how to determine if a vortex is laminar or turbulent In a cylindrical chamber with a high diameter-to-height ratio; a fluid is tangentially injected. there is an axial exit to the cylinder.
how do I determine if the vortex so formed is laminar or turbulent? what parameter will indicate this?
(this is a simulation; so I can measure whichever velocity/turbulent KE that is needed.)
 A: Firstly let us define what is meant by turbulent and laminar in a case such as the one you describe...
The Reynolds number of a flow gives a measure of the relative importance of inertial forces (associated with convective flow) and viscos forces. From experimental observations it is seen that for values of Re below the so-called critical Reynolds number the flow is smooth and adjacent layers of fluid slide past each other in an ordinary fashion. If the applied Boundary Conditions (BC) do not change with time the flow is steady and this is Laminar flow.
At values of Re above the critical Reynolds number a complicated series of events takes place which leads to a radical change in the flow character. In the final state the flows behaviour is random and chaotic. The motion in this case is intrinsically unsteady event with constant BCs - all flow characteristics vary in an random way and this is Turbulent flow.
Okay, now to address your problem directly. From what you have asked (I believe you have asked this in the right place) and from reading the comments, I think you have miss-understood what the Reynolds Average Navier-Stokes (RANS) models actually achieve. Taking first the standard $\mathrm{k}-\epsilon$ RANS model; this model (and all other RANS models based upon Reynolds Decomposition) provides a model for all turbulent length scales, even to the Komglarov scale (large eddies cascading down to smaller and smaller eddies until the dissipation length scale is reached). This is the same for the RNG $\mathrm{k}-\epsilon$ RANS model. 
Asside: the Renormalisation Group [$\mathrm{k}-\epsilon$] Model (RNG) uses statistical mechanics and a limited number of assumptions regarding the statistic of small-scale turbulence, to provide a rigorous basis for extension of eddy viscosity models.
The flow that you describe is a rotating flow with a curved Boundry Layer (BL). Now: 


*

*The $\mathrm{k}-\epsilon$ RANS model is very poor at resolving the turbulence generated in such flows, where the curved BL and swirling flow induces large extra strains (I would use a $\mathrm{k}-\omega$-type RANS model). 

*Secondly, in the flow that you describe the large scale rotational flow (vortex flow) could happily be resolved using a purely convective model (advection equation alone) and the dynamics of the flow will be dominated by advection, not (relatively small scale) turbulence. 
So, to answer your question: 
How do I determine if the vortex so formed is laminar or turbulent?

The flow in your case will have some 'lamina' features and be fully turbulent. To establish the dominating factor in this case should be obvious. It is the bulk motion of the flow that drives the type of vortex you describe, not turbulence. 
What paremeter will indiacate this?

For a case where you wanted to establish the length scale of turbulent eddys (relatively small-scale flow (possibly sub-grid level)), then with $\mathrm{k}-\epsilon$-type models we can define the velocity scale ($\vartheta$) and length scale ($l$) of the largest turbulent eddies via:
$$\vartheta = k^{1/2}, \;\; l=\frac{k^{3/2}}{\epsilon}$$.
You could question the validity of using the 'small eddy' variable $\epsilon$ to define the 'large eddy' scale $l$. This is reasonable because for large Re the rate at which large eddies extract energy from the mean flow is roughly matched to the rate of transfer of energy accross the energy-spectrum to small dissipating eddies if the flow is not changing rapidly (assuming you are doing steady-state simulations). if this is not the case then the energy at some turbulent scale could grow or deminish with out limit.
Finally, I would be careful in your selection of RANs model. The advantages and disadvantage for each one is well documented and should be addressed before selection. For your flow I would suggest the Menter SST $\mathrm{k}-\omega$ model, which uses $\mathrm{k}-\omega$ near wall and $\mathrm{k}-\epsilon$ in the free stream (with appropriate wall function treatment depending on your code).
I hope this helps.
