Chapter 4 Conclusions
4.2 Recommendation of Future Studies
Based on this study, future work is suggested as follows:
1. To develop a three-dimension virtual mesh refinement module on unstructured mesh in PDSC.
2. To apply the VMR module with PDSC to simulate realistic flow problem.
References
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[2] Bird, G. A., Molecular Gas Dynamics, Clarendon Press, Oxford, UK, 1976.
[3] Bird, G. A., “Monte Carlo Simulation in an Engineering Context”, Progr. Astro. Aero, 74, pp.239-255, 1981.
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[5] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, New York, 1994.
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Eeg. J., 2007 (in press).
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“Implementation of Unsteady Sampling Procedures for the Parallel Direct Simulation Monte Carlo Method”, J. of Computational Physics, 2007 (submitted).
[10] Ghia, U., Ghia, K.N., and Shin, C.T., “High –Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method”. J. of Computational Physics, 48, pp.387-411, 1982.
[11] Huang, J.-C., “A study of instantaneous starting cylinder and shock impinging over wedge flow”, in Proc. 10th National Computational Fluid Dynamics Conference, Hua-Lien, Taiwan, August 2003 (in Chinese).
[12] Kannenberg, K. and Boyd, I. D.,”Strategies for Efficient Particel Resolution in the Direct
Simulation Monte Carlo Method“, Journal of Computational Physics, 157, pp. 727-745, 2000.
[13] Karniadakis, G.E., and Beskok, A., Micro Flows. Fundamentals and Simulation, Springer, New York, 2001.
[14] Nanbu, K., “Theoretical Basis on the Direct Monte Carlo Method”, Rarefied Gas Dynamics, 1, Boffi, V. and Cercignani, C. (editor), Teubner, Stuttgart, 1986.
[15] Robinson, C. D., and Harvey, J. k., “ A parallel DSMC Implementation on Unstructured Meshes with Adaptive Domain Decomposition”, Proceeding of 20th International Symposium on Rarefied Gas Dynamics, pp. 227-232, Shen, C. (editor), Peking University Press, 1996.
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Algorithm and Results using Advanced Computers, pp. 469-476, 1997.
[17] Robinson, C. D., Particle Simulations on Parallel Computers with Dynamic Load Balancing, Imperial College of Science, Technology and Medicine, UK, Ph.D. Thesis, 1998.
[18] Tseng, K.-C., Cave, H. M., Wu, J.-S., Kuo, T.C., Lian, Y.-Y. and Jermy, M. C., “Implementation of a Transient Adaptive Sub-Cells for the Parallel DSMC Code Using Unstructured Grids”, Journal of Fluid Mechanics, 2008 (submitted)
[19] Wagner, W., “A convergence proof for Bird’s Direct simulation Monte Carlo method for the Boltzmann equation”, Journal State Physics, 66(3/4), pp. 1011-1044, 1992.
[20] Wu, J.-S. and Lian, Y.-Y., "Parallel Three-Dimensional Direct Simulation Monte Carlo Method and Its Applications," Computers & Fluids, Vol. 32, Issue 8, pp. 1133-1160, September 2003.
[21] Wu, J.-S., and Tseng, K.-C., “Parallel DSMC Method Using Dynamic Domain Decomposition”, International Journal for Numerical Methods in Engineering, Vol. 63, pp. 37-76, 2005.
[22] Wu, J.-S., Lee, W.-S., Lee, Fred and Wong, S.-C., “Pressure Boundary Treatment In Micromechanical Devices Using Direct Simulation Monte Carlo Method”, JSME International Journal, Series B, 44(3), pp. 439-450, 2001.
[23] Wu, J.-S., and Hsu Y.-L., “Derivation of Variable Soft Sphere Model Parameters in Direct-Simulation Monte Carlo Method Using Quantum Chemistry Computation”, Japanese
[24] Wu, J.-S., Tseng, K.-C., and Wu, F.-Y., “Parallel Three Dimensional Simulation Monte Carlo Method Using Adaptive Mesh and Variable Time Step“, Computer Physics Communications”, Vol. 162, No. 3, pp. 166-187, 2004.
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[26] Wu, J.-S., Lian, Y.-Y., Cheng, G., Koomullil, R. P., and Tseng, K.-C., “Development and Verification of a Coupled DSMC-NS Scheme Using Unstructured Mesh“, Journal of Computational Physics, Vol. 219, No. 2, pp. 579-607, 2006.
[27] Xu, D.Q., Honma, H., and Abe, T., “DSMC approach to nonstationary Mach reflection of strong incoming shock waves using a smoothing technique”, Shock Waves, 3(1), 67, 1993.
Tables
Table I The simulation condition of the upstream for 2-D hypersonic flow over a block.
Gas Mach No. Kn Velocity (m/s)
Temp.
(K)
Density (kg m/ 3)
Number density
( particles m ) / 3 λ (m)
Argon 12 0.05 1413 40 8.6043E-5 1.29E+21 0.001
Table II The simulation condition of different cases for 2-D flow over a block.
Case Cells Size per cell Average particles per cell
Total drag (kg m s⋅ / 2)
Benchmark 48000 1/4 mfp 36 3.06864
VMR 3000 1 mfp 36 3.14781
TAS 3000 1 mfp 36 2.85395
None 3000 1 mfp 36 2.88059
Table III The simulation condition of the upstream for 2-D hypersonic flow over a cylinder.
Gas Mach No. Kn Velocity (m/s)
Temp.
(K)
Density (kg m ) / 3
Number density
(particles m ) / 3 λ (m)
Argon 10 0.0091 2634.1 200 2.8507E-5 4.274E+20 0.003
Table IV The simulation condition of different cases with quadrilateral mesh for 2-D flow over a cylinder.
Case Geometry Cells Size per cell
Average particles per
cell
Total drag (kg m s⋅ / 2)
Computational time (hr)
Benchmark quadrilateral 195000 1/5~1/2 mfp 47 40.22456 15
VMR quadrilateral 7650 1~3 mfp 40 40.2736 3.5
TAS quadrilateral 7650 1~3 mfp 40 40.75246 0.5
None quadrilateral 7650 1~3 mfp 40 42.73472 0.5
Table V The simulation condition of different cases with triangular mesh for 2-D flow over a cylinder.
Case Geometry Cells Size per cell
Average particles per
cell
Total drag (kg m s⋅ / 2)
Computational time (hr)
Benchmark quadrilateral 195000 1/5~1/2 mfp 47 40.22456 15
VMR triangular 9802 1~3 mfp 43 40.0359 4.5
TAS triangular 9802 1~3 mfp 43 40.4769 0.667
None triangular 9802 1~3 mfp 43 41.14262 0.667
Table VI The simulation condition of different cases with mixed quadrilateral-trianglar mesh for 2-D flow over a cylinder.
Case Geometry Cells Size per cell
Average particles per
cell
Total drag (kg m s⋅ / 2)
Computational time (hr)
Benchmark quadrilateral 195000 1/5~1/2 mfp 47 40.22456 15
VMR mixed 12825 1~2 mfp 40 40.14168 5
TAS mixed 12825 1~2 mfp 40 40.37664 1
None mixed 12825 1~2 mfp 40 41.15758 1
Figure
Discrete Particle or Molecular
Model Boltzmann equation Collisionless
Boltzmann Equation
Continuum Model Equation Euler
Navier
Limit Local Kundsen Number Free-molecule
Limit
Fig. 2.1 Classifications of Flow Region.
move all molecules
Fig. 2.2 The flowchart of the standard DSMC method.
INITIALIZE
Fig. 2.3 Simplified flow chart of the parallel DSMC method for np processors.
Fig. 2.4 The additional schemes in the parallel DSMC code.
Fig. 2.5 Division of structured and unstructured elements into sub-cells.
Sampling cell boundary Sub-cell
Structured element Unstructured element
Sub-cell which will be empty
Sampling cell boundary Sub-cell
Fig. 2.6 The flowchart of DSMC simulation using virtual mesh refinement module.
Fig. 2.7 Division of structured and unstructured elements into refined cells. Refined background cell
Virtual cell
Sub-cell Virtual cell
Sub-cell
Virtual cell which will be empty
Structured element Unstructured element
Refined background cell
Fig. 2.8 Distribution of random number in the refined background cell.
(a)
(b)
(c)
Fig. 2.9 Evolution of domain decomposition using 64 processors during the simulation: (a) initial; (b) intermediate; (c) final.
Flow Condition:
Argon Gas Mach number: 12 Velocity: 1413 m/s Kn: 0.05
Temperature: 40K
Number density: 1.29E21 #/m3 Mean free path: 0.001 m
Fig. 3.1 Sketch of the computational domain of a argon hypersonic flow over a block (Ar gas, Kn∞=0.05,M∞=12, T∞=40 K, n∞=1.29E21 particles/m3)
Fig. 3.2 Computational domain of the benchmark (each cell size is 1/4 mean free path).
(a)
(b)
(c)
(d)
Fig. 3.3 Contours of computational results of the benchmark: (a) density; (b) temperature; (c) velocity in x-direction; (d) velocity in y-direction.
Fig. 3.4 Computation domain of VMR, TAS and None (each cell size is one mean free path).
Fig. 3.5 Compared contour of density of the benchmark, VMR, TAS and None.
Fig. 3.6 Compared contour of temperature of the benchmark, VMR, TAS and None.
Fig. 3.7 Compared contour of u-velocity of the benchmark, VMR, TAS and None.
Fig. 3.8 Compared contour of v-velocity of the benchmark, VMR, TAS and None.
(a)
(b)
(c)
(d)
Fig. 3.9 Contour of mcs/mpfs: (a) benchmark; (b) VMR; (c) TAS; (d) None.
(a) (b)
(c) (d)
Fig. 3.10 Profile of the benchmark, VMR, TAS, and None along x=0.01 m: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.11 Profile of the benchmark, VMR, TAS, and None along x=0.005 m: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.12 Profile of the benchmark, VMR, TAS, and None along x=0.0005 m: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.13 Profile of the benchmark, VMR, TAS, and None along y=0.02 m: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
Fig. 3.14 Compared local pressure coefficient along x=0 m on block.
Fig. 3.15 Compared local friction coefficient along x=0 m on block.
Fig. 3.16 Compared local pressure coefficient along y=0.01 m on block.
Fig. 3.17 Compared local friction coefficient along y=0.01 m on block.
Flow Condition:
Argon Gas Mach number: 10 Velocity: 2634.1 m/s Kn: 0.0091
Temperature: 200K
Number density: 4.274E20 #/m3 Mean free path: 0.003 m
Fig. 3.18 Sketch of the computational domain of a argon hypersonic flow over a cylinder (Ar gas, Kn∞=0.0091, M∞=10, T∞=200 K, n∞=4.274E20, particles/m3)
Fig. 3.19 Computational domain of the benchmark.
(a)
(b)
(c)
(d)
Fig. 3.20 Contours of computational results of the benchmark: (a) density; (b) temperature;
(c) velocity in x-direction; (d) velocity in y-direction.
Fig. 3.21 Using quadrilateral computation domain of VMR, TAS and None.
Fig. 3.22 Compared contour of density of the benchmark, VMR, TAS and None with quadrilateral mesh.
Fig. 3.23 Compared contour of temperature of the benchmark, VMR, TAS and None with quadrilateral mesh.
Fig. 3.24 Compared contour of u-velocity of the benchmark, VMR, TAS and None with quadrilateral mesh.
Fig. 3.25 Compared contour of v-velocity of the benchmark, VMR, TAS and None with quadrilateral mesh.
(a)
(b)
(c)
(d)
Fig. 3.26 Contour of mcs/mpfs with quadrilateral mesh: (a) benchmark; (b) VMR; (c) TAS;
(d) None.
(a) (b)
(c) (d)
Fig. 3.27 Profile of the benchmark, VMR, TAS, and None along x=0.005 m with quadrilateral mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.28 Profile of the benchmark, VMR, TAS, and None along x=0.4 m with quadrilateral mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.29 Profile of the benchmark, VMR, TAS, and None along x=0.5 m with quadrilateral mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.30 Profile of the benchmark, VMR, TAS, and None along y=0.2 m with quadrilateral mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c)
Fig. 3.31 Compared local coefficient along surface of cylinder with quadrilateral mesh: (a) pressure coefficient; (b) friction coefficient; (c) heat transfer coefficient.
Fig. 3.32 Using triangular computation domain of VMR, TAS and None.
Fig. 3.33 Compared contour of density of the benchmark, VMR, TAS and None with triangular mesh.
Fig. 3.34 Compared contour of temperature of the benchmark, VMR, TAS and None with triangular mesh.
Fig. 3.35 Compared contour of u-velocity of the benchmark, VMR, TAS and None with triangular mesh.
Fig. 3.36 Compared contour of v-velocity of the benchmark, VMR, TAS and None with triangular mesh.
(a)
(b)
(c)
(d)
Fig. 3.37 Contour of mcs/mpfs with triangular mesh: (a) benchmark; (b) VMR; (c) TAS; (d) None.
(a) (b)
(c) (d)
Fig. 3.38 Profile of the benchmark, VMR, TAS, and None along x=0.005 m with triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.39 Profile of the benchmark, VMR, TAS, and None along x=0.4 m with triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.40 Profile of the benchmark, VMR, TAS, and None along x=0.5 m with triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.41 Profile of the benchmark, VMR, TAS, and None along y=0.2 m with triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c)
Fig. 3.42 Compared local coefficient along surface of cylinder with triangular mesh: (a) pressure coefficient; (b) friction coefficient; (c) heat transfer coefficient.
Fig. 3.43 Using mixed quadrilateral-triangular computation domain of VMR, TAS and None.
Fig. 3.44 Compared contour of density of the benchmark, VMR, TAS and None with mixed quadrilateral-triangular mesh.
Fig. 3.45 Compared contour of temperature of the benchmark, VMR, TAS and None with mixed quadrilateral-triangular mesh.
Fig. 3.46 Compared contour of u-velocity of the benchmark, VMR, TAS and None with mixed quadrilateral-triangular mesh.
Fig. 3.47 Compared contour of v-velocity of the benchmark, VMR, TAS and None with mixed quadrilateral-triangular mesh.
(a)
(b)
(c)
(d)
Fig. 3.48 Contour of mcs/mpfs with mixed quadrilateral-triangular mesh: (a) benchmark; (b) VMR; (c) TAS; (d) None.
(a) (b)
(c) (d)
Fig. 3.49 Profile of the benchmark, VMR, TAS, and None along x=0.005 m with mixed quadrilateral-triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.50 Profile of the benchmark, VMR, TAS, and None along x=0.4 m with mixed quadrilateral-triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.51 Profile of the benchmark, VMR, TAS, and None along x=0.5 m with mixed quadrilateral-triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c) (d)
Fig. 3.52 Profile of the benchmark, VMR, TAS, and None along y=0.2 m with mixed quadrilateral-triangular mesh: (a) density; (b) temperature; (c) u-velocity; (d) v-velocity.
(a) (b)
(c)
Fig. 3.53 Compared local coefficient along surface of cylinder with mixed
quadrilateral-triangular mesh: (a) pressure coefficient; (b) friction coefficient; (c) heat transfer coefficient.
(a) (b)
(c)
Fig. 3.54 Compared local coefficient along surface of cylinder with different grids, include quadrilateral, triangular and mixed quadrilateral-triangular grids: (a) pressure coefficient; (b) friction coefficient; (c) heat transfer coefficient.