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

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 `<sub>12</sub> – 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<sup>0</sup>v<sup>0</sup>, and normal stresses u<sup>0</sup>u<sup>0</sup>, v<sup>0</sup>v<sup>0</sup>, w<sup>0</sup>w<sup>0</sup>) are analysed layer by layer, and similar four-layer formulae of the corresponding stress length functions `₁₁, `<sub>22</sub>, `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 `<sub>12</sub> and `₂₂ – which have the celebrated linear scalings in the logarithmic layer, i.e. `<sub>12</sub> ≈ κy and `₂₂ ≈ κ₂₂y. However, data show an invariant peak location for w⁰w⁰, which theoretically leads to an anomalous scaling in `<sub>33</sub> 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 −u⁰v⁰_p has a scaling transition from 5.7Re¹_τ^/3 to 1.5Re_τ¹/² at Re_τ ≈ 3000, with a 1 + u⁰v⁰⁺_p scaling transition from 8.5Re⁻_τ²/3 to 3.0Re⁻_τ¹/2 (Re_τ the friction Reynolds number); (3) the peak value w⁰w⁰⁺_p ≈ 0.84Re⁰_τ¹⁴(1 − 48/Re_τ); (4) the outer peak of u⁰u⁰ emerges above Re_τ ≈ 10⁴ with its location scaling as 1.1Re_τ¹/² and its magnitude scaling as 2.8Re⁰_τ⁰⁹; (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.