# Simcenter STAR-CCM+ How to calculate Turbulent Kinetic Energy in LES

2023-11-02T10:07:20.000-0400
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## Summary

While there is no explicit field function that reports turbulent kinetic energy of resolved motions in LES, it can be estimated through field variance values of the three velocity components.

## Details

Turbulent flows are unsteady and have a wide range of length scales. Large Eddy Simulation (LES) is a turbulence modelling approach for transient flows in which large scales of turbulence are solved, and the small-scale motions are modelled through subgrid-scale modelling. It is recommended that at least 80% of the large scale needs to be resolved in a Large Eddy Simulation, i.e. the mesh setup in a LES setup should be such that it resolves 80% of the large scale structure as shown in the figure below.

The process of separating the scales in a turbulent flow is shown in the above figure by the energy cascade concept.  E (k) is the energy spectrum of turbulence and
Δ꜀ is the mesh cutoff width, i.e. a mesh size smaller than this will resolve the flow, and if it is greater, the flow will be modelled.
In LES computations, one of the ways to determine the quality of  mesh and to see how well the flow is resolved is by looking at the ratio of SGS (subgrid-scale) turbulent kinetic energy to total turbulent kinetic energy (based on temporally averaged LES results). The ratio which gives the fraction of turbulent kinetic energy in the resolved motions is given as:

$M = \frac{k_{sgs}}{k_{sgs}+k_{res}}$

where k_sgs = SGS turbulent kinetic energy, k_res = resolved kinetic energy. The resolved turbulent kinetic energy is defined as:
$k_{res} = \frac{1}{2}\left \langle \left ( \widetilde{u}_{i} - \left \langle \widetilde{u}_{i} \right \rangle_{T} \right )\left ( \widetilde{u}_{i} - \left \langle \widetilde{u}_{i} \right \rangle_{T} \right )\right \rangle_{T}$

where  $\left \langle \right \rangle_{T}$ denotes the temporal average (mean velocity field), and u_i is the instantaneous velocity field.

While a value of M = 0 corresponds to DNS and indicates that the flow is very well resolved, M = 1 corresponds to RANS and indicates poor resolution for a coarse mesh. Based on this ratio, the mesh can be refined. This process of solution-adaptive grid refinement is known as adaptive LES.

While there is no explicit field function that reports turbulent kinetic energy of resolved motions in LES, it can be estimated through field variance values of the three velocity components.
• Create Monitors for Field Variance of Velocity Components by right-clicking on the Monitors node. Under the properties window, choose the velocity component as the field function.

• Variance of Velocity components will now become available as field functions
• Create a user-defined field function for resolved Turbulent Kinetic Energy through the Field Variance velocities as follows:
• A field function for SGS turbulent kinetic energy (SGSTurbulentKineticEnergy) is made available to the simulation when you activate a Ffowcs Williams-Hawkings Model along with a subgrid-scale model. (In previous versions was available by activation of an aeroacoustics or combustion model along with an LES turbulence model.)
• Now the ratio M can easily be created. While there is no set resolution which will be appropriate for every simulation, aiming for an M value in the range 0.05 <= M <= 0.10 is a good starting point.
• Lastly, one can also consider whether it is appropriate to allow the flow to develop before beginning to take variance statistics, e.g. - measure the TKE. To delay the calculation of statistics, you can change the "Start Time-step" property available in the sub-node under the field variance monitor.

• Create another field function.
This parameter, M, gives information about the large and small scales in the LES calculation.

In the figure below, a typical flow test case around a cylinder is considered to visualize the described process. As we can see from the distribution of M, the wake region is critical for this problem. While a value of M = 0 corresponds to DNS and indicates that the flow is very well resolved, M = 1 corresponds to RANS, indicating poor mesh resolution. Based on this criterion, the mesh should be refined for better accuracy. This process of solution-adaptive grid refinement is known as adaptive LES.