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      <page pageid="2766" ns="0" title="Realisability and Schwarz&#039; inequality">
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          <rev xml:space="preserve">{{Turbulence modeling}}Realisability is the minimum requirement to prevent a turbulence model generating non-physical results. For a model to be realisable the normal Reynolds stresses must be non-negative and the Schwarz' inequality must be satisfied between fluctuating quantities:

:&lt;math&gt;
\left\langle{u^'_\alpha u^'_\alpha}\right\rangle \geq 0
&lt;/math&gt;

:&lt;math&gt;
\frac{\left\langle{u^'_\alpha u^'_\beta}\right\rangle}{ \left\langle{u^'_\alpha u^'_\beta}\right\rangle \left\langle{u^'_\alpha u^'_\beta}\right\rangle} \leq 1
&lt;/math&gt;

where there is no summation over the indices. Some workers only apply the first inequality to satisfy realisability, or maintain non-negative vales of k and epsilon. This &quot;weak&quot; form of realisability is satisfied in non-linear models by setting &lt;math&gt;C_\mu=0.09&lt;/math&gt;.

==References==
{{reference-paper|author=Speziale, C.G.|year=1991|title=Analytical methods for the development of Reynolds-stress closures in turbulence|rest=Ann. Rev. Fluid Mechanics, Vol. 23, pp107-157}}</rev>
        </revisions>
      </page>
      <page pageid="1514" ns="0" title="Realisable k-epsilon model">
        <revisions>
          <rev xml:space="preserve">{{Turbulence modeling}}
== Transport Equations ==


:&lt;math&gt; \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_j} (\rho k u_j) = \frac{\partial}{\partial x_j} \left [ \left(\mu + \frac{\mu_t}{\sigma_k}\right) \frac{\partial k} {\partial x_j} \right ] + P_k + P_b - \rho \epsilon - Y_M + S_k &lt;/math&gt;


:&lt;math&gt; \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_j} (\rho \epsilon u_j) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_{\epsilon}}\right) \frac{\partial \epsilon}{\partial x_j} \right ] + \rho \, C_1 S \epsilon - \rho \, C_2 \frac{{\epsilon}^2} {k + \sqrt{\nu \epsilon}} + C_{1 \epsilon}\frac{\epsilon}{k} C_{3 \epsilon} P_b + S_{\epsilon} &lt;/math&gt;

Where &lt;br&gt;

&lt;math&gt; C_1  =  \max\left[0.43, \frac{\eta}{\eta + 5}\right] , \;\;\;\;\; \eta  =  S \frac{k}{\epsilon}, \;\;\;\;\; S =\sqrt{2 S_{ij} S_{ij}} &lt;/math&gt; &lt;br&gt;

In these equations, &lt;math&gt; P_k &lt;/math&gt; represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model. &lt;math&gt; P_b &lt;/math&gt; is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model.

== Modelling Turbulent Viscosity ==

:&lt;math&gt; \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} &lt;/math&gt; &lt;br&gt;
where &lt;br&gt;
&lt;math&gt; C_{\mu} = \frac{1}{A_0 + A_s \frac{k U^*}{\epsilon}} &lt;/math&gt; &lt;br&gt;
&lt;math&gt;  U^* \equiv \sqrt{S_{ij} S_{ij} + \tilde{\Omega}_{ij} \tilde{\Omega}_{ij}}  &lt;/math&gt;  ;&lt;br&gt;
&lt;math&gt; \tilde{\Omega}_{ij}  =  \Omega_{ij} - 2 \epsilon_{ijk} \omega_k     &lt;/math&gt; ; &lt;br&gt;
&lt;math&gt;  \Omega_{ij}  =  \overline{\Omega_{ij}} - \epsilon_{ijk} \omega_k     &lt;/math&gt; &lt;br&gt;

where &lt;math&gt; \overline{\Omega_{ij}} &lt;/math&gt;  is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity &lt;math&gt; \omega_k  &lt;/math&gt;. The model constants  &lt;math&gt; A_0  &lt;/math&gt; and &lt;math&gt; A_s  &lt;/math&gt; are given by: &lt;br&gt;
&lt;math&gt; A_0 = 4.04, \; \;  A_s = \sqrt{6}  \cos \phi &lt;/math&gt; &lt;br&gt;

&lt;math&gt;  \phi = \frac{1}{3} \cos^{-1} (\sqrt{6} W), \; \;    W = \frac{S_{ij} S_{jk} S_{ki}}{{\tilde{S}} ^3}, \; \; \tilde{S} = \sqrt{S_{ij} S_{ij}}, \; \; S_{ij} = \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i}  + \frac{\partial u_i}{\partial x_j} \right)   &lt;/math&gt;


==Model Constants ==

&lt;math&gt; C_{1 \epsilon} = 1.44,  \;\; C_2 = 1.9, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.2  &lt;/math&gt;


== References ==

See section [[K-epsilon_models#References|References]] in the parent page [[K-epsilon models]].

[[Category:Turbulence models]]</rev>
        </revisions>
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