Development of RANS equations

Boone · September 30, 2022, 09:00

Hello everyone,

I am learning the RANS equations for incompressible flows. I am not sure to understand the classical development of RANS equations as given in the attached figure.
1) Why eq(4) is simplified compared to eq(3) ?

2) I think that eq(4) is the conservative form of the equation while eq(3) is the non-conservative form. Is that true ?

3) The author introduces a "null term" to obtain eq(4). This null term is $u \left[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right]$ . I understand that $\frac{\partial v}{\partial y}$ is equal to 0 if we consider an established flow. However, if $\frac{\partial u}{\partial x}$ is null too it means that this term should then also be null in the left side of eq(3) and eq(4). Am I right ? So why not simplifying it from the begining ?

Thanks !

LuckyTran · September 30, 2022, 10:20

Yeah nothing is being simplified.

(3) is indeed the convective form and (4) would be the conservation form if the derivation is correct. If (3) & (4) is supposed to be the Navier-Stokes then your author has mixed up a u with a v, or a y with an x.

This null term is of course the divergence free incompressibility constraint. The sum must go to zero $\left[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right]=0$ , not the individual term $\frac{\partial v}{\partial y}=0$ .

FMDenaro · September 30, 2022, 12:03

Quote:

Originally Posted by Boone

Hello everyone,

I am learning the RANS equations for incompressible flows. I am not sure to understand the classical development of RANS equations as given in the attached figure.
1) Why eq(4) is simplified compared to eq(3) ?

2) I think that eq(4) is the conservative form of the equation while eq(3) is the non-conservative form. Is that true ?

3) The author introduces a "null term" to obtain eq(4). This null term is $u \left[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right]$ . I understand that $\frac{\partial v}{\partial y}$ is equal to 0 if we consider an established flow. However, if $\frac{\partial u}{\partial x}$ is null too it means that this term should then also be null in the left side of eq(3) and eq(4). Am I right ? So why not simplifying it from the begining ?

Thanks !

Where do you find these equations? To be honest, I don't like that...

I suppose they should illustrate the momentum equations in quasi-linear and divergence form but why the diffusive term has only one component?

LuckyTran · September 30, 2022, 12:33

Yeah I don't like it either. Either we are not looking at the navier-stokes or this is a really interesting derivation.

Not only is (3) questionable, they should be adding the entire continuity equation and not just the solenoidal condition if they want to go to the conservative form. And even then, where are all the other missing terms...?

September 30, 2022, 12:33		#4
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	Yeah I don't like it either. Either we are not looking at the navier-stokes or this is a really interesting derivation. Not only is (3) questionable, they should be adding the entire continuity equation and not just the solenoidal condition if they want to go to the conservative form. And even then, where are all the other missing terms...? FMDenaro likes this.

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September 30, 2022, 10:20		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	Yeah nothing is being simplified. (3) is indeed the convective form and (4) would be the conservation form if the derivation is correct. If (3) & (4) is supposed to be the Navier-Stokes then your author has mixed up a u with a v, or a y with an x. This null term is of course the divergence free incompressibility constraint. The sum must go to zero $\left[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right]=0$ , not the individual term $\frac{\partial v}{\partial y}=0$ .