Large-Reynolds-number asymptotics of the streamwise normal stress in zero-pressure-gradient turbulent boundary layers

A more poetic long title could be ‘A voyage from the shifting grounds of existing data on zero-pressure-gradient (abbreviated ZPG) turbulent boundary layers (abbreviated TBLs) to infinite Reynolds number’. Aided by the requirement of consistency with the Reynolds-averaged momentum equation, the ‘shifting grounds’ are sufficiently consolidated to allow some firm conclusions on the asymptotic expansion of the streamwise normal stress uu ⁺ , where the ⁺ indicates normalization with the friction velocity u_τ squared. A detailed analysis of direct numerical simulation data very close to the wall reveals that its inner near-wall asymptotic expansion must be of the form f₀(y⁺)-f₁(y⁺)/U∞⁺+O(U∞⁺)⁻² , where U∞⁺=U∞/u_τ , y⁺=yu_τ/ν and f₀ , f₁ are O(1) functions fitted to data in this paper. This means, in particular, that the inner peak of uu ⁺ does not increase indefinitely as the logarithm of the Reynolds number but reaches a finite limit. The outer expansion of uu ⁺ , on the other hand, is constructed by fitting a large number of data from various sources. This exercise, aided by estimates of turbulence production and dissipation, reveals that the overlap region between inner and outer expansions of uu ⁺ is its plateau or second maximum, extending to ybreak⁺=O(U∞⁺) , where the outer logarithmic decrease towards the boundary layer edge starts. The common part of the two expansions of uu ⁺ , i.e. the height of the plateau or second maximum, is of the form \,A∞-B∞/U∞⁺+⋯ \, with A∞ and B∞ constant. As a consequence, the logarithmic slope of the outer uu ⁺ cannot be independent of the Reynolds number as suggested by ‘attached eddy’ models but must slowly decrease as (1/U∞⁺) . A speculative explanation is proposed for the puzzling finding that the overlap region of uu ⁺ is centred near the lower edge of the mean velocity overlap, itself centred at y⁺=O(Re_δ1/2) with Re_δ the Reynolds number based on free stream velocity and displacement thickness. Finally, similarities and differences between uu ⁺ in ZPG TBLs and in pipe flow are briefly discussed.