Ray A. Berry

Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures

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, SIAM 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 waves 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.

Short note: Notes on the PCICE method: Simplification, generalization, and compressibility properties

The pressure-based PCICE numerical method [R.C. Martineau, R.A. Berry, The pressure-corrected ICE finite element method (PCICE-FEM) for compressible flows on unstructured meshes, J. Comput. Phys. 198 (2004) 659] for computing transient fluid flows of all speeds from nearly incompressible to high supersonic with strong shocks is simplified and generalized. Its behavior is examined in the nearly incompressible limit and in the fully compressible limit. In the nearly incompressible limit the PCICE algorithm is found to reduce to a generalization of the incompressible MAC method, which includes the density gradient as a driving function in the pressure Poisson equation. In the fully compressible regime, it is found to reduce to an expression equivalent to density-based methods for high-speed flow.

Particle-Based Direct Numerical Simulation of Contaminant Transport and Deposition in Porous Flow

This work describes an approach to porous flow modeling in which the “micro-length scale to macro-length scale” physical descriptions are addressed as Lagrangian, pore-level flow and transport. The flow features of the physical domain are solved by direct numerical simulation (DNS) with a grid-free, hybrid smoothed particle hydrodynamics (SPH) numerical method (Berry, 2002) based on a local Riemann solution. In addition to being able to handle the large deformation, fluid–fluid and fluid–solid interactions within the contorted geometries of intra- and inter-pore-scale modeling, this Riemann–SPH method should be able to simulate other complexities, such as multiple fluid phases and chemical, particulate, and microbial transport with volumetric and surface reactions. A simple model is presented for the transfer of a contaminant from a carrier fluid to solid surfaces and is demonstrated for flow in a simulated porous media.

RELAP-7 Theory Manual

This document summarizes the physical models and mathematical formulations used in the RELAP-7 code.

Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

In nuclear reactor safety and optimization there are key issues that rely on in-depth understanding of basic two-phase flow phenomena with heat and mass transfer. Within the context of multiphase flows, two bubble-dynamic phenomena – boiling (heterogeneous) and flashing or cavitation (homogeneous boiling), with bubble collapse, are technologically very important to nuclear reactor systems. The main difference between boiling and flashing is that bubble growth (and collapse) in boiling is inhibited by limitations on the heat transfer at the interface, whereas bubble growth (and collapse) in flashing is limited primarily by inertial effects in the surrounding liquid. The flashing process tends to be far more explosive (and implosive), and is more violent and damaging (at least in the near term) than the bubble dynamics of boiling. However, other problematic phenomena, such as crud deposition, appear to be intimately connecting with the boiling process. In reality, these two processes share many details.

Viscous Regularization for the Non-equilibrium Seven-Equation Two-Phase Flow Model

In this paper, a viscous regularization is derived for the non-equilibrium seven-equation two-phase flow model (SEM). This regularization, based on an entropy condition, is an artificial viscosity stabilization technique that selects a weak solution satisfying an entropy-minimum principle. The viscous regularization ensures nonnegativity of the entropy residual, is consistent with the viscous regularization for Euler equations when one phase disappears, does not depend on the spatial discretization scheme chosen, and is compatible with the generalized Harten entropies. We investigate the behavior of the proposed viscous regularization for two important limit-cases. First, a Chapman–Enskog expansion is performed for the regularized SEM and we show that the five-equation flow model of Kapila is recovered with a well-scaled viscous regularization. Second, a low-Mach asymptotic limit of the regularized seven-equation flow model is carried out whereby the scaling of the non-dimensional numbers associated with the viscous terms is determined such that an incompressible two-phase flow model, with a properly scaled regularization, is recovered. Both limit-cases are illustrated with one-dimensional numerical results, including two-phase flow shock tube tests and steady-state two-phase flows in converging-diverging nozzles. A continuous finite element discretization is employed for all numerical simulations.

Preliminary Study on the Suitability of a Second-Order Reconstructed Discontinuous Galerkin Method for RELAP-7 Thermal-Hydraulic Modeling

This document presents a preliminary study on the suitability of a second-order reconstructed discontinuous Galerkin (rDG) method for RELAP-7 thermal-hydraulic modeling. The document begins with a brief description of the governing equations for compressible, two-phase vapor and liquid flow, with a presentation of the seven-equation formulation details. A comparative study between the second-order rDG method and the RELAP-7’s finite element method (FEM) with a entropy viscosity method (EVM) based numerical stabilization scheme (namely FEM-EVM) over a series of benchmark test problems is demonstrated. The intent for this suite of test problems is to provide baseline comparison data that demonstrate the performance of 1) the rDG solution and 2) the RELAP-7’s FEM-EVM solution (with RELAP-7 code version dated August 15, 2017), on problems from singleto specific, limited two-phase flows. For all the test problems in this document, the rDG solutions were obtained with a second-order, two-step, explicit strong stability preserving Runge-Kutta time integration method. The computational results clearly indicate that the performance of the rDG method is superior to that of the RELAP-7’s FEM-EVM method in all the test problems presented. Therefore, as far as the test problems in this document are considered, the second-order rDG method is recommended as an improved solution method option for RELAP-7.

A Well-Posed Two Phase Flow Model and its Numerical Solutions for Reactor Thermal-Fluids Analysis

A 7-equation two-phase flow model and its numerical implementation is presented for reactor thermal-fluids applications. The equation system is well-posed and treats both phases as compressible flows. The numerical discretization of the equation system is based on the finite element formalism. The numerical algorithm is implemented in the next generation RELAP-7 code (Idaho National Laboratory (INL)’s thermal-fluids code) built on top of an other INL’s product, the massively parallel multi-implicit multi-physics object oriented code environment (MOOSE). Some preliminary thermal-fluids computations are presented.

RELAP-7 Progress Report. FY-2015 Optimization Activities Summary

This report summarily documents the optimization activities on RELAP-7 for FY-2015. It includes the migration from the analytical stiffened gas equation of state for both the vapor and liquid phases to accurate and efficient property evaluations for both equilibrium and metastable (nonequilibrium) states using the Spline-Based Table Look-up (SBTL) method with the IAPWS-95 properties for steam and water. It also includes the initiation of realistic closure models based, where appropriate, on the U.S. Nuclear Regulatory Commission’s TRACE code. It also describes an improved entropy viscosity numerical stabilization method for the nonequilibrium two-phase flow model of RELAP-7. For ease of presentation to the reader, the nonequilibrium two-phase flow model used in RELAP-7 is briefly presented, though for detailed explanation the reader is referred to RELAP-7 Theory Manual [R.A. Berry, J.W. Peterson, H. Zhang, R.C. Martineau, H. Zhao, L. Zou, D. Andrs, “RELAP-7 Theory Manual,” Idaho National Laboratory INL/EXT-14-31366(rev. 1), February 2014].

Refined Boiling Water Reactor Station Blackout Simulation with RELAP-7

This is a DOE milestone report to demonstrate refined BWR SBO simulations with the RELAP-7 code.