Cavitating/non-cavitating flows simulation by third-order finite volume scheme and power-law preconditioning method

doi:10.1007/s10483-013-1664-7

Abstract

Abstract: Equations of steady inviscid and laminar flows are solved by means of a third-order finite volume (FV) scheme. For this purpose, a cell-centered discretization technique is employed. In this technique, the flow parameters at the cell faces are computed using a third-order weighted averages procedure. A fourth-order artificial dissipation is used for stability of the solution. In order to achieve the steady-state situation, four-step Runge-Kutta explicit time integration method is applied. An advanced progressive preconditioning method, named the power-law preconditioning method, is used for faster convergence. In this method, the preconditioning matrix is adjusted automatically from the velocity and/or pressure flow-field by a power-law relation. Attention is directed towards accuracy and convergence of the schemes. The results presented in the paper focus on steady inviscid and laminar flows around sheet-cavitating and fully-wetted bodies including hydrofoils and circular/elliptical cylinder. Excellent agreements are obtained
when numerical predictions are compared with other available experimental and numerical results. In addition, it is found that using the power-law preconditioner significantly increases the numerical convergence speed.

Key words: hydrofoil, convergence, finite-volume (FV) scheme, cavitation, third-order accuracy, power-law preconditioner

P. AKBARZADEH. Cavitating/non-cavitating flows simulation by third-order finite volume scheme and power-law preconditioning method. Applied Mathematics and Mechanics (English Edition), 2013, 34(2): 209-228.

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References

[1] Rezgui, A., Cinnella, P., and Lerat, A. Third-order accurate finite volume schemes for Euler computations on curvilinear meshes. Computers & Fluids, 30, 875–901 (2001)

[2] Jameson, A., Schmidt, W., and Turkel, E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA Paper, 1259 (1981)

[3] Turkel, E. Accuracy of schemes with non-uniform meshes for compressible fluid flows. Applied Numerical Mathematics, 2, 529–550 (1985)

[4] Turkel, E., Yaniv, S., and Landau, U. Accuracy of Schemes for the Euler Equations with Nonuniform Meshes, ICASE Report, 85–59 (1985)

[5] Faille, E. A control volume method to solve an elliptic equation on a two-dimensional irregular mesh. Computational Methods in Applied Mechanical Engineering, 100, 275–290 (1992)

[6] Nejat, A. and Gooch, C. O. A high-order accurate unstructured finite volume Newton-Krylov algorithm for inviscid compressible flows. Journal of Computational Physics, 227, 2582–2609 (2008)

[7] Zingg, D., De-Rango, S., Nemec, M., and Pulliam, T. Comparison of several spatial discretizations for the Navier-Stokes equations. Journal of Computational Physics, 160, 683–704 (2000)

[8] Gooch, C. O. and Altena, M. V. A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation. Journal of Computational Physics, 181, 729–752 (2002)

[9] Lee, D. Local Preconditioning of the Euler and Navier-Stokes Equations, Ph. D. dissertation, University of Michigan, Michigan (1996)

[10] Volpe, G. Performance of compressible flow codes at low Mach numbers. AIAA Journal, 31, 49–56 (1993)

[11] Blazek, J. Computational Fluid Dynamics: Principles and Applications, Elsevier, Netherlands (2001)

[12] Chorin, A. J. A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics, 2, 12–26 (1967)

[13] Turkel, E. Preconditioning methods for solving the incompressible and low speed compressible equations. Journal of Computational Physics, 72, 227–298 (1987)

[14] Lee, W. T. Local Preconditioning of the Euler Equations, Ph. D. dissertation, University of Michigan, Michigan (1992)

[15] Choi, Y. H. and Merkle, C. L. Application of preconditioning in viscous flows. Journal of Computational Physics, 105, 207–223 (1993)

[16] Leer, V. B., Lee, W. T., and Roe, P. Characteristic time-stepping or local preconditioning of the Euler equations. AIAA Paper, 1552 (1991)

[17] Leer, V. B. and Lee, D. Progress in local preconditioning of the Euler and Navier-Stokes equations. AIAA Paper, 3328 (1993)

[18] Weiss, J. M. and Smith, W. A. Preconditioning applied to variable and constant density flows. AIAA Journal, 33, 2050–2057 (1993)

[19] Zaccanti, M. R. Analysis and Design of Preconditioning Methods for the Euler Equations, Ph. D. dissertation, College of Engineering Mississippi State, Mississippi (1999)

[20] Malan, A. G., Lewis, R. W., and Nithiarasu, P. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: part I. theory and implementation. International Journal for Numerical Methods in Engineering, 54, 695–714 (2002)

[21] Malan, A. G., Lewis, R. W., and Nithiarasu, P. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: part II. application. International Journal for Numerical Methods in Engineering, 54, 715–729 (2002)

[22] Esfahanian, V. and Akbarzadeh, P. Advanced investigation on design criteria of a robust, artificial compressibility and local preconditioning method for solving the inviscid incompressible flows. Proceeding of the Third International Conference on Modeling, Simulation and Applied Optimization (ICMSA09), Sharjah, U.A.E. (2009)

[23] Esfahanian, V. and Akbarzadeh, P. Local pressure preconditioning method for steady incompressible flows. International Journal of Computational Fluid Dynamics, 24, 169–186 (2010)

[24] Esfahanian, V. and Akbarzadeh, P. Numerical investigation on a new local preconditioning method for solving the incompressible inviscid, noncavitating and cavitating flows. Journal of the Franklin Institute, 348, 1208–1230 (2011)

[25] Esfahanian, V. and Akbarzadeh, P. A local power-law preconditioning method for steady incompressible flows. Second International Conference on Energy Conversion and Conservation (CICME10), Hammamet, Tunisia, 22–25 (2010)

[26] Esfahanian, V. and Akbarzadeh, P. An improved progressive preconditioning method for steady non-cavitating and sheet-cavitating flows. International Journal for Numerical Methods in Fluids, 68, 210–232 (2012)

[27] Xiang, Q., Wu, S. P., Li, C. X., and Cao, N. Low-diffusion preconditioning scheme for numerical simulation of low-speed flows past airfoil. Applied Mathematics and Mechanics (English Edition), 32(5), 613–620 (2011) DOI 10.1007/s10483-011-1443-8

[28] Lessani, B., Ramboer, J., and Lacor, C. Efficient large-eddy simulations of low Mach number flows using preconditioning and multigrid. International Journal of Computational Fluid Dynamics, 18, 221–233 (2004)

[29] Elmahi, I., Gloth, O., Hanel, D., and Vilsmeier, R. A preconditioned dual time-stepping method for combustion problems. International Journal of Computational Fluid Dynamics, 22, 169–181 (2004)

[30] Ahuja, V., Hosangadi, A., and Arunajatesan, S. Simulations of cavitating flows using hybrid unstructured meshes. Journal of Fluids Engineering, 123, 331–340 (2001)

[31] Coutier-Delgosha, O., Fortes-Patella, R., Reboud, J. L., Hakimi, N., and Hirsch, C. Numerical simulation of cavitating flow in 2D and 3D inducer geometries. International Journal for Numerical Methods in Fluids, 48, 135–167 (2005)

[32] Coutier-Delgosha, O., Reboud, J. L., and Delannoy, Y. Numerical simulation of the unsteady behaviour of cavitating flows. International Journal for Numerical Methods in Fluids, 42, 527–548 (2003)

[33] Huang, D. G. Preconditioned dual-time procedures and its application to simulating the flow with cavitations. Journal of Computational Physics, 223, 685–689 (2007)

[34] Kunz, R. F., Boger, D. A., Stinebring, D. R., Chyczewski, T. S., Lindau, J. W. H., Gibeling, J., Venkateswaran, S., and Govindan, T. R. A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction. Computers & Fluids, 29, 849–875 (2000)

[35] Merkle, C. L., Feng, J., and Buelow, P. E. O. Computational modelling of the dynamics of sheet cavitation. 3rd International Symposium on Cavitation, Grenoble, France (1998)

[36] Neaves, M. D. and Edwards, J. R. All-speed time-accurate underwater projectile calculations using a preconditioning algorithm. Journal of Fluids Engineering, 128, 284–296 (2006)

[37] Venkateswaran, S., Lindau, J. W., Kunz, R. F., and Merkle, C. L. Computation of multiphase mixture flows with compressibility effects. Journal of Computational Physics, 180, 54–77 (2002)

[38] Delannoy, Y. and Kueny, J. L. Two phase flow approach in unsteady cavitation modeling, cavitation and multiphase flow forum. ASME-FED, 98, 153–158 (1990)

[39] Esfahanian, V. and Akbarzadeh, P. The Jameson’s numerical method for solving the incompressible viscous and inviscid flows by means of artificial compressibility and preconditioning method. Applied Mathematics and Computation, 206, 651–661 (2008)

[40] Hoffmann, K. A. Computational Fluid Dynamics, Vol. I & II, EES, Wichita (2000)

[41] Turkel, E., Fiterman, A., and Leer, B. V. Preconditioning and the limit to the incompressible flow equations. ICASE Report, No. 191500, NASA, 93-42 (1993)

[42] Turkel, E., Vatsa, V. N., and Radespiel, R. Preconditioning methods for lowspeed flows. ICASE Report, No. 201605, NASA, 96-57 (1996)

[43] Morgan, K., Peraire, J., Peiro, J., and Zienkiewics, O. C. Adaptive remeshing applied to the solution of a shock interaction problem on a cylindrical leading edge. Computational Methods in Aeronautical Fluid Dynamics, Clarenden Press, Oxford, 327–344 (1990)

[44] Goncalves, E. and Patella, R. F. Numerical simulation of cavitating flows with homogeneous models. Computers & Fluids, 38, 1682–1696 (2009)

[45] Coutier-Delgosha, O., Deniset, F., Astolfi, J. A., and Leroux, J. B. Numerical prediction of cavitating flow on a two-dimensional symmetrical hydrofoil and comparison to experiments. Journal of Fluids Engineering, 129, 279–292 (2007)

[46] Coutier-Delgosha, O., Fortes-Patella, R., Reboud, J. L., Hakimi, N., and Hirsch, C. Stability of preconditioned Navier-Stokes equations associated with a cavitation model. Computers & Fluids, 34, 319–349 (2005)

[47] Coutier-Delgosha, O., Fortes-Patella, R., and Reboud, J. L. Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation. Journal of Fluids Engineering, 125, 38–45 (2003)

[48] Coutier-Delgosha, O., Fortes-Patella, R., and Reboud, J. L. Simulation of unsteady cavitation with a two-equation turbulence model including compressibility effects. Journal of Turbulence, 3, 058 (2002)

[49] Coutier-Delgosha, O., Fortes-Patella, R., Reboud, J. L., Hofmann, M., and Stoffel, B. Experimental and numerical studies in a centrifugal pump with two-dimensional curved blades in cavitating condition. Journal of Fluids Engineering, 125, 970–978 (2003)

[50] Leroux, J. B., Coutier-Delgosha, O., and Astolfi, J. A. A joint experimental and numerical study of mechanisms associated to instability of partial cavitation on two-dimensional hydrofoil. Physics of Fluids, 17, 052101 (2005)

[51] Reboud, J. L., Stutz, B., and Coutier, O. Cavitation: experiment and modeling of unsteady effects. 3rd International Symposium on Cavitation, Grenoble, France (1998)

[52] Song, C. and He, J. Numerical simulation of cavitating flows by single-phase flow approach. 3rd International Symposium on Cavitation, Grenoble, France (1998)

[53] Liu, T. G., Khoo, B. C., and Xie, W. F. Isentropic one-fluid modelling of unsteady cavitating flow. Journal of Computational Physics, 201, 80–108 (2004)

[54] Shin, B. R. and Ikohagi, T. Numerical analysis of unsteady cavity flows around a hydrofoil. Proceedings of the 1999 3rd ASME/JSME Joint Fluids Engineering Conference, FEDSM’99, San Francisco, California, 18–23 (1999)

[55] Song, C. S. Current status of CFD for cavitating flows. Proceeding of the 9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawai (2002)

[56] Ventikos, Y. and Tzabiras, G. A numerical method for the simulation of steady and unsteady cavitating flows. Computer & Fluids, 29, 63–88 (2000)

[57] Xie,W. F., Lu, T. G., and Khoo, B. C. Application of a one-fluid model for large scale homogeneous unsteady cavitation: the modified schmidt model. Computer & Fluids, 35, 1177–1192 (2006)

[58] Bernad, S., Susan-Resiga, R., Muntean, S., and Anton, I. Numerical analysis of the cavitating flows. Proceedings of the Romanian Academy, Series A, No. 12006, Romanian (2006)

[59] Hosangadi, A. and Ahuja, V. Numerical study of cavitation in cryogenic fluids. Journal of Fluids Engineering, 127, 267–281 (2005)

[60] Singhal, A. K., Athavale, M. M., Li, H., and Jiang, Y. Mathematical basis and validation of the full cavitation model. Journal of Fluids Engineering, 124, 617–624 (2002)

[61] Watanabe, S., Hidaka, T., Horiguchi, H., Furukawa, A., and Tsujimoto, Y. Steady analysis of the thermodynamic effect of partial cavitation using the singularity method. Journal of Fluids Engineering, 129, 121–127 (2007)

[62] Cheng, X. J. and Lu, C. J. On the partially cavitating flow around two dimensional hydrofoils. Applied Mathematics and Mechanics (English Edition), 21(12), 1450–1459 (2000) DOI 10.1007/BF02459224

[63] Stutz, B. and Reboud, J. L. Two-phase flow structure of sheet cavitation. Physics of Fluids, 9, 3678–3686 (1997)

[64] Gopalan, S. and Katz, J. Flow structure and modeling issues in the closure region of attached cavitation. Physics of Fluids, 12, 895–911 (1999)

[65] Coutier-Delgosha, O., Devillers, J. F., Pichon, T., Vabre, A., Woo, R., and Legoupil, S. Internal structure and dynamics of sheet cavitation. Physics of Fluids, 16, 017103 (2006)

[66] Jameson, A. Steady-state solution of the Euler equations for transonic flow, proceeding of transonic, shock, and multidimensional flows. Advances in Scientific Computing, Academic Press, 37–69 (1982)

[67] Manna, M. Three Dimensional High Resolution Compressible Flow Solver, Ph. D. dissertation, Catholic University of Louvain, Louvain (1992)

[68] Somers, D. M. Design and Experimental Results for the s809, Report of National Renewable Energy Laboratory, NREL/SR-440-6918, NREL (1997)

[69] Hafez, M., Shatalov, A., and Wahba, E. Numerical simulations of incompressible aerodynamic flows using viscous/inviscid interaction procedures. Computer Methods in Applied Mechanics and Engineering, 195, 3110–3127 (2006)

[70] Hafez, M., Shatalov, A., and Nakajima, M. Improved numerical simulations of incompressible flows based on viscous/inviscid interaction procedures. Computer & Fluids, 36, 1588–1591 (2007)

[71] Abdo, M. and Mateescu, D. Low-Reynolds number aerodynamics of airfoils at incidence. 43rd AIAA Aerospace Sciences Meeting and Exhibit-Meeting Papers, Reno, Nevada, 3927–3953 (2005)

[72] Shen, Y. T. and Dimotakis, P. E. The influence of surface cavitation on hydrodynamic forces. 22nd American Towing Tank Conference, St. Johns, 44–53 (1989)

[73] Deshpande, M., Feng, J., and Merkle, C. L. Cavity flow predictions based on the Euler equations. Journal of Fluids Engineering, 116, 36–44 (1994)

[74] Krishnaswamy, P. Flow Modelling for Partially Cavitating Hydrofoils, Ph. D. dissertation, Technical University of Denmark, Denmark (2000)