# Verification Studies for the Noh Problem using Non-ideal Equations of State and Finite Strength Shocks

## Abstract

We present that the Noh verification test problem is extended beyond the commonly studied ideal gamma-law gas to more realistic equations of state (EOSs) including the stiff gas, the Noble-Abel gas, and the Carnahan-Starling EOS for hard-sphere fluids. Self-similarity methods are used to solve the Euler compressible flow equations, which in combination with the Rankine-Hugoniot jump conditions provide a tractable general solution. This solution can be applied to fluids with EOSs that meet criterion such as it being a convex function and having a corresponding bulk modulus. For the planar case the solution can be applied to shocks of arbitrary strength, but for cylindrical and spherical geometries it is required that the analysis be restricted to strong shocks. Finally, the exact solutions are used to perform a variety of quantitative code verification studies of the Los Alamos National Laboratory Lagrangian hydrocode FLAG.

- Authors:

- Univ. of Maryland, College Park, MD (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1467323

- Report Number(s):
- LA-UR-17-21479

Journal ID: ISSN 2377-2158

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Verification, Validation and Uncertainty Quantification

- Additional Journal Information:
- Journal Volume: 3; Journal Issue: 2; Journal ID: ISSN 2377-2158

- Publisher:
- ASME

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 42 ENGINEERING

### Citation Formats

```
Burnett, Sarah Cassie, Honnell, Kevin Guy, Ramsey, Scott D., and Singleton, Robert Jr. Verification Studies for the Noh Problem using Non-ideal Equations of State and Finite Strength Shocks. United States: N. p., 2018.
Web. doi:10.1115/1.4041195.
```

```
Burnett, Sarah Cassie, Honnell, Kevin Guy, Ramsey, Scott D., & Singleton, Robert Jr. Verification Studies for the Noh Problem using Non-ideal Equations of State and Finite Strength Shocks. United States. doi:10.1115/1.4041195.
```

```
Burnett, Sarah Cassie, Honnell, Kevin Guy, Ramsey, Scott D., and Singleton, Robert Jr. Fri .
"Verification Studies for the Noh Problem using Non-ideal Equations of State and Finite Strength Shocks". United States. doi:10.1115/1.4041195. https://www.osti.gov/servlets/purl/1467323.
```

```
@article{osti_1467323,
```

title = {Verification Studies for the Noh Problem using Non-ideal Equations of State and Finite Strength Shocks},

author = {Burnett, Sarah Cassie and Honnell, Kevin Guy and Ramsey, Scott D. and Singleton, Robert Jr.},

abstractNote = {We present that the Noh verification test problem is extended beyond the commonly studied ideal gamma-law gas to more realistic equations of state (EOSs) including the stiff gas, the Noble-Abel gas, and the Carnahan-Starling EOS for hard-sphere fluids. Self-similarity methods are used to solve the Euler compressible flow equations, which in combination with the Rankine-Hugoniot jump conditions provide a tractable general solution. This solution can be applied to fluids with EOSs that meet criterion such as it being a convex function and having a corresponding bulk modulus. For the planar case the solution can be applied to shocks of arbitrary strength, but for cylindrical and spherical geometries it is required that the analysis be restricted to strong shocks. Finally, the exact solutions are used to perform a variety of quantitative code verification studies of the Los Alamos National Laboratory Lagrangian hydrocode FLAG.},

doi = {10.1115/1.4041195},

journal = {Journal of Verification, Validation and Uncertainty Quantification},

number = 2,

volume = 3,

place = {United States},

year = {2018},

month = {8}

}

Works referenced in this record:

##
On the Existence of Self-Similar Converging Shocks in Non-Ideal Materials

journal, August 2017

- Boyd, Z. M.; Ramsey, S. D.; Baty, R. S.
- The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 70, Issue 4

##
Numerical simulation of high pressure release and dispersion of hydrogen into air with real gas model

journal, May 2010

- Khaksarfard, R.; Kameshki, M. R.; Paraschivoiu, M.
- Shock Waves, Vol. 20, Issue 3

##
Noh's constant-velocity shock problem revisited

journal, October 1997

- Gehmeyr, M.; Cheng, B.; Mihalas, D.
- Shock Waves, Vol. 7, Issue 5

##
The covolume and equation of state of high-temperature real gases

journal, August 1986

- Baibuz, V. F.; Zitserman, V. Yu.; Golubushkin, L. M.
- Journal of Engineering Physics, Vol. 51, Issue 2

##
Equation of State for Nonattracting Rigid Spheres

journal, July 1969

- Carnahan, Norman F.; Starling, Kenneth E.
- The Journal of Chemical Physics, Vol. 51, Issue 2

##
The Sedov Blast Wave as a Radial Piston Verification Test

journal, June 2016

- Pederson, Clark; Brown, Bart; Morgan, Nathaniel
- Journal of Verification, Validation and Uncertainty Quantification, Vol. 1, Issue 3

##
Revisiting Wall Heating

journal, August 2000

- Rider, William J.
- Journal of Computational Physics, Vol. 162, Issue 2

##
Why does the Carnahan-Starling equation work so well?

journal, September 1989

- Song, Yuhua; Mason, E. A.; Stratt, Richard M.
- The Journal of Physical Chemistry, Vol. 93, Issue 19

##
A classification of equations of state

journal, May 1972

- Cagle, F. W.
- Journal of Chemical Education, Vol. 49, Issue 5

##
The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy

journal, October 1998

- Caramana, E. J.; Burton, D. E.; Shashkov, M. J.
- Journal of Computational Physics, Vol. 146, Issue 1

##
Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of Compressible Flow Similarity Solutions

journal, December 2015

- Ramsey, Scott D.; Ivancic, Philip R.; Lilieholm, Jennifer F.
- Journal of Verification, Validation and Uncertainty Quantification, Vol. 1, Issue 2

##
Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux

journal, September 1987

- Noh, W. F.
- Journal of Computational Physics, Vol. 72, Issue 1

##
Conduction Invariance in Similarity Solutions for Compressible Flow Code Verification

journal, December 2015

- Hendon, Raymond C.; Ramsey, Scott D.
- Journal of Verification, Validation and Uncertainty Quantification, Vol. 1, Issue 2

##
The RAGE radiation-hydrodynamic code

journal, October 2008

- Gittings, Michael; Weaver, Robert; Clover, Michael
- Computational Science & Discovery, Vol. 1, Issue 1

##
Formation and Penetration of High-Speed Collapse Jets

journal, January 1966

- Harlow, Francis H.
- Physics of Fluids, Vol. 9, Issue 10

##
A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws

journal, April 1978

- Sod, Gary A.
- Journal of Computational Physics, Vol. 27, Issue 1

##
On sub-linear convergence for linearly degenerate waves in capturing schemes

journal, July 2008

- Banks, J. W.; Aslam, T.; Rider, W. J.
- Journal of Computational Physics, Vol. 227, Issue 14

##
Solution of the Noh problem using the universal symmetry of the gas dynamics equations

journal, July 2016

- Ramsey, S. D.; Boyd, Z. M.; Burnett, S. C.
- Shock Waves, Vol. 27, Issue 3

##
Verification Assessment of Piston Boundary Conditions for Lagrangian Simulation of the Guderley Problem

journal, September 2017

- Ramsey, Scott D.; Lilieholm, Jennifer F.
- Journal of Verification, Validation and Uncertainty Quantification, Vol. 2, Issue 3

##
A Method for the Numerical Calculation of Hydrodynamic Shocks

journal, March 1950

- VonNeumann, J.; Richtmyer, R. D.
- Journal of Applied Physics, Vol. 21, Issue 3