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Title: Pressure evolution equation for the particulate phase in inhomogeneous compressible disperse multiphase flows

Authors:
; ; ;
Publication Date:
Sponsoring Org.:
USDOE
OSTI Identifier:
1342616
Grant/Contract Number:
NA0002378
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Physical Review Fluids
Additional Journal Information:
Journal Volume: 2; Journal Issue: 2; Related Information: CHORUS Timestamp: 2017-02-06 11:58:42; Journal ID: ISSN 2469-990X
Publisher:
American Physical Society
Country of Publication:
United States
Language:
English

Citation Formats

Annamalai, Subramanian, Balachandar, S., Sridharan, P., and Jackson, T. L. Pressure evolution equation for the particulate phase in inhomogeneous compressible disperse multiphase flows. United States: N. p., 2017. Web. doi:10.1103/PhysRevFluids.2.024301.
Annamalai, Subramanian, Balachandar, S., Sridharan, P., & Jackson, T. L. Pressure evolution equation for the particulate phase in inhomogeneous compressible disperse multiphase flows. United States. doi:10.1103/PhysRevFluids.2.024301.
Annamalai, Subramanian, Balachandar, S., Sridharan, P., and Jackson, T. L. Fri . "Pressure evolution equation for the particulate phase in inhomogeneous compressible disperse multiphase flows". United States. doi:10.1103/PhysRevFluids.2.024301.
@article{osti_1342616,
title = {Pressure evolution equation for the particulate phase in inhomogeneous compressible disperse multiphase flows},
author = {Annamalai, Subramanian and Balachandar, S. and Sridharan, P. and Jackson, T. L.},
abstractNote = {},
doi = {10.1103/PhysRevFluids.2.024301},
journal = {Physical Review Fluids},
number = 2,
volume = 2,
place = {United States},
year = {Fri Feb 03 00:00:00 EST 2017},
month = {Fri Feb 03 00:00:00 EST 2017}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1103/PhysRevFluids.2.024301

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  • For the simulation of light water nuclear reactor coolant flows, general two-phase models (valid for all volume fractions) have been generally used which, while allowing for velocity disequilibrium, normally force pressure equilibrium between the phases (see, for example, the numerous models of this type described in H. Städtke, Gasdynamic Aspects of Two-Phase Flow, Wiley-VCH, 2006). These equations are not hyperbolic, their physical wave dynamics are incorrect, and their solution algorithms rely on dubious truncation error induced artificial viscosity to render them numerically well posed over a portion of the computational spectrum. The inherent problems of the traditional approach to multiphasemore » modeling, which begins with an averaged system of (ill-posed) partial differential equations (PDEs) which are then discretized to form a numerical scheme, are avoided by employing a new homogenization method known as the Discrete Equation Method (DEM) (R. Abgrall and R. Saurel, Discrete Equations for Physical and Numerical Compressible Multiphase Mixtures, J. Comp. Phys. 186, 361-396, 2003). This method results in well-posed hyperbolic systems, this property being important for transient flows. This also allows a clear treatment of non-conservative terms (terms involving interfacial variables and volume fraction gradients) permitting the solution of interface problems without conservation errors, this feature being important for the direct numerical simulation of two-phase flows. Unlike conventional methods, the averaged system of PDEs for the mixture are not used, and the DEM method directly obtains a well-posed discrete equation system from the single-phase conservation laws, producing a numerical scheme which accurately computes fluxes for arbitrary number of phases and solves non-conservative products. The method effectively uses a sequence of single phase Riemann problem solutions. Phase interactions are accounted for by Riemann solvers at each interface. Non-conservative terms are correctly approximated. Some of the closure relations missing from the traditional approach are automatically obtained. Lastly, the continuous equation system resulting from the discrete equations can be identified by taking the continuous limit with weak-wave assumptions. In this work, this approach is tested by constructing a DEM model for the flow of two compressible phases in 1-D ducts of spatially varying cross-section with explicit time integration. An analytical equation of state is included for both water vapor and liquid phases, and a realistic interphase mass transfer model is developed based on interphase heat transfer. A robust compliment of boundary conditions are developed and discussed. Though originally conceived as a first step toward implict time integration of the DEM method (to relieve time step size restrictions due to stiffness and to achieve tighter coupling of equations) in multidimensions, this model offers some unique capabilities for incorporation into next generation light water reactor safety analysis codes. We demonstrate, on a converging-diverging two-phase nozzle, that this well-posed, 2-pressure, 2-velocity DEM model can be integrated to a realistic and meaningful steady-state with both phases treated as compressible.« less
  • Numerical approximation of the five-equation two-phase flow of Kapila et al. [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002–3024] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics 202(2) (2005) 664–698; R. Abgrall, V. Perrier, Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows, SIAMmore » Journal of Multiscale and Modeling and Simulation (5) (2006) 84–115; F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248]. However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave’s transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution. Following a strategy developed previously for another flow model [R. Saurel, R. Abgrall, A multiphase Godunov method for multifluid and multiphase flows, Journal of Computational Physics 150 (1999) 425–467], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214–2248] is adapted to the present formulation.« less
  • The relaxation-projection method developed in Saurel et al. [R. Saurel, E. Franquet, E. Daniel, O. Le Metayer, A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations, J. Comput. Phys. (2007) 822-845] is extended to the non-conservative hyperbolic multiphase flow model of Kapila et al. [A.K. Kapila, Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration to detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002-3024]. This model has the ability to treat multi-temperatures mixtures evolving with a single pressure and velocity and is particularly interesting formore » the computation of interface problems with compressible materials as well as wave propagation in heterogeneous mixtures. The non-conservative character of this model poses however computational challenges in the presence of shocks. The first issue is related to the Riemann problem resolution that necessitates shock jump conditions. Thanks to the Rankine-Hugoniot relations proposed and validated in Saurel et al. [R. Saurel, O. Le Metayer, J. Massoni, S. Gavrilyuk, Shock jump conditions for multiphase mixtures with stiff mechanical relaxation, Shock Waves 16 (3) (2007) 209-232] exact and approximate 2-shocks Riemann solvers are derived. However, the Riemann solver is only a part of a numerical scheme and non-conservative variables pose extra difficulties for the projection or cell average of the solution. It is shown that conventional Godunov schemes are unable to converge to the exact solution for strong multiphase shocks. This is due to the incorrect partition of the energies or entropies in the cell averaged mixture. To circumvent this difficulty a specific Lagrangian scheme is developed. The correct partition of the energies is achieved by using an artificial heat exchange in the shock layer. With the help of an asymptotic analysis this heat exchange takes a similar form as the 'pseudoviscosity' introduced by Von Neumann and Richtmyer [J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21 (1950) 232-237]. The present Lagrangian numerical scheme thus combines Riemann solvers and artificial heat exchanges. An Eulerian variant is then obtained by using the relaxation-projection method developed earlier by the authors for the Euler equations. The method is validated against exact solutions based on the multiphase shock relations as well as exact solutions of the Euler equations in the context of interface problems. The method is able to solve interfaces separating pure fluids or heterogeneous mixtures with very large density ratio and with very strong shocks.« less
  • Numerical approximation of the five-equation two-phase flow of Kapila et al. [A.K. Kapila, R. Menikoff, J.B. Bdzil, S.F. Son, D.S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations, Physics of Fluids 13(10) (2001) 3002-3024] is examined. This model has shown excellent capabilities for the numerical resolution of interfaces separating compressible fluids as well as wave propagation in compressible mixtures [A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problems, Journal of Computational Physics 202(2) (2005) 664-698; R. Abgrall, V. Perrier, Asymptotic expansion of a multiscale numerical scheme for compressible multiphase flows, SIAMmore » Journal of Multiscale and Modeling and Simulation (5) (2006) 84-115; F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214-2248]. However, its numerical approximation poses some serious difficulties. Among them, the non-monotonic behavior of the sound speed causes inaccuracies in wave's transmission across interfaces. Moreover, volume fraction variation across acoustic waves results in difficulties for the Riemann problem resolution, and in particular for the derivation of approximate solvers. Volume fraction positivity in the presence of shocks or strong expansion waves is another issue resulting in lack of robustness. To circumvent these difficulties, the pressure equilibrium assumption is relaxed and a pressure non-equilibrium model is developed. It results in a single velocity, non-conservative hyperbolic model with two energy equations involving relaxation terms. It fulfills the equation of state and energy conservation on both sides of interfaces and guarantees correct transmission of shocks across them. This formulation considerably simplifies numerical resolution. Following a strategy developed previously for another flow model [R. Saurel, R. Abgrall, A multiphase Godunov method for multifluid and multiphase flows, Journal of Computational Physics 150 (1999) 425-467], the hyperbolic part is first solved without relaxation terms with a simple, fast and robust algorithm, valid for unstructured meshes. Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness. The algorithm and model are compared to exact solutions of the Euler equations as well as solutions of the five-equation model under extreme flow conditions, for interface computation and cavitating flows involving dynamics appearance of interfaces. In order to deal with correct dynamic of shock waves propagating through multiphase mixtures, the artificial heat exchange method of Petitpas et al. [F. Petitpas, E. Franquet, R. Saurel, O. Le Metayer, A relaxation-projection method for compressible flows. Part II. The artificial heat exchange for multiphase shocks, Journal of Computational Physics 225(2) (2007) 2214-2248] is adapted to the present formulation.« less
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