The Discrete Equation Method (DEM) for Fully Compressible, Two-Phase Flows in Ducts of Spatially Varying Cross-Section
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 multiphase 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.
- Research Organization:
- Idaho National Lab. (INL), Idaho Falls, ID (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- DE-AC07-05ID14517
- OSTI ID:
- 1000532
- Report Number(s):
- INL/JOU-10-18479; TRN: US1100143
- Journal Information:
- Nuclear Engineering and Design, Vol. 240, Issue 11; ISSN 0029-5493
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
42 ENGINEERING
99 GENERAL AND MISCELLANEOUS
ALGORITHMS
BOUNDARY CONDITIONS
CLOSURES
CONSERVATION LAWS
COOLANTS
DUCTS
HEAT TRANSFER
HOMOGENIZATION METHODS
MASS TRANSFER
MIXTURES
PARTIAL DIFFERENTIAL EQUATIONS
REACTOR SAFETY
REACTORS
TRANSIENTS
TWO-PHASE FLOW
VELOCITY
VISCOSITY
WATER VAPOR
compressible two-phase flow
DEM
discrete equation method
nonequilibrium flow