C5 term in SSG Reynolds Stress Model

agustinvo · April 22, 2022, 07:00

Hi everyone,

I am checking the SSG implementation, concretely the C5 term. In the reference, the skwe of the velocity gradient is defined as:

$W_{ij} = \frac{1}{2}\left(\frac{\partial U_i}{\partial x_j} - \frac{\partial U_j}{\partial x_i}\right)$

When checking how this is implemented in OpenFOAM:

Code:

Omega(skew(gradU))

but, if the Programmer's Guide is right, the velocity gradient is defined as $\frac{\partial U_j}{\partial x_i}$ . When using the skew function $W =\frac{1}{2} \left(T - T^T\right)$ what whe should get is:
$W_{ij} = \frac{1}{2}\left(\frac{\partial U_j}{\partial x_i} - \frac{\partial U_i}{\partial x_j}\right)$
which is just the transpose of the expected skew tensor in the model.

Am I missing something? Is therefore properly implemented this term or am I right and it should be modified?

Tobermory · April 22, 2022, 09:39

Can you be more explicit please - I assume you are talking about the SSG Reynolds stress closure? If so, which term are you talking about and which reference?

Edit: ok, I see now. You are talking about the SSG model, and the term $C_5 K \left(b_{ik}W_{jk} + b_{jk}W_{ik} \right)$ from the Speziale paper which is coded in SSG.C (https://cpp.openfoam.org/v8/SSG_8C_source.html) as:

Code:

     volTensorField Omega(skew(gradU));
...
   + C5_*twoSymm(b&Omega)

Let me have a think on this.

Tobermory · April 22, 2022, 10:15

I think it comes down to semantics and the definition of the gradient in tensor notation (see an earlier thread, for example: discrepancy in fvc::grad(U)). In their JFM paper, eqn 23, Speziale etc write the

term of the velocity gradient as:

$\frac{\partial U_i}{\partial x_j} = S_{ij} + \omega_{ij}$

which is the inverse of the way that the OpenFOAM tensors are defined (again refer to the above thread ... if you have the patience!). To be clear, in OF, the

term of the velocity gradient is: $\frac{\partial U_j}{\partial x_i}$ . This means that the definition of

in the Speziale paper has inverted indices as well ... and so I think that the implemented model is probably correct. Hope that helps.

agustinvo · April 22, 2022, 14:29

Thanks Tobermory! It was impossible that after more than 9 OF versions nobody noticed it before.

As all of you discuss in that thread, it seems that OF knows already how we expect to operate with gradU. However, the programmer's guide equation (2.3) is confusing (that guide is quite old). Mathematically it is ok, but there is no link with the actual implementation of the gradient of a vector.

I used the OF programmer's guide to define some terms in custom turbulence models, whose results are ok. I have to review what the heck want on there.

After a test using Omega(skew(gradU)) and Omega(skew(gradU.T)) the original implmentation was the good one. As you say, maybe the transposed of Speciale et al and the transposed of OF cancel this effect.

April 22, 2022, 07:00	C5 term in SSG Reynolds Stress Model	#1
agustinvo Senior Member Agustín Villa Join Date: Apr 2013 Location: Alcorcón Posts: 313 Rep Power: 15	Hi everyone, I am checking the SSG implementation, concretely the C5 term. In the reference, the skwe of the velocity gradient is defined as: $W_{ij} = \frac{1}{2}\left(\frac{\partial U_i}{\partial x_j} - \frac{\partial U_j}{\partial x_i}\right)$ When checking how this is implemented in OpenFOAM: Code: Omega(skew(gradU)) but, if the Programmer's Guide is right, the velocity gradient is defined as $\frac{\partial U_j}{\partial x_i}$ . When using the skew function $W =\frac{1}{2} \left(T - T^T\right)$ what whe should get is: $W_{ij} = \frac{1}{2}\left(\frac{\partial U_j}{\partial x_i} - \frac{\partial U_i}{\partial x_j}\right)$ which is just the transpose of the expected skew tensor in the model. Am I missing something? Is therefore properly implemented this term or am I right and it should be modified?

April 22, 2022, 09:39		#2
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 668 Rep Power: 14	Can you be more explicit please - I assume you are talking about the SSG Reynolds stress closure? If so, which term are you talking about and which reference? Edit: ok, I see now. You are talking about the SSG model, and the term $C_5 K \left(b_{ik}W_{jk} + b_{jk}W_{ik} \right)$ from the Speziale paper which is coded in SSG.C (https://cpp.openfoam.org/v8/SSG_8C_source.html) as: Code: volTensorField Omega(skew(gradU)); ... + C5_*twoSymm(b&Omega) Let me have a think on this.

April 22, 2022, 14:29		#4
agustinvo Senior Member Agustín Villa Join Date: Apr 2013 Location: Alcorcón Posts: 313 Rep Power: 15	Thanks Tobermory! It was impossible that after more than 9 OF versions nobody noticed it before. As all of you discuss in that thread, it seems that OF knows already how we expect to operate with gradU. However, the programmer's guide equation (2.3) is confusing (that guide is quite old). Mathematically it is ok, but there is no link with the actual implementation of the gradient of a vector. I used the OF programmer's guide to define some terms in custom turbulence models, whose results are ok. I have to review what the heck want on there. After a test using Omega(skew(gradU)) and Omega(skew(gradU.T)) the original implmentation was the good one. As you say, maybe the transposed of Speciale et al and the transposed of OF cancel this effect. Tobermory likes this.

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April 22, 2022, 10:15		#3
Tobermory Senior Member Join Date: Apr 2020 Location: UK Posts: 668 Rep Power: 14	I think it comes down to semantics and the definition of the gradient in tensor notation (see an earlier thread, for example: discrepancy in fvc::grad(U)). In their JFM paper, eqn 23, Speziale etc write the term of the velocity gradient as: $\frac{\partial U_i}{\partial x_j} = S_{ij} + \omega_{ij}$ which is the inverse of the way that the OpenFOAM tensors are defined (again refer to the above thread ... if you have the patience!). To be clear, in OF, the term of the velocity gradient is: $\frac{\partial U_j}{\partial x_i}$ . This means that the definition of in the Speziale paper has inverted indices as well ... and so I think that the implemented model is probably correct. Hope that helps.