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By P. Bradshaw

ISBN-10: 0080166210

ISBN-13: 9780080166216

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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. These equations are for secondorder correlations, but contain third-order correlations on the righthand side (just as the turbulent energy equation contains q2v): if we derive the equations for the third-order correlations by multiplying the Navier-Stokes equations by second-order velocity products, we find fourth-order correlations on the right-hand side, and so on.

Probability Distributions and Intermittency The various mean products and space correlations are integral moments of the joint probability distribution for the velocity components at different points in space. The probability distribution for the single variable ut at a single point, P{ux), is defined (Fig. 9) by saying that P(Wi) d«! e. the probability that the variable lies between ux — \ du and ut + \ dux. ·, un) dux du2, . · , dun is the probability that the first variable lies between ux — \ dut and u i + i d«!

Although energy diffusion is not simply related EXAMPLES OF TURBULENT FLOWS 49 to the mean gradient of the turbulent energy, it will clearly be zero when the energy and length scales are everywhere the same: the longitudinal diffusion of turbulent energy in grid turbulence, with its very small gradient of turbulent energy in the x direction, can almost always be neglected. A more important shortcoming of grid turbulence is that it is not quite isotropic—the ratios of the turbulent intensities being 0-75w2 = v2 = w2 approximately.

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An Introduction to Turbulence and its Measurement by P. Bradshaw


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