the howarth kirwan pope -2 - max planck society · the howarth kirwan relation (see...

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The Howarth . Kirwan relation ( see Bonin - Saglom , vol -2 ; Pope ) ° fundamental statistical quantity of interest : velocity correlation tensor Rijlx , , x.at )= ( hill , ,t ) ujlxut ) > . evolution equation from Nse , = ( uiuj ' > of ( uiujstdkcuiuauj ) tai ( uiujui > = . 2 ; ( pujs - a :( pin ; > + r Qiluiujltudjicuiuj > Gi closure problem ! Go Statistical symmetries for homogeneous isotropic turbulence : homogeneity : ( uiuj ' )= Rijlr ) with 1=1 ' - I d×i= - On . %=2r ; isotropy : 1) pressure - velocity correlations : ( nip 's = ago ) : scalar function only isotropic depending our only tensor of rank I dri ( hip 's = air )r÷r÷ tar ) ( { . rig ) = a 'lr ) +2g acr ) t 0 ( incompressibility ) G is solved by alr ) = 0 acr ) = r - 2 ^ can be excluded on physical grounds because of divergence at origin

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Page 1: The Howarth Kirwan Pope -2 - Max Planck Society · The Howarth Kirwan relation (see Bonin-Saglomvol-2 Pope) ° fundamental statistical quantity of interest velocity correlation tensor

The Howarth . Kirwan relation ( see Bonin - Saglom ,vol -2 ; Pope )

°

fundamental statistical quantity of interest :

velocity correlation tensor Rijlx , , x.at )= ( hill, ,t ) ujlxut ) >

. evolution equation from Nse ,

= ( uiuj'

>

of ( uiujstdkcuiuauj )tai( uiujui >

= . 2 ; ( pujs - a :( pin; > + r Qiluiujltudjicuiuj >

Gi closure problem !Go Statistical symmetries for homogeneous isotropic turbulence :

homogeneity : ( uiuj'

)= Rijlr ) with 1=1'

- I ↳ d×i= - On. %=2r ;

isotropy : 1) pressure - velocity correlations : ( nip 's = ago):scalar function only isotropic

depending our only tensor of rank I

↳ dri ( hip 's = air )r÷r÷ tar ) ( { . rig )

= a 'lr ) +2g acr ) t 0 ( incompressibility)

G is solved by alr ) = 0 ✓ acr ) = r

- 2

^

can be excluded on physical groundsbecause of divergence at origin

Page 2: The Howarth Kirwan Pope -2 - Max Planck Society · The Howarth Kirwan relation (see Bonin-Saglomvol-2 Pope) ° fundamental statistical quantity of interest velocity correlation tensor

Gipressure - velocity covariance vanishes !

↳ A ( uiuj 's + dr..

[ ( uiujui ) - ( uiuuujy ] = Zvdni ( uiuj 's HI

.'

2) velocity covariance tensor Rijlr )= '±s' [ gir ) oijtffcn- gen) IYD]• pick i=j=l tree ,

G R"

( re , )= 431 finG. fk ) is the normalized longitudinal autocorrelation function

. pick i=j=2 I re ,G Rzz ( re ,)=k÷ '

gcr )

Cp glr ) is the normalized transverse autocorrelation function

^h{D ^u2c±+r± , ,correlation described by

> re'> > gcr) and flr)

Uik ) U,

( Itre , )

incompressibility : A: Rijk )=dj Rijcr ):Oimposes the relation gcr) = fchtlzrftr ) homework !

G 9 - component tensor is characterized by variance & single scalar function !

3) velocity triple correlation Bij, kk ) = ( uinjui. >

=f÷')"

[ 12 Hrtyrirgjkntiguttrty ( ri9÷triff

- ttoij⇒; 3

. pick i=j=k=l I = re ,B

" , ,= PT

Cp insertion into C* ) yields scalar equation in terms of FG) and TCD

Page 3: The Howarth Kirwan Pope -2 - Max Planck Society · The Howarth Kirwan relation (see Bonin-Saglomvol-2 Pope) ° fundamental statistical quantity of interest velocity correlation tensor

4 ( It % dr ) dtf = (

Itri. ) [ @, t4g ) rTt2r( dit 4g dr ) f

G integrate to obtain :

off = F,drr

" Tt 2¥,

drr " or f

von koiruiau - Howarth equation

non . trivial relation between longitudinal velocity autocorrelation

function and velocity triple correlations

The 415 - law

• prediction for inertial -

range behavior of third - order

structure function•

on of a few exact statistical results derived from NSE

. consider longitudinal structure functions

Such =LFalter) - ud ).IT >"

velocity fluctuations on scale r"

= ( veh )

:( Itr )

u*[

. relation to vk.tl relation can be expressed via

olf = rttzsz homework !

PT = to 5

Page 4: The Howarth Kirwan Pope -2 - Max Planck Society · The Howarth Kirwan relation (see Bonin-Saglomvol-2 Pope) ° fundamental statistical quantity of interest velocity correlation tensor

↳ von Kirwan - Howarth equation can be of expressed as

3M of Szt drr" § = 6v2rr4 qsz - 4 ( E > r4

^

from ofrk . } c e >

Gi integrate to obtain

3g, €4 of Sds ,

Holst § = Gvdrsz - 45 ( e > r ( * )

- -

= ° for statistical ,

a 0 in the

stationary fuqnqneinertial range

G)

Slr ) = - Esser kolmogorov's 45 law

° third order structure function is linear in the inertial range

• remember : Sslr ) = ( [( ucetr.) - ±kD.IT ) = Solve vs flue ; r )

G 45law predicts skewuus of velocity increment PDF!

Page 5: The Howarth Kirwan Pope -2 - Max Planck Society · The Howarth Kirwan relation (see Bonin-Saglomvol-2 Pope) ° fundamental statistical quantity of interest velocity correlation tensor

Dissipation rangebehaviour

4

Whathappens at small scales ?

uiktrieikutx) + FEWrittzftp.k.lritfddypcxr?+h.o.t.

↳ re=

Feretftp.ritfdodxpr?+h.o.t.

↳ sun= iris

.tt#.l2srittfd*.Yn)ritHWEzBri'

÷+ ÷ ( g÷Y÷

. yithai

4 sur .lt#x.t7ri ' HCo±atM "

insert into .

Scr ) = < vi >=#g÷B ri

(ettypyr .

www.24#zHr:tsseI*go (E) due to

isotropy

↳ eo÷p = . ul¥±third moment of E 0

velocity gradient

• velocity gradient PDF is skewed,

too !

• The probability of finding positre and negative velocity increments

of same magnitude differs !