A mentioned by Filippo Maria Denaro, ergodicity of a dynamical system is the primary requirement for the flow to be treated as 2D. This condition is not met, for instance, if the incoming flow has spanwise nonuniformities (i.e., boundary conditions are 3D) that can produce a measurable impact on the flow past a configuration with 2D geometry.

You can find an explanation also in the book of Wilcox. The key is if statistical (ensemble) averaging, spatial averaging and time averaging converge to the same result. If your 3D flow problem has a 2D geometry repeated "infinitely" in the spanwise direction, the RANS equations can be written for a 2D problem. This happens for example when you have periodic conditions.

Dear Vigneshwaran Sankar,

my modest recommendation:

In neutrally stratified flows we have full 3D character of the turbulence.

The The situation is quite different in stratified and convective flows. Namely stratified flows allow for two-dimensionalisation of the large-scale eddies and reduction of dissipation.

• With respect

A mentioned by Filippo Maria Denaro, ergodicity of a dynamical system is the primary requirement for the flow to be treated as 2D. This condition is not met, for instance, if the incoming flow has spanwise nonuniformities (i.e., boundary conditions are 3D) that can produce a measurable impact on the flow past a configuration with 2D geometry.

it will depends on the case. Because the turbulence is always a 3D phenomenon. If you calculates the rotational of all N-S equation - the vorticity governing equation, there is a term that represents the generation of vorticity, and ins 2D this source vanish. Hence, some differences can arise among 2D and 3D approach , it will depends the flow, and if these differences are acceptable in the analysis. Vortex streching will not happens in 2D. But a 2D approach is always good to a fist try, because changing the domain to 3D the physical setup will be the same.

Dear Ricardo Vicente de Paula Rezende

what you stated is correct in deterministic sense but not in statistical sense, that is the original question. The 2D statistical solution has all that is required in the vorticity dynamics. To convince yourself just immagine to have a full 3D DNS solution and perform the time averaging over a very long period. If the flow has homogeneity in the third direction you will see that such time-averaging will produce a 2D statistical field.

A follow up question to the researchers:

In the present case being discussed here, would it be correct to suggest that the Boussinesq approximation is responsible for re-creating the essentially 3-D effect of turbulence, in 2-D, by means of giving finite value to the Reynold's stress tensor even when strains exist in only x and y i.e. 2-D?

The treatment of Two-dimensional effect depends on the incoming flow so for simple case , try to solve the problem as two dimension and compare the predicted results with experimental measurements. You will decide which 2-D or 3-D.

Filippo Maria Denaro , I agree, because as I said " it will depends on the flow". To be homogeneous in then third direction is a specific case where a 2D run will agree with a 3D simulation in a statistical sense because it has the same physical meaning for such flow. But while in cases where this is not true? Such as fluidized beds with particle clusters and highly dynamic vortex stretching. This approach will fail because the flow nature is fully 3D. And how to determined if my flow have a homogeneity in the 3rd direction if I not know the flow a priori??? This is the answer what we are seeking. As Ahmed Abdelhameed has commented, a experimental comparison will say which is better. We can perform an average of anything, but the flow physics will say if this average has a physical meaning, because nature works in spite of how we try to describe it. If this physics is known or we have some idea about it, such the case you gave as example, OK, I agree totally with you, but not in a general sense. But despite this, I recommend to try 2D simulations due to computational costs. Some information is better than none, and it will be useful to a possible extension to 3D analysis if the results does not fit with the experimental data.