Mean-flow scaling of turbulent pipe flow

Abstract

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×103 to 35×106. Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60<y+<500 or y+<0.15R+, the outer limit depending on whether the Kármán number R+ is greater or less than 9×103; and a log law for 600<y+<0.07R+. The log law is only evident if the Reynolds number is greater than approximately 400×103 (R+>9×103). Von Kármán's constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600<y+<0.07R+, and the additive constant was shown to be 6.15 when the log law is expressed in inner scaling variables.A new theory is developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. A reasonable candidate for the outer velocity scale is the velocity deficit in the pipe, UCL−Ū, which is a true outer velocity scale, in contrast to the friction velocity which is a velocity scale associated with the near-wall region which is ‘impressed’ on the outer region. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity.The friction factor data at high Reynolds numbers were found to be significantly larger (>5%) than those predicted by Prandtl's relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×103 and 35×106, and includes a term to account for the near-wall velocity profile.

The Century of Turbulence Theory: The Main Achievements and Unsolved Problems

“…This makes an impression that at present the reached accuracy of the available data on near-wall turbulent profiles is simply insufficient for the obtaining of a convincing unique conclusion about the real form of the mean-velocity profile in the intermediate layer of not-too-small and not-too-large values of z. However it seems also that great (and continued to grow) scatter of experimental values for the coefficients A, B and B (1) and for the limits of the logarithmic layer (cf., e.g., the strongly differing results of [84] and [85] which both asserted that their data are precise), contradicts to the idea of an universal overlap layer with logarithmic velocity profile having always the same constant coefficients. Barenblatt et al [89] remarked in this respect that found in [85] too low value K = 0.38 of von Kärmän constant contradicts the logarithmic-law universality.…”

“…Experimental studies of nearwall turbulent flows continue to be popular and recently several such investigations claiming to be quite accurate were carried out but this did not clarify the situation. Here we will only mention often cited recent papers by Zagarola and Smits [84] and Österlund et al [85] which both stated that their data confirm the validity of the logarithmic law (1) and both gave rise to a controversy.…”

“…The compression decreases the kinematic viscosity v of air and thus make possible the study of pipe flows in a wide range of very high Reynolds numbers (data used in [84] covered the range 31xl0 3 < Re < 35xl0 6 where Re is based on the average flow velocity U av and pipe diameter 2R). The authors found that at Re > 4x10 5 logarithmic law (1) (with coefficients K= 1/A=0.436, B=6.15) was valid for values of z in the range 600/ w < z < 0.07R.…”

“…As to the velocity defect law (3), the authors recommended to replace in it the near-wall velocity scale u* by the outer velocity scale U 0 = U max -U av where U max is the mean velocity at the pipe axis. According to [84], this replacement makes the function f (2) really independent on Re while at large values of Re it changes nothing since then the ratio U(/«* takes constant value.…”

“…Panton considered only the first approximation of this method which he presented in a special form (corresponding to the uniformly valid so-called Poincare expansion), while the initial profile equation included in his analysis both the log low in the overlap layer and the wake law in a zone adjacent to this layer. Then he showed that the considered by him approximation leads to results describing with a good accuracy numerous experimental and DNA data [including, in particular, the data of papers [84,85]) on the mean velocity and Reynolds-stress profiles U(z) and x(z) = -(uw)(z)]. Obtained composite velocity profiles U{z) in a wide range of Re values agreed rather well with the available data and also with the logarithmic law within the traditional 'logarithmic layer' of z values (where the use of the conventional coefficients K = 0.41 and B = 5.25 did not lead in most of the cases to disagreement with the data).…”