Local Structure of the Turbulence Eddies that Underlie Generalized Linear Integral Scale Growth in the Surface Layer

In a recent publication (doi:10.1017/jfm.2025.292) we generalized the linear scaling of the integral length scale that underlies the formation of the classical inertial surface layer (SL) in the shear-dominated turbulent boundary layer (TBL) to grid-turbulence wall-bounded turbulent flows with zero mean shear, using particle image velocimetry data from two wind tunnel facilities. Detailed quantifications lead to the conclusion that linear scaling of the integral length scale with distance from the surface is a generalizable consequence of the blockage of vertical velocity fluctuations at the impermeable surface. In the current study we use the same datasets of instantaneous velocity fields (snapshots) on transverse and horizontal planes to analyze the space-time local structure of the turbulence motions that underlie the ensemble-averaged integral length scale of vertical velocity fluctuations, and its linear growth in the SL. The analytical framework centers on representing the integral length scale as a two-dimensional integral over the weighted joint PDF of normalized vertical fluctuating velocity, w', and its integral in the streamwise direction, I'. With this representation we determine that the range of instantaneous fluctuations in w' and I' underlying the integral length scale are entirely confined to localized “optimal” ellipses in quadrants 1 and 3 of joint PDF space, which implies that negative localized contributions in the other quadrants cancel with those outside the optimal ellipses in the ensemble average. Confining to the fluctuations within the optimized joint PDF ellipses we quantify “Local Integral Scale Regions” (LIRs) in the instantaneous snapshots that define the regions of vertical velocity fluctuation in instantaneous snapshots that contribute to the integral length scale, an ensemble average. The structure of the LIRs within the surface layer regions are visually and quantitatively analyzed. The contributions from the irregular and highly-variable LIRs contrast with the simplified 3D representations underlying “attached eddy” model representations of the TBL.