Consider parallel flow in the X direction over a 2D semi infinite flat plate. If turbulence is 2-D, in which axes should we expect the vortices to form. Also, are there any experimental/visualization evidence of 2-D vortical turbulent structures?
There are many examples of 2D turbulence. Many experiments involving 2D turbulence use flow in a thin film of soap. 2D turbulence is pretty easy to model too, and I've even seen online web-based examples: http://www.ibiblio.org/e-notes/webgl/gpu/fluid.htm
One super interesting thing about 2D turbulence is the "Inverse energy cascade." In traditional 3D turbulence big eddies break apart into smaller and smaller eddies:
Big whirls have little whirls That heed on their velocity, And little whirls have littler whirls And so on to viscosity. - L.F. Richardson
in 2D turbulence smaller eddies can and often do combine to form larger eddies.
You may consider reading this article: http://www.annualreviews.org/doi/pdf/10.1146/annurev-fluid-120710-101240