GPU Implementation for Solving in Compressible Two-Phase Flows
A one-step conservative level set method, combined with a global mass correction method, is developed in this study to simulate the incompressible two-phase flows. The present framework do not need to solve the conservative level set scheme at two separated steps, and the global mass can be exactly conserved. The present method is then more efficient than two-step conservative level set scheme. The dispersion-relation-preserving schemes are utilized for the advection terms. The pressure Poisson equation solver is applied to GPU computation using the pCDR library developed by National Center for High-Performance Computing, Taiwan. The SMP parallelization is used to accelerate the rest of calculations. Three benchmark problems were done for the performance evaluation. Good agreements with the referenced solutions are demonstrated for all the investigated problems.
[1] Anderson C. R., A vortex method for flows with slight density variations,
J. Comput. Phys., 61 (1985) 417-444.
[2] Boultone-Stone J. M., Blake J. R., Gas bubbles bursting at a free surface,
J. Fluid Mech., 254 (1993) 437-466.
[3] Hirt C. W., Nichols B. D., Volume of fluid method (VOF) for the
dynamics of free boundaries, J. Comput. Phys., 39 (1981) 201-225.
[4] Unverdi S., Tryggvason G., A front-tracking method for viscous, incompressible,
multi-fluid flows, J. Comput. Phys., 100 (1992) 25-37.
[5] Sussman M., Smereka P., Axisymmetric free boundary problems, J. Fluid
Mech., 341 (1997) 269-294.
[6] Sethian J. A., Smereka P., Level set methods for fluid interfaces, Annu.
Rev. Fluid Mech., 35 (2003) 341-372.
[7] Sussman M., Puckett E. G., A coupled level set and volume-of-fluid
method for computing 3D and axisymmetric incompressible two-phase
flows, J. Comput. Phys., 162 (2000) 301-337.
[8] Xiao F., Honma Y., Kono T., A simple algebraic interface capturing
scheme using hyperbolic tangent function, Int. J. Numer. Methods Fluid,
48 (2005) 1023-1040.
[9] Yokoi K., Efficient implementation of THINC scheme: A simple and
practical smoothed VOF algorithm, J. Comput. Phys. 226 (2007) 1985-
2002.
[10] Sun D. L., Tao W. Q.. A coupled volume-of-fluid and level set (VOSET)
method for computing incompressible two-phase flows, Int. J. Heat Mass
Tran. 53 (2010) 645-655.
[11] Olsson E., Kreiss G., A conservative level set method for two phase
flow, J. Comput. Phys., 210 (2005) 225-246.
[12] Kr¨uger J., Westermann R., Linear algebra operators for GPU implementation
of numerical algorithms, ACM Trans. Graphics 22 (3) (2003), pp.
908-916.
[13] Goodnight N., Woolley C., Lewin G., Luebke D., Humphreys G.,
A multigrid solver for boundary value problems using programmable
graphics hardware, Graphics Hardware (2003), pp. 1-11.
[14] Harris M. J., Fast fluid dynamics simulation on the GPU, GPU Gems
(2004), pp. 637-665 (Chapter 38).
[15] Hagen T.R., Lie K.A., Natvig J.R., Solving the Euler equations on
graphics processing units, Comput. Sci. ICCS 3994 (2006), pp. 220-227.
[16] Brandvik T., Pullan G., Acceleration of a two-dimensional Euler flow
solver using commodity graphics hardware, Proc. Inst. Mech. Engineers,
Pt C: J. Mech. Engrg. Sci. 221 (12) (2007), pp. 1745-1748.
[17] Corrigan A., Camelli F., L¨ohner R., Wallin J., Running unstructured grid
based CFD solvers on modern graphics hardware, AIAA Paper 2009-
4001, 19th AIAA Computational Fluid Dynamics, June 2009.
[18] Sheu T. W. H., Yu C. H., Chiu P. H., Development of a dispersively
accurate conservative level set scheme for capturing interface in twophase
flows, J. Comput. Phys., 228 (2009) 661-686.
[19] Brackbill J. U., Kothe D. B., Zemach C., A continuum method for
modeling surface tension, J. Comput. Phys., 100 (1992) 335-354.
[20] Tam C. K. W., Webb J. C., Dispersion-relation-preserving finite difference
schemes for computational acoustics, J. Comput. Phys., 107 (1993)
262-281.
[21] Chiu P. H., Sheu T. W.H., Lin R.K., Development of a dispersionrelation-
preserving upwinding scheme for incompressible Navier-Stokes
equations on non-staggered grids, Numer. Heat Transf., B Fundam, 48
(2005) 543-569.
[22] Chiu P. H., Sheu T. W. H., On the development of a dispersionrelation-
preserving dual-compact upwind scheme for convection-diffusion
equation, J. Comput. Phys., 228 (2009) 3640-3655.
[23] M. Peri'c, R. Kessler and G. Scheuerer, Comparison of finite-volume
numerical methods with staggered and colocated grids, Comput. Fluids,
16 (1988) 389-403.
[24] Golub G. H., Huang L. C., Simon H., Tang W. P., A fast Poisson
solver for the finite difference solution of the incompressible Navier-
Stokes equation, SIAM J. Sci. Comput., 19 (1998) 1606-1624.
[25] Martin J. C., Moyce W. J., An experimental study of the collapse of
fluid columns on a rigid horizontal plane, Philos. Trans. Roy. Soc. Lond.:
Ser. A, 244 (1952), 312-324.
[26] Guermond J. L., Quartapelle L., A projection FEM for variable density
incompressible flows, J. Comput. Phys., 165 (2000) 167-188.
[27] Ding H., Spelt P. D. M., Shu C.. Diffuse interface model for incompressible
two-phase flows with large density ratios, J. Comput. Phys. 226
(2007) 2078-2095.
[28] Kuo C. H., Hsieh C. W., Lin R. K., and Sheu W. H. , Solving Burgers-s
Equation Using Multithreading and GPU, (2010) LNCS 6082, pp. 297-
307.
[29] Brereton G., Korotney D., Coaxial and oblique coalescence of two rising
bubbles, In: Sahin, I., Tryggvason, G. (Eds.), Dynamics of Bubbles and
Vortices Near a Free Surface, AMD-vol. 119, 1991, ASME, New York.
[1] Anderson C. R., A vortex method for flows with slight density variations,
J. Comput. Phys., 61 (1985) 417-444.
[2] Boultone-Stone J. M., Blake J. R., Gas bubbles bursting at a free surface,
J. Fluid Mech., 254 (1993) 437-466.
[3] Hirt C. W., Nichols B. D., Volume of fluid method (VOF) for the
dynamics of free boundaries, J. Comput. Phys., 39 (1981) 201-225.
[4] Unverdi S., Tryggvason G., A front-tracking method for viscous, incompressible,
multi-fluid flows, J. Comput. Phys., 100 (1992) 25-37.
[5] Sussman M., Smereka P., Axisymmetric free boundary problems, J. Fluid
Mech., 341 (1997) 269-294.
[6] Sethian J. A., Smereka P., Level set methods for fluid interfaces, Annu.
Rev. Fluid Mech., 35 (2003) 341-372.
[7] Sussman M., Puckett E. G., A coupled level set and volume-of-fluid
method for computing 3D and axisymmetric incompressible two-phase
flows, J. Comput. Phys., 162 (2000) 301-337.
[8] Xiao F., Honma Y., Kono T., A simple algebraic interface capturing
scheme using hyperbolic tangent function, Int. J. Numer. Methods Fluid,
48 (2005) 1023-1040.
[9] Yokoi K., Efficient implementation of THINC scheme: A simple and
practical smoothed VOF algorithm, J. Comput. Phys. 226 (2007) 1985-
2002.
[10] Sun D. L., Tao W. Q.. A coupled volume-of-fluid and level set (VOSET)
method for computing incompressible two-phase flows, Int. J. Heat Mass
Tran. 53 (2010) 645-655.
[11] Olsson E., Kreiss G., A conservative level set method for two phase
flow, J. Comput. Phys., 210 (2005) 225-246.
[12] Kr¨uger J., Westermann R., Linear algebra operators for GPU implementation
of numerical algorithms, ACM Trans. Graphics 22 (3) (2003), pp.
908-916.
[13] Goodnight N., Woolley C., Lewin G., Luebke D., Humphreys G.,
A multigrid solver for boundary value problems using programmable
graphics hardware, Graphics Hardware (2003), pp. 1-11.
[14] Harris M. J., Fast fluid dynamics simulation on the GPU, GPU Gems
(2004), pp. 637-665 (Chapter 38).
[15] Hagen T.R., Lie K.A., Natvig J.R., Solving the Euler equations on
graphics processing units, Comput. Sci. ICCS 3994 (2006), pp. 220-227.
[16] Brandvik T., Pullan G., Acceleration of a two-dimensional Euler flow
solver using commodity graphics hardware, Proc. Inst. Mech. Engineers,
Pt C: J. Mech. Engrg. Sci. 221 (12) (2007), pp. 1745-1748.
[17] Corrigan A., Camelli F., L¨ohner R., Wallin J., Running unstructured grid
based CFD solvers on modern graphics hardware, AIAA Paper 2009-
4001, 19th AIAA Computational Fluid Dynamics, June 2009.
[18] Sheu T. W. H., Yu C. H., Chiu P. H., Development of a dispersively
accurate conservative level set scheme for capturing interface in twophase
flows, J. Comput. Phys., 228 (2009) 661-686.
[19] Brackbill J. U., Kothe D. B., Zemach C., A continuum method for
modeling surface tension, J. Comput. Phys., 100 (1992) 335-354.
[20] Tam C. K. W., Webb J. C., Dispersion-relation-preserving finite difference
schemes for computational acoustics, J. Comput. Phys., 107 (1993)
262-281.
[21] Chiu P. H., Sheu T. W.H., Lin R.K., Development of a dispersionrelation-
preserving upwinding scheme for incompressible Navier-Stokes
equations on non-staggered grids, Numer. Heat Transf., B Fundam, 48
(2005) 543-569.
[22] Chiu P. H., Sheu T. W. H., On the development of a dispersionrelation-
preserving dual-compact upwind scheme for convection-diffusion
equation, J. Comput. Phys., 228 (2009) 3640-3655.
[23] M. Peri'c, R. Kessler and G. Scheuerer, Comparison of finite-volume
numerical methods with staggered and colocated grids, Comput. Fluids,
16 (1988) 389-403.
[24] Golub G. H., Huang L. C., Simon H., Tang W. P., A fast Poisson
solver for the finite difference solution of the incompressible Navier-
Stokes equation, SIAM J. Sci. Comput., 19 (1998) 1606-1624.
[25] Martin J. C., Moyce W. J., An experimental study of the collapse of
fluid columns on a rigid horizontal plane, Philos. Trans. Roy. Soc. Lond.:
Ser. A, 244 (1952), 312-324.
[26] Guermond J. L., Quartapelle L., A projection FEM for variable density
incompressible flows, J. Comput. Phys., 165 (2000) 167-188.
[27] Ding H., Spelt P. D. M., Shu C.. Diffuse interface model for incompressible
two-phase flows with large density ratios, J. Comput. Phys. 226
(2007) 2078-2095.
[28] Kuo C. H., Hsieh C. W., Lin R. K., and Sheu W. H. , Solving Burgers-s
Equation Using Multithreading and GPU, (2010) LNCS 6082, pp. 297-
307.
[29] Brereton G., Korotney D., Coaxial and oblique coalescence of two rising
bubbles, In: Sahin, I., Tryggvason, G. (Eds.), Dynamics of Bubbles and
Vortices Near a Free Surface, AMD-vol. 119, 1991, ASME, New York.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51871", author = "Sheng-Hsiu Kuo and Pao-Hsiung Chiu and Reui-Kuo Lin and Yan-Ting Lin", title = "GPU Implementation for Solving in Compressible Two-Phase Flows", abstract = "A one-step conservative level set method, combined with a global mass correction method, is developed in this study to simulate the incompressible two-phase flows. The present framework do not need to solve the conservative level set scheme at two separated steps, and the global mass can be exactly conserved. The present method is then more efficient than two-step conservative level set scheme. The dispersion-relation-preserving schemes are utilized for the advection terms. The pressure Poisson equation solver is applied to GPU computation using the pCDR library developed by National Center for High-Performance Computing, Taiwan. The SMP parallelization is used to accelerate the rest of calculations. Three benchmark problems were done for the performance evaluation. Good agreements with the referenced solutions are demonstrated for all the investigated problems.
", keywords = "Conservative level set method, two-phase flow, dispersion-relation-preserving, graphics processing unit (GPU), multi-threading.", volume = "5", number = "4", pages = "525-9", }
