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Turbulence
Lecture 5B
If we have some function of
, say
u
( )
,
Guxt
±
e.g.
Gu
()
4
,sin
u
u
=
,
=
∫
±
G
;,
uPuxtd
u
±
The most common (popular) probability distribution function that occurs in turbulence
studies is the Gaussian (normal) p.d.f.
1
2
Puxt
πσ
=
±
exp
2
2
2
uu
σ
−−
,
x
t
=
 Mean, average, expectation
±
1
2
2
,
xt
u u
σσ
==
−
±
 Standard deviation
2
 variance
u
is the center
measures the width.
Note that if the process is known to be Gaussian distributed,
u
&
completely define the
statistical properties of the random variable
( )
,
uxt
±
for fixed ,
x t
±
.
In turbulence the velocity field is approximately a Gaussian random variable.
An important theorem – the Central Limit Theorem says many random variables are
Gaussian.
1
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View Full Document Area = Probability
()
{ }
,
uux
t u u
≤<
+
∆
±
( )
;,
Puxt
u
≈
∆
±
Properties of random variables
f
, constant
a
, averaging is a linear operator.
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This note was uploaded on 06/07/2011 for the course EGM 6341 taught by Professor Mei during the Spring '09 term at University of Florida.
 Spring '09
 MEI

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