By Grossi M., Lin C.-S., Prashanth S.
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Extra info for A uniqueness result for a Neumann problem involving the critical Sobolev exponent
Slowly changing functions of Re). Expanding these functions in powers of a small parameter (ln Re)−1 , the authors assumed that α(Re) = α1 / ln Re + O[(ln Re)−2 ], C(ln Re) = C0 + C1 / ln Re + O[(ln Re)−2 ] (constant term was omitted in the series for α(Re) to guarantee the validity of the K41 scaling when Re → ∞). For crude estimate of the function C(ln Re) the data by Praskovsky and Onsley  were used.
In the case of boundary layers the pipe-ﬂow equations (8) and (9) were used without any modiﬁcation but now the value of Re was determined as that leading to the best ﬁt of these equations with the available velocity data. ] In  it was found that the velocity proﬁles of turbulent boundary layers agree well with the power law (8) and (9) in the range of z-values extending from the upper edge of the viscous sublayer (located at u∗ z/ν = 70) to the upper edge of the whole boundary layer above which the homogeneous “free stream” begins.
Therefore, the old velocity-proﬁle problem which tortured Prandtl, Taylor and von K´ arm´ an in the ﬁrst quarter of the 20th century, now again became actual and apparently requires supplementary studies of physical mechanisms leading to possible violations of the logarithmic law and to reliably detected violations of related similarity laws for higher-order statistical characteristics of wall-bounded turbulent ﬂows. Before the appearance of much more accurate experimental (and/or DNS) data (and even after it too), better understanding of the main factors determining the form the velocity proﬁles in various turbulent ﬂows undoubtedly requires more direct use of the physical arguments concerning the mechanisms of turbulent mixing.