Reynolds number scaling of the peak turbulence intensity in wall flows

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, u', is thought to be among the few exceptions. In particular, the near-wall mean-square peak (u'u') over bar (+)(p), - where the superscript + indicates normalization by the friction velocity u(tau), the subscript p indicates the peak value and the overbar indicates time averaging - is known to increase with increasing Reynolds number. The existing explanations suggest a logarithmic growth with respect to Re, where Re is the Reynolds number based on u(tau) and the thickness of the wall flow. We show that this boundless growth calls into question the veracity of wall-scaling and so cannot be sustained, and we establish an alternative formula for the peak magnitude that approaches a finite limit (u'u') over bar (+)(infinity) owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.