Scaling of the streamwise velocity component in turbulent pipe flow

Abstract

Statistics of the streamwise velocity component in fully developed pipe flow are examined for Reynolds numbers in the range $5.5 \times 10^4 \le \hbox{\it Re}_D \le 5.7 \times 10^6$. 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 $(U_{cl}-\overline{U})$ 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 Reynolds-number ranges, one when $\hbox{\it Re}_D < 6 \times 10^4$, the other when $\hbox{\it Re}_D > 7 \times 10^4$. 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.