In high Reynolds number flows regions with velocity gradients are unstable and develop chaotic vortical motion (3-D) (see Bradshaw "Turbulence"). The chaotic motion involves unsteady (chaotic) motion transversal to the main (time averaged) velocity. This induces transport of momentum between the different layers of the time averaged fluid flow. This transport of fluid with high momentum to regions of fluid with lower momentum is "analogous" to molecular transport of momentum between layers of a laminar flow. In both cases one can speak of vorticity diffusion (or momentum diffusion). In laminar flow the diffusion coefficient is the kinematic viscosity nu. In turbulent flows the eddy diffusion coefficient is of the order of the length scale of the gradient L times the time averaged flow velocity U. Hence the turbulent transport is a factor UL/nu=Reynolds number larger than molecular diffusion over the length scale L. This Reynolds number is typically two to four orders of magnitude in turbulent flows. 

Next to momentum (or vorticity) the turbulence will transport chemical species (Diffusion) or heat (Heat transfer). Typically again this eddy diffusion is at least a few order of magnitudes larger than the molecular diffusion process. This is illustrated by the daily experience when pouring (carefully) milk into a cup with coffee (or thee). Without stirring with a spoon the cold and heavier milk will settle down at the bottom and stay there for the next few weeks (try with a transparent cup). By rigoroursly with a spoon you generate turbulence that mixes the milk with the coffee within seconds.

Dear Shashank,

I can follow your argumentation line. However, saying "turbulence is highly convective" leads into trouble.

Keeping ind mind that convection has vector character whereas diffusion is a scalar phenomena. Knowing that turbulence can be interpreted as a stability problem without any fixed/preferred direction, gives a hint on what of the both mentioned (convective/diffusive) concepts the turbulence is based. Moreover a real turbulent flow will be three dimensional, meaning its turbulent characteristics will travel/occur in all three room directions and leads to intense mixing effects. Beside this, the mean (!)  flow direction (described by convection) can be e.g. one dimensional. Again your can see the difference.

This is only my interpretation. However, i hope it will help you.

Bye

Hi Shashank,

The idea of 'higher diffusivity' is probably not a phrase to be taken very literally, in a technical sense. The intent, as you noted, is to highlight that you have enhanced transport due to the heat and mass being carried by both diffusion and eddy motion.

When you look at how turbulence is modelled (assuming you are not an LES or DNS person), you can see why 'higher diffusivity' might be a reasonable description. In the RANS equations, one of the more popular ways to account for the Reynolds stresses is to assume they are proportional to the velocity gradient. You end up with both kinematic viscosity and 'eddy viscosity' as coefficients for the diffusive term. This eddy viscosity is always positive, so for this model, the turbulence acts to increase the effective diffusivity of momentum.

When turbulent flows are described as (diffusive), it is meant that they enhance mixing and exchange of mass, momentum, and energy within the medium. This is the same effect caused by molecular diffusion but due to molecular action rather than to the strong flow motion in case of turbulence.

This is aligned with the fact that gradient-diffusion hypothesis is often used to model the turbulent terms in the partial differential equation as an additional diffusion term manifested in the magnification of the diffusion parameter (such as the viscosity in case of the momentum equation).

In high Reynolds number flows regions with velocity gradients are unstable and develop chaotic vortical motion (3-D) (see Bradshaw "Turbulence"). The chaotic motion involves unsteady (chaotic) motion transversal to the main (time averaged) velocity. This induces transport of momentum between the different layers of the time averaged fluid flow. This transport of fluid with high momentum to regions of fluid with lower momentum is "analogous" to molecular transport of momentum between layers of a laminar flow. In both cases one can speak of vorticity diffusion (or momentum diffusion). In laminar flow the diffusion coefficient is the kinematic viscosity nu. In turbulent flows the eddy diffusion coefficient is of the order of the length scale of the gradient L times the time averaged flow velocity U. Hence the turbulent transport is a factor UL/nu=Reynolds number larger than molecular diffusion over the length scale L. This Reynolds number is typically two to four orders of magnitude in turbulent flows. 

Next to momentum (or vorticity) the turbulence will transport chemical species (Diffusion) or heat (Heat transfer). Typically again this eddy diffusion is at least a few order of magnitudes larger than the molecular diffusion process. This is illustrated by the daily experience when pouring (carefully) milk into a cup with coffee (or thee). Without stirring with a spoon the cold and heavier milk will settle down at the bottom and stay there for the next few weeks (try with a transparent cup). By rigoroursly with a spoon you generate turbulence that mixes the milk with the coffee within seconds.

Shashank, turbulence is essentially a densely packed ensemble of chaotic but hirarchically organized vortex dipoles. They may collide and in case of colliding likewise rotating vortices they form the Kolmogorov cascade where the largest ("energy-containing") vortex is the lower end of the wavenumber spectrum. Mixing happens here, within this cascade, because only here the spatial scales go down to the Kolmogorov scale where molecular effects take over.

In case of colliding counter-rotating vortices nothing specifically happens because they simply form a new dipole, moving with the corresponding Helmholtz velocity. See attachment. One of the interesting results of this parameter-free mechanical picture of turbulence is the van-Karman constant as 1/SQRT(2*Pi) = 0.399 ...

Article Universal equations and constants of turbulent motion

the question is quite interesting.

Turbulence can be approached even in the inviscid limit, so why using such terminology? Actually, I think that defining "highly diffusive" the turbulence is not correct as it is a remind of rather a "modelling idea" not of its real nature. The viscosity of the fluid does not change at all. Hystorically, the observation that turbulence increases mixing (by a macroscopic point of view) is for sure the first issue.

But do not forget that real turbulence is a complex phoenomenon where large flow structures and  small foow structures cohexist interacting each other. Energy is exchanged from large to smallest scale (and possibly also from small to large!) in an inertial way, without any contribution of the diffusive terms. This effect has a macroscopic evidence in enhanced flow mixing and was one of the observation of the hystorical Bousinnesq assumption. That was used as basis for many turbulence models. Prandtl then developped the mixing lenght hypothesis.

However, you can see also differently this framework if you consider the local Reynolds number associated to the  characteristic lenght of the flow structures. When you consider a large structure the Re is high and the energy transfer is almost an inviscid-like mechanism. When the action of the stretching reduces the size of the structures, the associated local Re number decreases. When it is of O(1) that means that, at those scales, the convective terms are balanced by the viscous one, convective and diffusive time are of the same order. In this sense, the local nature of turbulence retains the highly diffusive character: mechanical energy is converted in internal energy by the action of the energy dissipation. That is an irreversible mechanism, associated to production of entropy.

In conclusion, any time you read "highly diffusive" think only in terms of a global modelling of the above depicted pysics.

Just a note, modern modelling for LES are no longer based only on the eddy viscosity concept.

Hi all,

I want to argument in a mathematical point of view. Checking the Navier-Stokes equations we will find the Cauchy stress tensor that can be split into deviatoric (shearing) and hydrostatic (pressure) part. The deviatoric part can be related to a diffusion term based on the velocity gradients (that was already mentioned here). The diffusion coefficient of this term is the molecular viscosity mu. If we model turbulence by RANS what happens now? The assumption is the eddy viscosity (also mentioned above); in other words we assume that the effective viscosity of the fluid is equal to the moleculare one plus the eddy viscosity. As you can see, the viscosity (diffusion coefficient) will increase and the flow gets more diffusive (in the mathematical point of view). The eddy viscosity is estimated using a length scale and a characteristic velocity. Normally we are using the turbulent kinetic energy k and omega / epsilon or other models.

In LES it is similar. Please consider the last notes from Filippo.