By P. Bradshaw
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Additional resources for An Introduction to Turbulence and its Measurement
Since for a given turbulent intensity the latter possibility would imply infinite turbulence dissipation rate but only a finite rate of energy production, it can be discounted, and it appears that the length scale must increase monotonically to infinity. The small amount of experimental data available supports this conclusion but, in practice, it is difficult to obtain at the same time a sufficiently high Reynolds number and a sufficiently small ratio of eddy size to test rig size. In any real shear flow an upper limit is set to the size of the turbulent eddies by the dimensions of the mean flow itself, and indeed this upper limit is nearly always reached, as in 50 TURBULENCE AND ITS MEASUREMENT a pipe flow where the largest eddies have wavelengths of the order of the pipe diameter.
Wave number spectra can be derived as the Fourier transforms of these and other cross-correlations. 9. g. those like w(x) v(x) v(x + r) transform into spectra giving the energy transfer from low wave numbers to high wave numbers]. Correlations of the fourth and higher orders are bound to be partly degenerate because there are only three velocity components to choose from. These correlations appear in "conservation" equations t Both signals must be filtered because a band-pass filter produces a phase shift.
12). If «! ω FIG. 12. Line spectrum. are always uncorrelated. 2, root-mean-squares are not useful because they do not add. Mathematically, the spectrum of asinco^ is (α2/2)<5(ω — ω^), where δ(ω — Wj) is the delta function, which is zero when ω Φ ω1 and goes to infinity at ω = ω± in such a way that f δ(ω — ω^ άω is unity: thus the integral of the spectrum of a sin ωχί over all ω is the mean square, a212. In practice, φ can be measured for any fluctuating quantity u with 32 TURBULENCE AND ITS MEASUREMENT zero mean by putting an electrical signal, proportional to u, through a "band-pass" filter, an arrangement of resonant circuits which, ideally, passes frequencies between ω± — %άω and ω1 + %άω only (Fig.
An Introduction to Turbulence and its Measurement by P. Bradshaw