Outer scales and parameters of adverse-pressure-gradient turbulent boundary layers

Abstract

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.

Effect of adverse pressure gradients on turbulent wing boundary layers

“…This in principle would also contradict the results by Spalart & Watmuff (1993), although it can be argued that at this particular location the viscous units may not be adequate to scale the profile in the NACA0012 case due to the fact that, as shown in figure 2(b), here, the friction Reynolds-number curve starts to show a decreasing trend. This suggests that, under these conditions, inner scaling may be inappropriate, as also reported by Maciel et al (2018). Therefore, for the matching β-Re τ case we will use the outer scaling based on U e .…”

Self-similar compressible turbulent boundary layers with pressure gradients. Part 2. Self-similarity analysis of the outer layer

“…A comparison between the most commonly used length and velocity scales (friction velocity, pressure velocity, ZS and wake scaling) has been presented recently by Maciel et al (2018) for general, non-equilibrium PG TBLs. In agreement with other studies, by Castillo et al (2004) or Maciel et al (2006) for instance, they consistently concluded that both u ZS and δ 99 are an excellent set of scales for equilibrium and non-equilibrium incompressible TBLs.…”

“…A comparison between the most commonly used length and velocity scales (friction velocity, pressure velocity, ZS and wake scaling) has been presented recently by Maciel et al. (2018) for general, non-equilibrium PG TBLs. In agreement with other studies, by Castillo et al.…”

“…Maciel et al. 2018). In addition to the definition of self-similarity for PG TBLs itself, also the extent to which those equilibrium boundary layers represent the general case is controversially discussed.…”

“…Fundamental questions are still open, such as the 'correct' choice of the length and velocity scales that characterize the outer layer of incompressible PG TBLs, as their choice has not been conclusively clarified (see e.g. Maciel et al 2018). In addition to the definition of self-similarity for PG TBLs itself, also the extent to which those equilibrium boundary layers represent the general case is controversially discussed.…”