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Mark Render June 16, 2000 04:16

Equations in full form

could someone give me some references where I can find the 3D Navier-Stokes equations and especially the k and epsilon equations in full form and not in tensor notation ? (in terms of U,V,W and not Ui, Uj)



John C. Chien June 16, 2000 09:25

Re: Equations in full form
(1). I think this is a good question. And I used to have the same problem. (2). But I think it is a good idea to do some journal paper search first, and then make some term-by-term comparisons, to make sure that the equations are definitely correct. (3). This is especially true when the equations are used in simplified environment, or 2-D cases. Even when the paper is dealing with modeling, in most cases it is 1-D problem. (4). Well, many years ago, I copied the Navier-Stokes equations from a reference book in 3-D, and later on, I found out that there were errors in the equation. Since then, I have to double check the equations first. (5). Many many years ago, W. Rodi had written a report with k-epsilon equations in various coordinates systems. But the eqautions in the report were hand-written. (6). So, the best way to get the correct equations is to first collect the equations from published papers, and then make the term-by-term comparison or derivation, to make sure that everything is correct. Then after fooling with these equations for a while, you may want to derive your own k-epsilon equations. Anyway, there are only convection, diffusion, and source terms. The problem could be in the source terms because the convection and the diffusion terms are rather standard forms. The source terms will include the production and the dissipation terms , or even extra terms. And there are a couple of different definitions for the epsilon. (6). Well, be patient and you will be rewarded by the correct set of equations.

clifford bradford June 16, 2000 15:10

Re: Equations in full form
for the N-S equations in full form you can try tannehill, Anderson and Pletcher's CFD book and/or John andersons CFD book. also other CFD books are like to have them in the popular conservative, column vector from which i personally find most useful. i think hirsch's CFD book has them in this form. if you can find it the conservative column vector form is probably most useful (at least i find to be that way) because the equations are in conservative from and the meaning of the terms can be more easily interpreted than any non-conservative form. personally i dislike tensor notation since i never formally learned it but it is difficult to find the k-e equations in any other form. also it is diffcult to find them written down in the compressible form it seems most turbulence books like to do everything in terms of the incompressible equations (at least that's my experience). i think you may be able to find the compressible and incompressible form of the k-e equations in Tannehill, Anderson, and pletcher. also yo can find the equation in the paper found at the description here is quite good and is part of a paper describing a 2d compressible N-S code. hopefully you'll find it useful. even though the k-e equation are not fully expanded they are in conservative form and you should be able to expand them readily.

Sergei Chernyshenko June 20, 2000 15:05

Re: Equations in full form
Hi, Mark,

Kghm, Ui notation is not a tensor notation, as you surely know. I mean, for example, that steady incompressible Navier-Stokes in tensor form looks like

u\nabla u = -\nabla p + (1/Re)\nabla^2 u

and in Ui form like

U_j (d U_i/d x_j) = - d p/d x_i+...

Tensor notation is of little use in CFD indeed, at least so far.

It is much easier to learn Ui notation than to use, understand from a glance or even to write down U,V,W form. U_i notation is just summation over the repeated indices. Learn it and

look at

It is a part of D.Apsley homepage which may be useful for you, too. I just did not read all the staff there and cannot say for sure.

The pdf file contains a specification of many turbulent models. As far as general equations, apart from any turbulence model, are concerned, try to find an old textbook. Fifty years ago tensors were not widely used in fluid dynamics.

Rgds, Sergei

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