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Transcript of Turbulence Part1
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Turbulence
T U R B U L E N C E
Lecture Note
2009 Fall
Parallel Computing Lab.
Hanyang Univ.
http://vortex.hanyang.ac.kr
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.
A random variable u is defined by :
A function which assigns a value to every outcome of the experiment)(u
Random variable u must have the same probability of generating a given outcome -> identically distributed
An ensemble average is defined by :(true avera e)
=
=N
i
i
N
uN
u1
1lim
But, in reality we never have an infinite # of siu
can never compute ensemble average
We can define an estimator for the average based on a finite # of siu
=N
iNu
Nu
1 itself a r.v.
=
Questions : Is this estimator unbiased?
(Dose it converge to correct answer?)
Does it converge at all?
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It is im ortant to be able to tell how a r.v. is distributed about the mean (ensemble avera e)
Define : Variance of r.v. u
{ } ( )22var uuu u
: standard deviation of uu
variance is also called 2nd central moment
Suppose 2 r.v.s are identically distributed, these must have the same variance
an e ne g er moments
nth moment : =
N
j
n
jN
nu
Nu
1
1lim
n=1 : mean
n=2 : mean square
central nth moment (first substract mean value)
=
Nn
jN
n uuN
uu1
)(1
lim)(
How does relate to ?{ }uvar 2u
Splitting u into mean and fluctuation partuuuu +=
mean fluc.
[ ]22 )( uuuu +=
22
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cf. Eas to see from def. of ensemble ave. that arithmatic o erations and avera in o arations commute
e.g. =
=
+=+
N
i
iN
N
i
iN
vN
uN
vu11
1lim
1lim
==N1
=
i
iiN N 1
222 )()(2 uuuuuuu ++=
{ }uuu var022 ++=
(mean square) = (square of man) + variance
or { } 22var uuu =
same mean , different variance
same mean , same variance
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Turbulence
Can we characterize the am litude distribution of si nal?
- make a frequency of occurance diagramConsider N samples
How many sample fall in a particular window
Let # of realizations increase while window size remain same
eve
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u
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Turbulence
= ,
0),(lim0
=
uuHu
But : Probability density function)(),(
lim0
uBu
uuH
u=
p. .
a double limit
N
0u
m t ng curve o stogram
Properties of PDF
B(u) 0
Prob {c u c+dc} B(c)dc- follows from def. of B(c) from histogram in limit as ,0c N
B(c)
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dcc
c
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c
)()(}{ 1221 cFcFcucp =
)()( cFdc
dcB = F(c)
1)( =
dccB
,1)( =F 0)( =F-
B(c)
c
Ex) 1. Drive pdf for a sine wave.
2. Drive pdf for a triangle wave.
3. Drive pdf for random square wave.
4. What happens to 1. 2. 3. if frequency of signal is changed?
Evaluation of moments from pdf
Ensemble average :
==
N
i
iN
u
N
u1
1lim uu
Nii ++=
)(
1lim
1111
= duuuB )(
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pro a y average an ensem e average are e same
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th
= duuBuu nn )(
nth central moment
= duuBuuuu nn )()()(
variance var{u} = 0)()(
duuBuu n (cf. )0)( uB
third central moment =
duuBuu )()( 3
)]()([2
1)]()([
2
1)( uBuBuBuBuB ++=
even odd
- variance can be zero only when (steady signal or all values exactly same))()( uuB =
- t r centra moment s zero u s even, can e non-zero on y u as an o part.
- As a measure of symmetry of pdf
3)( uu 2
3
}][var{u nondimensionalize
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Contribution of isbigger when
3)( uu 0
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Bivariate random variablesthe joint statistics how they interact with each other
v
o n momen s
Consider two r.v.`s u & v
2u 2v, - single
2)( uu - variance
2uv vu
2,joint ),( 11 vu
u
))(( vvuu (cross - ) correlation
(cross - ) covariance
u
1u
v
1v
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),,,(lim),(
00
vuvuHvuB
vu
N=
Properties of JPDF
0),( vuB
1),( =
dudvvuB
),(},{ 11 vuFvvuup = : joint prob. distribution function
vu
vuvuB
=
,),(
),( = uFFu
vFF =
marginal PDF = single variable P.D.F
v
== uF
dvvuBuBu
u ),()(
==v
FduvuBvB v
v ),()(
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),( vuB
u1u
1v
Suppose we want the statistics one var. given, a particular value of the other- conditional prob. Say the particular value of v is v1
( profiles )
/(,( vvuBvuB ==
=
== duuuBvudvBudududvvuuBu u )(),(),(
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m-nth joint moment
dudvvuBvuvu nmnm ),(
=
m-nth joint central moment
dudvvuBvvuuvvuu nmnm ),()()()()( =
for m=1, n=1
=== dudvvuBvvuuvuvvuuCuv ),())(())((
=
is equivalent to the product of inertiathus, a measure of asymmetry of B(u, v)
then we say & (or & ) are uncorrelateduv
vu
Note vuuvvuvuvvuuuv +++=++= ))((0 0
uncorrelated : vuuv =
If two r.v.`s have a non-zero mean
yield the product of their averages,even if they are uncorrelated.
Therefore the product of mean values has`
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no ng o o w w e er or no e r.v. sare correlated, only fluctuation part is of interested
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Define (cross-)correlation coefficient
vu
uvvu
vuvu
=
}var{}var{
if u & v perfectly correlated : 1uv
uncorrelated :
so must always 1uv
0uv
perfectly anti-correlated : 1=uv
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Statistical Inde endence
: occurrence of u in no way affects the probability of occurrence of v, and conversely
)()(),( vBuBvuB vu=
JPDF = product of MPDFs
Questions
1. If two r.v.s are statistically independent, are they always uncorrelated? yes
0((((((((
))()(,())((
===
=
dvvBvvduuBuududvvvuuvBuB
dudvvvuuvuBvvuupf.)
=0 =0
2. If two r.v.s are uncorrelation, are they necessarily statististical independence?
Assume two r.v.s & with . These are clearly not statistical independence0vu
No combination of these two r.v.s can be statistical independence.
However, we can find a combination which has zero correlation.
u v
22
22))((
,
,
vvuvuuvuvuyx
vuyvux
vuyvux
=
+=+=
=+=
=+=e.g.)
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22 =& are not statistical inde endence since & are not. If then and=u vx y
there is no correlation in spite of statistical independence.
uncorrelated Statistical independence Uncorrelated
statistical independence
c . var ate orma auss an str ut on
Suppose u & v are normally distributed r.v.s with standard deviation given by & ,
and with correlation coeff.u v
vuuv
vvuu
))((
=
2)( uu
Either r.v. taken separately has a Gaussian marginal distribution.
22
2)( u
u
u euB
=e.g.)
Bivariate Normal P.D.F
( )
+
=
2
2
2
2
2
)())((2)(
12
1exp
12
1),(
vvuuuvuvvu
vvvvuuuuvuB
cf. Central limit theorem (Law of large numbers)
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Estimation of statistical ro erties from a finite number of realization
- never have infinite number of realizations- so how good are our estimatiors based on a finite number
uuN N
Nu
Question : Does it converge to right answer?
i.e. as
taking the average of the estimator, e.g.
N
M
k
NM
uuM k
==
1
1lim or
NNuNNuduBuu
N
= )(
sampling P.D.F
Bias : Estimator is unbiased if the average of estimator yields the true averageno systematic errorconverge to correct value
uuNN
uN
uN
u ii
iN ==== =
111
1
This estimator is unbiased
N
{ } ( )2var NNN uuu
Does { } ?as0var NuN
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Define variability of estimator :
{ }22 var
uuN
if 0 as N, then estimator converges to true valuecan show that
2
2 }var{1
u
u
N=
i.e. variability of estimator is equal to variability of r.v. itself divided by the number of independentrea zat ons
error :uN
u1
=
N
Nu
NuB
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How many flips are required to measure expected value of to within 1% error?
Ex) Coin flip experiment
4
1)(}var{
2
2 ==
=
uuu
u
since
= 1
0
u
4
24
2
2
42
2
104
111
1001.0}var{
==
=
==
N
u
u
u
N
2
1
u
How well can we do with 100 flips?
%101.01100
11====
uN
u
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Stationar Random Process (stochastic Process)
u(t)
t
For stationary random process, its PDF and moments are time-independent (independent of the origin of time).
This will only approximate a real process, since stationary random process must go on forever.
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Ensemble average
dttuT
uT
T = 0 )(1
lim
can define the estimator (time average)
dttuT
uT
T = 0 )(1
itself random
dttudttuuTT
T == )(1)(1
for stationary random process
constutu ==)(
TT uudtu
T
u ==0
is unbiased estimator
Does as ? ( as ?)0}var{ tu T uuT T
))((1
)(1
)()(}var{
2
0
2
0
22
dtutu
T
udttu
T
uuuuu
TT
T
TTTT
=
=
==
)()'(')('])'(][)([
'])'(][)([1
0 02
Ctutuutuutu
dtdtutuutuT
T T
==
= : autocorrelation
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tt= 'where
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Why is a function of only?
This can not depend on time itself (i.e. t) since the process is assumed stationary.
)'(')(' tutu tt= '
Why does as ?0)( C
u(t)c()
t
and becomes uncorrelated as
)(' tu )(' +tu
.
- they have finite memories They become uncorrelated with themselves.
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What is a reasonable measure of how long a process is correlated with itself?
Define integral scale as a measure of the memory of process
c()
2/
I
( )
( ) constuuc
dcc
===
=
2/
0
}var{0
)(0 I
for stationary
2/
// )()(
)0(
)()(
u
tutu
c
c
+==
= )( dI
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Not all process have integral scale
if then
=0 0)( d
()
0=I
2effI
eff =01
or
Back to ,}var{ Tu =T T
T dtdtttcT
u0 0
//
2)(
1}var{
=T T
dtdtttT
u
0 0
//
2)(
}var{
After a partial integration ,
du
u
T
T
= 1)(}var{2
}var{
Since as)(1)(
T
T
}var{2
)(}var{2
}var{ udu
uT
T
I= (for T >> )I
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2 }var{2)var( uu cf. variability 22
uTu
Therefore asuuT T
Hence for stationary random processes we dont need to perform many experiments to determine statistics.
Compare2
2 }var{1
u
u
N= : N independent realization
2
2 }var{2
u
u
N
= : integral scale, T record timeI
Obviously the effective number of independent realization is
I2
TNeff =
The segments of our time record of two integral scales in length contribute to the average
as if they were statistically independent.
u(t)
t2I
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2 )(')(' rxuxu +x
DU
0.1xl 0 04x Spatial integral scale
l
cU
l=I where = local convection velocitycU
l 0.04x
EEc . .
Suppose we measure at x/D = 3
04.0 =xD .6.0 DUE
How long must we measure to obtain mean velocity to within 1%
01.199.0