Numerical simulations for non conservative hyperbolic system. Application to transient two-phase flow with cavitation phenomenon
Abstract
A numerical method for simulating transient flows of gas-liquid mixtures is proposed. The mathematical model, established for a suspension of gas bubbles in liquid, includes an equation taking into account the relative velocity between the gas and liquid. A numerical technique based on the MacCormack scheme combined with the method of characteristics is presented. Theoretical results for transients initiated by a rapid closing valves are compared with measurements. A good agreement is found particularly for large values of initial dissolved gas concentration.
Keywords:
non conservative system, MacCormack scheme, hyperbolic system, two phase flow, cavitationHow to Cite
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Copyright (c) 2019 The Author(s). Published by Vilnius Gediminas Technical University.
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References
R. Abgrall and S. Karni. A comment on the computation of non-conservative products. Comput. Phys., 229(8): 2759–2763, 2010. https://doi.org/10.1016/j.jcp.2009.12.015
A. Bernard-Champmartin, O. Poujade, J. Mathiaud and J. M. Ghidaglia. Modelling of an homogeneous equilibrium mixture model HEM. Acta Applicandae Mathematicae,129(1):1–21, 2014. https://doi.org/10.1007/s10440-013-9827-2
A. Biesheuvel and L. V. Wijngaarden. Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. Fluid Mech., 148(11):301–318, 1984. https://doi.org/10.1017/S0022112084002366
A. Canestrelli, A. Siviglia, M. Dumbser and E. F. Toro. Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed. Adv. Water Resour., 32(6):834–844, 2009. https://doi.org/10.1016/j.advwatres.2009.02.006
C. E. Castro and E. F. Toro. A Riemann solver and upwind methods for a twophase flow model in non-conservative form. Numer. Meth. Fluids, 50(3):275-307, 2006. https://doi.org/10.1002/fld.1055
P. J. Martínez Ferrer, T. Flåtten and S.T. Munkejord. On the effect of temperature and velocity relaxation in two-phase flow models. ESAIM: M2AN,
(2):411–442, 2012.
U. S. Fjordholm and S. Mishra. Accurate numerical discretizations of nonconservative hyperbolic systems. ESAIM: M2AN, 46(1):187–206, 2012. https://doi.org/10.1051/m2an/2011044
A. Qadi El Idrissi. Ecoulement Transitoire en Conduite avec Prise en Compte de Ph´enom`enes Diphasiques. Thesis, Institut National des Sciences Appliques de Lyon (France), 1996.
A. Jafarian and A. Pisheva. An exact multiphase Riemann solver for compressible cavitating flows. Multiphase Flow, 88:152–166, 2017. https://doi.org/10.1016/j.ijmultiphaseflow.2016.08.001
D. Liuzzi. Two-Phase Cavitation Modelling. Thesis, University of Rome - La Sapienza, 2012.
S. T. Munkejord, S. Evje and T. Flåtten. A Musta scheme for a nonconservative two-fluid model. SIAM J. Sci. Comput., 31(4):2587–2622, 2009. https://doi.org/10.1137/080719273
M. L. Munoz-Ruiz and C. Parés. Godunov method for nonconservative hyperbolic systems. Math. Model. Numer. Anal, 41(1):169–185, 2007. https://doi.org/10.1051/m2an:2007011
C. Parés. Numerical methods for nonconservative hyperbolic systems: A theoretical framework. SIAM J. Numer. Anal., 44(1):300–321, 2006. https://doi.org/10.1137/050628052
A. Prosperetti and L. V. Wijngaarden. On the characteristics of the equations of motion for a bubbly flow and the related problem of critical flow. Eng. Math., 10(2):153–162, 1976. https://doi.org/10.1007/BF01535658
O. V. Vasilyev R. S. Lagumbay and A. Haselbacher. Homogeneous equilibrium mixture model for simulation of multiphase/multicomponent flows. Numer. Meth. Fluids, 00:1–6, 2007.
N. D. Tam. Fluid Transients in Complex Systems with Air Entrainment. Thesis, National University of Singapore, 2009.
H. S. Tang and D. Huang. A second-order accurate capturing scheme for 1D inviscid flows of gas and water with vacuum zones. Comput. Phys., 128:301–318, 1996. https://doi.org/10.1006/jcph.1996.0212
D. C. Wiggert and M. J. Sundquist. The effect of gaseous cavitation on fluid transients. Fluids Eng., 101(1):79–86, 1979. https://doi.org/10.1115/1.3448739
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Copyright (c) 2019 The Author(s). Published by Vilnius Gediminas Technical University.
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This work is licensed under a Creative Commons Attribution 4.0 International License.