Law of the wall for small-scale streamwise turbulence intensity in high-Reynolds-number turbulent boundary layers

Following the dimensional analysis approach carried out in previous studies, it is hypothesized that the small-scale fluctuations should only depend on the inner scales, analogous to the Prandtl's law of the wall for the mean flow. This allows us to examine the high-frequency regime of the streamwise energy spectra where a “law of the wall” in spectra would hold. Observations in high-Reynolds-number turbulent boundary layer data indicate that a conservative estimate for the start of this law of the wall is $f^{+}=0.005$ (which corresponds to 200 viscous time units) across a range of wall-normal positions and Reynolds numbers. This is sufficient to capture the energetic viscous-scaled motions such as the near-wall streaks, which have a timescale of approximately 100 viscous units. This spectral collapse is consistent with the observations in internal flows and external flows in other studies. Furthermore, the spectral collapse leads to a universal scaling (based on skin-friction velocity and kinematic viscosity) for the small-scale streamwise turbulence variance (consistent with the hypothesis) across the entire boundary layer. A logarithmic variation of this small-scale variance is observed farther away from the wall.

Figure 1

[(a)–(f)] Inner-normalized energy spectra for four different Reynolds numbers at six different values of $y^{+}$. The figures show the value of $y^{+}$ and the variance in that selected wall-normal location across the Reynolds numbers. Lines are colored from lightest to darkest in order of increasing ${\text{Re}}_{\tau }$ (= 2800, 3900, 7300, and 14150). The solid black line shows the $f^{+}=0.005$, which is 200 inner time units, while the dashed black line shows $f^{+}=0.001$, which is 1000 inner time units.

Figure 2

(Left) Inner normalized large-scale (open squares) and small-scale (filled circles) variance of streamwise velocity for four different Reynolds numbers. The separation between large and small scale is based on a Fourier filter at $f^{+}=0.005$. The dashed black line shows the log decay of the small-scale variance beyond $y^{+}>100$. (Right) Compensated form of small-scale variance where the fitted log trend is subtracted from the data (symbols) and the log region in the mean flow (solid lines). The dashed lines show the extent of log region ($3\sqrt{{\text{Re}}_{\tau }}\le y^{+}\le 0.15{\text{Re}}_{\tau }$). Increasing darkness in all symbols and lines represents increasing Reynolds number.

Figure 3

Inner-normalized premultiplied spectra at $y^{+}\approx 15$ for different levels of free-stream turbulence (FST). Lines are colored from lightest to darkest in order of increasing turbulence intensity (the range is from 7% to 13%). The Taylor microscale of FST ranges from 300 to 760. The abscissa shows inner-normalized frequency $f^{+}$. Further details on the different turbulence intensities, integral length scales, and boundary layer characteristics can be found in Ref. [21].