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 groundsbecause of divergence at origin
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
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 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!
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 !