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A clear and consistent framework for the analysis of the outer region of adverse-pressure-gradient turbulent boundary layers is established in this paper based on basic principles and theory, and the help of six adverse-pressure-gradient turbulent boundary layer databases and a zero-pressure-gradient one. Outer velocity and length scales for the mean velocity defect and the Reynolds stresses are discussed first. The conditions of validity of four velocity scales are determined in terms of the shape factor, since one scale is restricted to small velocity-defect boundary layers (the friction velocity $u_{\unicode[STIX]{x1D70F}}$), one to large-defect ones (the pressure-gradient velocity $U_{po}$), while the two others are proper scales for all velocity-defect conditions (the Zagarola–Smits velocity $U_{zs}$ and the mixing-layer-type velocity $U_{m}$). The turbulent boundary layer equations are then used to bring out, in a consistent manner and without assuming any self-similar behaviour, a set of non-dimensional parameters characterizing the outer region of turbulent boundary layers with arbitrary pressure gradients. In terms of a generic outer length scale $L_{o}$ and velocity scale $U_{o}$, these non-dimensional parameters are the pressure-gradient parameter $\unicode[STIX]{x1D6FD}_{o}=L_{o}/(\unicode[STIX]{x1D70C}U_{o}^{2})\,\text{d}p_{e}/\text{d}x$, the Reynolds number $Re_{o}=U_{o}L_{o}/\unicode[STIX]{x1D708}(U_{o}/U_{e})$ and the inertial parameter $\unicode[STIX]{x1D6FC}_{o}=U_{e}V_{e}/U_{o}^{2}$, where $U_{e}$ and $V_{e}$ are respectively the streamwise and wall-normal components of mean velocity at the boundary layer edge. These parameters have a clear physical meaning: they are ratios of the order of magnitude of forces, with the Reynolds shear stress gradient (apparent turbulent force) as the reference force – inertial to apparent turbulent forces for $\unicode[STIX]{x1D6FC}_{o}$, pressure to apparent turbulent forces for $\unicode[STIX]{x1D6FD}_{o}$ and apparent turbulent to viscous forces for $Re_{o}$. We discuss at length their significance and determine under what conditions they retain their meaning depending on the outer velocity scale that is considered. The seven boundary layer databases are analysed and compared using the established framework. An astonishing and impressive result is obtained: by choosing $U_{o}=U_{zs}$, the streamwise evolution of the three ratios of forces in the outer region can be accurately followed with $\unicode[STIX]{x1D6FD}_{zs}$, $\unicode[STIX]{x1D6FC}_{zs}$ and $Re_{zs}$ in all these flows – not just the order of magnitude of these ratios. This cannot be achieved with $u_{\unicode[STIX]{x1D70F}}$ and $U_{po}$, and only imperfectly with $U_{m}$. Consequently, $\unicode[STIX]{x1D6FD}_{zs}$, $\unicode[STIX]{x1D6FC}_{zs}$ and $Re_{zs}$ can be used to follow – in a global sense – the streamwise evolution of the streamwise mean momentum balance in the outer region.

The characteristics of the coherent structures in a strongly decelerated large-velocity-defect boundary layer are analysed by direct numerical simulation.The simulated boundary layer starts as a zero-pressure-gradient boundary layer, decelerates under a strong adverse pressure gradient, and separates near the end of the domain, in the form of a very thin separation bubble. The Reynolds number at separation is Re θ = 3912 and the shape factor H = 3.43. The three-dimensional spatial correlations of (u, u) and (u, v) are investigated and compared to those of a zero-pressure-gradient boundary layer and another strongly decelerated boundary layer. These velocity pairs lose coherence in the streamwise and spanwise directions as the velocity defect increases. In the outer region, the shape of the correlations suggest that large-scale u structures are less streamwise elongated and more inclined with respect to the wall in large-defect boundary layers. The threedimensional properties of sweeps and ejections are characterized for the first time in both the zero-pressure-gradient and adverse-pressure-gradient boundary layers, following the method of Lozano-Durán et al. (J. Fluid Mech., vol. 694, 2012). Although longer sweeps and ejections are found in the zero-pressure-gradient boundary layer, with ejections reaching streamwise lengths of 5 boundary layer thicknesses, the sweeps and ejections tend to be bigger in the adverse-pressure-gradient boundary layer. Moreover, small near-wall sweeps and ejections are much less numerous in the large-defect boundary layer. Large sweeps and ejections that reach the wall region (wall-attached) are also less numerous, less streamwise elongated and they occupy less space than in the zero-pressure-gradient boundary layer.

Production and inter-component redistribution of turbulence in adverse pressure gradient (APG) turbulent boundary layers (TBLs) with small and large velocity defects are investigated, along with the structures playing a role in these energy transfer mechanisms. We examine the wall-normal and spectral distributions of energy, production and pressure-strain in APG TBLs, and compare these distributions with those in canonical flows. It is found that the spectral distributions of production and pressure-strain are not affected profoundly by an increase of the velocity defect, although the energy spectra change drastically in the inner layer of the large-defect APG TBL. In the latter, the signature of the inner-layer streaks is absent from the energy spectra. In the outer layer, energetic, production and pressure-strain structures appear to change from wall-attached to wall-detached structures with increasing velocity defect. Despite this, the two-dimensional spectral distributions have similar shapes and wavelength aspect ratios of the peaks in all these flows. Therefore, the conclusion is that the mechanisms responsible for turbulence production and inter-component energy transfer may remain the same within each layer in all these flows. It is the intensity of these mechanisms within one layer that changes with velocity defect, because of the local mean shear variation.