A simple and effective five-equation two-phase numerical model for liquid-vapor phase transition in cavitating flows
Abstract
Numerical difficulties, notably the non-monotonic behavior of the Wood speed of sound and the volume fraction positivity, associated with the reduced five-equation two-phase flow model of Kapila et al. (2001) [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, 2001. Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10), 3002–3024] have been resolved in the past through the introduction of a frozen speed of sound and an algebraic approach for mechanical relaxation afforded by a pressure non-equilibrium six-equation model proposed by [R. Saurel, F. Petitpas, R.A. Berry, 2009. Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixture, J. Comput. Phys. 228, 1678–1712]. By contrast, it is explored and demonstrated in this work that these difficulties can in fact still be resolved within the numerical scheme for solving the reduced five-equation model by numerically replacing the Wood speed of sound for the estimates of wave speeds in the approximate Riemann solver HLLC with the monotonic mixture speed of sound for a transport five-equation model. For shock interface (artificial mixture separating pure or nearly pure fluids) interaction problems, with the apparent advantage of monotonic behavior of the speedmore »
- Authors:
-
- Florida Institute of Technology, Melbourne, FL (United States). Dept. of Aerospace, Physics and Space Sciences
- Publication Date:
- Research Org.:
- Univ. of Florida, Gainesville, FL (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1850324
- Alternate Identifier(s):
- OSTI ID: 1646725
- Grant/Contract Number:
- NA0002378
- Resource Type:
- Accepted Manuscript
- Journal Name:
- International Journal of Multiphase Flow
- Additional Journal Information:
- Journal Volume: 132; Journal Issue: C; Journal ID: ISSN 0301-9322
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mechanics; Multiphase compressible flow models; Mixture speed of sound; Thermo-chemical relaxation; Cavitation; Phase transition; Five-equation model
Citation Formats
Zhang, Ju. A simple and effective five-equation two-phase numerical model for liquid-vapor phase transition in cavitating flows. United States: N. p., 2020.
Web. doi:10.1016/j.ijmultiphaseflow.2020.103417.
Zhang, Ju. A simple and effective five-equation two-phase numerical model for liquid-vapor phase transition in cavitating flows. United States. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103417
Zhang, Ju. Thu .
"A simple and effective five-equation two-phase numerical model for liquid-vapor phase transition in cavitating flows". United States. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103417. https://www.osti.gov/servlets/purl/1850324.
@article{osti_1850324,
title = {A simple and effective five-equation two-phase numerical model for liquid-vapor phase transition in cavitating flows},
author = {Zhang, Ju},
abstractNote = {Numerical difficulties, notably the non-monotonic behavior of the Wood speed of sound and the volume fraction positivity, associated with the reduced five-equation two-phase flow model of Kapila et al. (2001) [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, 2001. Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10), 3002–3024] have been resolved in the past through the introduction of a frozen speed of sound and an algebraic approach for mechanical relaxation afforded by a pressure non-equilibrium six-equation model proposed by [R. Saurel, F. Petitpas, R.A. Berry, 2009. Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixture, J. Comput. Phys. 228, 1678–1712]. By contrast, it is explored and demonstrated in this work that these difficulties can in fact still be resolved within the numerical scheme for solving the reduced five-equation model by numerically replacing the Wood speed of sound for the estimates of wave speeds in the approximate Riemann solver HLLC with the monotonic mixture speed of sound for a transport five-equation model. For shock interface (artificial mixture separating pure or nearly pure fluids) interaction problems, with the apparent advantage of monotonic behavior of the speed of sound in the interface, the effect of the numerical replacement is also confined to the interface. Differences other than the behavior of the speed of sound within the interface in the solutions due to the replacement diminish with increasing resolution when reasonable solution can be obtained with Wood speed of sound. For cavitating/expansion problems in physical fluid mixture, it is pointed out and explained why acoustics in the numerical solutions still propagate at the Wood speed of sound (therefore consistent with the reduced five-equation model) even though in some cases a much higher speed of sound like the numerical replacement above for solving the reduced five-equation model or the frozen speed of sound for solving a six-equation model is used for the estimates of wave speeds in the HLLC scheme. A variant of the five-equation two-phase flow model by Saurel et al. (2008) [R. Saurel, F. Petitpas, R. Abgrall, 2008. Modelling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607, 313–350] is then constructed for liquid-vapor phase transition in cavitating flows. The relaxation toward thermo-chemical equilibrium during phase transition is achieved by solving a simple system of algebraic equations for the equilibrium state variables for better efficiency, following Pelanti and Shyue (2014) [M. Pelanti, K.-M. Shyue, 2014. A mixture-energy-consistent six-equation two-phase numerical model for fluid with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259, 331–357]. Therefore, the current model retains both the simplicity afforded by the five-equation model and the efficiency of the algebraic relaxation solver. An alternative algebraic approach for handling the non-conservative term (the so-called K∇ · u term) in the reduced five-equation model for mechanical equilibrium of a liquid-vapor mixture is also explored by enforcing the thermal equilibrium at the same time. Finally, numerical results of sample tests in both one and two dimensions in the literature as well as that in three dimensions demonstrate the effectiveness and ability of the proposed model to simulate cavitating flows. An interesting mechanism of shock generation by acoustics in water due to phase transition is then found by the numerical simulations.},
doi = {10.1016/j.ijmultiphaseflow.2020.103417},
journal = {International Journal of Multiphase Flow},
number = C,
volume = 132,
place = {United States},
year = {Thu Jul 30 00:00:00 EDT 2020},
month = {Thu Jul 30 00:00:00 EDT 2020}
}
Works referenced in this record:
A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows
journal, April 1999
- Saurel, Richard; Abgrall, Rémi
- Journal of Computational Physics, Vol. 150, Issue 2
Finite-volume WENO scheme for viscous compressible multicomponent flows
journal, October 2014
- Coralic, Vedran; Colonius, Tim
- Journal of Computational Physics, Vol. 274
Growth-and-collapse dynamics of small bubble clusters near a wall
journal, June 2015
- Tiwari, A.; Pantano, C.; Freund, J. B.
- Journal of Fluid Mechanics, Vol. 775
Effects of air chemistry and stiffened EOS of air in numerical simulations of bubble collapse in water
journal, May 2017
- Zhang, J.; Jackson, T. L.; Jost, A. M. D.
- Physical Review Fluids, Vol. 2, Issue 5
Relaxation Two-Phase flow Models and the Subcharacteristic Condition
journal, December 2011
- FlÅTten, Tore; Lund, Halvor
- Mathematical Models and Methods in Applied Sciences, Vol. 21, Issue 12
Restoration of the contact surface in the HLL-Riemann solver
journal, July 1994
- Toro, E. F.; Spruce, M.; Speares, W.
- Shock Waves, Vol. 4, Issue 1
Numerical modeling of shock-to-detonation transition in energetic materials
journal, April 2012
- Zhang, Ju; Jackson, Thomas L.; Buckmaster, John D.
- Combustion and Flame, Vol. 159, Issue 4
A relaxation-projection method for compressible flows. Part II: Artificial heat exchanges for multiphase shocks
journal, August 2007
- Petitpas, Fabien; Franquet, Erwin; Saurel, Richard
- Journal of Computational Physics, Vol. 225, Issue 2
Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations
journal, October 2001
- Kapila, A. K.; Menikoff, R.; Bdzil, J. B.
- Physics of Fluids, Vol. 13, Issue 10
A general formulation for cavitating, boiling and evaporating flows
journal, April 2016
- Saurel, Richard; Boivin, Pierre; Le Métayer, Olivier
- Computers & Fluids, Vol. 128
The Second Gradient Method for the Direct Numerical Simulation of Liquid–Vapor Flows with Phase Change
journal, May 2001
- Jamet, D.; Lebaigue, O.; Coutris, N.
- Journal of Computational Physics, Vol. 169, Issue 2
On the behaviour of upwind schemes in the low Mach number limit
journal, January 1999
- Guillard, Hervé; Viozat, Cécile
- Computers & Fluids, Vol. 28, Issue 1
Low-Diffusion Flux-Splitting Methods for Real Fluid Flows with Phase Transitions
journal, September 2000
- Edwards, Jack R.; Franklin, Randall K.; Liou, Meng-Sing
- AIAA Journal, Vol. 38, Issue 9
A compressible flow model with capillary effects
journal, October 2005
- Perigaud, Guillaume; Saurel, Richard
- Journal of Computational Physics, Vol. 209, Issue 1
On the modeling and simulation of a laser-induced cavitation bubble: ON THE MODELING AND SIMULATION OF A LASER-INDUCED CAVITATION BUBBLE
journal, April 2013
- Zein, Ali; Hantke, Maren; Warnecke, Gerald
- International Journal for Numerical Methods in Fluids, Vol. 73, Issue 2
Isentropic one-fluid modelling of unsteady cavitating flow
journal, November 2004
- Liu, T. G.; Khoo, B. C.; Xie, W. F.
- Journal of Computational Physics, Vol. 201, Issue 1
Modelling evaporation fronts with reactive Riemann solvers
journal, May 2005
- Le Métayer, O.; Massoni, J.; Saurel, R.
- Journal of Computational Physics, Vol. 205, Issue 2
An interface capturing method for the simulation of multi-phase compressible flows
journal, September 2010
- Shukla, Ratnesh K.; Pantano, Carlos; Freund, Jonathan B.
- Journal of Computational Physics, Vol. 229, Issue 19
Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures
journal, March 2009
- Richard Saurel, ; Petitpas, Fabien; Berry, Ray A.
- Journal of Computational Physics, Vol. 228, Issue 5
Application of a one-fluid model for large scale homogeneous unsteady cavitation: The modified Schmidt model
journal, December 2006
- Xie, W. F.; Liu, T. G.; Khoo, B. C.
- Computers & Fluids, Vol. 35, Issue 10
A Five-Equation Model for the Simulation of Interfaces between Compressible Fluids
journal, September 2002
- Allaire, Grégoire; Clerc, Sébastien; Kokh, Samuel
- Journal of Computational Physics, Vol. 181, Issue 2
Hydrodynamic simulations of bubble collapse and picosecond sonoluminescence
journal, September 1994
- Moss, William C.; Clarke, Douglas B.; White, John W.
- Physics of Fluids, Vol. 6, Issue 9
A diffuse interface model with immiscibility preservation
journal, November 2013
- Tiwari, Arpit; Freund, Jonathan B.; Pantano, Carlos
- Journal of Computational Physics, Vol. 252
A five equation reduced model for compressible two phase flow problems
journal, January 2005
- Murrone, Angelo; Guillard, Hervé
- Journal of Computational Physics, Vol. 202, Issue 2
Modeling phase transition for compressible two-phase flows applied to metastable liquids
journal, April 2010
- Zein, Ali; Hantke, Maren; Warnecke, Gerald
- Journal of Computational Physics, Vol. 229, Issue 8
Sonoluminescence and bubble dynamics for a single, stable, cavitation bubble
journal, June 1992
- Gaitan, D. Felipe; Crum, Lawrence A.; Church, Charles C.
- The Journal of the Acoustical Society of America, Vol. 91, Issue 6
An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries
journal, December 1999
- Ye, T.; Mittal, R.; Udaykumar, H. S.
- Journal of Computational Physics, Vol. 156, Issue 2
Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium
journal, February 2012
- Faccanoni, Gloria; Kokh, Samuel; Allaire, Grégoire
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 5
Total variation diminishing Runge-Kutta schemes
journal, January 1998
- Gottlieb, Sigal; Shu, Chi-Wang
- Mathematics of Computation of the American Mathematical Society, Vol. 67, Issue 221
A critical comparison of smooth and sharp interface methods for phase transition
journal, November 2019
- Rajkotwala, A. H.; Panda, A.; Peters, E. A. J. F.
- International Journal of Multiphase Flow, Vol. 120
A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials
journal, November 1986
- Baer, M. R.; Nunziato, J. W.
- International Journal of Multiphase Flow, Vol. 12, Issue 6
Simplified Second-Order Godunov-Type Methods
journal, May 1988
- Davis, S. F.
- SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 3
Modelling phase transition in metastable liquids: application to cavitating and flashing flows
journal, June 2008
- Saurel, Richard; Petitpas, Fabien; Abgrall, Remi
- Journal of Fluid Mechanics, Vol. 607
Shock interaction with one-dimensional array of particles in air
journal, February 2015
- Sridharan, P.; Jackson, T. L.; Zhang, J.
- Journal of Applied Physics, Vol. 117, Issue 7
Multi-dimensional mesoscale simulations of detonation initiation in energetic materials with density-based kinetics
journal, December 2017
- Jackson, Thomas Luther; Jost, Antoine M. D.; Zhang, Ju
- Combustion Theory and Modelling, Vol. 22, Issue 2
A numerical model for multiphase liquid–vapor–gas flows with interfaces and cavitation
journal, April 2019
- Pelanti, Marica; Shyue, Keh-Ming
- International Journal of Multiphase Flow, Vol. 113
A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves
journal, February 2014
- Pelanti, Marica; Shyue, Keh-Ming
- Journal of Computational Physics, Vol. 259
Towards the direct numerical simulation of nucleate boiling flows
journal, November 2014
- Le Martelot, S.; Saurel, R.; Nkonga, B.
- International Journal of Multiphase Flow, Vol. 66
On the origin of sonoluminescence and sonochemistry
journal, September 1990
- Suslick, K. S.; Doktycz, S. J.; Flint, E. B.
- Ultrasonics, Vol. 28, Issue 5
Dynamic relaxation processes in compressible multiphase flows. Application to evaporation phenomena
journal, July 2013
- Le Métayer, O.; Massoni, J.; Saurel, R.
- ESAIM: Proceedings, Vol. 40