skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics

Abstract

We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification ofmore » the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below M=10-5, which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the AUSM + -up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.« less

Authors:
ORCiD logo [1]; ORCiD logo [1];  [1];  [1];  [1];  [1]; ORCiD logo [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Design Physics Dept.
  2. Univ. of California, Davis, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1562804
Report Number(s):
LLNL-JRNL-745590
Journal ID: ISSN 0938-1287; 900894; TRN: US2000762
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Shock Waves
Additional Journal Information:
Journal Volume: 29; Journal Issue: 5; Journal ID: ISSN 0938-1287
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; reconstructed/recovery discontinuous Galerkin method; all-speed AUSM+-up scheme; fully implicit; Jacobian-free Newton-Krylov; preconditioning

Citation Formats

Nourgaliev, R., Greene, P., Weston, B., Barney, R., Anderson, A., Khairallah, S., and Delplanque, J. -P. High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics. United States: N. p., 2018. Web. doi:10.1007/s00193-018-0871-8.
Nourgaliev, R., Greene, P., Weston, B., Barney, R., Anderson, A., Khairallah, S., & Delplanque, J. -P. High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics. United States. https://doi.org/10.1007/s00193-018-0871-8
Nourgaliev, R., Greene, P., Weston, B., Barney, R., Anderson, A., Khairallah, S., and Delplanque, J. -P. Wed . "High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics". United States. https://doi.org/10.1007/s00193-018-0871-8. https://www.osti.gov/servlets/purl/1562804.
@article{osti_1562804,
title = {High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics},
author = {Nourgaliev, R. and Greene, P. and Weston, B. and Barney, R. and Anderson, A. and Khairallah, S. and Delplanque, J. -P.},
abstractNote = {We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification of the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below M=10-5, which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the AUSM+ -up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.},
doi = {10.1007/s00193-018-0871-8},
url = {https://www.osti.gov/biblio/1562804}, journal = {Shock Waves},
issn = {0938-1287},
number = 5,
volume = 29,
place = {United States},
year = {2018},
month = {11}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share:

Works referenced in this record:

A pseudocompressibility method for the numerical simulation of incompressible multifluid flows
journal, July 2004


A Parallel Reconstructed Discontinuous Galerkin Method for Compressible Flows on Arbitrary Grids
conference, June 2012

  • Luo, Hong; Ali, Amjad; Nourgaliev, Robert
  • 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
  • https://doi.org/10.2514/6.2010-366

Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations
journal, October 2009


A Reconstructed Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations on Arbitrary Grids
conference, June 2012

  • Luo, Hong; Luo, Luqing; Nourgaliev, Robert
  • 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
  • https://doi.org/10.2514/6.2010-364

A Parallel, Reconstructed Discontinuous Galerkin Method for the Compressible Flows on Arbitrary Grids
journal, February 2011


A Projection Method for Low Speed Flows
journal, March 1999


Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)
conference, June 2012


On the behaviour of upwind schemes in the low Mach number limit
journal, January 1999


A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids
journal, June 2012


Effect of stepheight on the separated flow past a backward facing step
journal, March 1989


On physics-based preconditioning of the Navier–Stokes equations
journal, December 2009


Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change
journal, January 2016


Surface tension of binary metal—surface active solute systems under conditions relevant to welding metallurgy
journal, June 1988


Review of preconditioning methods for fluid dynamics
journal, May 1993


Jacobian-free Newton–Krylov methods: a survey of approaches and applications
journal, January 2004


Comparison of the AUSM+ and H-CUSP Schemes for Turbomachinery Applications
conference, June 2012


A Numerical Scheme for the Compressible Low-Mach Number Regime of Ideal Fluid Dynamics
journal, January 2017


A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids
journal, November 2012


Mesoscopic simulation model of selective laser melting of stainless steel powder
journal, November 2014


The numerical simulation of two-dimensional fluid flow with strong shocks
journal, April 1984


GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986


A sequel to AUSM, Part II: AUSM+-up for all speeds
journal, May 2006


Discontinuous Galerkin for Diffusion
conference, June 2012


Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics
journal, November 2019


The influence of surfactants on thermocapillary flow instabilities in low Prandtl melting pools
journal, June 2016


A Reconstructed Discontinuous Galerkin Method for the Compressible Euler Equations on Arbitrary Grids
conference, June 2012


A low-Mach number fix for Roe’s approximate Riemann solver
journal, June 2011


Choosing the Forcing Terms in an Inexact Newton Method
journal, January 1996


A numerical method for solving incompressible viscous flow problems
journal, August 1967


A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems
journal, August 1987


A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+-up scheme
journal, July 2007


Computational study of laminar heat transfer downstream of a backward-facing step
journal, February 1993


The lattice Boltzmann equation method: theoretical interpretation, numerics and implications
journal, January 2003


A reconstructed discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids
journal, September 2010


Contribution of the recent AUSM schemes to the OVERFLOW code - Implementation and validation
conference, February 2013


A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
journal, September 2008


Accurate Projection Methods for the Incompressible Navier–Stokes Equations
journal, April 2001


L 2 Roe: a low dissipation version of Roe's approximate Riemann solver for low Mach numbers : L
journal, September 2015


Low-Diffusion Flux-Splitting Methods for Flows at All Speeds
journal, September 1998


An algorithm for unsteady viscous flows at all speeds
journal, January 2000


A numerical fluid dynamics calculation method for all flow speeds
journal, October 1971


An evaluation of several differencing methods for inviscid fluid flow problems
journal, February 1968


Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods
journal, January 2014


Characteristic time-stepping or local preconditioning of the Euler equations
conference, February 2013


Numerical speed of sound and its application to schemes for all speeds
conference, February 2013


The design and application of upwind schemes on unstructured meshes
conference, February 2013


The Application of Preconditioning in Viscous Flows
journal, April 1993


Marangoni driven turbulence in high energy surface melting processes
journal, June 2016


Experimental and theoretical investigation of backward-facing step flow
journal, February 1983


Simulation of time-dependent pool shape during laser spot welding: Transient effects
journal, December 2003


On the accuracy of limiters and convergence to steady state solutions
conference, February 2013