## Abstract

We present new scaling expressions, including high-Reynolds-number (Re) 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 `_{12} – 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 −u^{0}v^{0}, and normal stresses u^{0}u^{0}, v^{0}v^{0}, w^{0}w^{0}) are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions `_{11}, `_{22}, `_{33} (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 `_{12} and `_{22} – which have the celebrated linear scalings in the logarithmic layer, i.e. `_{12} ≈ κy and `_{22} ≈ κ_{22}y. However, data show an invariant peak location for w^{0}w^{0}, which theoretically leads to an anomalous scaling in `_{33} in the log layer only, namely `_{33} ∝ y^{1−}γ with γ ≈ 0.07. Furthermore, another mesolayer modification of `_{11} yields the experimentally observed location and magnitude of the outer peak of u^{0}u^{0}. The resulting −u^{0}v^{0}, u^{0}u^{0}, v^{0}v^{0} and w^{0}w^{0} 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 y^{+} ≈ 12; (2) the location of peak value −u^{0}v^{0}_{p} has a scaling transition from 5.7Re^{1}_{τ}^{/}^{3} to 1.5Re_{τ}^{1}/^{2} at Re_{τ} ≈ 3000, with a 1 + u^{0}v^{0+}_{p} scaling transition from 8.5Re^{−}_{τ}^{2}/3 to 3.0Re^{−}_{τ}^{1}/2 (Re_{τ} the friction Reynolds number); (3) the peak value w^{0}w^{0+}_{p} ≈ 0.84Re^{0}_{τ}^{14}(1 − 48/Re_{τ}); (4) the outer peak of u^{0}u^{0} emerges above Re_{τ} ≈ 10^{4} with its location scaling as 1.1Re_{τ}^{1}/^{2} and its magnitude scaling as 2.8Re^{0}_{τ}^{09}; (5) an alternative derivation of the log law of Townsend (1976, The Structure of Turbulent Shear Flow, Cambridge University Press), namely, u^{0}u^{0+} ≈ −1.25 ln y + 1.63 and w^{0}w^{0+} ≈ −0.41 ln y + 1.00 in the bulk.

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

Number of pages | 38 |

Journal | Journal of Fluid Mechanics |

Volume | 850 |

DOIs | |

State | Published - 2017 |

## Keywords

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