## Abstract

We present new scaling expressions, including high-Reynolds-number ((Formula presented.)) predictions, for all Reynolds stress components in the entire flow domain of turbulent channel and pipe flows. In Part 1 (She et al., J. Fluid Mech., vol. 827, 2017, pp. 322–356), based on the dilation symmetry of the mean Navier–Stokes equation a four-layer formula of the Reynolds shear stress length (Formula presented.) – and hence also the entire mean velocity profile (MVP) – was obtained. Here, random dilations on the second-order balance equations for all the Reynolds stresses (shear stress (Formula presented.), and normal stresses (Formula presented.), (Formula presented.), (Formula presented.)) are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions (Formula presented.), (Formula presented.), (Formula presented.) (hence the three turbulence intensities) are obtained for turbulent channel and pipe flows. In particular, direct numerical simulation (DNS) data are shown to agree well with the four-layer formulae for (Formula presented.) and (Formula presented.) – which have the celebrated linear scalings in the logarithmic layer, i.e. (Formula presented.) and (Formula presented.). However, data show an invariant peak location for (Formula presented.), which theoretically leads to an anomalous scaling in (Formula presented.) in the log layer only, namely (Formula presented.) with (Formula presented.). Furthermore, another mesolayer modification of (Formula presented.) yields the experimentally observed location and magnitude of the outer peak of (Formula presented.). The resulting (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.) are all in good agreement with DNS and experimental data in the entire flow domain. Our additional results include: (1) the maximum turbulent production is located at (Formula presented.); (2) the location of peak value (Formula presented.) has a scaling transition from (Formula presented.) to (Formula presented.) at (Formula presented.), with a (Formula presented.) scaling transition from (Formula presented.) to (Formula presented.) ((Formula presented.) the friction Reynolds number); (3) the peak value (Formula presented.); (4) the outer peak of (Formula presented.) emerges above (Formula presented.) with its location scaling as (Formula presented.) and its magnitude scaling as (Formula presented.); (5) an alternative derivation of the log law of Townsend (1976, The Structure of Turbulent Shear Flow, Cambridge University Press), namely, (Formula presented.) and (Formula presented.) in the bulk.

Original language | English |
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Pages (from-to) | 401-438 |

Number of pages | 38 |

Journal | Journal of Fluid Mechanics |

DOIs | |

State | Accepted/In press - Jul 5 2018 |

## Keywords

- pipe flow boundary layer
- turbulence theory
- turbulent boundary layers