Second Order Entropy Consistent Modelling of Turbulent Compressible Flows

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)