How to calculate Reynolds Stress terms using the Boussinesq approximation?

Ashkan Kashani · July 7, 2023, 14:06

Hello all,

I need to define three field variables, namely UV, UU, and VV, which are supposed to provide estimates of Reynolds stress terms (i.e. $-\bar{u'v'}$ , $-\bar{u'u'}$ and $-\bar{v'v'}$ ) based on the Boussinesq approximation.

$-\bar{u'v'}\approx\frac{\mu_{turb}}{\rho}(\frac{\partial\bar{u}}{\partial y}+\frac{\partial\bar{v}}{\partial x})=UV$

$-\bar{u'u'}\approx\frac{\mu_{turb}}{\rho}(2\frac{\partial\bar{u}}{\partial x})-\frac{2}{3}k=UU$

$-\bar{v'v'}\approx\frac{\mu_{turb}}{\rho}(2\frac{\partial\bar{v}}{\partial y})-\frac{2}{3}k=VV$

where $\mu_{turb}$ and

are the eddy viscosity and the turbulence kinetic energy, respectively, which are known from the RANS model adopted in my study (i.e. GeKo model). So far so good.

However, since I'm using a homogeneous multiphase model (i.e. air and water in the domain), multiple combinations of the relevant variables are available in CFX-pre for substitution in the above equations.

$\rho=$ Density or Water.Density or Air.Density

$\mu_{turb}=$ Eddy Viscosity or Water.Eddy Viscosity or Air.Eddy Viscosity

Turbulence Kinetic Energy or Water.Turbulence Kinetic Energy or Air.Turbulence Kinetic Energy

$\bar{u}=$ Velocity u or Water.Velocity u or Water.Superficial Velocity X or Air.Velocity u or Air.Superficial Velocity X

and the same holds for $\bar{v}$ .

In addition, as shown in Figure 1 in the attachments, these additional variables can be defined on Water, Air or both.

I just need to compare the estimated Reynolds stress with experimental data. But I'm not sure what variables and settings are most relevant.

Also attached is the CCL for further details. I would really appreciate your help.

Regards,
Armin

ghorrocks · July 7, 2023, 19:53

Which variables are appropriate depends on many factors:
* The volume fractions - is one phase very high VF and one very low? Or both similar?
* The nature of the multiphase flow - bubbles, droplets, discrete free surface?
* What the paper you are comparing to say they have done
* How accurate you want to be

Ashkan Kashani · July 11, 2023, 15:14

Thank you for your comment.

Quote:

Originally Posted by ghorrocks

* The volume fractions - is one phase very high VF and one very low? Or both similar?
* The nature of the multiphase flow - bubbles, droplets, discrete free surface?

It is an open channel flow, so the two phases are largely separate except for some minor entrainment of air predicted by the simulation but not observed in the lab experiments.

Quote:

Originally Posted by ghorrocks

* What the paper you are comparing to say they have done

I am comparing the Reynolds stress terms predicted by the CFD model to those obtained from Particle Image Velocimetry (PIV) in the lab. Note the PIV analysis was applied to the water domain only (the tiny tracers were injected in the water, not air). The air flow is not deemed to have any effect and therefore not of any interest.

Taking all these into account, how can I set the field variables that are most comparable and relevant to the experimentally measured Reynolds stress terms? I couldn't find the exact definition of those variables in order to make a confident decision.

ghorrocks · July 11, 2023, 20:20

The Water.variable and Air.variable ones are obviously the variable of their respective phases. The "variable" one is the volume fraction weighted value.

But in your case as it is a free surface flow then the volume fraction will be 0 or 1 across almost the entire domain (with only a small region of intermediate VFs at the interface). This means the "variable" value will be the same as either Water.variable or Air.Variable depending on whether the location is in the air or water.

The choice of what variable to use to calculate your Reynolds stresses should be pretty obvious from that.

Ashkan Kashani · October 10, 2023, 15:40

Hello everyone. I have another relevant question, so I will post it here.

It is easy to compute the Reynold stress based on a linear eddy viscosity model in CFX; the Reynold stress terms can be defined as new field variables through simple short expressions following the Boussinesq assumption, i.e.

$\tau_{i j}=-\rho \overline{u_i^{\prime} u_j^{\prime}}=\mu_t (\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})-\frac{2}{3} \rho k \delta_{i j}$

However, when it comes to nonlinear models like EARSM which uses a much more complicated constitutive relation as follows

$\tau_{i j}=-\rho \overline{u_i^{\prime} u_j^{\prime}}=\left(-\frac{\beta_1}{2 C_\mu}\right) 2 \mu_t S_{i j}-\frac{2}{3} \rho k \delta_{i j}-a_{i j} \rho k$
with
$a_{i j}=\beta_3\left(\Omega^n_{i k} \Omega^n_{k j}-\frac{1}{3} \mathrm{II}_{\Omega^n} \delta_{i j}\right)+\beta_4\left(S^n_{i k} \Omega^n_{k j}-\Omega^n_{i k} S^n_{k j}\right)$
$+\beta_6\left(S^n_{i k} \Omega^n_{k l} \Omega^n_{l j}+\Omega^n_{i k} \Omega^n_{k l} S^n_{l j}-\frac{2}{3} \mathrm{IV} \delta_{i j}-\mathrm{II}_{\Omega^n} S^n_{i j}\right)$

This is not very straightforward to implement; I guess a piece of code is required. I am wondering if there is any easy way to compute the Reynolds stress based on the EARSM model. Could those terms be readily reported in the result file by any means?

I appreciate your help.

ghorrocks · October 10, 2023, 17:43

Have a look in the results file (including all the additional variables you can add in CFX-Pre) to see what is available - but I do not think that variable is reported.

So if you want to report it you will have to calculate it in user fortran or CEL.

July 7, 2023, 19:53		#2
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,703 Rep Power: 143	Which variables are appropriate depends on many factors: * The volume fractions - is one phase very high VF and one very low? Or both similar? * The nature of the multiphase flow - bubbles, droplets, discrete free surface? * What the paper you are comparing to say they have done * How accurate you want to be __________________ Note: I do not answer CFD questions by PM. CFD questions should be posted on the forum.

July 11, 2023, 20:20		#4
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,703 Rep Power: 143	The Water.variable and Air.variable ones are obviously the variable of their respective phases. The "variable" one is the volume fraction weighted value. But in your case as it is a free surface flow then the volume fraction will be 0 or 1 across almost the entire domain (with only a small region of intermediate VFs at the interface). This means the "variable" value will be the same as either Water.variable or Air.Variable depending on whether the location is in the air or water. The choice of what variable to use to calculate your Reynolds stresses should be pretty obvious from that. Ashkan Kashani likes this. __________________ Note: I do not answer CFD questions by PM. CFD questions should be posted on the forum.

October 10, 2023, 17:43		#6
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,703 Rep Power: 143	Have a look in the results file (including all the additional variables you can add in CFX-Pre) to see what is available - but I do not think that variable is reported. So if you want to report it you will have to calculate it in user fortran or CEL. Ashkan Kashani likes this. __________________ Note: I do not answer CFD questions by PM. CFD questions should be posted on the forum.

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Recommendation of paper for Reynolds Stress Model	FluidKo	Main CFD Forum	7	February 10, 2022 08:57
[ANSYS Meshing] Error in Workbench "Unable to attach to geometry file"	Jeremie84	ANSYS Meshing & Geometry	70	October 25, 2017 16:03
Reynold's Stress not shown	anjansir	OpenFOAM Post-Processing	0	December 16, 2011 01:55
Reynolds stress distribution (presentation)	quattro123	ANSYS	1	January 11, 2011 12:22
reynolds stress	tufito	Main CFD Forum	3	March 10, 2005 15:40

October 10, 2023, 15:40		#5
Ashkan Kashani Member Ashkan Kashani Join Date: Apr 2016 Posts: 46 Rep Power: 10	Hello everyone. I have another relevant question, so I will post it here. It is easy to compute the Reynold stress based on a linear eddy viscosity model in CFX; the Reynold stress terms can be defined as new field variables through simple short expressions following the Boussinesq assumption, i.e. $\tau_{i j}=-\rho \overline{u_i^{\prime} u_j^{\prime}}=\mu_t (\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})-\frac{2}{3} \rho k \delta_{i j}$ However, when it comes to nonlinear models like EARSM which uses a much more complicated constitutive relation as follows $\tau_{i j}=-\rho \overline{u_i^{\prime} u_j^{\prime}}=\left(-\frac{\beta_1}{2 C_\mu}\right) 2 \mu_t S_{i j}-\frac{2}{3} \rho k \delta_{i j}-a_{i j} \rho k$ with $a_{i j}=\beta_3\left(\Omega^n_{i k} \Omega^n_{k j}-\frac{1}{3} \mathrm{II}_{\Omega^n} \delta_{i j}\right)+\beta_4\left(S^n_{i k} \Omega^n_{k j}-\Omega^n_{i k} S^n_{k j}\right)$ $+\beta_6\left(S^n_{i k} \Omega^n_{k l} \Omega^n_{l j}+\Omega^n_{i k} \Omega^n_{k l} S^n_{l j}-\frac{2}{3} \mathrm{IV} \delta_{i j}-\mathrm{II}_{\Omega^n} S^n_{i j}\right)$ This is not very straightforward to implement; I guess a piece of code is required. I am wondering if there is any easy way to compute the Reynolds stress based on the EARSM model. Could those terms be readily reported in the result file by any means? I appreciate your help.