By P. Bradshaw
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Additional info for An Introduction to Turbulence and its Measurement
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. 13), and finding the mean square of the output, u2(co) say, as if it were a complete signal :φ is then 1/dco times this mean square value. Turbulent fluctuations have a broad spectrum (Fig. 13): sometimes fairly flat peaks can be distinguished but discrete frequencies occur only in the earlier stages of transition from laminar to turbulent flow.
Usually more important in practice are terms like —QÛv(dUldy) dy dxdz which represent work done against the shear stress — ρΰν to shear an element of fluid at the rate dU/dy [note that there is a second term involving uv, namely — qûvidVjàx) dxdy dz for the other constituent of the rate of strain (dU[dy) + (dV/dx) so that the sum of the two is —QÜv(dU/dy + dV/dx) dxdy dz, containing the product of the Reynolds shear stress and the rate of strain]. From the point of view of the turbulent motion, these are energy production terms; from the point of view of the mean motion, they are energy loss terms.
Of course, mean density changes certainly affect the mean flow and may also have some effect on the turbulence, but the latter effects are small unless strong streamwise density gradients occur (because of large pressure gradients or chemical reactions). For these reasons we shall ignore compressibility in the main part of the book: unfortunately, the process of making measurements is made much more difficult by the presence of temperature or density fluctuations, and these difficulties will be discussed in Chapter 7, with reference to high-speed and low-speed flows.
An Introduction to Turbulence and its Measurement by P. Bradshaw