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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 isotropicdepending 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

• Gi pressure - 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 fin G. 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÷t riff

- ttoij⇒ ; 3

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

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

• 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

• ↳ 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 fuqnqne inertial 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!

• Dissipation range behaviour

4

What happens 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