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

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5.3.2 Spatial auto- and cross-correlation functions
5.3 Statistical properties of the flow
It is obvious that the single point statistics, presented in the previous section, do not cover
any aspects associated with the organised flow motion outlined in chapter 1. However, it is
shown in section 5.1 that for the turbulent mixing in wall-bounded flows this coherent motion
must be of primary importance. This can be deduced from the presence of the single point
correlations terms · ¬ , ³ ¬ and especially Î · ³ in equation (5.8). For this reason the spatial
correlations of the velocity fluctuations, introduced in section 5.1, will be investigated in this
section. The investigation of the spatial correlations of the velocity fluctuations furnishes
quantitative information about the average dimensions of the turbulent flow structures and
allows to deduce several length and time scales with physical meaning such as the dissipation
or the integral length scales, which play a dominant role in the transport equations derived
in [90]. However, due to the anisotropic nature of wall-bounded flows according to figure 5.3,
the length scales are not uniquely defined as in the case of isotropic turbulence. In addition,
it should be kept in mind that the general investigation of the correlation function might be
quite complex in shear-flows because this function depends on the flow variables considered,
as well as their mutual location, separation and time delay. For simplicity, only the primary
spatial correlations ¼ÐÏ'Ï , ¼ÐÑÑ and ¼ÐÒHÒ of the velocity fluctuation will be considered in the
following, along with some double and triple correlations which are important for the turbulent
mixing according to equation (5.20). The dependence of the primary correlations in a turbulent
boundary layer along the three coordinates was extensively investigated by Grant in 1958 by
mm. ”Œ”
using a pair of hot-wire Ó probes m behind a trip fence »Ô Á4‘ÕÓz” at m/s Ö and
Although he could confirm the high degree of order present in the flow field by analysing the
large-scale dependence of various correlations along the principle coordinates, he was forced
to conclude that the various correlations are incompatible with the idea of energetic eddy
structures convecting downstream. Since it is impossible to predict a unique set of vortical
structures from the correlations, the motivation for this conclusion was based on the different
size of the primary correlations in stream-wise direction and the small extend ¼ÐÑÑ of along
the span-wise one. However, due to the small Ö thickness of the boundary layer, it is not
yet evident if this strong difference in the various correlations is a real shear-flow property or
maybe caused by the entraining motions, which do not belong to the main turbulence structure.
For this reason the present investigation focuses on the functional dependence of the spatial
correlations in the near-wall region and their variation with the Reynolds number. Of particular
interest is the correlation in the vicinity of the maximum, as this information is difficult to
achieve by using intrusive measurement techniques because the signal from a single probe is
affected by approaching the second probe and the values along the stream-wise axis is biased
because the measurement values of the down-stream probe are affected by the wake of the
upstream probe as pointed out in [24, 100].
In the following the size and shape of the primary correlation functions will be analysed
because these values furnish the information about the average dimensions of the moving flow
structures, their degree of organisation and their propagation direction relative to the mean
motion. This becomes obvious when figure 5.5 and figure 5.6 are considered which reveal the
primary correlation of the stream-wise velocity fluctuations for various wall distances of the
fixed point (compare location of the maximum or ÄHÅ the value in the legend) ¼× ÂŒz
at
and 15000. The contours of the following plots are spaced in intervals of 0.05 for the primary
correlation (excluding 0.05 and 1) and the position and amplitude of the minimum and maximum
are included for comparison. Continuous lines indicate positive correlation. Clearly
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