[Shabi]

Assume we have a 2D rectangular channel defined in the xy plane and the flow enters the domain from the left side and BC of the top and bottom is the no slip one.
Let us extrude the geometry in the z direction while keeping other conditions same(the z-component of the inlet velocity BC is zero).
Are the results obtained from the 2D simulation and the 3D one different If

1) We extrude in the z direction with only one layer of cells?
2) We extrude in the z direction with more layer of cells?
3) What are the possibilities for the BC of the front and back section? What is the difference intuitively and physically between symmetry boundary condition and Neumann BC in the context above?

In general, I appreciate any information about doing a 2D simulation in a 3D geometry...

[sbaffini]

Let's just first focus on the math side with no specific code implementation in mind.

The 3D simulation will replicate the 2D results if, for any layer of cells in the 3rd dimension, the solved equations will be identical to the 2D ones (with one practical caveat I'll get back to soon).

In order to achieve this you need all the extraneous terms (with respect to the 2D case) to disappear from the solved equations. Which, in turn, requires zero flow in the 3rd direction and zero extraneous derivatives. If you were simulating incompressible flows, a null divergence would have allowed for a constant velocity in the 3rd direction to be admissible just like the no flow condition, but that wouldn't work for general compressible cases.

Now, with Neumann condition we refer to the setting of the normal derivative of a variable to 0. With symmetry condition we imply using Neumann for all the variables and a no flow condition. So, a problem has a symmetry boundary condition, a single variable has Neumann condition.

In your case, replicating in 3D the 2D case requires the symmetry condition on both faces in the 3rd direction. The no flow or constant flow can be achieved either with a Dirichlet condition (assigning a value on the boundary) or with Neumann. In both cases a proper inlet value should be assigned.

How and if these work also from the practical side is, however, very code dependent, as it is the fact that a single layer of cells will suffice or if more will be needed. In theory one should just work as many.

However, as the number of cells in the third dimension grows, if the Re number is sufficiently high, even small numerical artifacts (like the grid coordinates not being perfectly aligned on the different planes) and the fact that the code is 3D, will eventually produce turbulence and a 3D flow, despite all that we said.