Wind Energy

12 Turbulence Modelling and Numerical Flow Simulation 69
to use a second filtering level � ∆ > ∆. Then the subgrid scale stress tensors
on the two filtering levels read
τij = uiuj − uiuj, Tij = �uiuj − ũi ũj. (12.7)
For both filtering levels one applies the Smagorinsky models
τij − 1
3 δijτkk =2Cs∆ 2 | S | Sij, Tij − 1
3 δijTkk =2Cs ˜ ∆ 2
| ˜ S | ˜ Sij. (12.8)
Similar to the subgrid scale stress tensor τij, the Leonard stress tensor
Lij = �uiuj − ũi ũj = Tij − ˜τij
(12.9)
is unknown in a simulation. But replacing Tij and τij by an approximation
like the Smagorinsky model of (12.6) leads to six equations for the unknown
Smagorinsky constant Cs.
Lij = −2Cs( ˜ ∆ 2
| ˜ S | ˜ Sij − ∆ 2 |
�
S | Sij) . (12.10)
� �� �
To obtain a single constant Cs Lilly [5] suggested to minimize the error tensor
εij applying the least square formulation in the sense
Mij
εij = Lij +2CsMij =⇒ Cs = − 1
2
LijMij
MijMij
. (12.11)
Unfortunately solving (12.11) results in a Cs field which strongly varies in
space and in time and with a significant number of negative values. In many
simulations large negative values tend to destabilize the numerical simulation
since they lead to a nonphysical growth of the resolved scale energy. To overcome
this, a statistical averaged 〈Cs〉 has been used for almost all reported
simulations.
12.6.2 Scale Similarity Modelling
Applying filtering with two filter widths Liu et al. [6] introduced the unknown
constant CL in a scale similarity model τij = CLLij. Later Wagner [8] proposed
to modify the dynamic process to determine CL. In his approach τij
and Lij in (12.9) are substituted by (12.8). Just as in Germano’s dynamic
model he obtains a set of equations for the unknown constant CL
∆ 2 | S | Sij = CL( ˜ ∆ 2
| ˜ S | ˜ Sij −
�
[∆ 2 | S | Sij]) (12.12)
which can be solved applying the least square method to minimize the error.
Applying his model in various LES of turbulent channel flow for different