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Derivation of ke model from momentum equation 

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May 3, 2006, 11:00 
Derivation of ke model from momentum equation

#1 
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Hello:
Could someone direct me to some notes/ books on ke 2 equation turbulence model derivation from momentum equation having a "source term".? I am not sure if I add a source to momentum, I should be adding a source to ke equations or is it taken into the governing equations by themselves? CFDtoy 

May 3, 2006, 23:12 
Re: Derivation of ke model from momentum equation

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The book by Pope is Ok for u


May 4, 2006, 13:02 
Re: Derivation of ke model from momentum equation

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It depends on the nature of the source term, or rather: on the physical process your source term is supposed to describe. Is it an unsteady or steady term? If unsteady: are the associated time scales close to turbulent time scales or not? If you can consider your source term as a time average (in the sense of the RANS equations), then you may be able to just add it to the timeaveraged momentum equation. However, if your source term is coupled to other terms, such that the Reynolds averaging process produces new terms, you'll have to go through the derivation to see which terms actually show up in the RA momentum and which (if any) would show up in the turbulence model.


May 9, 2006, 19:23 
Re: Derivation of ke model from momentum equation

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Hello Mani: Thanks for your reply. I am giving a brief description here of the problem. Please let me know your thoughts.
I have some Scalar Eq: d(alpha)/dt + Del. (alpha*U) = source(alpha) Momentum Equation: incompressible d(U)/dt+ Del. (UU) = Del(p) + Source (say Alpha*U) Now, I thought since kepsilon are derived based on momentum eqn, if I update the momentum and hence U , kepsilon would adjust itself to the new momentum source? Do I need to add an explicit source term due to this (alpha) scalar ? Regarding the time scales, I can assume alpha source acts at the same time scale as the turb kinetic energy etc... Thanks, CFDtoy 

May 10, 2006, 12:51 
Re: Derivation of ke model from momentum equation

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Just to make things a little more exact, let me write your momentum equation with density included:
rho*[d(U)/dt+ Del. (UU)] = Del(p) + vec_S where vec_S = (Sx, Sy, Sz) = source term vector If you are doing Reynolds averaging to derive ke eqns (u = [u] + u' ; [u'] = 0) then you may derive the keqn by taking the Reynoldsaverage of the momentum equation dotted with the velocity vector vec_U = (u,v,w) and then subtract from this the Reynoldsaveraged momentum equation dotted with the Reynoldsaveraged velocity vector: [momentum dot vec_U]  [momentum] dot [vec_U] In this case, you would have the extra term [Sx * u'] + [Sy * v'] + [Sz * w'] added to the keqn. If vec_S = rho*(gx,gy,gz) for example (g = gravity vector), then the extra term in the keqn is [rho' u'] gx + [rho' v'] gy + [rho' w'] gz If you are doing Favreaveraging rho = [rho] + rho' ; u = {u} + u" ; {rho u"} = 0 then deriving the keqn is a little more involved. In Wilcox's "Turbulence Modeling for CFD" book, he dots the instantaneous momentum equation with (u", v", w") and Favre averages, which would give you the extra term {Sx * u"} + {Sy * v"} + {Sz * w"} If again we had vec_S = rho*(gx, gy, gz), then we would get {Sx * u"} = {rho u" gx} = {rho u"} gx = 0 since {rho u"}=0 by definition of Favre averaging, and similarly for the v and w components. The effect of density fluctuations does "show up" in the keqn for the gravity body force (through the presure work term {u" dp/dx + v" dp/dy + w" dp/dz} since [rho] u" =  [rho' u'], etc...), but does not show explicity for Favre averaging, as it did for Reynolds averaging. I am hoping the above examples help guide your derivation of your k equation. 

May 10, 2006, 16:53 
Re: Derivation of ke model from momentum equation

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Greetings Moder: That example is very helpful. As posted before my source term is a function of the scalar (which is function of the flow field). I would like to use Reynolds averaging and hence, I guess the sources, as you explain, should show up explicitly in the kequation.
I am working on few more details on the source terms. shall get back to you soon in this concern. Thanks for your help. btw, is there any online PDF version derivation of ke models that can be accessed? If so, please direct me to the same. Thanks, CFDtoy 

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