Second Order Godunov-Type Scheme for Fluid-Structure Interaction Model of Liquid Piping Systems

MING CHEN1, GUANG-WEI JIAO1, SHAO-QI ZHOU1, QI-WEI YONG1

1Department of Petroleum Supply Engineering Logistical Engineering University of PLA

Chongqing 401311, China E-MAIL: chenarmy@126.com

Abstract—Second order explicit finite volume Godunov-Type scheme is formulated for simulation of Fluid-Structure Interaction (FSI) model in this paper. MUSCL-Hancock scheme is applied for calculation of numerical fluxes at cell interfaces. Slope limiter is introduced to obtain high resolution and stable solutions. At the boundaries of the piping systems, the virtual cells are established to restore miss information of parameters. So second order solutions can be obtain in the whole computational domain. The calculating instance shows that the results by the proposed scheme are in close agreements with experimental data.

Keywords- Godunov; Fluid-Structure Interaction; Model; Numerical simulation; Piping systems

I. INTRODUCTION Water hammer models of the piping systems belong to

hyperbolic partial differential equations and the method of characteristics (MOC) is the most widely accepted to obtain their numerical solutions. To four-equation model of water hammer considering FSI, there are two waves, i.e., axial stress wave and surge wave, propagating along pipe wall and liquid respectively. In computational grids of characteristics, the two waves propagate along two couple characteristics lines respectively. In general, only one couple characteristics lines can be ensured to pass all nodes while another couple can not. So interpolation method seems necessary in the whole process of calculation.

Obviously, artificial numerical dissipation will be introduced into solutions of model if interpolation method is used. To avoid the embarrassment, many alternative methods are proposed by experts at home and abroad. Here four representative schemes are listed as follow: 1) Wiggert, et al [1], applied MOC and modal representation to calculate fluid and pipe equations respectively. However, Poisson coupling effect is not considered in this method. 2) Lavooij, et al [2], suggested MOC-FEM procedure, i.e., MOC for fluid equations and FEM for pipe equations. 3) Zhang xiang lin, et al [3], proposed that both fluid equations and pipe equations are calculated by FEM. FEM is good at dealing with slow transient flow but easy to lead numerical divergence for strong transient flow. 4) Q.S. Li, et al [4], formulated their analytical method. By neglecting friction coupling effect, they obtained the exact analytical solutions of FSI four-equation model and verified their conclusions by experiments. However, the

analytical solutions can hardly be obtained when the effect of friction coupling can not be neglected.

A method using Riemann solutions to solve hyperbolic equations was put forward by Godunov in his doctoral dissertation. At present, Godunov method is used widely in various domains such as aerodynamics, open channel flow et al, because can simulate steep gradients flow and capture shock wave automatically. According to procedure of Godunov method, the discrete form of FSI four-equation model is formulated in this paper and high resolution computational scheme without spurious oscillations is obtained.

II. GOVERNING EQUATIONS OF FSI MODEL The system of FSI four-equation model can be written in

the following conservative form:

( ) 1

t x−∂∂ + =

∂ ∂ΦΦ F

A S (1)

whereVHUσ

=⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

Φ ; ( )Φ ΦF = C ; ( ) ( ) 12

f1 1K K D Eμ δ−∗ = + −⎡ ⎤⎣ ⎦ ;

( ) ( )

2 2

f f

P2

0 0 00 2 0

0 0 0 12 0 2 2 0

ga g a g

DK DK E

μρ

μ δ μ δ∗ ∗

−= −− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

C ;

( ) ( )

2

f

2

f P P

1 0 0 00 0 00 0 1 00 2 0 1

g a

g D E aρ μ δ ρ=

−

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

A ;

( )

( )r r

f P r r

2 sin0

8 sin0

fV V D g

fV V g

θ

ρ ρ δ θ

− +

= ⋅ +

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

S ;V is average velocity of

fluid; H is pressure of fluid;U is average axial velocity of the pipe; σ is average axial normal stress in the pipe wall; fρ is

fluid density; pρ is density of pipe wall material; fK is fluid bulk

modulus; rV is relative velocity between fluid and pipe wall; f is Darcy friction factor; D is internal diameter of pipe;δ is pipe wall thickness; E is Young’s modulus for pipe

2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.273

1103

wall material;θ is angle between pipe and horizontal line; μ is

Poisson’s ratio; g is gravity acceleration; fa is pressure wave

speed without FSI; pa is axial stress wave speed without FSI; x is axial direction and t is time.

III. DISCRETIZATION OF GOVERNING EQUATIONS Based on finite volume method, spatial discrete cells are

introduced and shown as in Fig. 1.

1n

i−Φ niΦ 1

ni+Φ

xΔ

i-1/2i-1 i i+1/2 i+1

Fi-1/2 Fi+1/2

n tΔ

( 1)n t+ Δ

Figure 1. Spatial discrete cells

The center of ith cell is named as node i and its control

volume interface extends from 1/ 2i − to 1/ 2i + . The parameters’ values of this cell are stored in its center and taken as average values of the cell.

Equation (1) is integrated with x from control surface 1/2i− to control surface 1/2i+ . The result is expressed as:

( )1/ 2 1/ 2 1

1/ 2 1/ 2d d

i i

i ix x

t x+ + −

− −

∂ ∂+ =

∂ ∂⎡ ⎤⎢ ⎥⎣ ⎦∫ ∫

Φ ΦFA S (2)

Take iΦ as an average value of cell i, i.e., 1/ 2

1/ 2

1 di

i ix

x+

−=

Δ ∫Φ Φ . Let iΦ in equation (2) and disperse it

further, so the following expression can be obtained:

( ) 1/ 21 1

1/ 2 1/ 2 1/ 2d

in n

i i i i i

t t xx x

++ −

+ − −

Δ Δ= − − +Δ Δ ∫Φ Φ F F A S (3)

where superscript n denotes that computational time t n t= Δ ,

1/ 2i +F and 1/ 2i −F are numerical fluxes at cell interfaces 1/ 2i + and 1/2i− respectively.

So the problem calculating 1n

i

+Φ is transformed into solving

the numerical fluxes 1/ 2i +F and 1/ 2i −F in equation (3).

IV. CALCULATION OF THE NUMERICAL FLUXES Godunov suggested that the fluxes at cell interfaces can be

determined from the exact solution of the Riemann problem. The Riemann problem corresponding to equation (1) is expressed as follows [5]:

( )0

t x∂∂ + =

∂ ∂ΦΦ F

, ( )n x =Φ L

R

n

n

ΦΦ

1/ 2

1/ 2

i

i

x xx x

+

+

<> (4)

where L

nΦ is the mean value of Φ at left side of interface 1/2i+

when t n t= Δ ; R

nΦ is the mean value of Φ at right side of interface 1/2i+ when t n t= Δ .

To interface 1/2i+ , the following expressions can be obtained based on Rankine-Hugoniot condition [6]:

( ) ( ) ( )1/ 2 L 1/ 2 L

n n

i iλ + +− = −Φ Φ Φ ΦF F (5)

( ) ( ) ( )1/ 2 R 1/ 2 R

n n

i iλ + +− = −Φ Φ Φ ΦF F (6) where λ is velocity of shock wave. To FSI four-equation model, λ has four unequal real values:

( )0.52 4 2 2

1,2 f f P0.5 4C q q a aλ = ± = ± − −⎡ ⎤⎣ ⎦ (7)

( )0.52 4 2 2

3,4 P f P0.5 4C q q a aλ = ± = ± + −⎡ ⎤⎣ ⎦ (8)

where 2 2 2 2 2

f P f f P( )q a a Daμ ρ δρ= + + .

Substitution equations (7), (8) and ( )Φ ΦF = C into equations (5) and (6), then the value of parameters at interface

1/2i+ is determined:

1/ 2

1/ 21/ 2 L R

1/ 2

1/ 2

i

n nii

i

i

VHUσ

+

++

+

+

= = +⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

Φ Φ ΦM N (9)

where ( )

f

f

P P

P P

1 0 01 1 0 0

0 0 1 120 0 1

g CC g

CC

ρρ

= −−

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

M ;

( )

f

f

P P

P P

1 0 01 1 0 0

0 0 1 120 0 1

g CC g

CC

ρρ

−−=⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

N .

Using equation (9), the numerical flux 1/ 2i +F at interface 1/2i+ is obtained:

1/ 2 1/ 2 L R

n n

i i+ += = +Φ Φ ΦF C CM CN (10)

According to equation (10), the computational precision of

1/ 2i +F is determined by the precision of L

nΦ and R

nΦ . However, the exact solutions of Riemann problem are usually very complex and each calculation includes so much iteration that it will spend a lot of CPU time. So it is necessary that L

nΦ and

R

nΦ are calculated by an effective approximate scheme.

V. SECOND ORDER SCHEME FOR FLUX TERMS In this paper, MUSCL-Hancock method with second order

in time and space is adopted. It includes three steps [7]:

Step 1, a sectioned linear function is constructed to approximate the exact solution when t = n tΔ :

( ) ( )n n

i i ix r x x= + −Φ Φ , 1/ 2 1/ 2i ix x x− +< < (11)

where ir denotes the slope of center at ith cell. Its left slope

L, 1( )n n

i i ir x−= − ΔΦ Φ and right slope R , 1( )n n

i i ir x+= − ΔΦ Φ .

Thus the approximate solutions at interface 1/ 2i ± can be expressed as follows:

L, 1/ 2 L , R ,0.5 ( , )n n

i i i ixMinmod r r+ = + ΔΦ Φ (12)

R , 1/ 2 L, R ,0.5 ( , )n n

i i i ixMinmod r r− = − ΔΦ Φ (13)

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where subscript L, 1/ 2i + denotes the left side of interface 1/2i+ and subscript R, 1/ 2i − denotes the right side of interface 1/2i− ; Minmod is a slope limiter function which can avoid the

fake numerical oscillation. Its expression is:

( , )Minmod a b = min( , )

0

a a a b⋅

if 0

if 0

ab

ab

>

≤

Step 2, the computational time is evolved by / 2tΔ and the approximate solutions are:

[ ]1/ 2

L , 1/ 2 L , 1/ 2 L, 1/ 2 R , 1/ 20.5n n n n

i i i it x+

+ + + −= − Δ − ΔΦ Φ F F (14)

[ ]1/ 2

R , 1/ 2 R , 1/ 2 L , 1/ 2 R , 1/ 20.5n n n n

i i i it x+

− − + −= − Δ − ΔΦ Φ F F (15)

where L, 1/ 2 L , 1 / 2

n n

i i+ += ΦF C ; R , 1/ 2 R , 1/ 2

n n

i i− −= ΦF C .

Step 3, let 1/ 2

L L, 1/ 2

n n

i

+

±=Φ Φ and 1/ 2

R R , 1/ 2

n n

i

+

±=Φ Φ , so the flux expression at interface 1/ 2i ± is:

1/ 2 1/ 2

1/ 2 L , 1 / 2 R , 1/ 2

n n

i i i

+ +

± ± ±= +Φ ΦF CM CN (16)

Then the numerical fluxes obtained by equation (16) have second order precision in time and space.

VI. CALCULATION OF PARAMETERS IN BOUNDARY CELLS To ensure second order, MUSCL-Hancock method still is

used to calculate parameters in boundary cells. But this method requires information on parameters in neighboring cells of boundary cells. To upstream boundary cell, the neighboring left cell is missing and left flux 1/ 2F can not be obtained. In the

same way, right flux 1/ 2N +F of downstream boundary cell can not be calculated. To solve the problem, virtual cells are introduced at upstream and downstream boundary to restore missing information of parameters, as in Fig. 2.

Figure 2. Virtual cells at the boundaries

Thus all conditions for MUSCL-Hancock method are obtained and detailed steps are as follows:

Step 1, begin calculation when t = 0 and let 0

0Φ and 0

1N +Φ

equal 0

1Φ and 0

NΦ respectively;

Step 2, L,ir at upstream boundary and R ,ir at downstream

boundary are calculated respectively. Then L, 1/ 2

n

i +Φ and R , 1/ 2

n

i−Φ can be obtained (i =1 or N);

Step 3, using equation (14) and (15) to calculate 1/ 2

L , 1 / 2

n

N

+

+Φ

and 1/ 2

R ,1/ 2

n+Φ respectively;

Step 4, using boundary conditions and equation (9), 1/ 2

L ,1 / 2

n+Φ at upstream boundary and 1/ 2

R , 1/ 2

n

N

+

+Φ at downstream boundary can be obtained;

Step 5, 1/ 2F and 1/ 2N +F can be determined by 1/ 2

L ,1 / 2

n+Φ , 1/ 2

R ,1/ 2

n+Φ , 1/ 2

L , 1 / 2

n

N

+

+Φ and 1/ 2

R , 1/ 2

n

N

+

+Φ ;

Step 6, using normal scheme to calculate numerical fluxes

3 / 2F and 1/ 2N −F . Then, substitution 1/ 2F , 3 / 2F , 1/ 2N −F and

1/ 2N +F respectively into Equation (3) to obtain 1

1

n+Φ and 1n

N

+Φ with second order;

Step 7, update parameters at the center of virtual cells, i.e., 1 1/ 2

0 L ,1 / 2

n n+ +=Φ Φ and 1 1/ 2

1 R , 1/ 2

n n

N N

+ +

+ +=Φ Φ . By doing so, it is assured that no non-physical oscillation will be introduced into computational domain;

Step 8, let t equals t t+ Δ and repeat Step 2～8 until calculation is ended.

VII. CALCULATION OF INTEGRATION TERM Inserting fluxes of all interfaces into equation (3) can

obtain the value of parameters in next time step. For

integration term1/ 2 1

1/ 2d

i

ix

+ −

−∫ A S in equation (3), Runge-Kutta

method [8] is used to ensure second order precision in time.

Step 1: ( )1

1/ 2 1/ 2

n n

i i i it x+

+ −= − Δ − ΔF FΦ Φ (17)

Step 2: ( )1 1 1 10.5n n n

i i it+ + − += + Δ A SΦ Φ Φ (18)

Step 3: ( )1 1 1 1n n n

i i it+ + − += + Δ A SΦ Φ Φ (19)

The time step should satisfy the CFL condition, i.e., ( )f PCr max , 1C C t x= Δ Δ ≤ .

VIII. INSTANCE VERIFICATION Experimental data are obtained from the experiment with

respect to FSI in a straight pipe provide by VARDY, A. E. and FAN, D. [9]. Experimental apparatus is composed of a steel pipe and a steel rod. As in Fig. 3, the steel pipe, which is suspended by wires, closed at both ends and filled with pressurized water, generates transients when subjected to axial impact by the steel rod at the impact end.

Figure 3. The sketch of experimental apparatus

Physical parameters of the experimental system are listed in Table I [9].

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TABLE I. PHYSICAL PARAMETERS OF EXPERIMENTAL SYSTEM

Steel pipe Length of pipe L = 4502 mm

Internal diameter D = 52.02 mm Pipe wall thickness δ = 3.945mm

Modulus of elasticity E = 168 GPa Density ρ = 7985 kg/m3

Poisson’s ratio μ = 0.29 Steel rod

Length of rod Lr = 5.006 m Diameter Dr = 50.74 mm

Modulus of elasticity Er = 200 GPa Density ρr = 7848 kg/m3

Impact time Ti = 1.98 ms Impact velocity V0 = 0.739 m/s

Plugs Length of impact end plug Li= 60 mm Mass of impact end plug mi =1.2866 kg

Length of remote end plug Le = 5 mm Mass of remote end plug me = 0.2925 kg

Parameters of water Bulk modulus Kf = 2.14 GPa

Density ρf = 999 kg/m3 Initial pressure P0 = 2 MPa

Dynamic viscosity coefficient υ = 0.001 Pa·s

Stress waves in the pipe wall and pressure waves in the water are generated simultaneously and influenced each other when the steel rod impacts one of the pipe ends. Fluid pressures, axial pipe-wall velocities and axial strains near the impact end of steel pipe are shown in Figure 4, 5 and 6 respectively.

Figure 4. Fluid pressures at impact end of the steel pipe

Figure 5. Axial pipe-wall velocities at impact end of the steel pipe

Figure 6. Axial strains at 0.57 m from impact end of the steel pipe

As in Figure 4, 5 and 6, the present method can agree with experiment very well, which proves the present method is correct. Change of correlative parameters with respect to FSI can be simulated efficiently. However, it also can be seen that there are some differences between the experiment and the calculation. The main reason is that the four-equation model adopts some simplified hypotheses and some nonlinear coupled terms are neglected.

IX. CONCLUSIONS The computational domain is dispersed by finite volume

method in this paper. Based on MUSCL-Hancock method, slope limiter function and Runge-Kutta method, second order numerical results of FSI four-equation model are obtained. For boundary cells, virtual cells are introduced to obtain second order solutions in the whole computational domain. The instance verification shows that the excellent agreement between the simulation and measurement is achieved. So the Godunov-type scheme proposed by this paper is efficient, robust and accurate.

REFERENCES [1] D. C. Wiggert, F. J. Hatfield and S. Stuckenbruck, “Analysis of Liquid

and Structural Transients in Piping by the Method of Characteristics”, Journal of Fluids Engineering, Vol 109, No.2, pp. 161-165, 1987.

[2] C. S. Lavooij and A. S. Tijsseling, “Fluid-Structure Interaction in Liquid-Filled Piping Systems”, Journal of Fluids and Structures, No. 5, pp. 573-595, 1991.

[3] X. L. Zhang, S. H. Huang, Y. G. Wang et al, “The Influence of Fluid-Structure Interaction on Pressure Transients”, Journal of HuaZhong University of Science and Technology, Vol 23, No. 3, pp. 30-33, 1995.

[4] Q. S. Li, K. Yang and L. X. Zhang, “Analytical Solution for Fluid-Structure Interaction in Liquid-Filled Pipes Subjected to Impact-Induced Water Hammer”, Journal of Engineering Mechanics, Vol 129, No. 12, pp. 1408-1417, 2003.

[5] E. F. Toro, “Riemann Solvers and Numerical Methods for Fluid Dynamics (2nd Edition)”, Springer-Verlag, Berlin, 1999.

[6] X. B. Lin, “Generalized Rankine-Hugoniot Condition and Shock Solutions for Quasilinear Hyperbolic Systems”, Journal of Differential Equations, Vol 168, No. 2, pp. 321-354, 2000.

[7] C. Berthon, “Why the MUSCL-Hancock scheme is L1-stable”, Numerische Mathematik, Vol 104, No.1, pp. 27-46, 2006.

[8] E. M. Wahba, “Runge-Kutta Time-Stepping Schemes with TVD Central Differencing for the Water Hammer Equations”, International Journal for Numerical Methods in Fluids, Vol 52, pp. 571-590, 2006.

[9] A. E. Vardy and D. Fan, “Flexural Waves in a Closed Tube”, Proceedings of the 6th International Conference on Pressure Surge, Cambridge, pp. 43-57, 1989.

P (M

Pa)

t (ms)

(m/s

)

t (ms)

axia

l

t (ms)

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