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Research Papers: Multiphase Flows

The Comparison of Viscous Force Approximations of Smoothed Particle Hydrodynamics in Poiseuille Flow Simulation

[+] Author and Article Information
Zhengang Liu

School of Power and Energy,
Northwestern Polytechnical University,
127 West Youyi Road,
Xi'an, Shaanxi 710072, China
e-mail: zgliu@nwpu.edu.cn

Zhenxia Liu

School of Power and Energy,
Northwestern Polytechnical University,
127 West Youyi Road,
Xi'an, Shaanxi 710072, China
e-mail: zxliu@nwpu.edu.cn

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received May 31, 2016; final manuscript received December 13, 2016; published online March 16, 2017. Assoc. Editor: Daniel Livescu.

J. Fluids Eng 139(5), 051302 (Mar 16, 2017) (13 pages) Paper No: FE-16-1335; doi: 10.1115/1.4035635 History: Received May 31, 2016; Revised December 13, 2016

Poiseuille flows at two Reynolds numbers (Re) 2.5 × 10−2 and 5.0 are simulated by two different smoothed particle hydrodynamics (SPH) schemes on regular and irregular initial particles' distributions. In the first scheme, the viscous stress is calculated directly by the basic SPH particle approximation, while in the second scheme, the viscous stress is calculated by the combination of SPH particle approximation and finite difference method (FDM). The main aims of this paper are (a) investigating the influences of two different schemes on simulations and reducing the numerical instability in simulating Poiseuille flows discovered by other researchers and (b) investigating whether the similar instability exists in other cases and comparing results with the two viscous stress approximations. For Re = 2.5 × 10−2, the simulation with the first scheme becomes instable after the flow approaches to steady-state. However, this instability could be reduced by the second scheme. For Re = 5.0, no instability for two schemes is found.

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References

Lucy, L. B. , 1977, “ A Numerical Approach to the Testing of the Fission Hypothesis,” Astron. J., 82, pp. 1013–1024. [CrossRef]
Gingold, R. A. , and Monaghan, J. J. , 1977, “ Smoothed Particle Hydrodynamics—Theory and Application to Non-Spherical Stars,” Mon. Not. R. Astron. Soc., 181(3), pp. 375–389. [CrossRef]
Monaghan, J. J. , 2012, “ Smoothed Particle Hydrodynamics and Its Diverse Applications,” Annu. Rev. Fluid Mech., 44(1), pp. 323–346. [CrossRef]
Shadloo, M. S. , Oger, G. , and Le Touzé, D. , 2016, “ Smoothed Particle Hydrodynamics Method for Fluid Flows, Towards Industrial Applications: Motivations, Current State, and Challenges,” Comput. Fluids, 136, pp. 11–34. [CrossRef]
Monaghan, J. J. , 1994, “ Simulating Free Surface Flows With SPH,” J. Comput. Phys., 110(2), pp. 399–406. [CrossRef]
Farrokhpanah, A. , Samareh, B. , and Mostaghimi, J. , 2015, “ Applying Contact Angle to a Two-Dimensional Multiphase Smoothed Particle Hydrodynamics Model,” ASME J. Fluids Eng., 137(4), p. 041303. [CrossRef]
Sadek, S. H. , and Yildiz, M. , 2013, “ Modeling Die Swell of Second-Order Fluids Using Smoothed Particle Hydrodynamics,” ASME J. Fluids Eng., 135(5), p. 051103. [CrossRef]
Marrone, S. , Colagrossi, A. , Antuono, M. , Colicchio, G. , and Graziani, G. , 2013, “ An Accurate SPH Modeling of Viscous Flows Around Bodies at Low and Moderate Reynolds Numbers,” J. Comput. Phys., 245, pp. 456–475. [CrossRef]
Sefid, M. , Fatehi, R. , and Shamsoddini, R. , 2015, “ A Modified Smoothed Particle Hydrodynamics Scheme to Model the Stationary and Moving Boundary Problems for Newtonian Fluid Flows,” ASME J. Fluids Eng., 137(3), p. 031201. [CrossRef]
Antoci, C. , Gallati, M. , and Sibilla, S. , 2007, “ Numerical Simulation of Fluid–Structure Interaction by SPH,” Comput. Struct., 85, pp. 879–890. [CrossRef]
Liu, G. R. , and Liu, M. B. , 2003, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing, Singapore.
Randles, P. W. , and Libersky, L. D. , 1996, “ Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications,” Comput. Methods Appl. Mech. Eng., 139, pp. 375–408. [CrossRef]
Chen, J. K. , Beraun, J. E. , and Carney, T. C. , 1999, “ A Corrective Smoothed Particle Method for Boundary Value Problems in Heat Conduction,” Int. J. Numer. Methods Eng., 46(2), pp. 231–252. [CrossRef]
Liu, M. B. , Xie, W. P. , and Liu, G. R. , 2005, “ Modeling Incompressible Flows Using a Finite Particle Method,” Appl. Math. Modell., 29(12), pp. 1252–1270. [CrossRef]
Liu, M. B. , and Liu, G. R. , 2006, “ Restoring Particle Consistency in Smoothed Particle Hydrodynamics,” Appl. Numer. Math., 56(1), pp. 19–36. [CrossRef]
Fatehi, R. , and Manzari, M. T. , 2011, “ Error Estimation in Smoothed Particle Hydrodynamics and a New Scheme for Second Derivatives,” Comput. Math. Appl., 61(2), pp. 482–498. [CrossRef]
Morris, J. P. , Fox, P. J. , and Zhu, Y. , 1997, “ Modeling Low Reynolds Number Incompressible Flows Using SPH,” J. Comput. Phys., 136(1), pp. 214–226. [CrossRef]
Sigalotti, L. D. G. , Klapp, J. , Sira, E. , Meleán, Y. , and Hasmy, A. , 2003, “ SPH Simulations of Time-Dependent Poiseuille Flow at Low Reynolds Numbers,” J. Comput. Phys., 191(2), pp. 622–638. [CrossRef]
Liu, M. B. , and Chang, J. Z. , 2010, “ Particle Distribution and Numerical Stability in Smoothed Particle Hydrodynamics Method,” Acta Phys. Sin.-Chin. Ed., 59, pp. 3654–3662.
Monaghan, J. J. , 2000, “ SPH Without a Tensile Instability,” J. Comput. Phys., 159(2), pp. 290–311. [CrossRef]
Meister, M. , Burger, G. , and Rauch, W. , 2014, “ On the Reynolds Number Sensitivity of Smoothed Particle Hydrodynamics,” J. Hydraul. Res., 52(6), pp. 824–835. [CrossRef]
Shadloo, M. S. , Zainali, A. , and Yildiz, M. , 2013, “ Simulation of Single Mode Rayleigh–Taylor Instability by SPH Method,” Comput. Mech., 51(5), pp. 699–715. [CrossRef]
Federico, I. , Marroneb, S. , Colagrossi, A. , Aristodemo, F. , and Antuonoc, M. , 2012, “ Simulating 2D Open-Channel Flows Through an SPH Model,” Eur. J. Mech. - B/Fluids, 34, pp. 35–46. [CrossRef]
Aristodemo, F. , Marrone, S. , and Federico, I. , 2015, “ SPH Modelling of Plane Jets Into Water Bodies Through an Inflow/Outflow,” Ocean Eng., 105, pp. 160–175. [CrossRef]
Marrone, S. , Antuono, M. , Colagrossi, A. , Colicchio, G. , Le Touzé, D. , and Graziani, G. , 2011, “ δ-SPH Model for Simulating Violent Impact Flows,” Comput. Methods Appl. Mech. Eng., 200, pp. 1526–1542. [CrossRef]
Shadloo, M. S. , and Yildiz, M. , 2011, “ Numerical Modeling of Kelvin–Helmholtz Instability Using Smoothed Particle Hydrodynamics,” Int. J. Numer. Methods Eng., 87(10), pp. 988–1006. [CrossRef]
Di Monaco, A. , Manenti, S. , Gallati, M. , Sibilla, S. , Agate, G. , and Guandalini, R. , 2011, “ SPH Modeling of Solid Boundaries Through a Semi-Analytic Approach,” Eng. Appl. Comput. Fluid Mech., 5(1), pp. 1–15.
Gómez-Gesteira, M. , and Dalrymple, R. A. , 2004, “ Using a Three-Dimensional Smoothed Particle Hydrodynamics Method for Wave Impact on a Tall Structure,” J. Waterw., Port, Coastal, Ocean Eng.-ASCE, 130(2), pp. 63–69. [CrossRef]
Monaghan, J. J. , and Kajtar, J. B. , 2009, “ SPH Particle Boundary Forces for Arbitrary Boundaries,” Comput. Phys. Commun., 180(10), pp. 1811–1820. [CrossRef]
White, F. M. , 2006, Viscous Fluid Flow, McGraw-Hill, New York.

Figures

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Fig. 1

The geometry model and boundary conditions

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Fig. 2

The implementation of boundary conditions

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Fig. 3

The regular initial particles' distribution

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Fig. 4

The x-velocity profiles simulated with scheme-2 on the regular initial particles' distribution for Re=2.5×10−2. (a) The numerical results at t=0.1 s are simulated with the smoothing lengths 1.75×10−5m, 2.50×10−5m, 2.70×10−5m, 2.75×10−5m, and 3.75×10−5m, respectively. (b) The numerical results at t=30 s are simulated with the smoothing lengths 2.50×10−5m, 2.70×10−5m, 2.75×10−5m, and 3.75×10−5m, respectively. (c)The numerical results at t=0.1 s are simulated with different resolutions Δx=Δy=5.00×10−5m, Δx=Δy=2.50×10−5m, and Δx=Δy=1.25×10−5m, and the smoothing length is kept as 2.75×10−5m. (d) The numerical results at t=30 s are simulated with different resolutions Δx=Δy=2.50×10−5m and Δx=Δy=1.25×10−5m, and the smoothing length is kept as 2.75×10−5m.

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Fig. 5

The x-velocity profiles for Re=2.5×10−2 at t=0.1 s, 0.2 s, 0.5 s, 1.0 s. The numerical results are simulated with scheme-1 on the regular initial particles' distribution.

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Fig. 6

The x-velocity profiles for Re=2.5×10−2 at t=2.0 s,3.0 s, 4.0 s. The numerical results are simulated with scheme-1 on the regular initial particles' distribution.

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Fig. 7

The x-velocity profiles for Re=2.5×10−2 at t=0.1 s, 0.2 s, 0.5 s, 1.0 s. The numerical results are simulated with scheme-2 on the regular initial particles' distribution.

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Fig. 8

The x-velocity profiles for Re=2.5×10−2 at t=10 s,20 s, 30 s. The numerical results are simulated with scheme-2 on the regular initial particles' distribution.

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Fig. 9

The irregular initial particles' distribution

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Fig. 10

The x-velocity profiles for Re=2.5×10−2 at t=30 s. The numerical results are simulated with scheme-2 on the irregular initial particles' distribution, and the smoothing lengths are 2.70×10−5m, 2.80×10−5m, and 3.75×10−5m, respectively.

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Fig. 11

The x-velocity profiles for Re=2.5×10−2 at t=0.1 s, 0.2 s, 0.5 s, 1.0 s. The numerical results are simulated with scheme-1 on the irregular initial particles' distribution.

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Fig. 12

The x-velocity profiles for Re=2.5×10−2 at t=0.1 s, 0.2 s, 0.5 s, 1.0 s. The numerical results are simulated with scheme-2 on the irregular initial particles' distribution.

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Fig. 13

The x-velocity profiles for Re=2.5×10−2 at t=10 s, 20 s, 30 s. The numerical results are simulated with scheme-2 on the irregular initial particles' distribution.

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Fig. 14

The numerical error of the maximum velocity at different times for Re=2.5×10−2. The numerical results are simulated with scheme-2 on both the regular and irregular initial particles' distributions.

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Fig. 15

The x-velocity profiles for Re = 5.0 at t=3000 s with three different smoothing lengths 2.70×10−5m, 2.75×10−5m, and 3.75×10−5m. The numerical results are simulated (a) with scheme-1 on the regular initial particles' distribution and (b) with scheme-2 on the regular initial particles' distribution.

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Fig. 16

The x-velocity profiles for Re=5.0 at t=10 s, 20 s, 100 s, 230 s. The numerical results are simulated with scheme-1 on the regular initial particles' distribution.

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Fig. 17

The x-velocity profiles for Re=5.0 at t=1000 s, 2000 s, 3000 s. The numerical results are simulated with scheme-1 on the regular initial particles' distribution.

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Fig. 18

The x-velocity profiles for Re=5.0 at t=10 s, 20 s,100s, 230 s. The numerical results are simulated with scheme-2 on the regular initial particles' distribution.

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Fig. 19

The x-velocity profiles for Re=5.0 at t=1000 s, 2000 s,3000 s. The numerical results are simulated with scheme-2 on the regular initial particles' distribution.

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Fig. 20

The x-velocity profiles for Re = 5.0 at t=3000 s with three different smoothing lengths 2.70×10−5m, 2.80×10−5m, and 3.75×10−5m. The numerical results are simulated (a) with scheme-1 on the irregular initial particles' distribution and (b) with scheme-2 on the irregular initial particles' distribution.

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Fig. 21

The x-velocity profiles for Re=5.0 at t=1000 s, 2000 s, 3000 s. The numerical results are simulated with scheme-1 on the irregular initial particles' distribution.

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Fig. 22

The x-velocity profiles for Re=5.0 at t=1000 s, 2000 s,3000 s. The numerical results are simulated with scheme-2 on the irregular initial particles' distribution.

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Fig. 23

The numerical error of the maximum velocity at different times for Re=5.0. The numerical results are simulated with scheme-1 and scheme-2 on both the regular and irregular initial particles' distributions.

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