Quantifying wall turbulence via a symmetry approach. Part 2. Reynolds stresses

Xi Chen, Fazle Hussain, Zhen Su She

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30 Scopus citations


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 −u0v0, and normal stresses u0u0, v0v0, w0w0) 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 ≈ κ22y. However, data show an invariant peak location for w0w0, which theoretically leads to an anomalous scaling in `33 in the log layer only, namely `33 ∝ y1−γ with γ ≈ 0.07. Furthermore, another mesolayer modification of `11 yields the experimentally observed location and magnitude of the outer peak of u0u0. The resulting −u0v0, u0u0, v0v0 and w0w0 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 −u0v0p has a scaling transition from 5.7Re1τ/3 to 1.5Reτ1/2 at Reτ ≈ 3000, with a 1 + u0v0+p scaling transition from 8.5Reτ2/3 to 3.0Reτ1/2 (Reτ the friction Reynolds number); (3) the peak value w0w0+p ≈ 0.84Re0τ14(1 − 48/Reτ); (4) the outer peak of u0u0 emerges above Reτ ≈ 104 with its location scaling as 1.1Reτ1/2 and its magnitude scaling as 2.8Re0τ09; (5) an alternative derivation of the log law of Townsend (1976, The Structure of Turbulent Shear Flow, Cambridge University Press), namely, u0u0+ ≈ −1.25 ln y + 1.63 and w0w0+ ≈ −0.41 ln y + 1.00 in the bulk.

Original languageEnglish
Pages (from-to)401-438
Number of pages38
JournalJournal of Fluid Mechanics
StatePublished - 2017


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


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