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21 over the full range of turbulent smooth wall flows

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Unformatted text preview: called the logarithmic-overlap layer. Thus by dimensional reasoning and physical insight we infer that a plot of u versus ln y in a turbulent-shear layer will show a curved wall region, a curved outer region, and a straight-line logarithmic overlap. Figure 6.9 shows that this is exactly the case. The four outer-law profiles shown all merge smoothly with the logarithmic-overlap law but have different magnitudes because they vary in external pressure gradient. The wall law is unique and follows the linear viscous relation u u u* yu* y (6.22) from the wall to about y 5, thereafter curving over to merge with the logarithmic law at about y 30. Believe it or not, Fig. 6.9, which is nothing more than a shrewd correlation of velocity profiles, is the basis for most existing “theory of turbulent-shear flows. Notice that we have not solved any equations at all but have merely expressed the streamwise velocity in a neat form. There is serendipity in Fig. 6.9: The logarithmic law (6.21), instead of just being a short overlapping link, actually approximates nearly the entire velocity profi...
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