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(tau) 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 f(0)(y(+)) - f(1)(y(+))/U-infinity(+) + O(U-infinity(+))(-2), where U-infinity(+) = U-infinity/u(tau), y(+) = yu(tau)/v and f(0), f(1) 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 y(break)(+) = O(U-infinity(+)), 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 infinity - B-infinity/U-infinity(+) + . . . with A(infinity) and B infinity. 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-infinity(+)). 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-delta*(1/2)) with Re-delta* 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.