BSA Flow Software Installation and User's Guide - CSI

Integral time-scale τI
Definition of the spectrum
2
{ () }
R( 0)
= E u t
(7-59)
The autocorrelation function can be interpreted as a measure of how well
future values of the data can be predicted from past observations. As you
might expect from this, R(τ) normally decrease with increasing lag time τ, as
does the autocovariance C(τ). The autocovariance generally approach 0 for
τ → ∞, whereas the autocorrelation approach (E{u})2.
The crosscorrelation can be interpreted similarly, but unlike the
autocorrelation it is not symmetrical around τ=0. This is easily understood
from a simple example:
Imagine crosscorrelation between inlet pressure and outlet velocity measured
on a length of pipe. Fluctuations in the inlet pressure will cause changes in
the outlet velocity, but there is a small delay, which can easily be calculated
from the speed of sound (c) and the length of the pipe (L). This will produce
a peak in the crosscorrelation at lag time τ=L/c, indicating that to some
extent future outlet velocities can be predicted from current inlet pressure.
If the crosscorrelation function was symmetrical, this would indicate, that
also future inlet pressure could be predicted from present outlet velocity,
which of course is impossible.
In the literature, the integral time-scale τI is usually defined on the basis of
autocorrelation R(τ), but this correlation is calculated from the fluctuating
part of the measured quantity, meaning that the mean-value has been
subtracted. Strictly speaking this is not correlation but covariance as
explained on page 7-122, so a more correct definition of the integral timescale
is based on autocovariance Cuu(τ):
τ
I
=
∞
∫
0
C
C
uu
uu
( τ)
dτ
( 0)
(7-60)
The integral time-scale τI is a rough measure of the longest connections in
the turbulent behaviour. Events more than 2 integral time scales apart, can be
considered independent.
The spectral density Suv(f) and the correlation Ruv(τ) form a Fourier
transform pair, defining the spectral density function:
∞
−i2πf
τ
Suv( f) = ∫ Ruv( τ) e dτ
−∞
∞
∫
R ( τ)
= S ( f) e
uv uv
−∞
i2πf τ
(Fourier transform) (7-61)
df (Inverse Fourier transform) (7-62)
Auto- & Cross-spectra Suv(f) is called cross-spectral density or simply cross-spectrum when u(t)
and v(t) represent two independent measurements, and auto-spectral density
when u(t) and v(t) represent the same physical quantity. In the latter case the
symbol S(f) is often used instead of Suu(f), and the auto-spectral density
function is frequently referred to as power spectral density (PSD), power
spectrum, or simply the spectrum.
BSA Flow Software:Reference guide 7-123