# An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows

## Abstract

Numerical solutions of the Euler equations using real gas equations of state (EOS) often exhibit serious inaccuracies. The focus here is the van der Waals EOS and its variants (often used in supercritical fluid computations). The problems are not related to a lack of convexity of the EOS since the EOS are considered in their domain of convexity at any mesh point and at any time. The difficulties appear as soon as a density discontinuity is present with the rest of the fluid in mechanical equilibrium and typically result in spurious pressure and velocity oscillations. This is reminiscent of well-known pressure oscillations occurring with ideal gas mixtures when a mass fraction discontinuity is present, which can be interpreted as a discontinuity in the EOS parameters. We are concerned with pressure oscillations that appear just for a single fluid each time a density discontinuity is present. As a result, the combination of density in a nonlinear fashion in the EOS with diffusion by the numerical method results in violation of mechanical equilibrium conditions which are not easy to eliminate, even under grid refinement.

- Authors:

- Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States)
- Aix Marseille Univ. and Univ. Institute of France, Marseille Cedex (France); RS2N SAS, Saint Zacharie (France)
- Ecole Centrale Paris, Chatenay-Malabry (France); Lab. d'Energetique Moleculaire et Macroscopique, Chatenay-Malabry (France)

- Publication Date:

- Research Org.:
- California Inst. of Technology (CalTech), Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA), Office of Defense Programs (DP) (NA-10)

- OSTI Identifier:
- 1343123

- Alternate Identifier(s):
- OSTI ID: 1352895; OSTI ID: 1411835

- Grant/Contract Number:
- NA0002382

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 335; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; shock-capturing methods; real equation of state; Van der Waals; nonconservative equations; pressure oscillations; 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Pantano, C., Saurel, R., and Schmitt, T.
```*An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows*. United States: N. p., 2017.
Web. doi:10.1016/j.jcp.2017.01.057.

```
Pantano, C., Saurel, R., & Schmitt, T.
```*An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows*. United States. doi:10.1016/j.jcp.2017.01.057.

```
Pantano, C., Saurel, R., and Schmitt, T. Wed .
"An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows". United States.
doi:10.1016/j.jcp.2017.01.057. https://www.osti.gov/servlets/purl/1343123.
```

```
@article{osti_1343123,
```

title = {An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows},

author = {Pantano, C. and Saurel, R. and Schmitt, T.},

abstractNote = {Numerical solutions of the Euler equations using real gas equations of state (EOS) often exhibit serious inaccuracies. The focus here is the van der Waals EOS and its variants (often used in supercritical fluid computations). The problems are not related to a lack of convexity of the EOS since the EOS are considered in their domain of convexity at any mesh point and at any time. The difficulties appear as soon as a density discontinuity is present with the rest of the fluid in mechanical equilibrium and typically result in spurious pressure and velocity oscillations. This is reminiscent of well-known pressure oscillations occurring with ideal gas mixtures when a mass fraction discontinuity is present, which can be interpreted as a discontinuity in the EOS parameters. We are concerned with pressure oscillations that appear just for a single fluid each time a density discontinuity is present. As a result, the combination of density in a nonlinear fashion in the EOS with diffusion by the numerical method results in violation of mechanical equilibrium conditions which are not easy to eliminate, even under grid refinement.},

doi = {10.1016/j.jcp.2017.01.057},

journal = {Journal of Computational Physics},

number = C,

volume = 335,

place = {United States},

year = {Wed Feb 01 00:00:00 EST 2017},

month = {Wed Feb 01 00:00:00 EST 2017}

}

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