Self-Similarity of Mean-Flow in Pipe Turbulence

Abstract

Based on our previous modi…ed log-wake law in turbulent pipe ‡ows, we invent two compound similarity numbers (Y;U), where Y is a combination of the inner variable y + and outer variable �, and U is the pure eect of the wall. The two similarity numbers can well collapse mean velocity pro…le data with dierent moderate and large Reynolds numbers into a single universal pro…le. We then propose an arctangent law for the buer layer and a general log law for the outer region in terms of (Y;U). From Milikan's maximum velocity law and the Princeton superpipe data, we derive the von Karman constant � = 0:43 and the additive constant B � 6. Using an asymptotic matching method, we obtain a self-similarity law that describes the mean velocity pro…le from the wall to axis; and embeds the linear law in the viscous sublayer, the quartic law in the bursting sublayer, the classic log law in the overlap, the sine-square wake law in the wake layer, and the parabolic law near the pipe axis. The proposed arctangent law, the general log law and the self-similarity law have been con…rmed with the high-quality data sets, with dierent Reynolds numbers, including those from the Princeton superpipe, Loulou et al., Durst et al., Perry et al., and den Toonder and Nieuwstadt. Finally, as an application of the proposed laws, we improve the McKeon et al. method for Pitot probe displacement correction, which can be used to correct the widely used Zagarola and Smits data set. Nomenclature A = constant in the arctangent law, Eq. (19) B = additive constant in the classic log law, Eq. (9) B1 = additive constant in Millikan's log law of the maximum velocity, Eq. (10) C = model constant in the self-similarity law, Eq. (26) Dp = diameter of Pitot probe in Eq. (30) f = generic function ki = coe¢ cients in the arctangent law, Eq. (19), where i = 1;2;3;��� U = similarity number representing the pure eect of the wall, Eq. (14e) u = mean velocity at distance y from the wall umax = maximum velocity at the pipe center u + = dimensionless maximum velocity, u + = umax=u� u + = dimensionless velocity scaled by the shear velocity, u + = u=u� u� = shear velocity R = radius of pipe Re = bulk Reynolds number Re� = Krmn number Y = similarity number combining the inner and outer variables, Eq. (14d) y = distance from the wall yc = true velocity at the center of the probe in Eq. (31) y + = inner variable, y + = yu?=�