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

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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

    velocity gradient

    • velocity gradient PDF is skewed, too !

    • The probability of finding positre and negative velocity increments

    of same magnitude differs !