By P. Bradshaw
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Additional info for An Introduction to Turbulence and its Measurement
44 TURBULENCE AND ITS MEASUREMENT derived from the Navier-Stokes equations by multiplying the latter by products of the velocities at different points before time averaging. For instance, the turbulent energy equation can be obtained by, multiplying the ^-component Navier-Stokes equation by the «-component velocity at the same point, adding the corresponding y and z equations to it and then taking the time average. If the Navier-Stokes equations at one point x are multiplied by velocity components at another point x + r we obtain an equation for the rate of change of a correlation with x or r: the simplest equation of this sort is the Karman-Howarth equation for decaying isotropic turbulence* 2,5) where the "rate of change" is strictly with respect to time.
5. Frequency Spectra The autocovariance of a quantity that varies as a sin ωγΐ is easily seen to be (a2/2) cos cor: this autocorrelation of a fluctuation at a single frequency does not decay to zero as x -► oo, unlike the typical correlation for turbulence quantities which cover a wide range of frequencies, so there is evidently some connection between the shape of the autocorrelation and the distribution of the fluctuation over different frequencies, the latter being called the spectrum. MEASURABLE QUANTITIES AND THEIR PHYSICAL SIGNIFICANCE 31 A quantity that varies sinusoidally at constant amplitude is said to have a line spectrum, since any definition we may choose for the "distribution of the fluctuation over different frequencies", φ say, will lead to a concentration of the spectral density φ at the frequency ω± (Fig.
The physical meaning of terms (1) to (5) is summarized in Fig. 10. The equation obtained by adding up all these terms and equating the sum to zero is the required conservation or "transport" equation for the turbulent energy or the sum of the Reynolds normal stresses. g. ref. 5, eqn. 10)]. Frequently, some of the terms can be neglected: usually, production and dissipation are the t The tensor form is — μ I—- + — - 1 which looks (but is not) half as large as 2 \ dxj dxi / the sum of the terms quoted here, because each term appears twice on summation.
An Introduction to Turbulence and its Measurement by P. Bradshaw