Reynolds-number dependence of streamwise velocity fluctuations in turbulent pipe flow

Statistics of the streamwise velocity component in fully-developed pipe flow are examined for Reynolds numbers in the range 5.5 × 10⁴ ≤ Reo ≤ 5.7 × 10⁶. The second moment exhibits two maxima: one in the viscous sublayer is Reynoldsnumber dependent while the other, near the lower edge of the log region, is also Reynolds-number dependent and follows roughly the peak in Reynolds shear stress. The behaviour of both peaks is consistent with the concept of inactive motion which increases with increasing Reynolds number and decreasing distance from the wall. No simple scaling is apparent, and in particular, so-called "mixed" scaling is no better than wall scaling in the viscous sublayer and is actually worse than wall scaling in the outer region. The second moment is compared with empirical and theoretical scaling laws and some anomalies are apparent. The scaling of spectra using y, R and u_τ is examined. It appears that even at the highest Reynolds number, they exhibit incomplete similarity only: while spectra do collapse with either inner or outer scales for limited ranges of wave number, these ranges do not overlap. Thus similarity may not be described as complete and any apparent k^-1₁ range does not attract any special significance and does not involve universal constants. It is suggested that this is because of the influence of inactive motion. Spectra also show the presence of very long structures close to the wall.