Quality Criteria for Large Eddy Simulation - Turbulence Mechanics ...

CHAPTER 3. TURBULENCE LENGTHSCALES 35
Fourier mode, while the components of Rij and Φij (the subscripts i and j)
give the directions of the velocities.
The two-point correlation and the velocity spectrum function can there-
fore be represented as Rij k and Φij k respectively, where the index k repre-
sents the direction in physical space.
A third part of the information is the lengthscale of the mode given by
l = 2π/|κ|.
Further, information regarding the velocity derivatives is contained in
Φij. This can be extracted, among other methods, by the relation
< δui δuj
δxk δxl
>=
∞
−∞
The dissipation rate is therefore given by
ε =
∞
−∞
κkκlΦij(κ)dκ (3.20)
2νκ 21
2 Φii(κ)dκ (3.21)
3.6 Energy spectrum
In comparison to the great deal of information contained in Φij, a more simple
yet less complete description is provided by the energy spectrum function
E(κ) given by
E(κ) ≡
∞
−∞
1
2 Φii(κ)δ(|κ| − κ)dκ (3.22)
This may be viewed as a non directional representation of the velocity
spectrum tensor Φij(κ) and is a very important quantity for qualitative dis-
cussions.
Integration of 3.22 over all scalar wave numbers κ yeilds the turbulent
kinetic energy,
k =
∞
0
E(κ)dκ = 1
2 Rii(0, t) = 1
2 (3.23)