg , εg = ,K = α= Vg + Vl Qg + Ql Ul

(9)

And the moment of inertia for internal flow If consist of two parts, fluid and gas, which are simply determined as:

Il = If × (1 − α), Ig = If × α

(10)

Monette and Pettigrew[15] proposed a new slip-ratio factor model to represent the characteristics of gasliquid flow in which the slip factor K was measured as a function of α based on the good agreement between the theory and the experimental results.

2 ∑ ∑ ∂2η ∂2η 1/2 ∂ η + 2 Γ β + Γk2 2 − k k ∂τ 2 ∂ξ∂τ ∂ξ k

k

2

Λ

∂ η ∂θ ∂η ∂ η +γ − γ(1 − ξ) 2 = 0 +Λ 2 ∂ξ ∂ξ ∂ξ ∂ξ

(1 − ξ)γ ∂η ∂ 2 θ 1 ∂ 2 θ Λ ∂η − θ) + ( − θ) = 0 (14) − − ( ∂τ 2 σ ∂ξ 2 σ ∂ξ σ ∂ξ

2.4 The boundary conditions and initial conditions It is necessary to know the boundary conditions of the transverse displace and bending angle. The pipe is assumed as a clamped-clamped pipe and the boundary conditions at both bottom and top of the pipe are:

η(0, τ ) = 0,

∂η(0, τ ) ∂η(1, τ ) = 0, η(1, τ ) = 0, = 0 (15) ∂ξ ∂ξ

θ(0, τ ) = 0, θ(1, τ ) = 0 ( K = α/(1 − α) =

εg 1 − εg

)1/2 (11)

2.3 Dimensionless The system may be rendered dimensionless by means of the following quantities:

√ t ξ = x/L, η = w/L, τ = [EI/(M + m)] 2 L √ √ Mk Mk Mk , βk = ∑ , Γk = Uk L uk = Uk L EI Mk + m EI k ( ) ∑ κGAL2 γ= Mk + m L3 g/EI, Λ = EI k

(ρP IP + ρf If ) T (L)L2 ) , ζL = σ=( ∑ EI Mk + m L2 (12) Substituting these terms into Eqs.(6-7) gives the dimensionless equations of motion:

(16)

Zero η initial condition is applied and a random noise with amplitude of order O(10−3 ) is applied as initial condition to ∂η ∂t , which is very small and induce the structure-fluid system to vibrate and to tend to stable gradually, similarly as Violette et al.[23] and Gu et al.[8] set.

η(ξ, 0) = 0,

∂η(ξ, 0) = O(10−3 ) ∂t

(17)

3 Integral transform solution In order to solve the two set of coupled equations of motions, one for the transverse displacement and one for the bending angel, the GITT technique is utilized to transform the nonlinear partial differential equation models to a set of ordinary differential equations. The adopted eigenfunctions for transverse displacement η(ξ, τ ) with clamped-clamped boundary condition is: { cos[λ

k

(13)

2

ϕi (ξ) =

i (ξ−0.5)] i (ξ−0.5)] − cosh[λ cos(λi /2) cosh(λi /2) sin[λi (ξ−0.5)] sinh[λi (ξ−0.5)] − sinh(λi /2) sin(λi /2)

for i odd, for i even,

(18)

where the eigenvalues are obtained through the transcendental equations:

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model

tanh(λi /2) =

{ − tan(λi /2) for i odd, tan(λi /2) for i even,

(19)

 ∫1 η˜i (τ ) = 0 ϕi (ξ)η(ξ, τ )dξ, transform ∞ ∑ ϕi (ξ)˜ ηi (τ ), inversion  η(ξ, τ ) =

(20)

The eigenfunctions satisfy the following orthogonality property ∫

1

ϕi (ξ)ϕj (ξ)dξ = δij Mi

(21)

0

where δij is Kronecker delta. And the norm is evaluated to yield: Mi = 1, i = 1, 2, 3, ...,

 ∫1 θ˜i (τ ) = 0 φi (ξ)θ(ξ, τ )dξ, transform ∞ ∑ φi (ξ)θ˜i (τ ), inversion  θ(ξ, τ ) =

φi (ξ) = sin(µi ξ), i = 1, 2, 3, ...,

(23)

with the eigenvalues µi = iπ, i = 1, 2, 3, ..., , which satisfies the following bounddary conditions

i=1

To perform the process of integral transformation of the original partial differential equation, the two sets of equations of motions for transverse displacement and ∫1 bending angel are multiplied by operator 0 ϕi (ξ)dξ and ∫1 φm (ξ)dξ, respectively, the inverse formula are ap0 plied and the transformed transverse equations yields the following set of ordinary differential equations.

∞ ∑ ∑ ∂ η˜j ∂ 2 ηi (τ ) 1/2 Γ β ( Aij + 2 )+ k k 2 ∂τ ∂τ j=1

(



k ∞ ∑

− Λ)(

Γk2

dφ2i (0) dφ2 (1) = 0, i 2 = 0 2 dξ dξ

k



∞ ∑

Aij ηj (τ ) − γ

+

while the norm is evaluated as

(26)

Therefore the normalized eigenfunctions for transverse displacement and bending angle coincide:

Dij ηj (τ ) = 0, i = 1, 2, 3, ......

j=1

∞ Λ˜ γ∑ θm (τ ) + Gmj η˜j (τ )− σ σ j=1

where the coefficients are determined as: ∫

1

ϕi (ξ)

∂ϕj (ξ) dξ ∂ξ

ϕi (ξ)

∂ 2 ϕj (ξ) dξ ∂ξ 2

0

∫ (27a,b)

For simplicity, the superpose tilde is dropped in the following article. The integral transform pair, the integral transformation and the inversion formula are as follows

(31)

∞ γ∑ Hmj θ˜j (τ ) = 0, m = 1, 2, 3, ...... σ j=1

Aij = ϕi (ξ) ϕ˜i (ξ) = 1/2 M φi (ξ) φ˜i (ξ) = 1/2 N

∞ ∑

∞ ∞ 1∑ Λ∑ ∂ 2 θ˜m (τ ) ˜ − E θ (τ ) − Fmj η˜j (τ ) mj j ∂τ 2 σ j=1 σ j=1

(25)

0

1 2

Cim θ˜m (τ )

m=1

(30)

1

Ni =

∞ ∑

(24)

and the eigenfunctions for bending angle satisfy the following orthogonality properties

φi (ξ)φj (ξ)dξ = δij Ni

Bij η˜j (τ )) + Λ

j=1

j=1



(29a,b)

(22)

Meanwhile, the chosen eigenvalue problem for the angle is

φi (0) = 0, φi (1) = 0,

(28a,b)

i=1

with boundary conditions dϕi (0) dϕi (1) ϕi (0) = 0, ϕi (1) = 0, = 0, =0 dξ dξ

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1

Bij = 0



1

Cim = ∫

0 1

∂φm (ξ) ϕi (ξ) dξ ∂ξ

ϕi (ξ)(1 − ξ)

Dij = 0

∂ 2 ϕj (ξ) dξ ∂ξ 2

(32a,b,c,d)

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Tianqi Ma et al.



1

Emj = ∫

φm (ξ)

∂ 2 φj (ξ) dξ ∂ξ 2

φm (ξ)

∂ϕj (ξ) dξ ∂ξ

0 1

Fmj = 0



1

Gmj = 0



∂ϕj (ξ) φm (ξ)(1 − ξ) dξ ∂ξ

(33a,b,c,d)

The geometrical and physical parameters of the flexible pipe are shown in Table 1. For the internal twophase flow, the densities of the water and the air are 1000 kg/m3 and 1.2 kg/m3 , respectively. In this work, the Euler-Bernoulli model conveying two-phase flow with clamped-clamped boundary condition is the same with An and Su[1], which is derived by Monette and Pettigrew[15] and Pa¨ıdoussis[16].

1

φm (ξ)(1 − ξ)φj (ξ)dξ

Hmj = 0

Similarly, initial conditions are integral transformed to eliminate the spatial coordinate, yielding

η˜i (0) = 0,

d˜ ηi (0) = dt



1

O(10−3 )ϕi (ξ)dξ, i = 1, 2, 3, ...,

0

(34) To calculate the η˜i (τ ) and θ˜i (τ ), the infinite expansions for the η and θ is truncated to finite orders N which satisfies the accuracy requirement. The truncated Eqs. (30-31) can be accurately calculated by the subroutine DIVPAG from IMSL Library[12] with automatic global accuracy control, which has been well proved to handle such problems. In this paper, the solution of the equations with error is selected as 10-6 to satisfy accuracy requirement. Once the system has been numerically solved, the dimensionless transverse displacement η(ξ, τ ) can be recovered from the inversion Eqs. (28b).

4 Results and discussion In this section, the vibration of pipe conveying air-water two-phase flow was calculated by using the GITT approach. For sufficient accuracy and computational efficiency, the truncation order N is chosen as 20 in this paper, which has been examined to be sufficient accurate by Gu et al.[5]. The time step size is set as 0.005 and the number of time steps is 6000. A series of numerical calculations in Fortran were performed. In order to study as nearly as possible the underlying physics of the dynamics of pipe conveying two-phase flow, the Timoshenko beam model and Euler-Bernoulli beam model have been compared with two vibration parameters, the dimensionless natural frequency and the dimensionless amplitude, for different aspect ratio from 2 to 50, different volumetric gas fractions from 0 to 1 and different volumetric flow rates Ql . The influences of the above three parameters on the vibration response have been discussed. The flexible pipe considered in present study takes the same parameters with that given by An and Su[1].

∑ ∂2w ∂2w ∂4w ∑ Mk Uk2 2 + 2 Mk Uk + + 4 ∂x ∂x ∂x∂t k k ) ( ∑ ∑ ∂2w Mk + m Mk + m)g + ( ∂t2 k k ( ) ∂ 2 w ∂w (L − x) 2 + =0 ∂x ∂x

EI

(35)

4.1 Vibration of pipe conveying two-phase flow The result of a flexible pipe with the aspect ratio of 50, the liquid volumetric flow rate of 0.0001 m3 /s, the volumetric gas fraction of 0.5, is described here. The dimensionless time history for τ ∈ [25, 30] of the transverse displacement η at the one fifth point of the pipe and the profiles of η at different timing are illustrated in Fig. 2 and Fig. 3. As seen in the Fig. 3, the modal displacement pattern is not classical normal mode any more but contains stationary wave and travelling wave components. Different from pipe conveying internal flow, pipe without internal flow vibrates in classical normal modes, which is depicted in Fig. 4, and the up and downstream propagating waves are symmetric. The phenomenon, same as Chen and Rosenberg[4] has discovered, is due to the destroy of the symmetry of up and downstream propagating waves, which has different phase speeds when U > 0. Through conducting Fast Fourier Transform amplitude spectrum of the dimensionless time history of the transverse displacement η, the frequency content can be identified. As Fig. 2(b) presented, the peaks in the spectral analysis are the corresponding natural frequencies, which means the dynamic response suffers multi-mode contributions. The peak value means the energy that each natural frequency component contributes. It can be gotten from Fig. 2(b) that the first two modes dominate the dynamic response. It can also be concluded from Fig. 5, which shows the mode contribution from 1 to 3, that the original deflections are dominated by mode 1 and mode 2, because the maximum deflections of the third mode is less than the first two modes with one order of magnitude.

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model

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4.2 The dimensionless natural frequency

The high flow rate and gas fraction accelerates the divergence of the dynamic system.

The natural frequency of flexible pipe is calculated based on Timoshenko beam theory and Euler-Bernoulli beam theory respectively. The comparison between the two theories have been conducted to analyse dimensionless natural frequencies at mode 1, 2 versus aspect ratio for 8 typical sets of volumetric flow rate Ql and volumetric gas fraction εg (Table 2). The results are shown in Fig. 6. The dash lines with triangle symbols represent the results based on Euler-Bernoulli beam model and the solid lines with rectangle symbols and circular symbols represent the first and second mode frequencies for Timoshenko beam model respectively. From Fig. 6, it is noticed that the natural frequencies from Timoshenko beam model are less than that of Euler Bernoulli beam model at low aspect ratio and approach to the value of Euler-Bernoulli beam model gradually at higher aspect ratio. In case 1, when L/D = 6, the first mode frequencies equal 16.96 and 22.38 for Timoshenko beam model and Euler-Bernoulli beam, respectively. While L/D = 50, the frequencies of the two models are very close, which are 35.19 and 35.34, respectively. The phenomenon is more remarkable at higher mode. That means the Timoshenko beam theory should be more suitable to short pipes or to the study of highmode dynamical behaviour of long pipes.

The dimensionless natural frequencies based on Timoshenko beam model with increasing volumetric flow rates are depicted in Fig. 7, when εg = 0.5, L/D = 50. As can be observed, the frequencies in the fundamental mode decrease as the volumetric flow rate increases, which indicates that the pipe will lose stability when the frequency approaches to zero and the correspond flow rate is the critical value. From the formulation inviscid fluid ) dynamic force FA = ( 2 of lateral ∂2w ∂ w 2 ∂2w −M ∂t2 + 2U ∂x∂t + U ∂x2 , it is obvious that the

It can also be observed that the influence of aspect ratio on the natural frequency does relate to the value of Ql and εg . When Ql is small, the ω is increasing with the increasing of aspect ratio in case 1,4,7. When the volumetric flow rate increases from 0.0001 to 0.0003, the increasing of frequency turns to a process of reducing, as the Fig. 6(iv), (v) and (vi) demonstrate. While the aspect ratio reaches 30 at Ql = 0.0003 m3 /s andεg = 0.5, the fundamental frequency approaches to zero, which means the dynamic system will lose its stability by divergence at higher aspect ratio. From the above analysis, it can be concluded that the flexible pipe has critical length at high flow rate. The same phenomenon is observed with high gas fractionεg , as shown in Fig. 6(viii), which indicates the critical length is smaller with higher gas fraction. Meanwhile, the influence of gas fraction on the relationship between natural frequency and aspect ratio depends on the value of Ql . The larger the volumetric flow rate Ql , the greater impact of gas fraction. When the volumetric flow rate is low, the frequency rises with the increase of aspect ratio regardless of the gas fraction, as Fig. 6(i), (iv) and (vii) show. When Ql is high, the frequencies increase with the aspect ratio increasing at low gas fraction (Fig. 6(ii)) and decrease with the aspect ratio at high gas fraction (Fig. 6(viii)).

2

centrifugal force M U 2 ∂∂xw2 acts in the same manner as a compressive load. With the increasing of U , the effective stiffness of the pipe diminished and the destabilizing centrifugal force may even become large enough to overcome the restoring flexural force, resulting in divergence, vulgarly known as buckling[4]. A similar behavior is observed by mode 2 and mode 3. To further analyze the impact of the aspect ratio and gas fraction on the relationship between frequencies and flow rates, the fundamental frequencies versus volumetric flow rates for L/D = 5, 10, 20, 50 and εg = 0, 0.5, 0.8 are also studied, as depicted in Fig. 8 and Fig. 9. As shown in Fig. 8, the decrease of frequencies with increasing volumetric flow rates is more remarkable at higher aspect ratio. When volumetric flow rates increases from 0 to 0.00024, the frequencies based on Timoshenko beam model at gas fraction equals 0.5, drop from 33.62 to 5.03 for L/D = 50 and from 15.71 to 15.39 for L/D = 5. The dynamic system has a critical value of volumetric flow rates and the larger the aspect ratio, the smaller value of critical volumetric flow rates. Apart from these, it can also be seen the critical volumetric flow rates for the Timoshenko beam model is less than the ones of Euler Bernoulli beam model at low aspect ratio. If the flexible pipe is not sufficient long, the Timoshenko beam model should be taken into consideration to calculate the critical volumetric flow rate. As the Fig. 9 shows, at higher gas fraction, the frequency drops faster with the increase of volumetric flow rates and the critical velocities are smaller. The natural frequencies of the pipe versus volumetric gas fractions 0 < εg < 1 for different volumetric flow rates Ql = 0.0001, 0.0002, 0.0003m3 /s and aspect ratios L/D = 5, 50 are calculated, as shown in Fig. 10. The black dot lines represent the results from An et al.[1] based on Euler-Bernoulli beam model at L/D = 62.9 (L = 1(m), D = 15.9(mm)). The fundamental frequency decreases with the volumetric gas fraction at L/D = 50, the same with that concluded by An et al.[1]. When the pipe with L/D = 50, conveys

8

liquid only, the fundamental frequencies equal 35.19, 29.53 and 19.16 for Ql = 0.0001, 0.0002, 0.0003 m3 /s respectively. With the internal flow changes to two-phase flow, the fundamental frequencies decrease and are in close proximity to zero when the volumetric gas fraction approaches to a critical value εg = 0.97, 0.76, 0.16 for Ql = 0.0001, 0.0002, 0.0003m3 /s, respectively. The behavior is more obviously for higher flow rate. While for L/D = 5, the frequencies, based on Timoshenko beam model and Euler-Bernoulli beam model, present small variations although the gas fraction increases considerably. This suggests the two phase flow has little influence on the results when the pipe is short. To sum up, these phenomena show that the two-phase conveyed in the pipe accelerates the instability and influences more significantly for higher aspect ratio and higher internal flow rate.

4.3 The vibration amplitude It is of interest to investigate the effect of aspect ratio, flow rate and two-phase flow parameter on the vibration amplitudes of pipe. Fig. 11 shows the amplitude, which is the maximum absolute value over the calculated time-history response at the central point of the flexible pipe, under various aspect ratios. The aspect ratio varying from 2 to 50 was selected in the amplitude analysis. In order to deeply analyze the evolution of amplitude with aspect ratio due to the effect of internal flow, the amplitudes with increasing aspect ratio for 8 typical sets of volumetric flow rate Ql and volumetric gas fraction εg as listed in Table 2 has been calculated. From Fig. 11, it is obvious that the amplitudes of Timoshenko beam model are larger than the ones of Euler-Bernoulli beam model especially at low aspect ratio and drop to that of Euler-Bernoulli beam model with the increasing of aspect ratio. Same with natural frequencies, the evolution of amplitudes varying with aspect ratio does relate to the value of flow rate and gas fraction. From Fig. 11(i), (ii), (iv) and (vii) it can be noted that the amplitudes diminish with aspect ratio increasing. While for high flow rate or high gas fraction, as shown in Fig. 11(vi) and (viii), the evolution of amplitudes varying with aspect ratio is entirely contrary. At high flow rate or high gas fraction, the system is unstable and the amplitudes increase with the aspect ratio increasing. For case 6 and case 8, when aspect ratio reaches a critical value, L/D = 30 for Ql = 0.0003 m3 /s, εg = 0.5and L/D = 42 for Ql = 0.0002 m3 /s, εg = 0.8 respectively, the vibration deflection increases dramatically, which means the instability will occur. The critical value of

Tianqi Ma et al.

aspect ratio equals to that shown in Fig. 4(vi) and (viii). Apart from this, it is also perceived that, when the pipe is going to lose stability, the amplitude for Timoshenko beam model suggests a worse agreement with the result of Euler-Bernoulli beam model at larger aspect ratio. The amplitude of Timoshenko beam model increases more quickly than that of Euler-Bernoulli beam model. Therefore, it could come to a conclusion that the Timoshenko beam model should be adopted when the instability is going to occur. The maximum deflection of the flexible pipe versus internal flow rate under various aspect ratios is evaluated in Fig. 12. The maximum amplitude increases slowly as the flow rate increases, then it increases dramatically to infinity when the internal flow rate approaches to a critical value. It is also noticed that the amplitude based on Timoshenko beam model is larger than that calculated from Euler-Bernoulli beam model and the phenomenon is more noticeable at lower aspect ratio as well as in the condition that the dynamic system is going to lose stability. The influence of volumetric gas fractions on the amplitude is studied for increasing gas fraction at different aspect ratio with Ql = 0.0002 m3 /s, which is illustrated in Fig. 13. In the vicinity of the critical gas fraction, the amplitude rises rapidly with the increasing of gas fraction, which indicates that the pipe is going to be unstable.

5 Conclusions The dynamic behavior of pipes conveying two-phase flow based on Timoshenko beam model and the slipratio factor model has been calculated using the generalized integral transform technique. The dynamic response has been analysed through parametric study of dimensionless frequencies and amplitudes over various aspect ratios, internal fluid flow rates and volumetric gas fractions. The conclusions are summed up as follows: (i) The frequencies of Timoshenko beam model are less than that of Euler-Bernoulli beam model at low aspect ratio especially for high mode. The amplitude for Timoshenko beam model is larger at low aspect ratio and increases more quickly than that of EulerBernoulli beam model when the pipe is going to lose stability. These phenomena suggest that the Timoshenko beam model should be taken into account at low aspect ratio or when the pipe is excited at higher mode or is going to lose stability. (ii) The influence of aspect ratio on the natural frequency does relate to the values of Ql and εg . When

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model

Ql and εg is small, the ω increases with the increasing of aspect ratio. When the volumetric flow rate or gas fraction increases, the increasing of frequency turns to a process of reducing. The pipe has a critical length at high flow rate and gas fraction, the larger the flow rate and gas fraction the smaller value of critical length. (iii) The frequencies decrease with increasing volumetric flow rates. The phenomenon is more remarkably at higher aspect ratio or higher gas fraction, in which case the critical flow rate is smaller. (iv) The fundamental frequency decreases with the volumetric gas fraction at high aspect ratio and presents small variations at low aspect ratio. The decrease is more obviously for higher flow rate. The two phase flow accelerates the instability and has significant influence when the pipe is long and the internal liquid flows fast. (v) At low flow rate or gas fraction, amplitudes of pipe conveying two phase flow diminish with the increasing of aspect ratio. While for high flow rate or high gas fraction, the pipe is unstable with amplitudes increasing with the aspect ratio and rising dramatically when the aspect ratio reaches a critical value. Moreover, the amplitude increases as the flow rate and gas fraction increases, and it increases dramatically to infinity when the two parameters approach to a critical value. Acknowledgements The authors acknowledge gratefully financial support provided by the National Natural Science Foundation of China (Grant No. 51379214, 51409259), the Science Foundation of China University of Petroleum, Beijing (No.C201602, 2462013YJRC004) for the financial support of this research.

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References 1. An, C., Su, J.: Dynamic behavior of pipes conveying gascliquid two-phase flow. Nuclear Engineering and Design 292, 204–212 (2015). DOI 10.1016/j.nucengdes.2015.06.012. Item number: S0029549315002629 identifier: S0029549315002629 2. Bourrires, F.J.: Sur un phnomne doscillation autoentretenue en mcanique des fluids rels. Publications Scientifiques et Techniques du Ministre de l’Air (147) (1939) 3. Cargnelutti, M.F., Belfroid, S.P.C., Schiferli, W.: Twophase flow-induced forces on bends in small scale tubes. In: ASME 2009 Pressure Vessels and Piping Conference, ASME 2009 Pressure Vessels and Piping Conference, pp. 369–377. Prague, Czech Republic (2009) 4. Chen, S.S., Rosenberg, G.S.: Vibration and stability of a tube conveying fluid. Tech. rep., Argonne National Laborary Report ANL-7762 (1971) 5. Gu, J., An, C., Duan, M., Levi, C., Su, J.: Integral transform solutions of dynamic response of a clampedcclamped pipe conveying fluid. Nuclear Engineering and Design 254, 237–245 (2013).

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DOI 10.1016/j.nucengdes.2012.09.018. Item number: S0029549312005006 identifier: S0029549312005006 GU, J., AN, C., LEVI, C., SU, J.: Prediction of vortexinduced vibration of long flexible cylinders modeled by a coupled nonlinear oscillator: Integral transform solution. Journal of Hydrodynamics, Ser. B 24(6), 888–898 (2012). DOI 10.1016/S1001-6058(11)60317-X. Item number: S100160581160317X identifier: S100160581160317X Gu, J., Ma, T., Duan, M.: Effect of aspect ratio on the dynamic response of a fluid-conveying pipe using the timoshenko beam model. Ocean Engineering 114, 185–191 (2016). DOI 10.1016/j.oceaneng.2016.01.021. Item number: S0029801816000329 identifier: S0029801816000329 Gu, J., Wang, Y., Zhang, Y., Duan, M., Levi, C.: Analytical solution of mean top tension of long flexible riser in modeling vortex-induced vibrations. Applied Ocean Research 41, 1–8 (2013). DOI 10.1016/j.apor.2013.01.004 Huang, Y., Liu, Y., Li, B., Li, Y., Yue, Z.: Natural frequency analysis of fluid conveying pipeline with different boundary conditions. Nuclear Engineering and Design 240(3), 461–467 (2010). DOI 10.1016/j.nucengdes.2009.11.038. Item number: S0029549309006293 identifier: S0029549309006293 Ibrahim, R.A.: Over view of mechanics of pipes conveying fluids. parti. fundamental studies. J. Pressure Vessel Technol.: Trans. ASME 132(3) (2010) Ibrahim, R.A.: Mechanics of pipes conveying fluids part ii: Applications and fluidelastic problems. J. Pressure Vessel Technol.: Trans. ASME 133(2) (2011) IMSL: IMSL Fortran Library version 5.0, MATH/LIBRARY. Visual Numerics, Inc., Houston, TX (2003) Laithier, B., Pa¨ıdoussis, M.: The equations of motion of initially stressed timoshenko tubular beams conveying fluid. Journal of Sound and Vibration 79(2), 175–195 (1981). 1981 Li, B., Gao, H., Zhai, H., Liu, Y., Yue, Z.: Free vibration analysis of multi-span pipe conveying fluid with dynamic stiffness method. Nuclear Engineering and Design 241(3), 666–671 (2011). DOI 10.1016/j.nucengdes.2010.12.002. Item number: S0029549310008058 identifier: S0029549310008058 Monette, C., Pettigrew, M.J.: Fluidelastic instability of flexible tubes subjected to two-phase internal flow. Journal of Fluids and Structures 19(7), 943–956 (2004). DOI 10.1016/j.jfluidstructs.2004.06.003. Item number: S0889974604000969 identifier: S0889974604000969 Pa¨ıdoussis, M.P.: FluidCStructure Interactions: Slender Structures and Axial Flow. Academic Press, Inc, San Diego, CA (1998) Pa¨ıdoussis, M.P.: The canonical problem of the fluidconveying pipe and radiation of the knowledge gained to other dynamics problems across applied mechanics. Journal of Sound and Vibration 310(3), 462–492 (2008). DOI 10.1016/j.jsv.2007.03.065. Item number: S0022460X07002428 identifier: S0022460X07002428 Pa¨ıdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration 33(3), 267–294 (1974). 1974/4/8/ Pa¨ıdoussis, M.P., Laithier, B.E.: Dynamics of timoshenko beams conveying fluid. Journal of Mechanical Engineering Science 18(4), 210–220 (1976). DOI 10.1243/JMES JOUR 1976 018 034 02 Pettigrew, M.J., Taylor, C.E.: Two-phase flow-induced vibration: An overview (survey paper). J. Pressure Vessel Technol.: Trans. ASME 116(3), 233–253 (1994)

10 21. Pontaza, J.P., Menon, R.G.: Flow-induced vibrations of subsea jumpers due to internal multi-phase flow. In: ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering, ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering, vol. 7, pp. 585–595. Rotterdam, The Netherlands (2011) 22. Sinha, J.K., Rao, A.R., Sinha, R.K.: Prediction of flowinduced excitation in a pipe conveying fluid. Nuclear Engineering and Design 235(5), 627–636 (2005). DOI 10.1016/j.nucengdes.2004.10.001. Item number: S0029549304003383 identifier: S0029549304003383 23. Violette, R., de Langre, E., Szydlowski, J.: Computation of vortex-induced vibrations of long structures using a wake oscillator model: Comparison with dns and experiments. Computers & Structures 85(11C14), 1134–1141 (2007). DOI 10.1016/j.compstruc.2006.08.005. 2007/7// 24. Yu, D., Wen, J., Zhao, H., Liu, Y.: Flexural vibration band gap in a periodic fluid-conveying pipe system based on the timoshenko beam theory. Journal of Vibration and Acoustics 133, 1–3 (2011) 25. Zhai, H., Wu, Z., Liu, Y., Yue, Z.: Dynamic response of pipeline conveying fluid to random excitation. Nuclear Engineering and Design 241(8), 2744–2749 (2011). 2011/8// 26. Zhang, M., Xu, J.: Effect of internal bubbly flow on pipe vibrations. Science China Technological Sciences 53(2), 423–428 (2010). DOI 10.1007/s11431-009-0405-9. Identifier: 405

Tianqi Ma et al.

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model

11

 = 0.2

2 1

 (10-3 )

Z

0 -1 -2 25

25.5

26

26.5

27

1.5

27.5 (a)  PSD

28

28.5

29

29.5

30

1

0.5

8

0 0

b T

/

wZ  w[

wZ  w[

20

40

60

80 (b) f

100

120

140

160

Fig. 2 Time history of simulation for flexible pipe at L/D = 50, Ql = 0.0001 m3 /s, εg = 0.5 (a)The dimensionless time history for τ ∈ [25, 30] of the transverse displacement at the one fifth point of the pipe; (b) amplitudes spectrum of the structural response at τ ∈ [0, 50]

5      

4 3

-3  (10 )

2

=3 =6 =9 = 12 = 15 = 18

1 0 -1 -2

[

-3 -4

Fig. 1 A clamped-clamped pipe conveying gas-liquid twophase flow

-5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



Fig. 3 Transverse displacement profiles for flexible pipe with dimensionless time interval ∆τ = 3 at L/D = 50, Ql = 0.0001 m3 /s, εg = 0.5

12

Tianqi Ma et al.

1 =3 =6 0.9 =9 = 12 0.8 = 15 = 18 0.7



     

1

1

1

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

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0 -5

0 original  (10-3 )

0 -5

5

0 mode 1  (10-3 )

5

0 -1

0 mode 2  (10-3 )

1

0 -2

0 mode 3  (10-4 )

2

Fig. 5 Mode separation of transverse displacement η with dimensionless time interval ∆τ = 3 at L/D = 50, Ql = 0.0001 m3 /s, εg = 0.5  

PRGH PRGH PRGH

5      

4 3

  

1 0





Ȧ 

-3  (10 )

2

=3 =6 =9 = 12 = 15 = 18



-1 -2



-3



-4

 

-5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



Fig. 4 Transverse displacement profiles of flexible pipe without internal flow at L/D = 50











4 O Fig. 7 Dimensionless natural frequencies of flexible pipe at mode 1(black), mode 2(red), mode 3(blue) based on Timoshenko beam model with increasing volumetric flow rates at L/D = 50, εg = 0.5

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model LL˖FDVH

L˖FDVH 7LPRVKHQNRVWPRGH 7LPRVKHQNRQGPRGH (%VWPRGH (%QGPRGH



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Fig. 6 Dimensionless natural frequencies of flexible pipe at mode 1(black), 2(red) versus aspect ratio for eight typical sets of values of volumetric flow rate Ql and volumetric gas fractionεg

14

Tianqi Ma et al.

50 45

AN et al. L/D=62.9

Ql=0.0001

Ql=0.0002

Ql=0.0003

Timoshenko L/D=50

Ql=0.0001

Ql=0.0002

Ql=0.0003

Timoshenko L/D=5

Ql=0.0001

Ql=0.0002

Ql=0.0003

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40

L/D=5

Ql=0.0001

Ql=0.0002

Ql=0.0003

35 30

ω

25 20 15 10 5 0 0.0

0.2

0.4

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0.6

0.8

1.0

Fig. 10 The fundamental natural frequencies of flexible pipe conveying air-water two phase flow versus volumetric gas fraction 0 < εg < 1at volumetric flow rate Ql = 0.0001, 0.0002, 0.0003 m3 /sand aspect ratio L/D = 5, 50

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model



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Fig. 11 Dimensionless amplitude of flexible pipe versus aspect ratio L/Dbased on Timoshenko beam model and EulerBernoulli beam model for eight typical sets of values of volumetric flow rate Ql and volumetric gas fractionεg ; black line with rectangle symbol: Timoshenko beam model; red line with triangle symbol: Euler-Bernoulli beam model

16

Tianqi Ma et al. 40

0.05 L/D=5 Timoshenko L/D=5 E-B L/D=10 Timoshenko L/D=10 E-B L/D=20 Timoshenko L/D=20 E-B L/D=50 Timoshenko L/D=50 E-B

35 30

0.04

0.03

20

η

ω

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L/D=5 Timoshenko L/D=5 E-B L/D=20 Timoshenko L/D=20 E-B L/D=50 Timoshenko L/D=50 E-B

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0.01 5 0 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018

0.00 0.0000

0.0005

0.0010

0.0015

Ql

Ql

Fig. 12 Maximum deflections of flexible pipe based on Timoshenko beam model (rectangle symbol) and Euler-Bernoulli beam model (triangle symbol) versus internal flow rate at aspect ratio L/D = 5, 20, 50, when εg = 0.5: black line L/D = 5; red line L/D = 20; blue line L/D = 50

Fig. 8 Fundamental frequencies of flexible pipe based on Timoshenko beam model(rectangle symbol with solid line) and Euler-Bernoulli beam model(cross symbol with dash line) with increasing volumetric flow rates for L/D = 5(black), L/D = 10(red), L/D = 20(blue), L/D = 50(green) at εg = 0.5

0.08

 

7LPRVKHQNRHJ 

L/D=5 L/D=10 L/D=20 L/D=50

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(%HJ  (%HJ 

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0.2

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0.6

0.8

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 4O







Fig. 9 Fundamental frequencies of flexible pipe based on Timoshenko beam model (rectangle symbol) and EulerBernoulli beam model (triangle symbol) with increasing volumetric flow rates for εg = 0(black solid line), εg = 0.5(red dash line) and εg = 0.8 (blue dot line) at L/D = 5

Fig. 13 Dimensionless amplitude of flexible pipe versus volumetric gas fractions when Ql = 0.0002 m3 /s at L/D = 5, 10, 20, 50 Table 2 The eight typical sets of values of volumetric flow rate and volumetric gas fraction case

Table 1 The main geometrical and physical parameters of the flexible pipe Parameter

Flexible pipe

Youngs modulus E(Mpa) Poissons ratioν Outer diameter D(m) Inner diameter d(m) Density of material ρp (kg/m3 )

3.6 0.4 0.0159 0.0127 1180

Case Case Case Case Case Case Case Case

1 2 3 4 5 6 7 8

Volumetric flow rate Ql (m3 /s)

Volumetric fraction εg

0.0001 0.0002 0.0003 0.0001 0.0002 0.0003 0.0001 0.0002

0 0 0 0.5 0.5 0.5 0.8 0.8

gas

17

Appendices

S$

+

7

4

The dynamic equations of Timoshenko beam are derived through )G [ TG [ TG [ following closely the work by Pa¨ıdoussis and Issid[18] and b wZ  w[ applying Newtons second law. In the derivation process, the F wZ  w[ G [ T small deflection approximation was adopted. Then the curvi+  +G [ )G [ linear coordinate s may be interchanged by the coordinate x. Conduct the force analysis of the two elements δs of fluid and 7  7G [ wZ  w[ pipe, as shown in Fig. A1 $ S  wS For fluid element, it is subjected to pressure p, which is P JG [ 0 JG [ measured above the ambient pressure and due to the friction loss, reaction force F of the pipe on the fluid normal to the )OXLGHOHPHQW 3LSHHOHPHQW fluid, wall shear stress q between the fluid and the pipe tanFig. A1 Forces acting on fluid element and pipe element gential to the fluid element as well as gravity force M g. Then the force and moment equilibrium equations in the x and w directions yield Adding Eg.(A1) and Eg.(A6), one can obtained −A

∂w ∂p − q + Mg + F = M af x ∂x ∂x

(A1)

∂ ∂w ∂w (p )−q = M af w ∂x ∂x ∂x

(A2)

Pa¨ıdoussis[16] intergrated from x to L and derived the expression T − pA at x = L, which yields

(A3)

T − pA = T¯ − p¯A(1 − 2νδ)+

−F − A

pAΥ = If θ¨

where af x and af w are the acceleration in the x and w direction of the fluid element, which were derived by Pa¨ıdoussis[16] with the assumption that the fluid flow was approximated as a plug flow, yield

af x

dU = dx [

af w =

∂ ∂ +U ∂t ∂x

(A4) ]2 w=

∂2w ∂2w ∂ 2 w dU ∂w +2U + +U 2 2 ∂t ∂x∂t ∂x2 dt ∂x (A5)

For the pipe element, it subjected to reactive force F and shear stress q from fluid element as well as gravity force mg, longitudinal tension T , transverse shear force Q, bending moment H and damping due to friction with surrounding fluid c ∂w . Projection of the forces on the x and w direction and ∂t consideration of moments, gives ∂T ∂w + q + mg − F =0 ∂x ∂x

(A6)

∂Q ∂ ∂w ∂w ∂w ∂2w +F + (T )+q −c =m 2 ∂x ∂x ∂x ∂x ∂t ∂t

(A7)

∂2θ ∂H + Q − T Υ = Ip 2 ∂x ∂t

(A8)

Combining Eqs.(A2 A7) and the equation of shear force, it can be obtained that ′

k GAp ( −c

∂ ∂w ∂θ ∂2w )+ [(T − pA) ] − ∂x2 ∂x ∂x ∂x

∂w = M af w + maf w ∂t

(A9)

∂ dU (T − pA) = M − (M + m)g ∂x dt

[(M + m)g − M (dU/dt)](L − x)

(A10)

(A11)

Substitution of Eq.(A11) into Eq.(A9) gives the equation of Timoshenko beam conveying fluid in w direction. ) ( 2 2 ∂w ∂ w ∂2w ∂2w 2∂ w + c + 2U + M + U ∂t2 ∂t ∂t2 ∂x∂t ∂x2 2 ′ ∂ w ∂θ = k GAp ( 2 − ) + {[T¯ − p¯A(1 − 2νδ)]− ∂x ∂x ∂w dU ∂2w [M − (M + m)g](L − x)} 2 − (M + m)g dt ∂x ∂x m

(A12)

Adding Eq.(A3) and Eq.(A8) and substituting the bending moment and transverse shear force, the moment equation can be derived. ′ ∂2θ ∂w (ρp IP + ρf If )θ¨ = EIp 2 + κ GAp ( − θ) ∂x ∂x − {[T¯ − p¯A(1 − 2νδ)]−

dU ∂w [M − (M + m)g](L − x)}( − θ) dt ∂x

(A13)

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