Richard C. Martineau

Tightly Coupled Multiphysics Algorithms for Pebble Bed Reactors

Abstract We have developed a tightly coupled multiphysics simulation tool for the pebble bed reactor (PBR) concept, a specific type of very high temperature gas-cooled reactor. The simulation tool PRONGHORN takes advantage of the Multiphysics Object-Oriented Simulation Environment library and is capable of solving multidimensional thermal-fluid and neutronics problems implicitly with a Newton-based approach. Expensive Jacobian matrix formation is alleviated via the Jacobian-free Newton-Krylov method, and physics-based preconditioning is applied to minimize Krylov iterations. Motivation for the work is provided via analysis and numerical experiments on simpler multiphysics reactor models. We then provide detail of the physical models and numerical methods in PRONGHORN. Finally, PRONGHORN’s algorithmic capability is demonstrated on a number of PBR test cases.

On physics-based preconditioning of the Navier-Stokes equations

We develop a fully implicit scheme for the Navier-Stokes equations, in conservative form, for low to intermediate Mach number flows. Simulations in this range of flow regime produce stiff wave systems in which slow dynamical (advective) modes coexist with fast acoustic modes. Viscous and thermal diffusion effects in refined boundary layers can also produce stiffness. Implicit schemes allow one to step over the fast wave phenomena (or unresolved viscous time scales), while resolving advective time scales. In this study we employ the Jacobian-free Newton-Krylov (JFNK) method and develop a new physics-based preconditioner. To aid in overcoming numerical stiffness caused by the disparity between acoustic and advective modes, the governing equations are transformed into the primitive-variable form in a preconditioning step. The physics-based preconditioning incorporates traditional semi-implicit and physics-based splitting approaches without a loss of consistency between the original and preconditioned systems. The resulting algorithm is capable of solving low-speed natural circulation problems (M~10^-^4) with significant heat flux as well as intermediate speed (M~1) flows efficiently by following dynamical (advective) time scales of the problem.

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.

Diffusion Acceleration Schemes for Self-Adjoint Angular Flux Formulation with a Void Treatment

Abstract A Galerkin weak form for the monoenergetic neutron transport equation with a continuous finite element method and discrete ordinate method is developed based on self-adjoint angular flux formulation. This weak form is modified for treating void regions. A consistent diffusion scheme is developed with P0 projection. Correction terms of the diffusion scheme are derived to reproduce the transport scalar flux. A source iteration that decouples the solution of all directions with both linear and nonlinear diffusion accelerations is developed and demonstrated. One-dimensional Fourier analysis is conducted to demonstrate the stability of the linear and nonlinear diffusion accelerations. Numerical results of these schemes are presented.

A flexible nonlinear diffusion acceleration method for the SN transport equations discretized with discontinuous finite elements

Abstract This work presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form is based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. While NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.

A Second Order JFNK-Based IMEX Method for Single and Multi-Phase Flows

Abstract We present a second order time accurate IMplicit/EXplicit (IMEX) method for solving single and multi-phase flow problems. The algorithm consists of a combination of an explicit and an implicit blocks. The explicit block solves the non-stiff parts of the governing system whereas the implicit block operates on the stiff terms. In our self-conisstent IMEX implementation, the explicit part is always executed inside the implicit block as part of the nonlinear functions evaluation making use of the Jacobian-freeNewton Krylov (JFNK) method [7]. This leads to an implicitly balanced algorithm in that all non-linearities due to the coupling of different time terms are consistently converged. In this paper, we present computational results when this IMEX strategy is applied to single/multi-phase incompressible flow models. Samet

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.

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.