[Serouj]

Hello people, I've been trying to plot a trajectory on the anisotropic invariant map of a Lumley triangle for a turbulent flow in a pipe by calculating the II-III components of the graph. I've attached a picture of the map and the corresponding equations used to calculate the 2 components in order to plot a trajectory in a turbulent pipe flow for example. My question is how to get the values of u_i*u_j^mean and the value of the trace of the matrix out of my simulation data. Does anyone know of any sources that mention guides and examples about this? So far I was unsuccessful to find anything related to this topic. Some help or guidance would be much appreciated!

[LuckyTran]

What you need is the anisotropic Reynold's stress tensor everywhere in your pipe. Then calculate the invariants and you can built this plot.


If you need more examples for how to calculate the invariants of tensors.


Just get the Reynolds stress tensor first and that is 90% of the work. Just to be clear, that's: uu, uv, uw, vv, ww, vw, 6 thingies you need to determine, at every x,y,z location in your pipe.


What software you are using? And are you doing RANS/LES/DNS? There may be some built-in tools to get the Reynolds stresses. The invariants you can clculate in-software if you hard-code the equations for the invariants, but you'll likely do this in post-processing software like matlab or tecplot (using the same equations).

[Serouj]

Originally Posted by LuckyTran

What you need is the anisotropic Reynold's stress tensor everywhere in your pipe. Then calculate the invariants and you can built this plot.


If you need more examples for how to calculate the invariants of tensors.


Just get the Reynolds stress tensor first and that is 90% of the work. Just to be clear, that's: uu, uv, uw, vv, ww, vw, 6 thingies you need to determine, at every x,y,z location in your pipe.


What software you are using? And are you doing RANS/LES/DNS? There may be some built-in tools to get the Reynolds stresses. The invariants you can clculate in-software if you hard-code the equations for the invariants, but you'll likely do this in post-processing software like matlab or tecplot (using the same equations).

Hello, thank you for your answer with clarifying some things which were vague to me. I'm running a RANS simulation on Ansys Fluent, that's why I dont have access to the equations that are running in the background and therefore cannot extract any data from the equations themselves and need to rely on the Post-Processing step in order to do this. However, in the Post-Processing step, if I extract from a certain control Volume the 3 velocity components, build my Reynold's stress tensor, calculate the anisotropy of turbulence tensor (a_ij) which is of the same shape as the Reynold's tensor and then go on and calculate the invariants II and III, and finally repeat this for a Line-Section in the pipe, will this work? I don't think so, because the velocities in this case would be the averaged mean velocities and not the averaged velocity fluctuations which are required to build my Reynold's stress tensor. Any further thoughts about this subjects?
Best Regards,
Serouj Djabraian

[LuckyTran]

You don't get the Reynolds stresses from just the mean velocity components, no. What you need is the strain rate tensor (often denoted S and closely related to the velocity gradient).

From the strain-rate tensor you can get your Reynolds stress tensor from your eddy viscosity model. But there's no actual need to do this since they are proportional via the turbulent viscosity. So really you could just calculate the invariants of the strain rate tensor and call it a day. Do this and you'll find that all your flow falls into one particular section of Lumley's triangle. It is a cute little result.

[Serouj]

Originally Posted by LuckyTran

You don't get the Reynolds stresses from just the mean velocity components, no. What you need is the strain rate tensor (often denoted S and closely related to the velocity gradient).

From the strain-rate tensor you can get your Reynolds stress tensor from your eddy viscosity model. But there's no actual need to do this since they are proportional via the turbulent viscosity. So really you could just calculate the invariants of the strain rate tensor and call it a day. Do this and you'll find that all your flow falls into one particular section of Lumley's triangle. It is a cute little result.

Yes thank you, it all makes sense now! Thank you for your time and effort in clarifying things. I will proceed with your suggestion!