The significance of coherent flow structures for the turbulent mixing ...

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5 Investigation of the xy-plane
(5.17), into turbulent fluctuations is described by the following equation, which follows from
equation (5.8) after the multiplication with õ ,
õ Ê
õ Ê
õ ;
6 ;
Çõ> ?
Ê ( Ê)C ;
( ;
õ]Ê
ced f
b
loss of kinetic energy of the mean motion
cd f
b
production of turbulence
Ê õ Ê
ced f
b
transfer of kinetic energy
cd f b
direct dissipation
Ê
9hg
( Ê CZi
Ø ¾ (5.18)
( C
é í ;
:jA
: Ê
:)ï
and the generation, transport and decay ( of can be derived from the Navier-Stokes equation
when the equation for the
component multiplied by before averaging.
component is multiplied by ( and added to the equation for the (
(5.19)
õ ;
6 ;
( Ê ;
cd f
b
convection
cd f
b
redistribution
ced f b
production
ö†í ;
9 ï
§1" §
" §
A b ced f :
diffusion
ced f
b
dissipation
k Ê ( C
(4k Ê
;
( Ê
ökà[÷ C
Unfortunately, the remaining mathematical difficulty, associated with the statistical approach,
is that the dynamical equations are under-determined due to the non-linearity in the momentum
equations. In case of equation (5.8), for example, a relation between the õ
mean §QDF§lG motion
and all dominant components of the general Reynolds-stress tensor have to be assumed
before a numerical integration can be performed. From the present point of view it seems
unlikely that this so called closure problem can be solved from first principles, because most
of the developed mathematical approximation methods fail for two reasons. Firstly, turbulent
shear flows appear as a non-homogeneous, non-isotropic, multi-scale problem without any real
scaling similarity ranking from the largest to the smallest scales. Secondly, turbulent flows are
generally global and not local, which means that the local properties are not governed by the
mean motion. For these reasons, it is evident that the functional dependence and §QDF§lG
statistical
structure of must be determined experimentally. This will be the prime issue in the
following.
5.2 Experimental set-up
The principle set-up of the double stereo-scopic PIV system utilised for the investigation in the
wind-tunnel displayed on page 8, is shown in figure 5.1. Because of the size of the large-scale
structures in stream-wise direction and the relatively small velocity variations which must be
properly resolved, the system is installed side by side. The actual observation â angles and
distances along with the exact coordinates of each camera sensor, relative to the centre
?nm
of the corresponding observation area, are displayed in table 5.2. For the registration and
storage of the particle images four Peltier cooled high resolution PCO cameras with 1280
72