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

CHAPTER 3. TURBULENCE LENGTHSCALES 34
3.5 Velocity spectra
In case of homogeneous isotropic turbulence, the two-point correlation Rij(r)
can be expressed in terms of a wavenumber spectrum.
Consider the spatial Fourier mode
e iκ·x = cos(κ · x) + i sin(κ · x) (3.16)
This function varies sinusoidally in the direction of the wavenumber vector
κ with wavelength l = 2π/|κ|.
The velocity spectrum tensor Φij(κ) is defined as the Fourier transform
of the two-point correlation
Φij(κ) = 1
(2π) 3
∞
e
−∞
−iκ·r Rij(r)dr (3.17)
where dr is written for dr1dr2dr3.
Consequently, the two-point correlation can be obtained from the velocity
spectrum tensor by employing the inverse Fourier transform as
Rij(r) =
∞
−∞
where dκ is written for dκ1dκ2dκ3.
e iκ·r Φij(κ)dκ (3.18)
For r = 0, from equations 3.4 and 3.18, we have the relation
Rij(0) = =
∞
−∞
Φij(κ)dκ (3.19)
Consequently, Φij(κ) represents the contribution of the velocity modes
with wavenumber κ to the autocovariance .
The two-point correlation as well as the velocity spectrum function con-
tain two different kinds of directional information. The dependence of Rij(r)
and Φij(κ) on r and κ respectively give the direction in physical space of the