By P. Bradshaw
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Additional info for An Introduction to Turbulence and its Measurement
FIG. 11. 5. Physically, the correlation with separation r is a measure of the strength of eddies whose length in the direction of the vector r is greater than the magnitude of r (since eddies smaller than this will not contribute to the correlation): as a warning that this is an imprecise concept, note that the correlations for the three velocity components will in general be different for the same r. 4. When we speak of the length scale of the energy00 containing eddies we mean a length of order j Rar (actually called the "integral scale").
Again, it is the non-linearity of the Navier-Stokes equations (p. 11) that is responsible. Books on statistical mathematics and such linear random processes as electrical noise tend to concentrate on Gaussian theory, sometimes to the extent of presenting results that are valid only for Gaussian processes without making this restriction entirely clear. Also, elementary books usually start with discrete processes (successive events like the throwing of dice) which are even less relevant than continuous Gaussian processes to the statistics of turbulence.
This cannot be quite correct: when discussing the smaller eddies we can define the "mean" energy transfer by an average over, say, 100 wavelengths or periods of the smaller eddies, and this average will itself vary at the much larger wavelength or period of the energy-containing eddies, so that the properties of the small-scale eddies have a "spotty" distribution in space and time. According to plausible theoretical estimates, this does not greatly affect the simple dimensional results given above—for instance, the —5/3 (—1-67) power law is replaced by a —1-71 power law—but the phenomenon has to be borne in mind when considering the smaller-scale motion in more detail.
An Introduction to Turbulence and its Measurement by P. Bradshaw