The constrained interpolation profile method for multiphase analysis. (English) Zbl 1047.76104

Summary: We present a review of the constrained interpolation profile (CIP) method that is known as a general numerical solver for solid, liquid, gas, and plasma problems. This method is a kind of semi-Lagrangian scheme and has been extended to treat incompressible flows in the framework of compressible fluids. Since it uses primitive Euler representation, it is suitable for multiphase analysis. The recent version of this method guarantees the exact mass conservation, even in the framework of a semi-Lagrangian scheme. We provide a comprehensive review of the strategy of CIP method, which has a compact support and subcell resolution, including a front-capturing algorithm with functional transformation, a pressure-based algorithm, and other miscellaneous physics such as the elastic-plastic effect and surface tension. Some practical applications are also reviewed, such as milk crown or coronet, laser-induced melting, and turbulent mixing layer on liquid-gas interface.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76T10 Liquid-gas two-phase flows, bubbly flows


Full Text: DOI


[1] Amsden, A.A.; Harlow, F.H., A simplified MAC technique for incompressible fluid flow calculations, J. comput. phys., 6, 322, (1970) · Zbl 0206.55002
[2] Aoki, T., Multi-dimensional advection of CIP (cubic-interpolated propagation) scheme, Cfd j., 4, 279, (1995)
[3] Bell, J.B.; Marcus, D.L., A second-order projection method for variable-density flows, J. comput. phys., 101, 334, (1992) · Zbl 0759.76045
[4] Bermejo, R.; Staniforth, A., The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes, Mon. weather rev., 120, 2622, (1992)
[5] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modeling surface tension, J. comput. phys., 100, 335, (1992) · Zbl 0775.76110
[6] Colella, P.; Woodward, P.R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. comput. phys., 54, 174, (1984) · Zbl 0531.76082
[7] Ghia, U.; Ghia, K.N.; Shin, C.T., High-resolution for incompressible flow using the Navier-Stokes equations and a multi-grid method, J. comput. phys., 48, 387, (1982) · Zbl 0511.76031
[8] Harlow, F.H.; Shannon, J.P., The splash of a liquid drop, J. appl. phys., 38, 3855, (1967)
[9] Harlow, F.H.; Amsden, A.A., Numerical simulation of almost incompressible flow, J. comput. phys., 3, 80, (1968) · Zbl 0172.52903
[10] Harlow, F.H.; Welch, J.E., Numerical calculation of time-dependent viscous incompressible flow with free surface, Phys. fluids, 8, 2182, (1965) · Zbl 1180.76043
[11] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. phys., 39, 201, (1981) · Zbl 0462.76020
[12] Issa, R.I., Solution of the implicitly discretised fluid flow equations by operator-splitting, J. comput. phys., 62, 40, (1985) · Zbl 0619.76024
[13] D. Jacqmin, A variational approach to deriving smeared interface surface tension models, in, Barriers and Challenges in CFD, edited by, V. Venkatakrishnan, et al., Kluwer Academic, Dordrecht/Norwell, MA, 1998, p, 231.
[14] Karki, K.C.; Patankar, S.V., Pressure-based calculation procedure for viscous flows at all speeds in arbitrary configurations, Aiaa j., 27, 1167, (1989)
[15] Karni, S., Multicomponent flow calculations by a consistent primitive algorithm, J. comput. phys., 112, 31, (1994) · Zbl 0811.76044
[16] Kothe, D.B.; Rider, W.J.; Mosso, S.J.; Brock, J.S., Volume tracking of interfaces having surface tension in two and three dimensions, (1996)
[17] Kudoh, T.; Matsumoto, R.; Shibata, K., Magnetically driven jets from accretion disks. III. 2.5-dimensional nonsteady simulations for thick disk case, Astrophys. J., 508, 186, (1998)
[18] C. E. Leith, Numerical simulation of the earth’s atmosphere, in, Methods in Computational Physics, Vol. 4, edited by, B. Alder, et al., Academic Press, New York, 1965, p, 1.
[19] McMaster, W.H., 1984 computer codes for fluid-structure interactions, (1984)
[20] Mutsuda, H.; Yasuda, T., Physical mechanism based on numerical simulations of strongly turbulent air-water mixing layer within surf-zone, Proceedings 27th int. conf. on coastal eng., Sydney, (1999)
[21] Nakamura, T.; Yabe, T., Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space, Comput. phys. commun., 120, 122, (1999) · Zbl 1001.82003
[22] Von Neumann, J.; Richtmyer, R.D., A method for the numerical calculation of the hydrodynamic shocks, J. appl. phys., 21, 232, (1950) · Zbl 0037.12002
[23] Ogata, Y.; Yabe, T., Shock capturing with improved numerical viscosity in primitive Euler representation, Comput. phys. commun., 119, 179, (1999) · Zbl 1175.76112
[24] Osher, S.; Sethian, J.A., Front propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. comput. phys., 79, 12, (1988) · Zbl 0659.65132
[25] Pan, Y.; Banerjee, S., Numerical simulation of particle interactions with wall turbulence, Phys. fluids, 8, 2733, (1996)
[26] Pan, Y.; Banerjee, S., Numerical investigation of the effects of large particles on wall-turbulence, Phys. fluids, 9, 3786, (1997)
[27] Patanker, S.V., Numerical heat transfer and fluid flow, (1980)
[28] Patanker, S.V.; Spalding, D.B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. heat mass transfer, 15, 1787, (1972) · Zbl 0246.76080
[29] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.K.; Kang, M., A PDE-based fast local level-set method, J. comput. phys., 155, 410, (1999) · Zbl 0964.76069
[30] Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220, (1977) · Zbl 0403.76100
[31] Peskin, C.S.; McQueen, D.M., A three-dimensional computational model for blood flow in the heart. I. immersed elastic fibers in viscous incompressible fluid, J. comput. phys., 81, 372, (1989) · Zbl 0668.76159
[32] Puckett, E.G.; Almgren, A.S.; Bell, J.B.; Marcus, D.L.; Rider, W.J., A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. comput. phys., 130, 269, (1997) · Zbl 0872.76065
[33] Puckett, E.G.; Saltzman, J.S., A 3D adaptive mesh refinement algorithm for multimaterial gas dynamics, Physica D, 60, 84, (1992) · Zbl 0779.76059
[34] Purnell, D.K., Solution of the advective equation by upstream interpolation with a cubic spline, Mon. weather rev., 104, 42, (1975)
[35] Rider, W.J.; Kothe, D.B., Stretching and tearing interface tracking methods, (1995)
[36] Shapiro, S.L.; Teukolsky, S.A., Black holes, white dwarfs, and neutron stars, (1983)
[37] Staniforth, A.; Côtè, J., Semi-Lagrangian integration scheme for atmospheric model—A review, Mon. weather rev., 119, 2206, (1991)
[38] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146, (1994) · Zbl 0808.76077
[39] Sweby, P.K., High-resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. numer. anal., 21, 995, (1984) · Zbl 0565.65048
[40] Takewaki, H.; Nishiguchi, A.; Yabe, T., The cubic-interpolated pseudo-particle (CIP) method for solving hyperbolic-type equations, J. comput. phys., 61, 261, (1985) · Zbl 0607.65055
[41] Takewaki, H.; Yabe, T., Cubic-interpolated pseudo particle (CIP) method—application to nonlinear or multi-dimensional problems, J. comput. phys., 70, 355, (1987) · Zbl 0624.65079
[42] Tanaka, R.; Nakamura, T.; Yabe, T.; Wu, H., A class of conservative formulation of the CIP method, Cfd j., 8, 1, (1999)
[43] Tanaka, R.; Nakamura, T.; Yabe, T., Constructing an exactly conservative scheme in a non-conservative form, Comput. phys. commun., 126, 232, (2000) · Zbl 0959.65097
[44] Toro, E.T., Riemann solvers and numerical methods for fluid dynamics, (1997)
[45] Unverdi, S.O.; Tryggvasson, G.A., A front-tracking method for viscous, incompressible, multi-fluid flows, J. comput. phys., 100, 25, (1992) · Zbl 0758.76047
[46] Utsumi, T.; Kunugi, T.; Aoki, T., Stability and accuracy of the cubic interpolated propagation scheme, Comput. phys. commun., 101, 9, (1996)
[47] Wakisaka, T.; Takeuchi, S.; Chung, J.H., A numerical study on the mixture formation process in fuel injection engines by means of the CIP method, Cfd j., 8, 82, (1999)
[48] Wang, P.Y.; Yabe, T.; Aoki, T., A general hyperbolic solver—the CIP method—applied to curvilinear coordinates, J. phys. soc. Japan, 62, 1865, (1993) · Zbl 0972.65510
[49] Wilkins, M.L., Use of artificial viscosity in multidimensional fluid dynamic calculations, J. comput. phys., 36, 281, (1980) · Zbl 0436.76040
[50] Wilkins, M.L., Computer simulation of dynamic phenomena, (1999) · Zbl 0926.76001
[51] Williamson, D.L.; Rasch, P.J., Two-dimensional semi-Lagrangian transport with shape-preserving interpolation, Mon. weather rev., 117, 102, (1989)
[52] Woodward, P.R.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984) · Zbl 0573.76057
[53] Xiao, F., Numerical scheme for advection equation and multi-layered fluid dynamics, (1996)
[54] Xiao, F.; Yabe, T.; Ito, T., Constructing an oscillation-preventing scheme for the advection equation by a rational function, Comput. phys. commun., 93, 1, (1996) · Zbl 0921.76118
[55] Xiao, F.; Yabe, T.; Nizam, G.; Ito, T., Constructing a multi-dimensional oscillation-preventing scheme for the advection equation by a rational function, Comput. phys. commun., 94, 103, (1996) · Zbl 0926.65100
[56] Xiao, F., An efficient model for driven flow and application to a gas circuit breaker, Comput. model. sim. eng., 1, 235, (1996)
[57] Xiao, F.; Yabe, T.; Ito, T.; Tajima, M., An algorithm for simulating solid objects suspended in stratified flow, Comput. phys. commun., 102, 147, (1997)
[58] Xiao, F.; Yabe, T., Computation of complex flows containing rheological bodies, Cfd j., 8, 43, (1999)
[59] Xu, X.; Nadim, A., Deformation and orientation of an elastic slender body sedimenting in viscous liquid, Phys. fluids, 6, 2889, (1994) · Zbl 0829.76029
[60] Yabe, T.; Takei, E., A new higher-order Godunov method for general hyerbolic equations, J. phys. soc. Japan, 57, 2598, (1988)
[61] Yabe, T.; Aoki, T., A universal solver for hyperbolic-equations by cubic-polynomial interpolation. I. one-dimensional solver, Comput. phys. commun., 66, 219, (1991) · Zbl 0991.65521
[62] Yabe, T.; Ishikawa, T.; Wang, P.Y.; Aoki, T.; Kadota, Y.; Ikeda, F., A universal solver for hyperbolic-equations by cubic-polynomial interpolation. II. two- and three-dimensional solvers, Comput. phys. commun., 66, 233, (1991) · Zbl 0991.65522
[63] Yabe, T.; Wang, P.Y., Unified numerical procedure for compressible and incompressible fluid, J. phys. soc. Japan, 60, 2105, (1991)
[64] Yabe, T.; Xiao, F., Description of complex and sharp interface during shock wave interaction with liquid drop, J. phys. soc. Japan, 62, 2537, (1993)
[65] Yabe, T.; Xiao, F., Description of complex and sharp interface with fixed grids in incompressible and compressible fluid, Computer math. applic., 29, 15, (1995) · Zbl 0816.76075
[66] Yabe, T., Interface capturing and universal solution of solid, liquid and gas by CIP method, Proceedings conf. high-performance computing on multi-phase flow, (1997)
[67] Yabe, T.; Zhang, Y.; Xiao, F., A numerical procedure—CIP—to solve all phases of matter together, Lecture note in physics, 439, (1998)
[68] T. Yabe, Simulation of laser-induced melting and evaporation dynamics by the unified solver CIP for solid, liquid and gas, in, Mathematical Modeling of Weld Phenomena 4, edited by, H. Cerjak, cambridge Univ. Press, Cambridge, U.K, 1998, p, 26.
[69] Yabe, T.; Xiao, F.; Zhang, Y., Strategy for unified solution of solid, liquid, gas and plasmas, (1999)
[70] Yoon, S.Y.; Yabe, T., The unified simulation for incompressible and compressible flow by the predictor – corrector scheme based on the CIP method, Comput. phys. commun., 119, 149, (1999) · Zbl 1175.76119
[71] D. L. Youngs, Time-dependent multi-material flow with large fluid distortion, in, Numerical Methods for Fluid Dynamics, edited by, K. W. Morton and M. J. Baines, Academic Press, New York, 1982, p, 273.
[72] Zhang, Y.; Yabe, T., Effect of ambient gas on three-dimensional breakup in coronet formation, Cfd j., 8, 378, (1999)
[73] Zienkiewicz, O.C.; Wu, J., A general explicit or semi-explicit algorithm for compressible and incompressibe flows, Int. J. num. meth. eng., 35, 457, (1992) · Zbl 0764.76040
[74] Zalesak, S.T., Fully multidimensional flux-corrected transport algorithm for fluids, J. comput. phys., 31, 335, (1979) · Zbl 0416.76002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.