Research Papers

Conduction Invariance in Similarity Solutions for Compressible Flow Code Verification

[+] Author and Article Information
Raymond C. Hendon

Computational Science Department,
Middle Tennessee State University,
MTSU Box 210,
Murfreesboro, TN 37132
e-mail: rchendon@lanl.gov

Scott D. Ramsey

Los Alamos National Laboratory,
P.O. Box 1663, MS T082,
Los Alamos, NM 87545
e-mail: ramsey@lanl.gov

1Corresponding author.

Manuscript received January 13, 2015; final manuscript received November 5, 2015; published online December 14, 2015. Assoc. Editor: Christopher J Roy. The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Verif. Valid. Uncert 1(2), 021004 (Dec 14, 2015) (10 pages) Paper No: VVUQ-15-1003; doi: 10.1115/1.4032017 History: Received January 13, 2015; Revised November 05, 2015

In 1991, Coggeshall published a series of 22 closed-form solutions of the Euler compressible flow equations with a heat conduction term included. A remarkable feature of some of these solutions is invariance with respect to conduction; this phenomenon follows from subtle ancillary constraints wherein a heat flux term is assumed to be either identically zero or nontrivially divergence-free. However, the solutions featuring the nontrivial divergence-free heat flux constraint can be shown to be incomplete, using a well-known result most commonly encountered in elementary electrostatic theory. With this result, the application of the divergence operator to the heat flux distributions exhibited by many of the solutions yields a delta function source term instead of identically zero. In theory, the relevant solutions will be conduction invariant only if the appropriate source term is included. This result has important implications for the use of the Coggeshall similarity solutions as code verification test problems for simulation codes featuring coupled compressible fluid flow and heat conduction processes. Computational reproduction of the conduction invariance property represents a conceptually simple check for verifying the robustness of a multiphysics algorithm. In this work, it is demonstrated in the context of various computational instantiations of Coggeshall solution #8 (Cog8) that to maintain any semblance of conduction invariance, a heat source term must be included even with a simple nonlinear heat conduction process. The efficacy of the heat source term is shown to depend not only on values of the various free parameters included in the Coggeshall mathematical model but also the representation of heat sources in multiphysics simulation codes of interest.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Sedov, L. , 1959, Similarity and Dimensional Methods in Mechanics, Academic Press, Boca Raton, FL.
Barenblatt, G. , 1996, Scaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics, Cambridge University Press, Cambridge, MA.
Sachdev, P. , 2004, Shock Waves and Explosions, Chapman & Hall/CRC, Boca Raton, FL.
Coggeshall, S. , 1991, “ Analytic Solutions of Hydrodynamics Equations,” Phys. Fluids A, 3(5), pp. 757–769. [CrossRef]
Oberkampf, W. , Trucano, T. , and Hirsh, C. , 2004, “ Verification, Validation, and Predictive Capability in Computational Engineering and Physics,” ASME Appl. Mech. Rev., 57(5), pp. 345–384. [CrossRef]
Roy, C. , 2005, “ Review of Code and Solution Verification Procedures for Computational Simulation,” J. Comput. Phys., 205(1), pp. 131–156. [CrossRef]
Marshak, R. , 1958, “ Effect of Radiation on Shock Wave Behavior,” Phys. Fluids, 1(1), pp. 24–29. [CrossRef]
Barrero, A. , and Sanmartin, J. , 1977, “ Self-Similar Motion of Laser Fusion Plasmas. Absorption in an Unbounded Plasma,” Phys. Fluids, 20(7), pp. 1155–1163. [CrossRef]
Sanmartin, J. , and Barrero, A. , 1978, “ Self-Similar Motion of Laser Half-Space Plasmas. I. Deflagration Regime,” Phys. Fluids, 21(11), pp. 1957–1966. [CrossRef]
Boudesocque-DuBois, C. , Gauthier, S. , and Clarisse, J.-M. , 2008, “ Self-Similar Solutions of Unsteady Ablation Flows in Inertial Confinement Fusion,” J. Fluid Mech., 603, pp. 151–178.
Murakami, M. , 2011, “ Self-Similar Hydrodynamics With Heat Conduction,” Heat Conduction—Basic Research, V. Vikhrenko , ed., InTech, Rijeka, Croatia.
Noh, W. , 1987, “ Errors for Calculations of Strong Shocks Using an Artificial Viscosity and an Artificial Heat Flux,” J. Comput. Phys., 72(1), pp. 78–120. [CrossRef]
Rider, W. , 2000, “ Revisiting Wall Heating,” J. Comput. Phys., 162(2), pp. 395–410. [CrossRef]
Axford, R. , 2000, “ Solutions of the Noh Problem for Various Equations of State Using Lie Groups,” Lasers Part. Beams, 18(1), pp. 93–100. [CrossRef]
Clover, M. , 2007, “ Analytic Test Problem Setups in Crestone,” Los Alamos National Laboratory, Report No. LA-UR-07-05424.
Timmes, F. , Clover, M. , Kamm, J. , and Ramsey, S. , 2009, “ On a Cell-Averaged Solution of the Coggeshall #8 Problem,” Los Alamos National Laboratory, Report No. LA-UR-09-04438.
Doebling, S. , and Ramsey, S. , 2013, “ Impact of Artificial Viscosity Models on Verification Assessment of a Lagrangian Hydrodynamics Code Using the Sedov Problem,” Los Alamos National Laboratory, Report No. LA-UR-13-23559.
Marcath, M. , Wang, M. , and Ramsey, S. , 2012, “ Development and Implementation of Radiation Hydrodynamics Verification Test Problems,” Los Alamos National Laboratory, Report No. LA-UR-12-24269.
Ramsey, S. , Ivancic, P. , and Lilieholm, J. , 2014, “ Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of Compressible Flow Similarity Solutions,” Los Alamos National Laboratory, Report No. LA-UR-15-20171.
Pollack, G. , and Stump, D. , 2002, Electromagnetism, Addison-Wesley, Boston, MA.
Coggeshall, S. , and Axford, R. , 1986, “ Lie Group Invariance Properties of Radiation Hydrodynamics Equations and Their Associated Similarity Solutions,” Phys. Fluids, 29(8), pp. 2398–2420. [CrossRef]
Burton, D. , 1990, “ Conservation of Energy, Momentum, and Angular Momentum in Lagrangian Staggered-Grid Hydrodynamics,” Lawrence Livermore National Laboratory, Report No. UCRL-JC-195926.
Caramana, E. , Burton, D. , Shashkov, M. , and Whalen, P. , 1998, “ The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy,” J. Comput. Phys., 146(1), pp. 227–262. [CrossRef]
Caramana, E. , Shashkov, M. , and Whalen, P. , 1998, “ Formulations of Artificial Viscosity for Multi-Dimensional Shock Wave Computations,” J. Comput. Phys., 144(1), pp. 70–97. [CrossRef]
Drake, R. , 2006, High-Energy-Density Physics, Springer, New York.
Ramsey, S. , and Hendon, R. , 2014, “ Modeling Classical Heat Conduction in FLAG,” Los Alamos National Laboratory, Report No. LA-UR-15-20196.
Taylor, G. , 1950, “ The Formation of a Blast Wave by a Very Intense Explosion. I. Theoretical Discussion,” Proc. R. Soc. London, Ser. A, 201(1065), pp. 159–174. [CrossRef]
Korobeĭnikov, V. , 1991, Problems of Point Blast Theory, Springer, New York.
Roache, P. , 1998, Verification and Validation in Computational Science and Engineering, Hermosa, Socorro, NM.


Grahic Jump Location
Fig. 1

Cog8-case 1 temperature simulation results: top left—Δr = 0.04, top right—Δr = 0.02, bottom left—Δr = 0.01, and bottom right—Δr = 0.005

Grahic Jump Location
Fig. 2

Cog8-case 2 temperature simulation results: top left—Δr = 0.04, top right—Δr = 0.02, bottom left—Δr = 0.01, and bottom right—Δr = 0.005



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In