Alexander, the Reynolds eqns need a closure and you have chosen k-eps, which is a semi-empirical closure. Better you the attached equations which are a continuous description of an actually discrete-vortex approach, formulated only in terms of probabilities. One spectacular result is the missing of any empirical elements but predicition of all relevant universal constants like Karman as 1/sqrt(2*pi) = 0.399, You can use the idea to couple Reynolds eqns with a stochastic particle approach,  i.e. direct simulation of a vortex ensemble. Of course, it would be an approximation, but also FDM a la A.A. Samarskij is or ony other numerical approach is an approximation to the real contiuous theory. But I guess that you simply want to know finite-element or finite-difference schemes for k-eps. However I recommend to use the exact turbulence theory rather than outdated k-eps.

Helmut, I need exactly k-eps model for my research. I'm trying to solve this

Sorry Alexander, k-eps has internal contradictions and is outdated. E.g. eps is not a conserved quantity.It is a semi-empirical model and cannot predict the universal constants of high-Re turbulence.

Is it true that your boss calls for using k-eps? Then he possibly does not know the most recent literature. Ask people in Barnaul and Novosibirsk from IWEP about the case. They know me well. Do you have stratification in your problem? I strongly recommed NOT to use k-eps ... People who are familiar with numerics of k-eps are a former PhD student, Hans Burchard, and his group of the General Ocean Turbulence Model (www.GOTM.NET). It goes back to an older paper of Hans and me which I attach. When we wrote it we didnt yet know the flaws of k-eps which I could overcome in the book MARINE TURBULENCE, Cambridge U. Press, 2005. But the final theory I have send you attached to the last message.

Thank's a lot, Helmut! And what do you thing about Lattice Bolzmann method? I forgot to mention that i investigate natural convection in closed areas in conjugate formulation. For this problems k-eps model is enough.

Lattice Boltzmann is a fine method for LES. "For this.. k-eps.. is enough": Well, k-eps is simply NONSENSE if we may agree that eps is not a conservation quantity (rather than TKE and r.m.s. vorticity). The theory I have send you is based on conservations laws for TKE and r.m.s. vorticity and is even somewhat simpler than k-eps, in computational terms. It would perfectly fit into an convective LES framework because there you do not need to speculate about the role of stratification, internal waves, and internal-wave turbulence.

I attach another paper about k-eps giving a comparison with real-world data. I used a scheme recommended by Samarskij. Yes, it works not that bad, but again, the physics is actually not correct! Please simply compare the situation with the geocentric and the heliocentric system. Initially the latter was more accurate because these monk-anstronomers had incorporated more data over the centuries. Nevertheless heliocentrics is clear nonsense, seen from today. Besides, nice that people still read books by Samarskij. I have a whole collection here in Russian and learned a lot about numerics for PDEs, he is a good writer, I have had the honor to meet him personally.

Dear  Alexander 

If you would like to use the k-eps model, you may find the discretized equations in many papers. A basic but good source for theoretical concepts is the monograph of W. Rodi about turbulence models.

After you did finish with discretization and developing you code, as you have correctly mentioned you need to verify your model. To do so the best is to compare the simulated results with the experimental measured values of the turbulence parameters that you may find in the literature. However, before that and to quickly check your results you may check the hydrodynamic results through comparing your predictions with the measured  values of the same applications in the literature. If you would like to discuss this further please let me know what application you would be using the k-eps model for, so that I could introduce you the more appropriate references.

Dear Kourosh,

First of all, thank you very much for your reasonable answer. As i have already mentioned, i investigate conjugate heat transfer in a closed rectangular area in natural convection mode. Boundary value problem is formulated in dimensionless variables "Vorticity - Stream function - Temperature" (i need exactly this problem formulation). I'm  using FDM. The scheme of approximation is a locally one scheme of A.A. Samarskiy. I attached a paper with equations which i wanted to solve  

Alexander,

I started with turbulence about 30 years ago, not occasionally in the Inst. Gidrodinamiki AH CCCP in Novosibirsk, Later at Weierstraß Inst in Berlin and Univ Hamburg and Oldenburg I continued and we met with Wolfgang Rodi and Guri Martshuk and George Mellor (the inventor of the Mellor-Yamada model) as our dear guests we began to systematically study the world of two-equation turbulence models and realized that there are many, TOO many! I posed to ask a silly question: whether a model can PREDICT universal constants like Karmans 0.4.

They could not, not a single one. Rather, they needed many empirical parameters and they failed in certain applications which I will not list here. If you bring ONE expert who is able to scientifically justify the choice of the balance equation for eps in the k-eps model today, I would owe you not only one good German beer.

Note that the Sheremet paper you attached uses a specifically augmented form of k-eps which principally collides with the idea of LES. Because LES of convection already RESOLVES the energy-containing scales of convection but - other than DNS - it does NOT resolve the scales further below are no longer influenced by buayancy and have to be described by a closure for neutral stratification. The closure used by Sheremet is dedicated to be used in a RANS environment for a "global" circulation picture.

The universal turbulence theory is structurally identical with what has been shown to be the empirically best two-eqn model: k-omega (first formulated by Wilcox, I mention him in my text). For his k-omega exist "well trained" numerics because it is used by Boeing. The "small" difference to the exact theory is the slight differences in the parameters, and that Wilcox cannot predict the universal constants but takes them from measurements. He cannot physically interprete the nature of his equations and cannot predict the coefficients used.

So for the numerics you may simply take Samarskij and apply it to the eqns I have sent you. For my part I close the discussion at this point because I don't need to repeat myself.

If you have specific questions, e.g. numerics, where you may directly write in Russian language and refer to the classic "Bbedenie B Teoriyu Pasnoctnych Cxem" (Hayka 1971) then please contact me directly by mail [email protected] Good luck!

Dear  Alexander,

I looked at the paper. It has presented all the original equations and the normalized ones including the turbulence model that have been used in the model which I guess you have been through them and are clear for you. The numerical solution however has been briefly discussed on page 112 in the 3rd paragraph. If you have discretized the equations then this part would help to check your procedure otherwise you may use the relevant references there and that might be helpful.

But from what I understood these are not your problems. You would like to check whether your code is working correctly. If this it the main reference you are using I would suggest you go through its tests and results provided therein to avoid digging other suitable data. Once you get good results from these tests you could further test your code with other sources.

If you don't get good results in the first attempt, try to check whether the model without the turbulence model is working and then concentrate on the turbulence model.