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adding viscous dissipation to openfoam solver

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Old   January 19, 2018, 09:31
Default adding viscous dissipation to openfoam solver
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Hi
I want to add viscous dissipation to energy eqn. but my fluid is not Newtonian and has a special stress tensor. So the work of stress tensor field is "tau && grad(U)"

1- is it a correct relation for work of stress tensor field? or I should write:
tau && [grad(U) + transpose of (grad(U))]

2- is it correct to write:

solve
(
fvm::div(phi, T)
- fvm::laplacian(alphaEff, T)
- tau && grad(U)
);

thank you
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Old   January 22, 2018, 18:01
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Hi,

I would expect to have the shear heating given as:

\boldsymbol \tau \boldsymbol \colon (\nabla \otimes \textbf{U})

However, I am not sure about Non-Newtonian fluids. The only change should be in the viscosity (which does not follow a linear profile). The stress tensor itself should be similar to those of Newtonian fluids. I might be wrong in that case and have to proof it somewhere in literature. The equation of the shear-heating is correct.
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Old   January 23, 2018, 02:34
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Originally Posted by Tobi View Post
Hi,

I would expect to have the shear heating given as:

\boldsymbol \tau \boldsymbol \colon (\nabla \otimes \textbf{U})

However, I am not sure about Non-Newtonian fluids. The only change should be in the viscosity (which does not follow a linear profile). The stress tensor itself should be similar to those of Newtonian fluids. I might be wrong in that case and have to proof it somewhere in literature. The equation of the shear-heating is correct.
Thank you, I found it. Yes, that is correct. but we should add transpose of grad(U) to your formula:
tau && [grad(U) + transpose of (grad(U))]
this relation is valid for viscoelastic fluid, just we should use viscoelastic stress tensor.
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Old   January 23, 2018, 03:21
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Thanks for the reply. Can you give me a reference for proof? Thus, I can add this information to my book.
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Old   January 23, 2018, 10:48
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Hi,
this is incorrect.
If Tau is Cauchy stress tensor then (does not matter what kind, i.e., viscoelastic fluid etc.)
\boldsymbol{\tau} : \boldsymbol{\nabla} \boldsymbol{v} = 
\boldsymbol{\tau} : \frac{1}{2} \left( \boldsymbol{\nabla} \boldsymbol{v} + (\boldsymbol{\nabla} \boldsymbol{v})^T \right),
but note that by "additing" transpose of gradient of the velocity to the formula, you really gives less information in the formula as
\frac{1}{2} \left( \boldsymbol{\nabla} \boldsymbol{v} + (\boldsymbol{\nabla} \boldsymbol{v})^T \right),
is symmetric part of the gradient of the velocity. This is indeed used in many proofs.

Best Ondra
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Old   January 23, 2018, 13:00
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I don't get your point. Who is wrong and what is correct now?
In general the Cauchy stress tensor is defined as \boldsymbol \sigma while the shear tensor is \boldsymbol \tau such as:

\boldsymbol \sigma = \boldsymbol \tau - p\textbf{I}

while \boldsymbol \tau is the deviatoric part of the Cauchy stress tensor and the negative pressure represents the hydrostatic part such as:

\boldsymbol \sigma= \boldsymbol \sigma^\mathrm{dev} + \boldsymbol \sigma^\mathrm{hyd}

Based on Bird et al. the viscose heating is related to:

\boldsymbol \tau \boldsymbol \colon (\nabla \otimes \textbf{U})

but it is mentioned that you have to have an symmetric shear tensor (\tau_{xy} = \tau_{yx}). Thus, your relation is only valid for symmetric ones; if this was the thing you wanted to point out, I just clearyfied the things. If there is some mistake, please don't hesitate to correct me.

only valid for symmetric shear rate tensor \textbf{D}

while \textbf{D} is given as:

\textbf{D} = \frac{1}{2} [\nabla \otimes \textbf{U} + (\nabla \otimes \textbf{U})^\mathrm{T}]
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Old   January 23, 2018, 13:24
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Quote:
Originally Posted by Tobi View Post
I don't get your point. Who is wrong and what is correct now?
In general the Cauchy stress tensor is defined as \boldsymbol \sigma while the shear tensor is \boldsymbol \tau such as:

\boldsymbol \sigma = \boldsymbol \tau - p\textbf{I}

while \boldsymbol \tau is the deviatoric part of the Cauchy stress tensor and the negative pressure represents the hydrostatic part such as:

\boldsymbol \sigma= \boldsymbol \sigma^\mathrm{dev} + \boldsymbol \sigma^\mathrm{hyd}
you mean the gradient of the velocity. Note that

Based on Bird et al. the viscose heating is related to:

\boldsymbol \tau \boldsymbol \colon (\nabla \otimes \textbf{U})

but it is mentioned that you have to have an symmetric shear tensor (\tau_{xy} = \tau_{yx}). Thus, your relation is only valid for symmetric ones; if this was the thing you wanted to point out, I just clearyfied the things. If there is some mistake, please don't hesitate to correct me.

only valid for symmetric shear rate tensor \textbf{D}

while \textbf{D} is given as:

\textbf{D} = \frac{1}{2} [\nabla \otimes \textbf{U} + (\nabla \otimes \textbf{U})^\mathrm{T}]

Sorry, I forget to cite alimea post. My post was reaction mainly to point-out that it does not matter if the Newtonian fluid is under consideration.

I defined Cauchy stress as Tau not Sigma, but it really does not matter in our discussion. Cauchy stress is always symmetric and the D is defined as symmetric part of the gradient of the velocity. So the formula holds always within classical mechanics of continuum.
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Old   January 23, 2018, 14:55
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I agree, the nomenclature does not play a big deal here. However, do you have any example in which the Cauchy stress tensor, or the general formulation - given in Gurtin et al. - The Mechanics and Thermodynamics of Continua - is not symmetric?

PS: Did you comment in the quotes? I did not mean the gradient. What did you mean there?
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Old   January 23, 2018, 15:09
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Quote:
Originally Posted by Tobi View Post
I agree, the nomenclature does not play a big deal here. However, do you have any example in which the Cauchy stress tensor, or the general formulation - given in Gurtin et al. - The Mechanics and Thermodynamics of Continua - is not symmetric?
No, as long as conservation of angular momentum holds the Cauchy stress tensor is symmetric.
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Old   March 22, 2019, 00:54
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Dear alimea
Do you add viscous dissipation to energy equation? I have the same problem, if favor me, I pleasure. thanks
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Old   March 22, 2019, 13:48
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Originally Posted by Hgholami View Post
Dear alimea
Do you add viscous dissipation to energy equation? I have the same problem, if favor me, I pleasure. thanks
Hi
what's your problem?
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Old   March 22, 2019, 14:17
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Dear alimea
I want to add a viscous heating term to energy equation as below
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Old   March 23, 2019, 17:21
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Originally Posted by Hgholami View Post
Dear alimea
I want to add a viscous heating term to energy equation as below
ok. what's the problem?
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Old   March 24, 2019, 00:05
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I want to add last term to energy equation, So I uses
Quote:
tau=rho_*fluid.nu()*(twoSymm(fvc::grad(U_)))
I22=(tau && fvc::grad(U_)); //for I2/2
Is it true?
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Old   March 24, 2019, 01:09
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Originally Posted by Hgholami View Post
I want to add last term to energy equation, So I uses Is it true?
Yes. But don't forget that there is a viscosity in your first formula. in fact:

stressWork = (1.0/(rho*Cp))*( tau && fvc::grad(U) );

Regards
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Old   March 24, 2019, 01:19
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Originally Posted by alimea View Post
Yes. But don't forget that there is a viscosity in your first formula. in fact:

stressWork = (1.0/(rho*Cp))*( tau && fvc::grad(U) );

Regards
In first Formula, fluid.nu() is dynamic viscosity that multiply with density to give kinematic viscosity?
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