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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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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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