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-   -   Calculating divDevReff (http://www.cfd-online.com/Forums/openfoam-solving/58214-calculating-divdevreff.html)

jposunz December 11, 2008 22:57

Hi All, I'm using simpleFoa
 
Hi All,

I'm using simpleFoam and turbFoam with k-epsilon and will be trying different turbulence models in the future. Before that though, I'm trying to understand the implementation exactly.

I've searched a lot and I don't understand why the calculation for turbulence->divDevReff(U) is:

divDevReff(U) =
- fvm::laplacian(nuEff(), U)
- fvc::div(nuEff()*dev(fvc::grad(U)().T()))

instead of:

divDevReff(U) =
- fvm::laplacian(nuEff(), U)
- fvc::div(nuEff()*fvc::grad(U)().T())

ie. why do we take the deviatoric?

I've read several papers including Hrvoje's "A tensorial approach to computational continuum mechanics using object-oriented techniques" and still haven't found an answer.

Is there another paper that explains the implementation?

Thanks very much
John

jposunz December 14, 2008 18:48

Anyone? Or should I have been
 
Anyone? Or should I have been able to find this in the literature somewhere already?

louisgag July 16, 2009 16:59

Hi John,

did you find the answer to that? If so I'd be curious to know what it is. Thanks!

-Louis

jet November 14, 2009 14:09

Hi,

have you already found an answer to your question? I tried to understand this equation, too. But I failed!

louisgag November 30, 2009 23:04

I think it has to do with the fact that

dev(fvc::grad(U)().T()))

is, at convergence, the same as

(fvc::grad(U)().T()))

for some fluids

-Louis

4xF December 2, 2009 12:39

fvc::div(nuEff()*dev(fvc::grad(U)().T())) and fvc::div(nuEff()*fvc::grad(U)().T()) are the same. The reason is that the missing term is the diagonal of the tensor fvc::grad(U)().T() which is the trace of fvc::grad(U)().T() which is div(phi). For an incompressible fluid, this value is strictly zero by virtue of mass conservation.

mchurchf December 22, 2009 17:45

John,

I don't know if anyone has answered your question yet, but I've come across the same thing, and I have questions about the incompressible implementation of divDevReff also. As you said, in the incompressible models, turbulence->divDevReff(U) is:

- fvm::laplacian(nuEff(), U)
- fvc::div(nuEff()*dev(fvc::grad(U)().T()))

Shouldn't dev2 be used instead of dev? In the compressible models, that is the case as turbulence->divDevRhoReff is:

- fvm::laplacian(muEff(), U)
- fvc::div(muEff()*dev2(fvc::grad(U)().T()))

In src\OpenFOAM\primitives\Tensor\TensorI.H, dev and dev2 are defined as dev(A) = A-1/3*I*trace(A) and dev2(A) = A-2/3*I*trace(A).

The stress term in the incompressible momentum equation is as follows:
d/dx_j {nu_eff*[(du_i/dx_j + du_j/dx_i)-2/3*du_k/dx_k delta_ij]},

which is the same as:
d/dx_j {nu_eff*(du_i/dx_j+du_j/dx_i)} - 2/3*d/dx_i {nu_eff(du_k/dx_k)}

which is in vector notation:
laplacian(nu_eff*U)
+ div{nu_eff*transpose[grad(U)]}
- div{2/3*nu_eff*I*trace[transpose(grad(U))]}

and reduces to:
laplacian(nu_eff*U) + div{nu_eff*dev2(transpose[grad(U)])}

Notice that dev2 is here instead of dev, which agrees with the compressible coding. However, in the incompressible coding, dev is used, which seem incorrect.

Now, the du_k/dx_k = trace[transpose(grad(U))] should be equal to zero in incompressible flow due to the continuity equation, so it seems that it should not matter if dev or dev2 is used. However, in the PISO algorithm, velocity may be predicted from a field that does not satisfy continuity due to initial conditions or insufficient pressure correction, so it seems like it would be wise to keep du_k/dx_k.

I am wondering if dev2 is an addition to OpenFOAM 1.6 and the incompressible divDevReff (and divDevBeff for LES) didn't get updated to reflect the change. Can someone with the correct knowledge please help?

Thank you,

Matt






Quote:

Originally Posted by jposunz (Post 183644)
Hi All,

I'm using simpleFoam and turbFoam with k-epsilon and will be trying different turbulence models in the future. Before that though, I'm trying to understand the implementation exactly.

I've searched a lot and I don't understand why the calculation for turbulence->divDevReff(U) is:

divDevReff(U) =
- fvm::laplacian(nuEff(), U)
- fvc::div(nuEff()*dev(fvc::grad(U)().T()))

instead of:

divDevReff(U) =
- fvm::laplacian(nuEff(), U)
- fvc::div(nuEff()*fvc::grad(U)().T())

ie. why do we take the deviatoric?

I've read several papers including Hrvoje's "A tensorial approach to computational continuum mechanics using object-oriented techniques" and still haven't found an answer.

Is there another paper that explains the implementation?

Thanks very much
John


christju January 2, 2010 12:04

Hello,
In fact, I've come across the same thing and I found the same conclusion than you Matthew...
The reynolds stress tensor is defined as :
-rho u'_i u'_j = 2 mu_t S_ij - 2/3 rho k delta_ij

where k = (u'_i u'_i) /2 is the kinetic energy, and S_ij is the deviatoric part of the mean strain rate :
S_ij = 1/2 (du_i/dx_j + du_j/dx_i) - 1/3 d_uk/dx_k delta_ij

As you said Matthew, the stress term in the incompressible momentum equation is as follows:
d/dx_j {nu_eff*[(du_i/dx_j + du_j/dx_i)-2/3*du_k/dx_k delta_ij]},

And in fact, from my point of view, a 2 is missing in the implementation of turbulence->divDevReff(U) of the incompressible model....

My other question is :
Where the term "- 2/3 rho k delta_ij" of the Reynold stress is taken into account?
Is it through a modified pressure defined as P = p/rho+2/3k and solved in the pressure equation?
And then, what is writen in the output files? this modified pressure?

If you have any idea....
Thank you in advance for your help,
Julien

poplar August 25, 2010 00:51

Hi,mchurchf.

The complete form of div(StressTensor)is:
div(muEff()*dev(fvc::grad(U)())) + div(muEff()*dev(fvc::grad(U)().T()))
So -1/3*I*trace(A) is lncluded in both dev(grad(U)) and dev(grad(U).T()),

in fact,
laplacian(nuEff(), U) = div(muEff()*grad(U)) =div(muEff()*dev(fvc::grad(U))) + 1/3*I*trace(A)

as a result, one 1/3*I*trace(A) is addded in laplacian(nuEff(), U), so a minus one must be subtracted in the later term.

then,dev2(A) = A-2/3*I*trace(A).

du_k/dx_k in incompressible fluids may be removed. If one retains it for correction, 1/3 or 2/3 is only a scale factor.

Quote:

Originally Posted by mchurchf (Post 240765)
John,

I don't know if anyone has answered your question yet, but I've come across the same thing, and I have questions about the incompressible implementation of divDevReff also. As you said, in the incompressible models, turbulence->divDevReff(U) is:

- fvm::laplacian(nuEff(), U)
- fvc::div(nuEff()*dev(fvc::grad(U)().T()))

Shouldn't dev2 be used instead of dev? In the compressible models, that is the case as turbulence->divDevRhoReff is:

- fvm::laplacian(muEff(), U)
- fvc::div(muEff()*dev2(fvc::grad(U)().T()))

In src\OpenFOAM\primitives\Tensor\TensorI.H, dev and dev2 are defined as dev(A) = A-1/3*I*trace(A) and dev2(A) = A-2/3*I*trace(A).

The stress term in the incompressible momentum equation is as follows:
d/dx_j {nu_eff*[(du_i/dx_j + du_j/dx_i)-2/3*du_k/dx_k delta_ij]},

which is the same as:
d/dx_j {nu_eff*(du_i/dx_j+du_j/dx_i)} - 2/3*d/dx_i {nu_eff(du_k/dx_k)}

which is in vector notation:
laplacian(nu_eff*U)
+ div{nu_eff*transpose[grad(U)]}
- div{2/3*nu_eff*I*trace[transpose(grad(U))]}

and reduces to:
laplacian(nu_eff*U) + div{nu_eff*dev2(transpose[grad(U)])}

Notice that dev2 is here instead of dev, which agrees with the compressible coding. However, in the incompressible coding, dev is used, which seem incorrect.

Now, the du_k/dx_k = trace[transpose(grad(U))] should be equal to zero in incompressible flow due to the continuity equation, so it seems that it should not matter if dev or dev2 is used. However, in the PISO algorithm, velocity may be predicted from a field that does not satisfy continuity due to initial conditions or insufficient pressure correction, so it seems like it would be wise to keep du_k/dx_k.

I am wondering if dev2 is an addition to OpenFOAM 1.6 and the incompressible divDevReff (and divDevBeff for LES) didn't get updated to reflect the change. Can someone with the correct knowledge please help?

Thank you,

Matt


T.D. October 13, 2010 12:54

new nuEff()
 
Hi guys
i need to add (implement) and use a new nuEff2(), how to do that?
where to add the new nuEff() ?

i tried but it's hard, since it was only one original "nu" defined in all the transportModels and viscosityModels, and is hard to change in all the files.

help please

haze_1986 October 26, 2012 09:14

I am curious about this too

haze_1986 January 25, 2013 05:27

Quote:

Originally Posted by poplar (Post 272612)
Hi,mchurchf.

The complete form of div(StressTensor)is:
div(muEff()*dev(fvc::grad(U)())) + div(muEff()*dev(fvc::grad(U)().T()))
So -1/3*I*trace(A) is lncluded in both dev(grad(U)) and dev(grad(U).T()),

in fact,
laplacian(nuEff(), U) = div(muEff()*grad(U)) =div(muEff()*dev(fvc::grad(U))) + 1/3*I*trace(A)

as a result, one 1/3*I*trace(A) is addded in laplacian(nuEff(), U), so a minus one must be subtracted in the later term.

then,dev2(A) = A-2/3*I*trace(A).

du_k/dx_k in incompressible fluids may be removed. If one retains it for correction, 1/3 or 2/3 is only a scale factor.

Hi, that sounds correct, but I do not understand why for the compressible case divDevRhoReff(U), dev2 is used instead?
http://www.cfd-online.com/Forums/ope...sonicfoam.html

MatzeS March 31, 2014 08:22

Hello,

although it's some time ago and several replies were posted, I still don't understand, if the implementation is correct...

Don't we have to use dev2() instead of dev() for the incompressible solvers (at least until convergence is reached)?
Or did I just miss the correct reply?

Regards,
Matthias

galap November 6, 2014 04:31

Quote:

Originally Posted by christju (Post 241306)
Hello,
In fact, I've come across the same thing and I found the same conclusion than you Matthew...
The reynolds stress tensor is defined as :
-rho u'_i u'_j = 2 mu_t S_ij - 2/3 rho k delta_ij

where k = (u'_i u'_i) /2 is the kinetic energy, and S_ij is the deviatoric part of the mean strain rate :
S_ij = 1/2 (du_i/dx_j + du_j/dx_i) - 1/3 d_uk/dx_k delta_ij

As you said Matthew, the stress term in the incompressible momentum equation is as follows:
d/dx_j {nu_eff*[(du_i/dx_j + du_j/dx_i)-2/3*du_k/dx_k delta_ij]},

And in fact, from my point of view, a 2 is missing in the implementation of turbulence->divDevReff(U) of the incompressible model....

My other question is :
Where the term "- 2/3 rho k delta_ij" of the Reynold stress is taken into account?
Is it through a modified pressure defined as P = p/rho+2/3k and solved in the pressure equation?
And then, what is writen in the output files? this modified pressure?

If you have any idea....
Thank you in advance for your help,
Julien

Exactly reprint of my doubts. The issue in the incompressible is not the problem for me - since I am dealing with compressible flows. There the implementation seems to be correct. But what I don't see:

Where the term "- 2/3 rho k delta_ij" of the Reynold stress is taken into account? Has anyone find the answer?

Tobi July 15, 2015 15:07

divDevRhoReff and divDevReff
 
Hi all,

as I read before, there are still some people who are confused about the equations and the behavior of the implemented code. Therefore, I want to make the derivations and want to show you that the implementation is correct for both cases. We consider compressible and incompressible fluids. Further more I only take into account laminar flow and explain the way to get to divDevRhoReff and divDevReff.

Momentum equation

Lets start with the obvious momentum equation:

\frac{\partial \rho \textbf{U}}{\partial t} + \nabla \bullet (\rho  \textbf{U} \textbf{U}) = \nabla \bullet \boldsymbol \sigma + \textbf{f}

Due to the fact that everyone are more or less familiar with the equations, I do not mention the variables meaning. For the further proceeding I do not consider body forces. If we have newtonian fluid the stress tensor \boldsymbol \sigma can be expressed as:

\boldsymbol \sigma = - \left(p + \frac{2}{3} \mu \nabla \bullet \textbf{U}\right) \textbf{I} + \mu \left[ \nabla \textbf{U} + (\nabla \textbf{U})^T\right]

Some modifications and we end up with:

\boldsymbol \sigma = - p\textbf{I} - \frac{2}{3} \mu \nabla \bullet  \textbf{U} \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla  \textbf{U})^T

Combining both, and neglect all other body forces, we end up with:

\frac{\partial \rho \textbf{U}}{\partial t} + \nabla \bullet (\rho   \textbf{U} \textbf{U}) = \nabla \bullet \left[ - p\textbf{I} - \frac{2}{3} \mu \nabla \bullet  \textbf{U} \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla  \textbf{U})^T\right]

Lets focus only only to the RHS (step by step):

\nabla \bullet  (- p\textbf{I}) + \nabla \bullet \left[-\frac{2}{3} \mu \nabla \bullet   \textbf{U} \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla   \textbf{U})^T\right]

Now it can be shown that the first term is the gradient of the pressure:

\nabla \bullet  (- p\textbf{I}) = -\nabla p

The analytic proof:

\nabla \bullet  (- p\textbf{I}) = \nabla \bullet \left[ -p 
\begin{pmatrix}
1 & 0 &0 \\ 
0 & 1 &0 \\ 
 0 & 0 & 1
\end{pmatrix}
\right]
 =
\nabla \bullet
\begin{pmatrix}
-p & 0 &0 \\ 
0 & -p &0 \\ 
 0 & 0 & -p
\end{pmatrix}
= \begin{pmatrix}
 (-\frac{\partial p}{\partial x}) + 0 + 0\\ 
0 + (-\frac{\partial p}{\partial y}) + 0\\
0 + 0 + (- \frac{\partial p}{\partial x})
\end{pmatrix} =

=
\begin{pmatrix}
 (-\frac{\partial p}{\partial x})\\ 
(-\frac{\partial p}{\partial y})\\
(-\frac{\partial p}{\partial x})
\end{pmatrix}
=- \nabla p

The second term at the RHS is often symboled with \boldsymbol \tau and stand for the viscose part of the stress tensor (normally without the divergence symbol):

\nabla \bullet \boldsymbol \tau = \nabla \bullet \left[-\frac{2}{3} \mu \nabla \bullet   \textbf{U}  \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla   \textbf{U})^T\right]

This guy is calculated by calling the function divDevRhoReff(U) or divDevReff(U). Now I will show that the implemented functions are correct.


Compressible

In compressible flow we have the full viscose stress tensor. Now I demonstrate that this term is exactly the divDevRhoReff(U) function. Lets start to modify the equation (again step by step). But first I want to introduce the necessary math operation. \textbf{a} denotes an arbitary vector:

\nabla \bullet
\textbf{a} = \mathrm{tr}(\nabla\textbf{a}) = \mathrm{tr}(\nabla \textbf{a})^T

using this formula and sorting the equation, we get the following:

\nabla \bullet \left[\mu \nabla \textbf{U} + \mu  \left((\nabla   \textbf{U})^T -\frac{2}{3} \mathrm{tr}(\nabla \textbf{U})^T   \textbf{I} \right)\right]

\nabla \bullet (\mu \nabla \textbf{U}) 
+ \nabla \bullet 
\left( \mu  \left[(\nabla   \textbf{U})^T - \frac{2}{3}  \mathrm{tr}(\nabla  \textbf{U})^T   \textbf{I} \right]\right)

Thats it. If we call turbulence->divDevRhoReff() we will exactly get this equation. Using a substituion for the gradient of the transponed velocity field, the name of the function dev2 gets clearer:

(\nabla   \textbf{U})^T = \textbf{A}

\nabla \bullet (\mu \nabla \textbf{U}) 
+ \nabla \bullet 
\left( \mu  \left[(\textbf{A} - \frac{2}{3}\mathrm{tr}(\textbf{A})   \textbf{I} \right]\right)

The deviatoric part of a matrix is defined as:

\mathrm{dev}(\textbf{A}) = \textbf{A} - \frac{1}{3}\mathrm{tr} (\textbf{A})\textbf{I}

Now we see, that the second term include the deviatoric part of the matrix (gradU). In compressible we call the dev2 due to the fact that the factor 2 is in the trace. To proove the equation, we can check out the code in the compressible tubulence file (laminar.C): http://foam.sourceforge.net/docs/cpp...ce.html#l00219

Code:

  219 tmp<fvVectorMatrix> laminar::divDevRhoReff(volVectorField& U) const   220 {
  221    return
  222    (
  223      - fvm::laplacian(muEff(), U)
  224      - fvc::div(muEff()*dev2(T(fvc::grad(U))))
  225    );
  226 }



\nabla \bullet (\mu \nabla \textbf{U})

is equal to (muEff = mu for laminar):
Code:

fvm::laplacian(muEff, U)
\nabla \bullet 
\left( \mu  \left[(\nabla   \textbf{U})^T - \frac{2}{3}  \mathrm{tr}(\nabla  \textbf{U})^T   \textbf{I} \right]\right)

is equal to
Code:

  224      - fvc::div(muEff()*dev2(T(fvc::grad(U))))
At least dev2 calculates exactly the deviatoric part. The code is here: http://foam.sourceforge.net/docs/cpp/a08631_source.html

Code:

  303 //- Return the deviatoric part of a symmetric tensor  304 template<class Cmpt>
   305 inline SymmTensor<Cmpt> dev2(const SymmTensor<Cmpt>& st)
  306 {
  307    return st - SphericalTensor<Cmpt>::twoThirdsI*tr(st);
  308 }

This is equal to:

\textbf{A} - \frac{2}{3}\mathrm{tr} (\textbf{A})\textbf{I} = \left[(\nabla   \textbf{U})^T - \frac{2}{3}  \mathrm{tr}(\nabla  \textbf{U})^T   \textbf{I} \right]

Check :) or no check :confused:, I think, its obvious.


Incompressible
For incompressible fluids (devide by \rho) we again start with the viscous stress tensor \boldsymbol \tau:

\nabla \bullet \boldsymbol \tau = \nabla \bullet \left[-\frac{2}{3} \nu  \underline{\nabla \bullet   \textbf{U}}  \textbf{I} + \nu \nabla \textbf{U} + \nu   (\nabla   \textbf{U})^T\right]

Hence, the density is constant the first term (underlined) at the RHS is zero due to the mass conservation equation:

\nabla\bullet \textbf{U} = 0

\nabla \bullet \boldsymbol \tau = \nabla \bullet \left[\nu \nabla \textbf{U} + \nu   (\nabla   \textbf{U})^T\right]

After a huge amount of mathematics we get :D

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet (\nu   (\nabla   \textbf{U})^T)

So there is no deviatoric part till now. But in FOAM we calculate the viscouse part of the stress tensor using divDevReff. Now we add the hydrostatic part multiply by the viscosity and substract it again:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu   (\nabla   \textbf{U})^T + \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I} -  \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right)

it follows:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[  (\nabla    \textbf{U})^T + \frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right] -   \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right)

The last term is zero due to mass conservation and can be removed:

-   \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I} =  -   \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})\textbf{I} =  -   \nu \nabla \bullet \textbf{U}

We end up with:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[  (\nabla    \textbf{U})^T + \frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right]\right)

If we substitute the transponed gradient U matrix with A:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[ A + \frac{1}{3}\mathrm{tr}(A)\textbf{I}\right]\right)

Again we have the laplacian term and again the deviatoric part (here calcated with oneThird, so the correct definition of dev).

And this is exactly what we get here:

Code:

  tmp<fvVectorMatrix> laminar::divDevReff(volVectorField& U) const
{
    return
    (
      - fvm::laplacian(nuEff(), U)
      - fvc::div(nuEff()*dev(T(fvc::grad(U))))
    );
}

The code for the deviatoric part (dev()) is equal to dev2. The only difference is the factor 2/3 which is introduced by the divergence term as shown before.

I hope everything is clear now and that there are no mistakes.
Additionally I hope it makes stuff clearer and help anybody to get a better understanding how things work.

Kind regards


Thanks to Alexander Vakhrushev for all support!

ssherman July 16, 2015 09:10

Tobi,

Thanks for the clear and concise explanation. However, in the incompressible case, there are a few places where mass conservation can be invoked to set terms to zero, but are not. Why do we set the terms that we do to zero?

I ask because nonNewtonianIcoFoam does not compute the viscous terms through divDevReff. Instead it goes through a couple more steps (that appear valid to me) to get to an alternate expression:
Code:

- fvm::laplacian(fluid.nu(), U) - (fvc::grad(U) & fvc::grad(fluid.nu())
Is nonNewtonianIcoFoam 'wrong'? What is the motivation for the different terms?

In the thread below, I show that the two forms yield noticeably different results for a model problem on a highly non-Newtonian fluid.
http://www.cfd-online.com/Forums/ope...tml#post555712

Thanks,
Steve

Tobi July 24, 2015 02:17

Hi,

why we set some term (that could be set to zero) not to zero is the way I derivated the equations (:
If we will go straight forward there is no need to do this. See below.

Tobi August 6, 2015 03:23

Hi all,

the derivation is correct but not straight forward from the beginning. If I have time I will make a PDF where everything is straight forward. Then you can see why once we have 1/3 and the other time 2/3 and its not due to the fact that we extend a equation (like I did in the incompressible case). Of course, like I showed it is possible to derivate the 2/3 and 1/3 like I did but as I mentioned it is not straight forward.

galap August 6, 2015 03:48

I would highly appreciate it! Thank you - good work

syavash October 23, 2015 08:35

Quote:

Originally Posted by Tobi (Post 555624)
Hi all,

as I read before, there are still some people who are confused about the equations and the behavior of the implemented code. Therefore, I want to make the derivations and want to show you that the implementation is correct for both cases. We consider compressible and incompressible fluids. Further more I only take into account laminar flow and explain the way to get to divDevRhoReff and divDevReff.

Momentum equation

Lets start with the obvious momentum equation:

\frac{\partial \rho \textbf{U}}{\partial t} + \nabla \bullet (\rho  \textbf{U} \textbf{U}) = \nabla \bullet \boldsymbol \sigma + \textbf{f}

Due to the fact that everyone are more or less familiar with the equations, I do not mention the variables meaning. For the further proceeding I do not consider body forces. If we have newtonian fluid the stress tensor \boldsymbol \sigma can be expressed as:

\boldsymbol \sigma = - \left(p + \frac{2}{3} \mu \nabla \bullet \textbf{U}\right) \textbf{I} + \mu \left[ \nabla \textbf{U} + (\nabla \textbf{U})^T\right]

Some modifications and we end up with:

\boldsymbol \sigma = - p\textbf{I} - \frac{2}{3} \mu \nabla \bullet  \textbf{U} \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla  \textbf{U})^T

Combining both, and neglect all other body forces, we end up with:

\frac{\partial \rho \textbf{U}}{\partial t} + \nabla \bullet (\rho   \textbf{U} \textbf{U}) = \nabla \bullet \left[ - p\textbf{I} - \frac{2}{3} \mu \nabla \bullet  \textbf{U} \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla  \textbf{U})^T\right]

Lets focus only only to the RHS (step by step):

\nabla \bullet  (- p\textbf{I}) + \nabla \bullet \left[-\frac{2}{3} \mu \nabla \bullet   \textbf{U} \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla   \textbf{U})^T\right]

Now it can be shown that the first term is the gradient of the pressure:

\nabla \bullet  (- p\textbf{I}) = -\nabla p

The analytic proof:

\nabla \bullet  (- p\textbf{I}) = \nabla \bullet \left[ -p 
\begin{pmatrix}
1 & 0 &0 \\ 
0 & 1 &0 \\ 
 0 & 0 & 1
\end{pmatrix}
\right]
 =
\nabla \bullet
\begin{pmatrix}
-p & 0 &0 \\ 
0 & -p &0 \\ 
 0 & 0 & -p
\end{pmatrix}
= \begin{pmatrix}
 (-\frac{\partial p}{\partial x}) + 0 + 0\\ 
0 + (-\frac{\partial p}{\partial y}) + 0\\
0 + 0 + (- \frac{\partial p}{\partial x})
\end{pmatrix} =

=
\begin{pmatrix}
 (-\frac{\partial p}{\partial x})\\ 
(-\frac{\partial p}{\partial y})\\
(-\frac{\partial p}{\partial x})
\end{pmatrix}
=- \nabla p

The second term at the RHS is often symboled with \boldsymbol \tau and stand for the viscose part of the stress tensor (normally without the divergence symbol):

\nabla \bullet \boldsymbol \tau = \nabla \bullet \left[-\frac{2}{3} \mu \nabla \bullet   \textbf{U}  \textbf{I} + \mu \nabla \textbf{U} + \mu  (\nabla   \textbf{U})^T\right]

This guy is calculated by calling the function divDevRhoReff(U) or divDevReff(U). Now I will show that the implemented functions are correct.


Compressible

In compressible flow we have the full viscose stress tensor. Now I demonstrate that this term is exactly the divDevRhoReff(U) function. Lets start to modify the equation (again step by step). But first I want to introduce the necessary math operation. \textbf{a} denotes an arbitary vector:

\nabla \bullet
\textbf{a} = \mathrm{tr}(\nabla\textbf{a}) = \mathrm{tr}(\nabla \textbf{a})^T

using this formula and sorting the equation, we get the following:

\nabla \bullet \left[\mu \nabla \textbf{U} + \mu  \left((\nabla   \textbf{U})^T -\frac{2}{3} \mathrm{tr}(\nabla \textbf{U})^T   \textbf{I} \right)\right]

\nabla \bullet (\mu \nabla \textbf{U}) 
+ \nabla \bullet 
\left( \mu  \left[(\nabla   \textbf{U})^T - \frac{2}{3}  \mathrm{tr}(\nabla  \textbf{U})^T   \textbf{I} \right]\right)

Thats it. If we call turbulence->divDevRhoReff() we will exactly get this equation. Using a substituion for the gradient of the transponed velocity field, the name of the function dev2 gets clearer:

(\nabla   \textbf{U})^T = \textbf{A}

\nabla \bullet (\mu \nabla \textbf{U}) 
+ \nabla \bullet 
\left( \mu  \left[(\textbf{A} - \frac{2}{3}\mathrm{tr}(\textbf{A})   \textbf{I} \right]\right)

The deviatoric part of a matrix is defined as:

\mathrm{dev}(\textbf{A}) = \textbf{A} - \frac{1}{3}\mathrm{tr} (\textbf{A})\textbf{I}

Now we see, that the second term include the deviatoric part of the matrix (gradU). In compressible we call the dev2 due to the fact that the factor 2 is in the trace. To proove the equation, we can check out the code in the compressible tubulence file (laminar.C): http://foam.sourceforge.net/docs/cpp...ce.html#l00219

Code:

  219 tmp<fvVectorMatrix> laminar::divDevRhoReff(volVectorField& U) const   220 {
  221    return
  222    (
  223      - fvm::laplacian(muEff(), U)
  224      - fvc::div(muEff()*dev2(T(fvc::grad(U))))
  225    );
  226 }



\nabla \bullet (\mu \nabla \textbf{U})

is equal to (muEff = mu for laminar):
Code:

fvm::laplacian(muEff, U)
\nabla \bullet 
\left( \mu  \left[(\nabla   \textbf{U})^T - \frac{2}{3}  \mathrm{tr}(\nabla  \textbf{U})^T   \textbf{I} \right]\right)

is equal to
Code:

  224      - fvc::div(muEff()*dev2(T(fvc::grad(U))))
At least dev2 calculates exactly the deviatoric part. The code is here: http://foam.sourceforge.net/docs/cpp/a08631_source.html

Code:

  303 //- Return the deviatoric part of a symmetric tensor  304 template<class Cmpt>
   305 inline SymmTensor<Cmpt> dev2(const SymmTensor<Cmpt>& st)
  306 {
  307    return st - SphericalTensor<Cmpt>::twoThirdsI*tr(st);
  308 }

This is equal to:

\textbf{A} - \frac{2}{3}\mathrm{tr} (\textbf{A})\textbf{I} = \left[(\nabla   \textbf{U})^T - \frac{2}{3}  \mathrm{tr}(\nabla  \textbf{U})^T   \textbf{I} \right]

Check :) or no check :confused:, I think, its obvious.


Incompressible
For incompressible fluids (devide by \rho) we again start with the viscous stress tensor \boldsymbol \tau:

\nabla \bullet \boldsymbol \tau = \nabla \bullet \left[-\frac{2}{3} \nu  \underline{\nabla \bullet   \textbf{U}}  \textbf{I} + \nu \nabla \textbf{U} + \nu   (\nabla   \textbf{U})^T\right]

Hence, the density is constant the first term (underlined) at the RHS is zero due to the mass conservation equation:

\nabla\bullet \textbf{U} = 0

\nabla \bullet \boldsymbol \tau = \nabla \bullet \left[\nu \nabla \textbf{U} + \nu   (\nabla   \textbf{U})^T\right]

After a huge amount of mathematics we get :D

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet (\nu   (\nabla   \textbf{U})^T)

So there is no deviatoric part till now. But in FOAM we calculate the viscouse part of the stress tensor using divDevReff. Now we add the hydrostatic part multiply by the viscosity and substract it again:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu   (\nabla   \textbf{U})^T + \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I} -  \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right)

it follows:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[  (\nabla    \textbf{U})^T + \frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right] -   \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right)

The last term is zero due to mass conservation and can be removed:

-   \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I} =  -   \nu\frac{1}{3}\mathrm{tr}(\nabla \textbf{U})\textbf{I} =  -   \nu \nabla \bullet \textbf{U}

We end up with:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[  (\nabla    \textbf{U})^T + \frac{1}{3}\mathrm{tr}(\nabla \textbf{U})^T\textbf{I}\right]\right)

If we substitute the transponed gradient U matrix with A:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[ A + \frac{1}{3}\mathrm{tr}(A)\textbf{I}\right]\right)

Again we have the laplacian term and again the deviatoric part (here calcated with oneThird, so the correct definition of dev).

And this is exactly what we get here:

Code:

  tmp<fvVectorMatrix> laminar::divDevReff(volVectorField& U) const
{
    return
    (
      - fvm::laplacian(nuEff(), U)
      - fvc::div(nuEff()*dev(T(fvc::grad(U))))
    );
}

The code for the deviatoric part (dev()) is equal to dev2. The only difference is the factor 2/3 which is introduced by the divergence term as shown before.

I hope everything is clear now and that there are no mistakes.
Additionally I hope it makes stuff clearer and help anybody to get a better understanding how things work.

Kind regards


Thanks to Alexander Vakhrushev for all support!

Dear Tobias,

I suppose that a deviatoric tensor should be trace-free, while you have added the trace to the original tensor in the following part:

\nabla \bullet (\nu \nabla \textbf{U}) + \nabla \bullet \left(\nu \left[ A + \frac{1}{3}\mathrm{tr}(A)\textbf{I}\right]\right)


Please correct me if I am wrong!!

Thanks,
Syavash


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