Turbulence Kinetic Energy vs Velocity Correlation
 

Dear colleagues and CFD experts
Can anybody help me understanding the difference between the following quantities, which I thought should have been the same thing :P :
1. Turbulence Kinetic Energy
2. 0.5*(Velocity Correlation uu+Velocity Correlation vv+Velocity Correlation ww)

Solution data:
- Rotating frame of reference
- k-omega SST turbulence model

Thanks!
Bruno

they are the same if u,v,w is the spatially resolved & temporally resolved velocity from DNS. For not DNS, u may be a filtered velocity and not be exactly the same at tke.

In fluid dynamics, specifically in turbulence modeling, the concepts of Turbulence Kinetic Energy (TKE) and Velocity Correlation are important aspects to understand.

1. **Turbulence Kinetic Energy (TKE):**
- Turbulence Kinetic Energy is a measure of the energy associated with the random, chaotic motion of fluid particles in a turbulent flow. It represents the energy per unit mass in turbulent motion. TKE is a key parameter in turbulence models and is often denoted by the symbol \(k\).

- The TKE equation is typically expressed as:
\[ k = \frac{1}{2} \left( u'^2 + v'^2 + w'^2 \right) \]
where \(u'\), \(v'\), and \(w'\) are the turbulent fluctuations in the three velocity components.

2. **Velocity Correlation:**
- Velocity correlation refers to the statistical relationship between the velocities of fluid particles at different points in a turbulent flow. It quantifies how the velocity at one point in space correlates with the velocity at another point.

- In mathematical terms, the velocity correlation function \(R\) between two points \(x\) and \(x+r\) is defined as:
\[ R(r) = \langle u(x) u(x+r) \rangle \]
where \(\langle \cdot \rangle\) denotes an ensemble average.

- The correlation function provides information about the length scales and structure of turbulence within the flow. For example, if the correlation decreases rapidly with increasing separation distance \(r\), it indicates that the turbulence structures are small and quickly dissipate.

In summary, Turbulence Kinetic Energy is a measure of the overall energy associated with turbulence, considering the fluctuations in all three velocity components. Velocity correlation, on the other hand, describes the statistical relationship between velocities at different points in space, providing insights into the spatial structure of turbulence. Both concepts play crucial roles in turbulence modeling and understanding the complex dynamics of turbulent flows.

Hi Babyjons

I just thought the six "Velocity Correlation" were the components of the Reynolds Stress Tensor <u'u'>, hence the misunderstanding.
On the other hand, the k-omega SST does not solve for the whole tensor but only for TKE and Eddy Viscosity.
In this case the stress tensor should be

Regards
Bruno

Keep in mind that you don't need the entire Reynolds stress tensor to solve a RANS problem, only the divergence of it. And this should be no surprise because RANS deals with only statistics. Hence, the two-point velocity correlation functions contain much more information (i.e. much more turbulence) than what is needed for RANS.


Likewise, turbulence kinetic energy is also a time accurate quantity and the k in k-omega model is actually only the mean turbulent kinetic energy.


Given that you are asking about two equation models, it is contextually clear (to me) that you are asking about statistics of k and statistics of two-point correlation functions. But maybe that is the reason for confusion...

My goal here would be to try and reconstruct the entire stress tensor, as I need to calculate the shear stress on non-wall surfaces.
I implemented the formula I posted earlier in the thread and tried to validate it against the given "Wall Shear" output variable, but my vector, although nicely aligned with Wall Shear, is approximately half in magnitude. Am I missing some constant somewhere?

edit: I missed to specify that I'm working in Ansys CFX :D

Not sure about your goal, do you want to derive the Reynolds stress tensor from a solution where it was modelled with an eddy viscosity assumption?

Hi Dr. Denaro
I'm aware of the fact that any eddy viscosity-based model does not aim at deriving the six individual components of the Reyonlds tensor, but I was thinking of using the equation above to rebuild the (whole) stress tensor starting from eddy viscosity and tke. Is this conceptually wrong?
Bruno

What makes sense is to evaluate the deviatoric part, this way you have the the stress tensor that contains the eddy viscosity model.
Nothing to do with the Reynolds tensor.

Thanks for your hint.With respect to the original goal and to CFX solution data, is it possible to evaluate shear stress in any point of the domain? (i.e. not just on wall boundaries)

In general yes, from the mean velocity components you can evaluate any derivative you want in any location into the domain.
However, any software has its proper method, for the specific details about how accessing data in CFX I suggest posting the question in the sub-section of the forum.

The whole talk arises from this: whether I miss a lot by considering mean velocity and not taking into account turbulent stresses!

That depends also on the formulation. But you have to realize that the mean velocity v_mean is a result of the action of the modelled stress, too. Thus, if you compute the mean stress, it has also the action of the modelled turbulent stresses.
It is your choice if you want to compute only the stress 2*mu*Grad v_mean or you want to add also 2*mu_t*Grad v_mean.