Derivation of k-omega SST turbulence model.

dweaver123 · August 7, 2020, 04:14

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

.

sbaffini · August 7, 2020, 07:36

I suggest to give a look here
http://www.tfd.chalmers.se/~lada/pos...port_lowre.pdf

and to the original references

dweaver123 · August 7, 2020, 13:12

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.

August 7, 2020, 04:14	Derivation of k-omega SST turbulence model.	#1
dweaver123 New Member Dustin Weaver Join Date: Dec 2016 Posts: 4 Rep Power: 9	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. quentin_carre likes this.

August 7, 2020, 07:36		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	I suggest to give a look here http://www.tfd.chalmers.se/~lada/pos...port_lowre.pdf and to the original references jonasa97 likes this.

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August 7, 2020, 13:12		#3
dweaver123 New Member Dustin Weaver Join Date: Dec 2016 Posts: 4 Rep Power: 9	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.