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

6 Investigation of the xz-plane
processes associated with the organised motion in span-wise direction in more detail. Especially
the value of maximum correlation should be noted for different wall distances and with
respect to à Ñ$ and ÃË . In order to interprete the various correlation patterns qualitatively,
it is necessary to assume a pure flow motion indicated by the first subscript (motion towards
the wall in case of à ê Ë ë3ìÌí ê í ) at the origin of the correlation plane. Using this method it becomes
evident that a motion towards the wall is on average associated with a span-wise motion
away from the structure while the opposite holds for an organised motion away from the wall
(compare the sign of the structures). The location of the maxima, on the other hand, implies
that a phase relation exists between the two motions. When the flow direction is taken into
account, it becomes apparent that the span-wise motion can be considered as the footprint of
the wall-ward motion. Thus, the span-wise motion can be considered as a secondary motion.
It is clear that the processes described are associated with two stream-wise vortices whose
length is determined by the flow region moving in ­ -direction. For clarity the main features of
the non-conditional functions have been summarised in figure 6.9 in form of one-dimensional
graphs extracted at the location of the maximum of correlation in Î - and Ï -direction. This
representation allows to compare quantitatively the height of the correlation and the location
of the maximum in wall-units.
6.3 Spatio-temporal buffer layer statistics
In this section the spatio-temporal behaviour of the flow structures present in the buffer layer
will be analysed quantitatively to examine their convection velocity and temporal decay. The
first investigation of this type was performed by Favre et al (1957) who studied the spatiotemporal
structure of the stream-wise velocity Ã Ñ˜Ñ component at ½ü©À and ÂĥŠ¿ ª Þ ÀÀ ,
by using a pair of spatially separated hot wire probes [19, 20]. Their measurements were
performed ­® ¤ È Æ
m and ª ¤ Æ © m behind a tripping device at free-stream velocities of ª ¨
m/s with
%
À
ª©È mm and % Ê «© mm. Here, the structures at ­ ®é¿ «¢À and below will be investigated
Ê
ÂĥŠ¿ È$É¢À¢À at by analysing the primary (Ã Ñ˜Ñ Á–ÃËÌË correlations à & and ), the different crosscorrelations
ÃË and ) and conditional correlations taking into account the sign
(Ã Ñ ËkÁ–Ã Ñ$
of the (Ã ® Ñ Ë$Á Ã î Ñ Ë ,Ã ® Ë Á–Ã î Ë fluctuations and others). The conditional correlations yield
information about the space-time structure of the bursting phenomenon and allow to estimate
the mean convection velocity of the coherent velocity structures in the near-wall region. This
becomes important for proving the validity of Taylor’s hypothesis in the near-wall region of
a turbulent boundary layer flow. In other words, when the characteristic velocity of the large
scale structure motions differs from the velocity of the small scale structures, the space-time
'§ ')( ¿ ¦'§ ¸ 'Î transformation , usually applied when spatial derivatives are measured by
¸
using a hot-wire, becomes questionable.
All correlations considered so far were calculated at a fixed location. Here, in contrast,
the spatio-temporal correlation between separated planes is considered. The spacing
is always 10 wall-units but the position of the lower measurement plane is altered ­® between
Ê ª¥À and 20 wall-units. When the temporal delay between a pair of measurements is
­ ®
varied, this approach allows to investigate the structures moving toward and away from the
wall, depending on the temporal order of the measurements. In other words, when a structure
is moving towards the wall, this structure can be investigated best when the first measurement
is performed Í ­
at Í ­ ½ À with and the second at while flow-structures
moving away from the wall yield higher correlation values when the first measurement is
­ ­+*
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