# Second Order Entropy Consistent Modelling of Turbulent Compressible Flows

• G. Brun
• J. M. Hérard
• L. Leal De Sousa
• M. Uhlmann
Conference paper
Part of the Fluid Mechanics and its Applications book series (FMIA, volume 36)

## Abstract

We examine herein the suitability of some second order closures to describe turbulent compressible flows with shocks, applying the standard Favre averaging technique. The basic set of equations reads:
$$\left( \rho \right),\text{t + }\left( {\rho \text{U}_\text{i} } \right),\text{i = 0}$$
(1)
setting:
$$\left( {\rho \text{U}_\text{i} } \right),\text{t + }\left( {\rho \text{U}_\text{i} \text{U}_\text{j} } \right),_\text{j} + \left( {\rho \;\delta _{\text{ij}} } \right),_\text{j} + \left( {\text{R}_{\text{ij}} } \right),_\text{j} = - \left( {\mathop \Sigma \nolimits_{\text{ij}}^{\text{visc}} } \right),\text{j}$$
(2)
$$\left( \text{E} \right),_\text{t} \text{ + }\left( {\text{E U}_\text{j} } \right),_\text{j} \text{ + }\left( {\text{U}_\text{i} \left( {\rho \text{ }\delta _{\text{ij}} + \text{R}_{\text{ij}} } \right)} \right),\text{j = - }\left( {\text{U}_\text{i} \left( {\mathop \Sigma \nolimits_{\text{ij}}^{\text{visc}} } \right)} \right),\text{j + }\left( {\sigma \text{E}\left( {\frac{\text{p}}{\rho }} \right)_\text{j} } \right),_\text{j}$$
(3)
$$\left( {\text{R}_{\text{ij}} } \right),_\text{t} + \;\left( {\text{R}_{\text{ij}} \;\text{U}_\text{k} } \right),_\text{k} + \;\text{R}_{\text{ik}} \text{U}_{\text{j,k}} + \;\text{R}_{\text{jk}} \text{U}_{\text{i,k}} = \Phi _{i\text{j}} - \frac{2}{3}\left( {\frac{ \in }{\text{I}}} \right)\;\text{trace}\;\text{(R) }\delta _{\text{ij}} + \left( {\mathop \Phi \nolimits_{i\text{j}}^\text{k} } \right),\text{k}$$
(4)
setting:
$${{{\text{R}}}_{{{\text{ij}}}}} = \;\user1{\& }\;\rho \;{{{\text{u''}}}_{{\text{i}}}}\;{{{\text{u''}}}_{{\text{j}}}} > ;\;2{\text{k}}\;{\text{ = }}\;{\text{I}}\;{\text{ = }}\;{\text{trace(R)}}\;{\text{ = }}{{{\text{R}}}_{{{\text{ii}}}}}$$
(5)
$$\mathop \Sigma \nolimits_{\text{ij}}^\text{v} = - \mu \left( {\text{U}_{\text{i,j}} + \text{U}_{\text{i,j}} - \frac{2}{3}\text{U}_{\text{1,1}} \delta _{i\text{j}} } \right);\;\text{p = }\left( {\gamma \text{ - 1}} \right)(\text{E - }\frac{{\rho \text{U}_\text{j} \text{U}_\text{j} }}{\text{2}} - \frac{1}{2}\text{R}_{\text{jj}} )$$
(1)

## Keywords

Riemann Problem Riemann Solver Moment Closure Approximate Riemann Solver Entropy Weak Solution
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• G. Brun
• 2
• J. M. Hérard
• 1
• L. Leal De Sousa
• 1
• M. Uhlmann
• 2
1. 1.EDF/DER/AEE/LNH/GRETChatouFrance
2. 2.METRAFLUEcullyFrance