A symmetry approach to quantify wall turbulence: Reynolds stresses

We (Chen, Hussain & She, JFM 2018) 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. Here, by extending our previous symmetry approach, we apply random dilations on the second-order balance equations for all the Reynolds stresses (-u'v', u'u', v'v', w'w'), and obtain four-layer formulae (similar to that for l₁₂ in She, Chen & Hussain, JFM 2017) of the corresponding stress length functions l₁₁, l₂₂, l₃₃. Direct numerical simulation (DNS) data are shown to agree well with our l₁₂ and l₂₂. However, data show an invariant peak location for w'w', implying an anomalous scaling in l₃₃ in the log layer only. Furthermore, another meso-layer modification of l₁₁ yields the experimentally observed location and magnitude of the outer peak of u'u'. The resulting Reynolds stresses are all in good agreement with DNS and experimental data in the entire flow domain. Our additional results include the location of peak -u'v'_p has a scaling transition from Re^1/3_τ to Re^1/2_τ at Re_τ≈3000; the peak value w'w'⁺_p≈0.84Re^0.14_τ (1-48/Re_τ ); and an alternative derivation of the log law of Townsend, namely, u'u'⁺≈-1.25lny +1.63 and w'w'⁺≈-0.41 lny+1.00 in the bulk.