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Published June 10, 2004 | public
Journal Article Open

Scaling of the streamwise velocity component 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 x 10^4 ≤ ReD ≤ 5.7 x 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 ReD < 6 x 10^4, the other when ReD > 7 x 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.

Additional Information

Copyright © 2004 Cambridge University Press. Reprinted with permission. (Received 7 March 2003 and in revised form 30 January 2004). Published online by Cambridge University Press 03 June 2004. The support of ONR under Grant Nos. N00014-98-1-0525 and N00014-99-1-0340 is gratefully acknowledged. J.F.M. is indebted to the Engineering and Physical Sciences Research Council (grants GR/M64536/01 and GR/R48193/01), the Royal Academy of Engineering (England), and the Leverhulme Trust (grant F/07058/H) for financial support.


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