Energy equation in weird format

asda3D · May 11, 2019, 18:44

Hi everyone,

I am confused on the NS and RANS format of the energy equation (attached below) I do not understand why there are only two terms on the LHS of the equation. The first term on the LHS represents the Fourier law of thermal conductivity.

I have researched and found that q''' represents the volumetric heat generation. I cannot find any equation online that bears similar resemblance.

Also, Why is the thermal conductivity term expanded to produce a 'Reynolds stress' for the energy equation?
I am presuming a model like the mixing length model will interpret this the same way as it would for a momentum Reynolds stress term.

If anyone can explain the terms in these equations, you are a true CFD genius!!!!

Any help much appreciated.

FMDenaro · May 12, 2019, 04:22

The term q''' seems just a production term in the internal energy equation, for example due to chemical reaction.
The RANS equations are deduced when you introduce the decomposition of temperature and velocity in average and residual terms. The convective terms produces the unresolved term that must be modelled. For the temperature equation it can be debated that a eddy diffusivity-like term is physically meaningful

asda3D · May 12, 2019, 07:16

Thank you for the response kind sir.

So is it safe to say then the turbulence models are used to solve the 'reynold stress' in the energy equation. As I cannot seem to find any models that solve the blue part in the energy equation

Regards

FMDenaro · May 12, 2019, 07:21

Quote:

Originally Posted by asda3D

Thank you for the response kind sir.

So is it safe to say then the turbulence models are used to solve the 'reynold stress' in the energy equation. As I cannot seem to find any models that solve the blue part in the energy equation

Regards

In the internal energy equation the unresolved terms are a stress-like term as in the momentum equation. They are often modelled a a supplementary heat flux. Have a look to the book of Wilcox.

asda3D · May 12, 2019, 07:28

I have read some literature that describes the temperature term in the RANS as a turbulent stress. I will look at the book to see if i can find a derivation of how a turbulence model interprets the energy equation

Thank you for the quick response

LuckyTran · May 13, 2019, 00:39

The 1st term on the RHS of the Reynolds averaged equation looks very similar to the original term in the time-accurate equation (the red term) except with $\bar u$ instead of

. The blue term actually comes from the 2nd term on the left hand side. It has nothing to do with thermal conductivity. That is, start with:

$u_j\frac{\partial T}{\partial x_j}$

Put
$u=\bar u+u'$ and $T=\bar T+T'$
and then take the Reynolds average and the blue term will pop out. The negative (minus sign) comes from moving this term from the LHS to the RHS.

The blue term is modeled most commonly using a turbulent Prandtl number and most often using a constant turbulent Prandtl number which is slightly less than 1.

Also these are not weird forms for the energy equation, this is how it is supposed to look. =)
Whatever sources you use might derive one or more variants of the RANS, but it is very standard. Not weird. I've personally written down this equation on napkins probably a hundred times and thrown them into the wastebin.

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May 12, 2019, 04:22		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,769 Rep Power: 71	The term q''' seems just a production term in the internal energy equation, for example due to chemical reaction. The RANS equations are deduced when you introduce the decomposition of temperature and velocity in average and residual terms. The convective terms produces the unresolved term that must be modelled. For the temperature equation it can be debated that a eddy diffusivity-like term is physically meaningful

May 12, 2019, 07:16		#3
asda3D New Member anonymous Join Date: Apr 2018 Posts: 26 Rep Power: 8	Thank you for the response kind sir. So is it safe to say then the turbulence models are used to solve the 'reynold stress' in the energy equation. As I cannot seem to find any models that solve the blue part in the energy equation Regards

May 12, 2019, 07:28		#5
asda3D New Member anonymous Join Date: Apr 2018 Posts: 26 Rep Power: 8	I have read some literature that describes the temperature term in the RANS as a turbulent stress. I will look at the book to see if i can find a derivation of how a turbulence model interprets the energy equation Thank you for the quick response

May 13, 2019, 00:39		#6
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,675 Rep Power: 66	The 1st term on the RHS of the Reynolds averaged equation looks very similar to the original term in the time-accurate equation (the red term) except with $\bar u$ instead of . The blue term actually comes from the 2nd term on the left hand side. It has nothing to do with thermal conductivity. That is, start with: $u_j\frac{\partial T}{\partial x_j}$ Put $u=\bar u+u'$ and $T=\bar T+T'$ and then take the Reynolds average and the blue term will pop out. The negative (minus sign) comes from moving this term from the LHS to the RHS. The blue term is modeled most commonly using a turbulent Prandtl number and most often using a constant turbulent Prandtl number which is slightly less than 1. Also these are not weird forms for the energy equation, this is how it is supposed to look. =) Whatever sources you use might derive one or more variants of the RANS, but it is very standard. Not weird. I've personally written down this equation on napkins probably a hundred times and thrown them into the wastebin.