Multiscale Geometry and Scaling of the Turbulent-Nonturbulent Interface in High Reynolds Number Boundary Layers

The scaling and surface area properties of the wrinkled surface separating turbulent from nonturbulent regions in open shear flows are important to our understanding of entrainment mechanisms at the boundaries of turbulent flows. Particle image velocimetry data from high Reynolds number turbulent boundary layers covering three decades in scale are used to resolve the turbulent-nonturbulent interface experimentally and, for the first time, determine unambiguously whether such surfaces exhibit fractal scaling. Box counting of the interface intersection with the measurement plane exhibits power-law scaling, with an exponent between $-1.3$ and $-1.4$. A complementary analysis based on spatial filtering of the velocity fields also shows power-law behavior of the coarse-grained interface length as a function of filter width, with an exponent between $-0.3$ and $-0.4$. These results establish that the interface is fractal-like with a multiscale geometry and fractal dimension of $D_f\approx 2.3–2.4$.

Figure 1

Experimental setup for the planar PIV measurements in the High Reynolds Number Boundary Layer Wind Tunnel at the University of Melbourne, and table of relevant flow properties, where $\eta$ and ${\lambda }_T$ are the Kolmogorov and Taylor microscales calculated near the interface and ${\tau }_w$ is the wall stress [26].

Figure 2

A snapshot of the velocity field relative to the free stream at ${\mathrm{Re}}_{\tau }=14500$ from planar PIV. Every 10th vector is plotted in each direction. Color contours represent kinetic energy $K=(1/2)(u^2+v^2)$, where velocities are in a frame moving with the free stream. The solid black line indicates the location of the TNTI computed as an isokinetic energy surface using a threshold of $K={10}^{-3}\mathbf{(}(1/2)U_{\infty }^2\mathbf{)}$. The inset shows a magnified view of the TNTI, showing vectors at the original resolution, and from $1.34<x/\delta <1.39$ and $0.69<y/\delta <0.74$. A velocity magnitude of $8\mathrm{m}/\mathrm{s}$ corresponds to a length of $0.1\delta$ in the main vector map, and $0.03\delta$ in the inset.

Figure 3

Plot of the number of boxes $N(r)$ needed to cover the TNTI at a kinetic energy threshold of 0.1%. Squares: ${\mathrm{Re}}_{\tau }=7900$ and circles: ${\mathrm{Re}}_{\tau }=14500$. For comparison, dashed lines show slopes of $-1$ and $-2$. Vertical lines show the limits used for the fit for each set of data. The inset shows the local slopes; the solid and dashed lines ($c_0=0.05\%$ and 0.2%), and symbols (0.1%) which correspond to the TNTI detected using different thresholds, where, $K=c_0\mathbf{(}(1/2)U_{\infty }^2\mathbf{)}$.

Figure 4

TNTI determined from the filtered velocity fields at a kinetic energy threshold of 0.1%, with filter scales (a) $\Delta =0.02\delta$, (b) $\Delta =0.08\delta$, and (c) $\Delta =0.18\delta$. Color contours represent the filtered kinetic energy, $K^{\Delta }=(1/2)({\tilde{u}}^2+{\tilde{v}}^2)$ in logarithmic units, in a frame moving with the free stream.

Figure 5

Average measured interface length as function of filter scale. (a) Normalized with viscous units for data at ${\mathrm{Re}}_{\tau }\approx 7900$ (open squares) and ${\mathrm{Re}}_{\tau }\approx 14500$ (open circles). At ${\mathrm{Re}}_{\tau }\approx 14500$, three thresholds are considered, namely 0.05%, 0.1%, and 0.2% of $(1/2)U_{\infty }^2$ and indicated by right triangles, open circles, and left triangles, respectively. (b) Average measured interface length normalized by the length of the PIV image ($L_x$) and $\delta$, for a 0.1% kinetic energy threshold. Note that $L_x$ decreases with increasing $\Delta$ as a larger region on the edges has to be excluded. Closed squares and closed circles: local slope computed using central differences. The slope uncertainty of $\pm 0.02$ in Fig. 5b is determined by the maximum deviation of the slope in the range $6\times {10}^{-3}<\Delta /\delta <3\times {10}^{-2}$ for both data sets.