Turbulence
Lecture 22
Smaller scale motions cause the lack of large scale motion; nonlinearity counteracts this.
In the grid experiment far downstream where the nonlinearities are very weak – “final period of
decay.”
( )
0
0
i
i
i
i
i
i
ii
u
pu
pu
p
r
x
x
P
r
∂
∂
∂
′ =
=
′
′
∂
∂
∂
=
±
=
Pressure forces have no effect on
ii
R
, but only on each component.
The pressure forces
redistribute the energy among the components.
The Inertial terms
ij
I
at zero spatial lag.
( )
0
0
ij
j
i
k
i
k
j
i
j
k
k
k
k
I
u
u u
u u
u
u u u
x
x
x
∂
∂
∂
= −
−
= −
=
∂
∂
∂
±
by homogeneity.
( )
0
ij
I
=
±
ii
0
, the inertial terms don’t affect overall content of the energy spectra (Fourier transform
of
R
)
The nonlinear terms conserve energy, change the scales of energy.
Pressure terms also conserve energy, they don’t change the scales, but they change directions.
The equation for
(
,
ij
)
R
r t
±
or its Fourier Transform are the starting point for a number of theories.
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 Spring '09
 MEI
 Fourier Series, Fundamental physics concepts, ui uk uj

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