BSA Flow Software Installation and User's Guide - CSI

As you might expect from (7-61), the cross-spectrum Suv(f) is generally
complex, but with real time functions u(t) and v(t), the crosscorrelation
Ruv(τ) will be real, and consequently the cross-spectrum will fulfil:
* ( ) ( )
S − f = S f ⇒
uv uv
[ uv( − ) ] =
uv(
)
[ ]
⎧Mod
S f Mod S f
⎪
⎨
⎪
⎩
Arg S f Arg S f
[ uv( − ) ] = − uv(
)
[ ]
(even)
(7-63)
(odd )
The autospectral density function Suu(f) can be considered a special case of
the cross-spectrum, where the real and even autocorrelation Ruu(τ) lead to
Suu(f) also being real and even:
( ) ( )
S − f = S f
(7-64)
uu uu
Inserting (7-57) in (7-61) it is obvious that the spectral density can also be
calculated from the covariance Cuv(τ):
∞
∫ ( )
−i2πf
τ
S ( f) = C ( τ) + uv e dτ
uv uv
−∞
∞
∫
( )
−i2πf
τ
= C ( τ) e dτ+ uvδ f
−∞
uv
(7-65)
If the delta-function δ(f) in (7-65) is omitted, the only deviation between the
spectrum calculated from Ruv(τ) and the one based on Cuv(τ) will be at f=0,
where the calculation based on Cuv(τ) will yield Suv(0)=0, since the DCcomponent
of u(t) and v(t) has been removed in the calculation of
covariance. This is used to increase the accuracy of the spectrum-calculation,
since floating point numbers in computer-calculations are represented with a
fixed size mantissa, and large DC-components thus may drown small
fluctuations if they are included.
The autospectral density function can be interpreted as the frequency
distribution of the turbulent energy in the flow, and periodic phenomena will
produce peaks at frequencies corresponding to the dominating
frequency/frequencies of the phenomena investigated.
Estimating correlations and spectra using finite Fourier transforms
Correlation theorem To reduce calculation time correlations and spectra are estimated using the
correlation theorem, which states that the spectral density Suv(f) can be
calculated directly from the separate Fourier transforms of u(t) and v(t):
1
Suv( f)
= lim EU [ ( −fT
, ) VfT ( , ) ]
(7-66)
T→∞
T
–where E[...] represent the expectation (mean) value, and U(f, T) and V(f, T)
represent the finite Fourier transforms of u(t) and u(t) respectively:
T
∫
( )
−i2πf
t
UfT) ( , = ut ( ) − ue dt
0
T
∫
( )
−i2πf
t
VfT) ( , = vt ( ) − ve dt
0
(7-67)
7-124 BSA Flow Software: Reference guide