i 66
IiJAl.MAW FRANZ
with the itutocorrelations K as herriels of known shape and the turbulence
intensity 1. Of course there are two relations like Eys. ( I ) and (2). one for the
longitudinal and one for the two lateral Eulerian correlations. Finally. the
explicit expression can be arrived at by the numerical elimination of
the unknown distribution function from the equations above in the general
form
The transfer function T is determined by the autocorrelations only
. *
K J z ) =
T(s)Kb,(sz; I ) ds.
Recause the turbulence intensity is ; 1 pure Eulerian quantity, the transfer
function T has to depend on 1. The structure of the actual turbulence field is
contained in the distribution function D.
Some general properties of T are very convenient for the mathematical
treat men t :
'0
(4 1
( 5 )
T( -s) = T ( s )
(6)
1; ~ ( s ) s " ds + x for all 11 2 o
especially
(7)
The following equations hold for every relation between the Lagrangian and
Eulerian correlation functions representable in the general form of Eqs. ( I )
and (2).
Substituting Re in Eq. (3) by its cosine transform. one gets the Lagrangian
correlation expressed by the Eulerian power spectrum 5:
1
R J f ) = 1 L ( \ ~ ~ ) P , ( ~ ) ~ ~ ~
' 0
(8)
with
L ( r ) = T ( s ) cos(rs) ds.
Thc cosine transform of Eq. (3) yields
7
P&) = 1 s- ' T ( s - I ) P E ( S V ) ds,
'0