THE EULERIAN-LAGRANGIAN
RELATIONSHIP RESULTING FROM A
TURBULENCE MODEL
HJALMAR FRANZ
Deutsches Hpdrographisrhus Iiisti~ut, Hamhrrrg, Germuiiy
In two papers (Franc 1970. 1973) the author has published a relation
between the Lagrangian and the Eulerian correlation functions. Here the
consequences will be presented for the power spectra and other functions
related to the correlations. For a better understanding of the physics involved, a short review of the main assumptions of the model is given.
(1) The turbulent velocity fluctuations are regarded to be caused by vortices penetrating each other. Each vortex is displaced by the mean current '
and by all the other vortices.
(2) The time dependency of the vortices is taken into account bccause
otherwise the Lagrangian integral time scale would tend to zero for large
timcs.
(3) The correlations between different vortices are assumed to be small in
thc mean compared with the autocorrelations.
(4) In each system the mean correlation is found by weighting the autocorrelation with a distribution function common for both systems. That
distribution function of unknown shape depends above all on a characteristic freyuencj and on a characteristic length.
( 5 ) That frequency- -characteristic for the Lagrangian autocorrelation-and th;it Icnglh-,-charactcristic for thc Eulerian autocorrelation-are
related in the mean. As a consequence the distribution function depends
further on one sc:ilc parameter only.
Especially the last assumption is necessary to follow a relation between
the Lagrangian and the Eulerian mean correlations R. The relation is given
implicitly by integrals over the distribution function D
U
R,.(t) = f D(co)K,(rot) d~
' 0
(1)
(2)
1
R,(t) = [ D(w)KE(cot; I) do)
* 0
16.5
RELATIONSHIP RESULTING FROM A
TURBULENCE MODEL
HJALMAR FRANZ
Deutsches Hpdrographisrhus Iiisti~ut, Hamhrrrg, Germuiiy
In two papers (Franc 1970. 1973) the author has published a relation
between the Lagrangian and the Eulerian correlation functions. Here the
consequences will be presented for the power spectra and other functions
related to the correlations. For a better understanding of the physics involved, a short review of the main assumptions of the model is given.
(1) The turbulent velocity fluctuations are regarded to be caused by vortices penetrating each other. Each vortex is displaced by the mean current '
and by all the other vortices.
(2) The time dependency of the vortices is taken into account bccause
otherwise the Lagrangian integral time scale would tend to zero for large
timcs.
(3) The correlations between different vortices are assumed to be small in
thc mean compared with the autocorrelations.
(4) In each system the mean correlation is found by weighting the autocorrelation with a distribution function common for both systems. That
distribution function of unknown shape depends above all on a characteristic freyuencj and on a characteristic length.
( 5 ) That frequency- -characteristic for the Lagrangian autocorrelation-and th;it Icnglh-,-charactcristic for thc Eulerian autocorrelation-are
related in the mean. As a consequence the distribution function depends
further on one sc:ilc parameter only.
Especially the last assumption is necessary to follow a relation between
the Lagrangian and the Eulerian mean correlations R. The relation is given
implicitly by integrals over the distribution function D
U
R,.(t) = f D(co)K,(rot) d~
' 0
(1)
(2)
1
R,(t) = [ D(w)KE(cot; I) do)
* 0
16.5