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3.7: The Smallest Eddies

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  • The ratio between the largest eddies and the smallest eddies in turbulent pipe flow is of the magnitude of the Reynolds number to the power of 3⁄4. Assuming that the largest eddies are of the magnitude of the pipe diameter, then this gives for the diameter of the smallest eddies:

    \[\ \mathrm{d}_{\mathrm{e}}=\frac{\mathrm{D}_{\mathrm{p}}}{\mathrm{R} \mathrm{e}^{\mathrm{3} / 4}}=\frac{\mathrm{D}_{\mathrm{p}}^{\mathrm{1} / 4} \cdot \mathrm{v}_{\mathrm{l}}^{\mathrm{3} / 4}}{\mathrm{v}_{\mathrm{l}}^{3 / 4}}\]

    Using the Blasius equation for the Darcy Weisbach friction factor, this gives for the ratio between the diameters of the smallest eddies to the thickness of the viscous sub layer:

    \[\ \frac{\mathrm{d}_{\mathrm{e}}}{\delta_{\mathrm{v}}}=\mathrm{0 . 0 1 7} \cdot \mathrm{R} \mathrm{e}^{\mathrm{1/8}}\]

    For Reynolds numbers ranging from 100,000 for small pipe diameters to 10,000,000 for large pipe diameters this gives a ratio of 0.072 to 0.127, so about 10%.

    Figure 3.7-1: The Darcy-Weisbach friction factor λl for smooth pipes as a function of the line speed vls.

    Screen Shot 2020-07-07 at 1.42.50 PM.png

    Figure 3.7-2: The Darcy-Weisbach friction factor λl for smooth pipes as a function of the pipe diameter Dp.

    Screen Shot 2020-07-07 at 1.45.54 PM.png