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Derivation of k-omega SST turbulence model.

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Old   August 7, 2020, 04:14
Post Derivation of k-omega SST turbulence model.
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Does anyone know a detailed derivation of the k-omega SST turbulence model? I can't seem to find a fully derivation anywhere and yet a lot of people use this model quite frequently. I am having issues with going from the k-epsilon to the "transformed" k-epsilon in terms of omega.


The kinetic energy equation transformed derivation is straight forward but I'm struggling with dissipation.


With the dissipation in the standard k-epsilon equation given by:
{{D\varepsilon } \over {Dt}} = {C_{\varepsilon 1}}{\varepsilon  \over k}{\tau _{ij}}{{\partial {u_i}} \over {\partial {x_j}}} - {C_{\varepsilon 2}}{{{\varepsilon ^2}} \over k} + {\partial  \over {\partial {x_j}}}\left[ {\left( {\nu  + {{{\nu _t}} \over {{\sigma _\varepsilon }}}} \right){{\partial \varepsilon } \over {\partial {x_j}}}} \right] (eq1)

and using the ratio given by:

\varepsilon  = {C_\mu }\omega k (eq2)

We now obtain
{{D\varepsilon } \over {Dt}} = {C_\mu }\omega {{Dk} \over {Dt}} + {C_\mu }k{{D\omega } \over {Dt}} (eq3)


Plugging eq2 and eq3 into eq1 while putting a placeholder for epsilon constants with phi we obtain

{C_\mu }\omega {{Dk} \over {Dt}} + {C_\mu }k{{D\omega } \over {Dt}} = {C_{\phi 1}}{C_\mu }\omega {\tau _{ij}}{{\partial {u_i}} \over {\partial {x_j}}} - {C_{\phi 2}}{C_\mu }^2{\omega ^2}k + {\partial  \over {\partial {x_j}}}\left[ {\left( {\nu  + {{{\nu _t}} \over {{\sigma _\varepsilon }}}} \right){{\partial \left( {{C_\mu }\omega k} \right)} \over {\partial {x_j}}}} \right] (eq4)


Some reducing and rearranging...
{{D\omega } \over {Dt}} = {C_{\phi 1}}{\omega  \over k}{\tau _{ij}}{{\partial {u_i}} \over {\partial {x_j}}} - {C_{\phi 2}}{C_\mu }{\omega ^2} + {1 \over {{C_\mu }k}}{\partial  \over {\partial {x_j}}}\left[ {\left( {\nu  + {{{\nu _t}} \over {{\sigma _\phi }}}} \right){{\partial \left( {{C_\mu }\omega k} \right)} \over {\partial {x_j}}}} \right] - {\omega  \over k}{{Dk} \over {Dt}} (eq5)


I'm now stuck here and I believe just the last two terms are incomplete. My calculus is pretty rusty and not sure quite where to go. The transformed k-epsilon model stated by Menter,1994 is given by

{{D\omega } \over {Dt}} = {{{\gamma _2}} \over {{\nu _t}}}{\tau _{ij}}{{\partial {u_i}} \over {\partial {x_j}}} - {\beta _2}{\omega ^2} + {\partial  \over {\partial {x_j}}}\left[ {\left( {\nu  + {\sigma _{\omega 2}}{\nu _t}} \right){{\partial \omega } \over {\partial {x_j}}}} \right] + 2{\sigma _{\omega 2}}{1 \over \omega }{{\partial k} \over {\partial {x_j}}}{{\partial \omega } \over {\partial {x_j}}}


I've seen a lot of people just say that merely the addition of the last cross-diffusion term makes it an exact copy of the k-epsilon equation (that's what menter,1994 states as well) but I haven't seen a full derivation of this anywhere and so I'm not sure how that's the case.


This being said, I feel I'm getting closer but my calculus is rusty and so any help will be greatly appreciated.
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Old   August 7, 2020, 07:36
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I suggest to give a look here
http://www.tfd.chalmers.se/~lada/pos...port_lowre.pdf

and to the original references
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Old   August 7, 2020, 13:12
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This is a good reference and I'll go through this and post a reply. I already checked the original references and many others and they just say that the exact transformed k-epsilon equation merely has the addition of the cross-diffusion term. It seems that there is also a turbulent diffusion term that gets dropped out. I'm really surprised that I haven't found a full derivation when so many people use this model.
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