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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

Mem. ASME
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.

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Figures

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

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