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Publication# Revisiting the quest for a universal log-law and the role of pressure gradient in "canonical" wall-bounded turbulent flows

Abstract

The trinity of so-called "canonical" wall-bounded turbulent flows, comprising the zero pressure gradient turbulent boundary layer, abbreviated ZPG TBL, turbulent pipe flow, and channel/duct flows has continued to receive intense attention as new and more reliable experimental data have become available. Nevertheless, the debate on whether the logarithmic part of the mean velocity profile, in particular the Karman constant kappa, is identical for these three canonical flows or flow-dependent is still ongoing. In this paper, the asymptotic matching requirement of equal. in the logarithmic overlap layer, which links the inner and outer flow regions, and in the expression for the centerline/free-stream velocity is reiterated and shown to preclude a universal logarithmic overlap layer in the three canonical flows. However, the majority of pipe and channel flowstudies at friction Reynolds numbers Re-tau below approximate to 10(4) extract from near-wall profiles the same kappa of 0.38-0.39 as in the ZPG TBL. This apparent contradiction is resolved by a careful reanalysis of high-quality mean velocity profiles in the Princeton "Superpipe" and other pipes, channels, and ducts, which shows that the mean velocity in a near-wall region extending to around 700 "+" units in channels and ducts and 500 "+" units in pipes is the same as in the ZPG TBL. In other words, all the "canonical" flow profiles contain the lower end of the ZPG TBL log-region, which starts at a wall distance of 150-200 "+" units with a universal kappa of kappa(ZPG) approximate to 0.384. This interior log-region is followed by a second logarithmic region with a flow specific. > kappa(ZPG), which increases monotonically with pressure gradient. This second, exterior log-layer is the actual overlap layer matching up to the outer expansion, which implies equality of the exterior. and kappa(CL) obtained from the evolution of the respective centerline velocity with Reynolds number. The location of the switch-over point implies furthermore that this second log-layer only becomes clearly identifiable, i.e., separated from the wake region, for Re-tau well beyond 10(4) (see Fig. 1). This explains the discrepancies between the Karman constants of 0.38-0.39, extracted from near-wall pipe profiles below Re-tau approximate to 10(4) and the kappa's obtained from the evolution of the centerline velocity with Reynolds number. The same analysis is successfully applied to velocity profiles in channels and ducts even though experiments and numerical simulations have not yet reached Reynolds numbers where the different layers have even started to clearly separate.

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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.

The paper presents an in depth assessment of different similarity laws for the mean velocity profile in zero pressure gradient (ZPG) turbulent boundary layers (TBL's) in comparison with mostly experimental and few computational data. The emphasis is on the descriptions which are complete in the sense that a full representation of the mean velocity profile, its streamwise evolution and all integral parameters, including the friction factor and the shape factor, are provided as a function of Reynolds number. The first such complete description is the classical two-layer theory with its characteristic logarithmic mean velocity profile in the region where the two layers overlap, henceforth referred to as the "log law." The main alternative scalings which have been proposed over the last decade have led to power law descriptions of the turbulent mean velocity profile. Since the different descriptions were calibrated with different data sets, the controversy over the relative merits of the different approaches has lingered on. The purpose of the present paper is to measure the principal competing theories against the same vast data set of more than 300 mean velocity profiles from more than twenty different sources. The results confirm the conclusions of numerous authors that the log law provides a fully self-consistent and accurate description of all the mean quantities and demonstrates conclusively that the same cannot be achieved by the competing power law theories. Along the way, it is also argued that the traditional description of the outer velocity profile in terms of a wall-normal coordinate normalized to unity at a hypothetical boundary layer "edge" delta and a "wake parameter" Pi is not robust with respect to the fit of the outer velicity profile and should therefore not be used in theoretical arguments. (C) 2008 American Institute of Physics.

2008Charles Vivant Ignacio Meneveau, Marc Parlange

A scale-dependent dynamic subgrid model based on Lagrangian time averaging is proposed and tested in large eddy simulations sLESd of high-Reynolds number boundary layer flows over homogeneous and heterogeneous rough surfaces. The model is based on the Lagrangian dynamic Smagorinsky model in which required averages are accumulated in time, following fluid trajectories of the resolved velocity field. The model allows for scale dependence of the coefficient by including a second test-filtering operation to determine how the coefficient changes as a function of scale. The model also uses the empirical observation that when scale dependence occurs ssuch as when the filter scale approaches the limits of the inertial ranged, the classic dynamic model yields the coefficient value appropriate for the test-filter scale. Validation tests in LES of high Reynolds number, rough wall, boundary layer flow are performed at various resolutions. Results are compared with other eddy-viscosity subgrid-scale models. Unlike the Smagorinsky–Lilly model with wall-damping swhich is overdissipatived or the scale-invariant dynamic model swhich is underdissipatived, the scale-dependent Lagrangian dynamic model is shown to have good dissipation characteristics. The model is also tested against detailed atmospheric boundary layer data that include measurements of the response of the flow to abrupt transitions in wall roughness. For such flows over variable surfaces, the plane-averaged version of the dynamic model is not appropriate and the Lagrangian averaging is desirable. The simulated wall stress overshoot and relaxation after a jump in surface roughness and the velocity profiles at several downstream distances from the jump are compared to the experimental data. Results show that the dynamic Smagorinsky coefficient close to the wall is very sensitive to the underlying local surface roughness, thus justifying the use of the Lagrangian formulation. In addition, the Lagrangian formulation reproduces experimental data more accurately than the planar-averaged formulation in simulations over heterogeneous rough walls.

2005