Abstract
To calculate flows of a gas-liquid mixture, a modified inverse method of characteristics is proposed. An additional fractional time step is introduced in its algorithm, which makes it possible to carry out calculations with a large time step without loss of accuracy and stability. A formulation of boundary conditions on curvilinear walls is discussed in relation to a multidimensional nodal method of characteristics which is based on splitting along the coordinate directions of the original system of equations into a number of one-dimensional subsystems. For the boundary points located on curvilinear impenetrable surfaces, a calculation method based on a method of fictitious nodes is proposed. When testing the modified method, a supersonic interaction of a homogeneous dispersed flow with a barrier is calculated for a flow regime with an attached shock wave. Problems of steady mixture flows near an external obtuse angle, as well as near a cone, which are analogues of Prandtl–Meyer and Busemann flows in gas dynamics, are solved. The calculation results are compared with available self-similar solutions, and a satisfactory agreement is reached.
Similar content being viewed by others
REFERENCES
Surov, V.S., One-Velocity Model of a Heterogeneous Medium with a Hyperbolic Adiabatic Kernel, Comput. Math. Math. Phys., 2008, vol. 48, no. 6, pp. 1048–1062, https://doi.org/10.1134/ S0965542508060146.
Kapila, A., Menikoff, R., Bdzil, J., Son, S., and Stewart, D., Two-Phase Modeling of Deflagration-to-Detonation Transition in Granular Materials: Reduced Equations, Phys. Fluids, 2001, vol. 13, pp. 3002–3024.
Murrone, A. and Guillard, H., A Five Equation Reduced Model for Compressible Two Phase Flow Problems, J. Comput. Phys., 2005, vol. 202, pp. 664–698.
Ma, Z.H., Causon, D.M., Qian, L., Gu, H., Mingham, C.G., and Ferrera, P.M., A GPU Based Compressible Multiphase Hydrocode for Modeling Violent Hydrodynamic Impact Problems, Comput. Fluids, 2015, vol. 120, pp. 1–23.
Surov, V.S., Multidimensional Nodal Method of Characteristics for Hyperbolic Systems, Komp. Issled. Model., 2021, vol. 13, no. 1, pp. 19–32; https://doi.org/10.20537/2076-7633-2021-13-1-19-32.
Surov, V.S., Nodal Method of Characteristics in Multifluid Hydrodynamics, J. Eng. Phys. Thermophys., 2013, vol. 86, no. 5, pp. 1151–1159, https://doi.org/10.1007/s10891-013-0937-5.
Nakamura, T., Tanaka, R., Yabec, T., and Takizawa, K., Exactly Conservative Semi-Lagrangian Scheme for Multi-Dimensional Hyperbolic Equations with Directional Splitting Technique, J. Comput. Phys., 2001, vol. 174, no. 1, pp. 171–207.
Biryukov, V.A., Miryakha, V.A., Petrov, I.B., and Khokhlov, N.I., Simulation of Elastic Wave Propagation in Geological Media: Intercomparison of Three Numerical Methods, Comput. Math. Math. Phys., 2016, vol. 56, no. 6, pp. 1086–1062.
Magomedov, K.M. and Kholodov, A.S., Setochno-kharateristicheskie chislennye metody: uchebnoe posobie dlya vuzov (Grid-Characteristic Numerical Methods: Textbook for Higher Education Institutions), 2nd ed., Moscow: Yurait, 2020.
Sauerwein, H., Numerical Calculations of Multidimensional and Unsteady Flows by the Method of Characteristics, J. Comput. Phys., 1967, vol. 1, pp. 406–432.
Sauer, R., Nichtstationare Probleme der Gasdynamik, Berlin: Springer-Verlag, 1966.
Parpia, I.H., Kentzer, C.P., and Williams, M.H., Multidimensional Time Dependent Method of Characteristics, AIAA J., 1985, vol. 23, no. 10, pp. 1497–1505.
Marcum, D.L. and Hoffman, J.D., Calculation of Three-Dimensional Flowfields by the Unsteady Method of Characteristics, Comput. Fluids, 1988, vol. 16, no. 1, pp. 105–117.
Surov, V.S., Calculation of Heat-Conducting Vapor-Gas-Drop Mixture Flows, Num. An. Appl., 2020, vol. 13, no. 2, pp. 165–179; https://doi.org/10.1134/S199542392002007X.
Kulikovskii, A.G., Pogorelov, N.V., and Semenov, A.Yu., Matematicheskie voprosy chislennogo reshcheniya giperbolicheskikh sistem uravnenii (Mathematical Problems of Numerical Solutions of Hyperbolic Equation Systems), Moscow: Fizmatlit, 2012.
Roache, P., Vychislitel’naya gidrodinamika (Computational Fluid Dynamics), Moscow: Mir, 1980.
Anderson, D.A., Tannehill, J.C., and Pletcher R.H., Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, 1984.
Godunov, S.K., Zabrodin, A.V., Ivanov, M.Ya., Kraiko, A.N., and Prokopov, G.P., Chislennoe reshenie mnogomernykh zadach gazovoi dinamiki (Numerical Solution of Multidimensional Problems of Gas Dynamics), Moscow: Nauka, 1976.
Grishin, Yu.A., Zenkin, V.A., and Khmelev, R.N., Boundary Conditions for Numerical Calculation of Gas Exchange in Piston Engines, J. Eng. Phys. Thermophys., 2017, vol. 90, no. 4, pp. 965–970.
Surov, V.S. On the Problem of Boundary Conditions in the Multidimensional Nodal Method of Characteristics, J. Eng. Phys. Thermophys., 2021, vol. 94, no. 3, pp. 695–701; https://doi.org/10.1007/s10891-021-02346-1.
Surov, V.S., Latent Waves in Heterogeneous Media, J. Eng. Phys. Thermophys., 2014, vol. 87, no. 6, pp. 1463–1468; https://doi.org/10.1007/S10891-014-1151-9.
Surov, V.S., The Godunov Method for Calculating Multidimensional Flows of a One-Velocity Multicomponent Mixture, J. Eng. Phys. Thermophys., 2016, vol. 89, no. 5, pp. 1227–1240; https://doi.org/ 10.1007/s10891-016-1486-5.
Toro, E.F., Riemann Solvers with Evolved Initial Condition, Int. J. Num. Meth. Fluids, 2006, vol. 52, pp. 433–453.
Surov, V.S., Shock Adiabat of a One-Velocity Heterogeneous Medium, J. Eng. Phys. Thermophys., 2006, vol. 79, no. 5, pp. 886–892; https://doi.org/10.1007/s10891-006-0179-x.
Surov, V.S., Certain Self-Similar Problems of Flow of a One-Velocity Heterogeneous Medium, J. Eng. Phys. Thermophys., 2007, vol. 80, no. 6, pp. 1237–1246; https://doi.org/10.1007/s10891-007-0160-3.
Surov, V.S., The Busemann Flow for a One-Velocity Model of a Heterogeneous Medium, J. Eng. Phys. Thermophys., 2007, vol. 80, no. 4, pp. 681–688; https://doi.org/10.1007/s10891-007-0092-y.
Surov, V.S., Hyperbolic Models in the Mechanics of Heterogeneous Media, Comput. Math. Math. Phys., 2014, vol. 54, no. 1, pp. 148–157; https://doi.org/10.1134/S096554251401014X.
Surov, V.S., Calculation of the Elastic Plastic Deformation of a Solid Body by Multidimensional Nodal Method of Characteristics, Vych. Tekhnol., 2021, vol. 26, no. 4, pp. 39–52; https://doi.org/10.25743/ ICT.2021.26.4.005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Zhurnal Vychislitel’noi Matematiki, 2023, Vol. 26, No. 4, pp. 431-450. https://doi.org/10.15372/SJNM20230407.
Publisher’s Note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Surov, V.S. Calculation of Flows of Gas-Liquid Mixtures by a Modified Nodal Method of Characteristics. Numer. Analys. Appl. 16, 359–374 (2023). https://doi.org/10.1134/S1995423923040079
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995423923040079