The measurements by Zagarola & Smits (1998) of mean velocity profiles in fully developed turbulent pipe flow are repeated using a smaller Pitot probe to reduce the uncertainties due to velocity gradient corrections. A new static pressure correction (McKeon & Smits 2002) is used in analysing all data and leads to significant differences from the Zagarola & Smits conclusions. The results confirm the presence of a power-law region near the wall and, for Reynolds numbers greater than $230\,{\times}\,10^3$ ($R^+\,{>}\,5\,{\times}\,10^3$), a logarithmic region further out, but the limits of these regions and some of the constants differ from those reported by Zagarola & Smits. In particular, the log law is found for $600\,{<}\, y^+\,{<}\,0.12R^+$ (instead of $600\,{<}\,y^+\,{<}\,0.07R^+$), and the von Kármán constant $\kappa$, the additive constant $B$ for the log law using inner flow scaling, and the additive constant $B^*$ for the log law using outer scaling are found to be $0.421 \pm 0.002$, $5.60 \pm 0.08 $ and $1.20 \pm 0.10$, respectively, with 95% confidence level (compared with $0.436 \pm 0.002$, $6.15 \pm 0.08$, and $1.51 \pm 0.03$ found by Zagarola & Smits). The data also confirm that the pipe flow data for Re$_D\,{\le}\,13.6\,{\times}\,10^6$ (as a minimum) are not affected by surface roughness.

Statistics of the streamwise velocity component in fully developed pipe flow are examined for Reynolds numbers in the range 5.5 × 10 4 6 Re D 6 5.7 × 10 6 . Probability density functions and their moments (up to sixth order) are presented and their scaling with Reynolds number is assessed. The second moment exhibits two maxima: the one in the viscous sublayer is Reynolds-number dependent while the other, near the lower edge of the log region, follows approximately the peak in Reynolds shear stress. Its locus has an approximate (R + ) 0.5 dependence. This peak shows no sign of 'saturation', increasing indefinitely with Reynolds number. Scalings of the moments with wall friction velocity and (U cl − U ) are examined and the latter is shown to be a better velocity scale for the outer region, y/R > 0.35, but in two distinct Reynoldsnumber ranges, one when Re D < 6 × 10 4 , the other when Re D > 7 × 10 4 . Probability density functions do not show any universal behaviour, their higher moments showing small variations with distance from the wall outside the viscous sublayer. They are most nearly Gaussian in the overlap region. Their departures from Gaussian are assessed by examining the behaviour of the higher moments as functions of the lower ones. Spectra and the second moment are compared with empirical and theoretical scaling laws and some anomalies are apparent. In particular, even at the highest Reynolds number, the spectrum does not show a self-similar range of wavenumbers in which the spectral density is proportional to the inverse streamwise wavenumber. Thus such a range does not attract any special significance and does not involve a universal constant.

Spectra of the streamwise velocity component in fully developed turbulent pipe flow are presented for Reynolds numbers up to 5.7 3 10 6 . Even at the highest Reynolds number, streamwise velocity spectra exhibit incomplete similarity only: while spectra collapse with both classical inner and outer scaling for limited ranges of wave number, these ranges do not overlap. Thus similarity may not be described as complete, and a region varying with the inverse of the streamwise wave number, k 1 , is not expected, and any apparent k 21 1 range does not attract any special significance and does not involve a universal constant. Reasons for this are suggested. DOI: 10.1103/PhysRevLett.88.214501 PACS numbers: 47.27.Jv, 02.50. -r, 05.45. -a, 47.27.Nz Reynolds number similarity is an essential concept in describing the fundamental properties of turbulent wallbounded flow. Unlike the drag coefficient for bluff bodies, that for a turbulent boundary layer continues to decrease with increasing Reynolds number because the small-scale motion near the surface is directly affected by viscosity at any Reynolds number. Therefore Reynolds number similarity is very important in design and is a vital tool for the engineer, who, plied with information from either direct numerical simulations or wind-tunnel tests (or both), may well have to extrapolate over several orders of magnitude in order to estimate quantities such as drag at engineering or even meteorological Reynolds numbers. Perhaps the most well-known example of Reynolds number similarity is the region of log velocity variation (the log law) found in wall-bounded flows which, at sufficiently high Reynolds numbers, exists regardless of the nature of the surface boundary condition or the form of the outer imposed length scale. The log law may be derived by an overlap argument, where the overlap is said to occur between a near-wall region described by "wall" variables, that is, a velocity scale u t and a length scale n͞u t (the superscript "1" denotes nondimensionalization with wall variables), and an outer layer that depends on "outer" variables, that is, a velocity scale u t and a length scale that is, for pipe flow, the radius R. Here, u t p t w ͞r, t w is the wall shear stress, r is the density, and n is the kinematic viscosity [1].For turbulence at sufficiently large distances from the wall, Kolmogorov's famous theories express the most important way in which Reynolds number similarity is used [2]. Yet, of increasing importance is whether or not the large scales also exhibit Reynolds number similarity when suitably scaled. In the context of wall-bounded turbulent flow, there is growing interest in using these ideas to develop subgrid models and boundary conditions for large-eddy simulations (LES). In LES, only the large scales are resolved so that in the near-wall region where all the eddies are "small," there is a need to model not only the energy drain from the resolved scales but also to provide an "off-the-surface" condition for the simulation. In this context, self-sim...

Mean flow measurements taken in fully developed turbulent pipe flow over a wide Reynolds number range are used to evaluate current methods of correcting Pitot probe data. Based on this evaluation, a new form for the displacement correction is proposed which appears to be more accurate over a wider range of conditions than those currently available. The difficulty of obtaining the true near-wall velocity profile near the wall is explored.

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