Defining the mean turbulent boundary layer thickness based on streamwise velocity skewness

TL;DR

The paper defines a threshold-free turbulent boundary layer thickness, $\delta_S$, as the wall-normal location where the outer-region skewness of streamwise velocity fluctuations satisfies $\overline{\overline{u^{3}}}(z=\delta_S)=0$, tying the thickness to TNTI physics. It validates $\delta_S$ across large-scale experiments, published datasets, and attached-eddy modelling, using two estimation methods: linear interpolation and a Fourier representation of the outer skewness profile. Results show $\delta_S$ closely aligns with traditional thickness measures in canonical cases (e.g., $\delta_S \approx (1.2-1.3)\delta_{99}$ for ZPG and $\delta_S \approx \delta_{TNTI}$ within a few percent), while remaining robust in non-canonical flows, including APG and rough-wall TBLs. The approach yields a physically meaningful, threshold-free edge indicator that can be retroactively applied to existing single-point data and is practical for calculating integral thicknesses, albeit with caveats for very high freestream turbulence or limited data resolution.

Abstract

A new statistical definition for the mean turbulent boundary layer thickness is introduced, based on identification of the point where the streamwise velocity skewness changes sign, from negative to positive, in the outermost region of the boundary layer. Importantly, this definition is independent of arbitrary thresholds, and broadly applicable, including to past single-point measurements. Further, this definition is motivated by the phenomenology of streamwise velocity fluctuations near the turbulent/non-turbulent interface, whose local characteristics are shown to be universal for turbulent boundary layers under low freestream turbulence conditions (i.e., with or without pressure gradients, surface roughness, etc.) through large-scale experiments, simulations and coherent structure-based modelling. The new approach yields a turbulent boundary layer thickness that is consistent with previous definitions, such as those based on Reynolds shear stress or `composite' mean velocity profiles, and which can be used practically e.g., to calculate integral thicknesses. Two methods are proposed for estimating the turbulent boundary layer thickness using this definition: one based on simple linear interpolation and the other on fitting a generalised Fourier model to the outer skewness profile. The robustness and limitations of these methods are demonstrated through analysis of several published experimental and numerical datasets, which cover a range of canonical and non-canonical turbulent boundary layers. These datasets also vary in key characteristics such as wall-normal resolution and measurement noise, particularly in the critical turbulent/non-turbulent interface region.

Figures (11)

• Figure 1: (a) Schematic of modified Melbourne large wind tunnel facility, adapted from Deshpande_APG_2023. (b) Schematic of PIV setup adapted from Marusic_APGTNTI_2024. Snapshots of instantaneous streamwise velocity for (c) ZPG and (d) APG cases across the full TBL, made possible by stitching individual flow fields from the four PIV cameras (C1-C4).
• Figure 2: (a) Profile of streamwise velocity skewness in the outer region of a TBL. (b,c) Instantaneous LKE interface (black lines) imposed on the instantaneous streamwise velocity field (colours) with instantaneous fluctuation vectors (arrows) overlaid. (d,e) Probability distribution of the interface location in the outer region of a TBL with contours of zero skewness overlaid. Data are from the MELB1 (a,b,d) ZPG and (c,e) APG cases. (f) Schematic of instantaneous flow phenomenology associated with the characteristic wall-normal variation of streamwise velocity as shown in (a,b,c).
• Figure 3: (a) Skewness profiles and (c) a snapshot of the instantaneous velocity field from Deshpande_AEM_2021. (b) Filtered skewness profiles from lozier2024. (d) Filtered PIV snapshot of the instantaneous velocity field from figure \ref{['fig2']}(b).
• Figure 4: Comparison of experimental and numerical ZPG TBL statistics with varying wall-normal resolutions. (a) Diagnostic style plot used to find $\delta_{D}$ following the methodology of Vinuesa_LengthScale_2016 (analogous to $\delta_{99}$). (b) Relationship between turbulence intensity and skewness of streamwise velocity fluctuations in the outer region of the TBL. Wall-normal profiles of skewness normalised by $\delta_{S}$ in (c) logarithmic scaling and (d) linear scaling. The magenta curve represents a generalised form of the normalised skewness profile \ref{['eq:FM']} fit to DNS data from Sillero_ZPGDNS_2013.
• Figure 5: Two-dimensional fields of relevant statistics from PIV measurements of (top) ZPG and (bottom) APG TBLs with solid contours of the TBL thickness overlaid based on (a) $\delta_{99}$, $\Delta_{1.25}$, (b) $\delta_{uw}$, (c) $\delta_{\mathrm{TNTI}}$, and (d) $\delta_{S}$ definitions. The black dotted lines overlaid in (c) represent p.d.f.s of the TNTI height \ref{['eq:pdf']}.