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

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6 Investigation of the xz-plane
boundary layer approximation and the discussion on page 93. However, the displacement
of the maximum 6 to negative -values, with decreasing wall distance, is not evident for
þ¡
both
PDF’s. This implies u.v ; that at most of the structures (or the large ones) show a slightly
negative stream-wise velocity fluctuation with a rather weak amplitude, as can be seen from
the location of the maximum. As the mean value of all fluctuations must be zero by definition,
the amplitudes of the structures with a positive stream-wise velocity fluctuation are necessarily
larger in order to compensate the maximum 6 at negative . This explains the fact that the
in the lower right of figure 6.1 result from the positive fluctuations and not
þ¡
uv ;
maxima at
from the negative as expected. The power-of-two operation, required for calculating the rmsvalues,
increases the weight of the positive large-scale fluctuations when the average is taken
and decreases the contribution of the negative high frequency fluctuations (the argumentation
can be reversed for the uv ; Ó measurements at ). As the asymptotic behaviour of the spanwise
velocity amplitude is weaker with respect 7 to the -component, the displacement of the
maximum can be seen more clearly in the lower row of figure 6.3 where the joint probability
density function of the velocity fluctuations in stream-wise and span-wise direction is shown.
Another interesting feature, appearing in the lower row, is the asymmetry for the large-scale
fluctuations, expressed PDF{|6 PDF{Î6 t~} by which is visible in the right graph. The
fact that large-scale fluctuations of the span-wise t component are frequently associated with
t‚}„ƒ
positive stream-wise fluctuations and t small-scale -fluctuations with 6 negative -fluctuations
may be an effect associated with organised motions in the flow field. This will be considered
later. Figure 6.4 reveals the same result but in a different representation in order to highlight
the displacement of the maximum in wall-units and the amplitude of the signal as a function
of the parameter indicated in the legend. The graph with the thick line-width in the upper right
plot shows the distribution of the wall-normal velocity component when the fluctuation in the
stream-wise direction is exactly zero (e.g. the instantaneous velocity component is identical
with the time averaged one). The function seems to be symmetrical and Gaussian in shape.
If the stream-wise fluctuations become negative, the distribution of the wall-normal velocity
component decreases slowly while the shape of the graphs becomes non Gaussian and the
spectrum around the maximum decreases. In connection with the sign of the the fluctuations,
this again implies that low-speed structures are more frequently associated with a positive
wall-normal velocity component whereas the probability of finding strong high-momentum
structures is related to a motion toward the wall. This can be explained simply by assuming
that turbulent structures coming from large wall distances uZY u at conserve their momentum
while travelling wall-ward [91]. Furthermore, the different decay of the maximum with
increasing magnitude of 6 should be noted as this indicates that the flow structures associated
with positive wall-normal motion seem to be more coherent than the other structures. Finally,
the fact is important that the magnitude of the largest stream-wise and wall-normal fluctuations
are approximately twice as large as the corresponding rms value measured at the same
u.v -position because this point supports the arguments used to explain the two maxima in the
lower right plot of figure 6.1. Whereas the graphs in the lower plots, which represent the
statistical structure of 7…t the fluctuations, are nearly Gaussian and symmetrical in this representation,
PDF{²Î7 t‚} W PDF{|7 t~} e.g. , the graphs of the centre row possess more or less
pronounced shoulders which indicate a complex interaction of the structures in this region,
but the nature of the processes cannot be identified from the single-point statistics. To deduce
the typical flow angles associated with the transport of low and high momentum fluid towards
and away from the wall, the probability density function of the absolute angle between the
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