Reynolds number scaling of pocket events in the viscous sublayer

Recent findings [X. Wu et al., Proc. Natl. Acad. Sci. USA 114, E5292 (2017)] reinforce earlier assertions [e.g., R. Falco, Philos. Trans. R. Soc. London A 336, 103 (1991)] that the sublayer pocket motions play a distinctly important role in near-wall dynamics. In the present study, smoke visualization and axial velocity measurements are combined in order to establish the scaling behavior of pocket events in the viscous sublayer of the turbulent boundary layer. In doing so, an identical analysis methodology is employed over an extensive range of friction Reynolds numbers $388\le {\delta }^{+}\le 2.2\times {10}^5$. Both the pocket width $W$ and time interval between pocket events $T$ increase logarithmically with Reynolds number when normalized by viscous units. Normalization of $W$ and $T$ by the Taylor microscales evaluated at a wall-normal location of about 100 viscous units, however, appears to successfully remove this Reynolds-number dependence. The present results are discussed in the context of motion formation owing to the three dimensionalization of the near-wall vorticity field and, concomitantly, the recurring perturbation of the viscous sublayer.

Figure 1

Idealization of streaks and pocket structures as made visible by smoke visualization of the viscous sublayer: (a) side view, streamwise wall-normal plane, (b) plan view, streamwise-spanwise plane. Note that pockets are quasisymmetrical regions devoid of the tracer, often (but not always) occurring between two streaks. (Adapted from Ref. [7].)

Figure 2

Sample snapshots of sublayer smoke visualization with several pocket events identified: wind tunnel (left) and atmospheric surface layer (right). The corresponding physical scale is indicated in each case. Flow is from top to bottom.

Figure 3

Inner normalized profiles of the Taylor microscales: Taylor length scale (left) and Taylor time scale (right). Closed symbols $\blacksquare , •$, and $▴$ represent the field data from Metzger [40] acquired over three different years; gray open symbols are laboratory data from Vincenti et al. [41] and black open symbols are laboratory data from Klewicki [42].

Figure 4

Reynolds-number dependence of the inner normalized pocket width $W^{+}$ and time interval between pockets $T^{+}$ from the present work, $\circ$; Ref. [7], $\diamond$; and Ref. [15], $\square$, with logarithmic curve fits.

Figure 5

Scaling the pocket width (left) and time interval between pockets (right). Circles and triangles indicate laboratory data and squares indicate field data. The dashed lines represent power-law curve fits.

Figure 6

Slope of the curve fits represented by (7) and (8) as a function of $y^{+}$ for $W/\lambda$ (left) and $T/{\lambda }_t$ (right), respectively. The vertical dashed lines delineate $y^{+}=100$, the wall-normal location used in the Taylor normalization of Fig. 5.

Figure 7

Inner normalized Kolmogorov length versus inner normalized distance from the wall. Markers are the same as in Fig. 3, except all of the atmospheric data at ${\delta }^{+}=2.2\times {10}^5$ are plotted as black circles here. The dashed line represents ${\eta }^{+}={\left(\kappa y^{+}\right)}^{1/4}$.

Figure 8

Schematic of an eddy that initiates from the inner layer (at $y^{+}\approx 100$) and propagates toward the wall with a velocity ${\vec{V}}_p^{+}$. For a given characteristic size ${\ell }^{+}$ of the eddy, the footprint at the wall (in the form of a pocket event) is expected to extend a length $L^{+}$ due to the acute angle of propagation $\theta$.