Vorticity is a key ingredient of turbulence, so irrotational flows cannot be turbulent flows. They can be random, such as the irrotational motion of fluid outside a jet that is pushed around by the eddies in the jet, and they can certainly be high Re, as in inviscid irrotational flow. But without vorticity there is no viscous dissipation, and dissipation is essential to the establishment of a wide range of length scales-the most distinguishing feature of turbulence. The introductory text  on turbulence by Tennekes and Lumley discusses these and other properties that a flow must have in order to be called 'turbulent'. Note that there are several field in which 'turbulence' is used much more loosely. For example, in biofluids any flow that is not laminar and approximately uni-directional is often called 'turbulent'.

Vorticity is a key ingredient of turbulence, so irrotational flows cannot be turbulent flows. They can be random, such as the irrotational motion of fluid outside a jet that is pushed around by the eddies in the jet, and they can certainly be high Re, as in inviscid irrotational flow. But without vorticity there is no viscous dissipation, and dissipation is essential to the establishment of a wide range of length scales-the most distinguishing feature of turbulence. The introductory text  on turbulence by Tennekes and Lumley discusses these and other properties that a flow must have in order to be called 'turbulent'. Note that there are several field in which 'turbulence' is used much more loosely. For example, in biofluids any flow that is not laminar and approximately uni-directional is often called 'turbulent'.

A turbulent flow is characterized by some features such as having irregular flow, diffusive flow, rotational flow, and dissipation. As Ronald J. Adrian said, irrotational flows have zero vorticity value. Hence, irrotational flow is not turbulent. You might also want to study Fluid Mechanics Book by Frank M. White; chapter 4, Differential Relations for Fluid Flow; 4.8. Vorticity and Irrotationality.

We call her turbulence, but nobody knows in detail at which space and time she was born and who are her parents, just the granny should be Brownian

if you analyse the vorticity  transport equation, the generation of vorticity is inherently 3D, in 2D this term vanish. As Ronald  correctly pointed, there is no viscous dissipation without vorticity. For example, a 2D laminar boundary layer has vorticity because there is a u=u(y),  and the vorticity is w=w(z); and the boundary layer exists due the viscous dominance at the wall which causes a pressure loss. But this vorticity does not imply in turbulence! In a free laminar stream, with a U=cte. Is irrotational, and without  vorticity.Other case, a  really turbulent boundary layer, the vorticity is present again. And in the first and last cases we have a rotational flow. Hence, rotational flow implies in vorticity,which in its turn,  in viscous dissipation, but not all flow with vorticity are turbulent, but all turbulent flows have vorticity field and generation of vorticity.

The energy cascade explained by Kolmogorov is a dissipation  chain, with vortex inside vortex, in scales  from the mean flow to the viscous,  where the energy is injected, and passed for each flow structure - the vortex - until  the smallest scale where the effective Rel ~1, where l is the Kolmogorov's length scale. Ronald and Navid have  indicated two good references. 

The "irrotational turbulence" outside of a boundary layer (or jet) is probably not very well named.  The imprecise terminology may be confusing you (and many others). The flow (or turbulence) in these  "irrotational regions" is not "irrotational"  in the sense that the word is used everywhere else in fluid dynamics.

Most importantly.  Classical irrotational (free from vorticity) flows are governed by linear PDEs (where superposition works) and classical  turbulence is nonlinear.   In the classical meaning of the two words, irrotational and turbulent are exclusive of each other (as many of the comments above point out).  

It should be called 'non-local' turbulence or something like that.  The turbulence outside a jet or B.L.  is turbulence.  It is caused by the vorticity inside the jet acting at a long distance so it can be a bit different than 'normal' turbulence where the generation is happening by vortices that are very close by.  It is not mathematically an irrotational flow outside the turbulent jet (since the flow motion is a direct result of vorticity).  The poor terminology perhaps comes from the fact that it might be getting "closer" to irrotational than what is happening inside the jet (if "closer" could be defined - which it probably can't).   Quasi-irrotational would be a slightly better word, but is still missing the real issue (which is non-local).  

Mathematically, a flow domain cannot be irrotational in some regions and rotational in others. The PDE for incompressible flow couples all regions spatially and voids any such delineation being defined in any precise way.  

For example, 3 point vortices in a 2D box produce nonlinear and chaotic flow EVERYwhere in the box.  Even though at any location in the box you will see zero vorticity (with 100% probability) at that location.    With 100 point vortices you will get something indistinguishable from 2D turbulence.  The flow in between the vortices is NOT considered to be irrotational just because the vorticity is zero at that location.   The same applies in 3D (using vortex line rings).  If the flow in between a collection of point vortices is not irrotational - then the flow nearby is not irrotational either. 

Now if you have a giant box and you put all your vortices in one corner you will get something like the jet case.  Far from the vortices the flow is still governed (entirely) by the vortices.  Far from the vortices the flow is still non-linear.  Still turbulent.  (Still rotational in my book).   But  ... lots of cancellation can happen over the long distance causing only certain aspects of the 'turbulence' to survive at that long distance.  

That is confusing, but somehow clarifies the doubt. Thank you all for answering.

Just to add two cents... An interesting (and somehow curious) case of a model for turbulence-like without vorticity is the 1D "Burgulence" that Burgers proposed several decades ago.

It produces a characteristic energy cascade, with inertial and dissipative ranges.

I agree with Ricardo and Perot, all flows with vorticity are not turbulent, but it is other way around all turbulent flows will accompany vorticity. Within boundary layer, even laminar flows have vorticity (rotational flows), this is proved in the past.

Take two infinitely close trajectories. If the dispersion will be large, then this is turbulence. All other discussions lead to nowhere

Why is this discussion going on? Ronald Adrian gave the answer some time ago!

I would take Prof. Adrian's answer without reading them, because I know he is one of the pioneers of this field (my turbulence course teacher around 1992).

AS RONALD ANDRIAN'S ANSWER

A problem arises sometimes with this if you take engineering courses that are divorced from fluid mechanics.

In lecture 1 you see a picture of a laminar and a turbulent flow and the latter is clearly rotational. You are then told about the Reynolds number that characterises this difference and are told that, by and large, you will be working with turbulent flows.

However, rather than then working with time-averaged (RANS) models for the Navier-Stokes (NS) equations, or spatially filtered (LES) models, or direct numerical simulations, you start working with simplifications that eliminate all the cross-terms and hence, impose irrotation on a high Reynolds number flow, leading to the need to add in an unphysical eddy viscosity term to make the turbulent flow "slow down" and not be a laminar fluid.

IRROTATIONAL FLOWS ARE LAMINAR FLOWS , NOT  TURBULENT FLOW .

BECAUSE TURBULENCE FLOWS ARE CAUSED DUE TO VORTICITY , THAT MEANS ROTATIONAL FLOW. SO IRROTATIONAL FLOWS ARE LAMINAR ONLY