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Research Papers: Techniques and Procedures

# On the Approximation of Two-Dimensional Transient Pipe Flow Using a Modified Wave Propagation Algorithm

[+] Author and Article Information

Department of Civil Engineering,
University of Birjand,
South Khorasan Province,
Birjand 97175615, Iran

Soroosh Sharifi

School of Engineering,
Department of Civil Engineering,
University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK
e-mail: S.Sharifi@bham.ac.uk

Pourya Omidvar

Department of Mechanical Engineering,
Yasouj University,
Daneshjoo Ave,
Yasouj 7591874934, Iran
e-mail: Omidvar@yu.ac.ir

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 30, 2017; final manuscript received January 9, 2018; published online March 16, 2018. Assoc. Editor: Sergio Pirozzoli.

J. Fluids Eng 140(7), 071402 (Mar 16, 2018) (7 pages) Paper No: FE-17-1202; doi: 10.1115/1.4039248 History: Received March 30, 2017; Revised January 09, 2018

## Abstract

In this study, a second-order accurate Godunov-type finite volume method is used for the solution of the two-dimensional (2D) water hammer problem. The numerical scheme applied here is well balanced and is able to treat the unsteady friction terms, together with the convective terms, within the differences between fluxes of neighboring computational cells. In order to consider the effect of unsteady friction terms during the water hammer process, $k−ε$ and $k−ω$ turbulence models are employed. The performance of the proposed method with the choice of different turbulence models is evaluated using experimental data obtained from one low and one high Reynolds-number turbulent test cases. In addition to velocity and pressure distributions, the turbulence characteristics of each variant of the model, including eddy viscosity, dissipation rate, and turbulent kinetic energy during the water hammer process are fully analyzed. It is found that the inclusion of the convective inertia terms leads to more accurate pressure profiles. The results also show that using a relatively high Courant–Friedrichs–Lewy (CFL) number close to unity, the introduced numerical solver with both choices of turbulence models provides reasonable and acceptable predictions for the studied flows.

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## References

Riasi, A. , Nourbakhsh, A. , and Raisee, M. , 2009, “Unsteady Velocity Profiles in Laminar and Turbulent Water Hammer Flows,” ASME J. Fluids Eng., 131(12), p. 121202.
Zhao, M. , and Ghidaoui, M. S. , 2006, “Investigation of Turbulence Behavior in Pipe Transient Using a k–∈Model,” J. Hydraul. Res., 44(5), pp. 682–692.
Shamloo, H. , and Mousavifard, M. , 2015, “Numerical Simulation of Turbulent Pipe Flow for Water Hammer,” ASME J. Fluids Eng., 137(11), p. 111203.
Vardy, A. E. , and Brown, J. M. B. , 1995, “Transient, Turbulent, Smooth Pipe Friction,” J. Hydraul. Res., 33(4), pp. 435–456.
Fan, S. , Lakshminarayana, B. , and Barnett, M. , 1993, “Low-Reynolds-Number k-Epsilon Model for Unsteady Turbulent Boundary-Layer Flows,” AIAA J., 31(10), pp. 1777–1784.
Wahba, E. M. , 2006, “Runge–Kutta Time-Stepping Schemes With TVD Central Differencing for the Water Hammer Equations,” Int. J. Numer. Methods Fluids, 52(5), pp. 571–590.
Wahba, E. M. , 2008, “Modelling the Attenuation of Laminar Fluid Transients in Piping Systems,” Appl. Math. Modell., 32(12), pp. 2863–2871.
Wahba, E. M. , 2015, “On the Propagation and Attenuation of Turbulent Fluid Transients in Circular Pipes,” ASME J. Fluids Eng., 138(3), p. 031106.
Baldwin, B. , and Lomax, H. , 1978, “Thin-Layer Approximation and Algebraic Model for Separated Turbulent Flows,” AIAA Paper No. 78-257.
Zhao, M. , and Ghidaoui, M. S. , 2004, “Godunov-Type Solutions for Water Hammer Flows,” J. Hydraul. Eng., 130(4), pp. 341–348.
Hwang, Y.-H. , and Chung, N.-M. , 2002, “A Fast Godunov Method for the Water-Hammer Problem,” Int. J. Numer. Methods Fluids, 40(6), pp. 799–819.
León, A. , Ghidaoui, M. , Schmidt, A. , and García, M. , 2008, “Efficient Second-Order Accurate Shock-Capturing Scheme for Modeling One- and Two-Phase Water Hammer Flows,” J. Hydraul. Eng., 134(7), pp. 970–983.
Toro, E. F. , 1997, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Tokyo, Japan.
Toro, E. F. , 2001, Shock Capturing Methods for Free Surface Shallow Flows, Wiley, Chichester, UK.
LeVeque, R. J. , 2002, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, New York.
Mahdizadeh, H. , Stansby, P. K. , and Rogers, B. D. , 2011, “On the Approximation of Local Efflux/Influx Bed Discharge in the Shallow Water Equations Based on a Wave Propagation Algorithm,” Int. J. Numer. Methods Fluids, 66(10), pp. 1295–1314.
Mahdizadeh, H. , Stansby, P. K. , and Rogers, B. D. , 2012, “Flood Wave Modeling Based on a Two-Dimensional Modified Wave Propagation Algorithm Coupled to a Full-Pipe Network Solver,” J. Hydraul. Eng., 138(3), pp. 247–259.
Mahdizadeh, H. , 2010, “Modelling of Flood Waves Based on Wave Propagation Algorithms With Bed Efflux and Influx Including a Coupled Pipe Network Solver,” Ph.D. thesis, University of Manchester, Manchester, UK.
George, D. L. , 2008, “Augmented Riemann Solvers for the Shallow Water Equations Over Variable Topography With Steady States and Inundation,” J. Comput. Phys., 227(6), pp. 3089–3113.
Wilcox, D. C. , 2006, Turbulence Modeling for CFD, DCW Industries, La Canada, CA.
Zhao, M. , and Ghidaoui, M. S. , 2003, “Efficient Quasi-Two-Dimensional Model for Water Hammer Problems,” J. Hydraul. Eng., 129(12), pp. 1007–1013.
Ghidaoui, M. S. , Zhao, M. , McInnis, D. A. , and Axworthy, D. H. , 2005, “A Review of Water Hammer Theory and Practice,” ASME Appl. Mech. Rev., 58(1), pp. 49–76.
LeVeque, R. J. , 1998, “Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm,” J. Comput. Phys., 146(1), pp. 346–365.
Bale, D. S. , Leveque, R. J. , Mitran, S. , and Rossmanith, J. A. , 2002, “A Wave Propagation Method for Conservation Laws and Balance Laws With Spatially Varying Flux Functions,” SIAM J. Sci. Comput., 24(3), pp. 955–978.
Launder, B. E. , and Sharma, B. I. , 1974, “Application of the Energy-Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc,” Lett. Heat and Mass Transfer, 1(2), pp. 131–137.
Bergant, A. , Vítkovsky, J. , Simpson, A. , and Lambert, M. , 2001, “Valve Induced Transients Influenced by Unsteady Pipe Flow Friction,” Tenth International Meeting of the Work Group on the Behaviour of Hydraulic Machinery Under Steady Oscillatory Conditions, Trondheim, Norway, June 26–28.
Marcinkiewicz, J. , Adamowski, A. , and Lewandowski, M. , 2008, “Experimental Evaluation of Ability of Relap5, Drako®, Flowmaster2™ and Program Using Unsteady Wall Friction Model to Calculate Water Hammer Loadings on Pipelines,” Nucl. Eng. Des., 238(8), pp. 2084–2093.

## Figures

Fig. 1

Steady-state velocity profile computed using the k−ε and k−ω turbulence models: (a) test case 1 (Re = 5600) and (b) test case 2 (Re = 15,800)

Fig. 2

Comparison between the pressure head computed using the MFWC and MFW methods using the k−ε turbulence model and experimental data for test case 1 (Re = 5600) at (a) the pipe midpoint and (b) the location of the valve

Fig. 3

Comparison between the pressure head values computed using the MFWC and MFW methods using the k−ω turbulence model and experimental data for test case 1 (Re = 5600) at (a) the pipe midpoint and (b) the location of the valve

Fig. 4

Comparison between the pressure head values computed using the MFWC and MFW methods using the k−ε turbulence model and experimental data for test case 2 (Re = 15,800) at (a) the pipe midpoint and (b) the location of the valve

Fig. 5

Comparison between the pressure head values computed using the MFWC and MFW methods using the k−ω turbulence model and experimental data for test case 2 (Re = 15,800) at (a) the pipe midpoint and (b) the location of the valve

Fig. 6

Kinematic eddy viscosity profiles at the pipe midpoint for different water hammer time cycles using the k−ε turbulence model for (a) test case 1 (Re = 5600) and (b) test case 2 (Re = 15,800)

Fig. 7

Kinematic eddy viscosity profiles at the pipe midpoint for different water hammer time cycles using the k−ω turbulence model for (a) test case 1 (Re = 5600) and (b) test case 2 (Re = 15,800)

Fig. 8

Turbulent kinetic energy profiles at the pipe midpoint for different water hammer time cycles using the k−ε turbulence model and MFWC approach for (a) test case 1 (Re = 5600) and (b) test case 2 (Re = 15,800)

Fig. 9

Calculated dissipation rates at the pipe midpoint for different water hammer time cycles using the k−ε turbulence model and MFWC approach for (a) test case 1 (Re = 5600) and (b) test case 2 (Re = 15,800)

Fig. 10

Turbulent kinetic energy calculations at the pipe midpoint for different water hammer time cycles using the k−ω turbulence model and MFWC approach for (a) test case 1 (Re = 5600) and (b) test case 2 (Re = 15,800)

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