Energy contributions by inner and outer motions in turbulent channel flows

The wall-bounded turbulent flows are populated by coherent motions with different scales. The inner motions (near-wall cycle) have been well recognized in wall turbulence. Since the last decade, it has also been reported that a distinct outer peak will emerge in the premultiplied spectra of streamwise velocity fluctuations in canonical-wall-flow configurations at high Reynolds numbers. It is an important energetic signature of the very-large-scale motions or the outer motions herein. However, at low to moderate Reynolds numbers, both the wall-normal locations and length scales of the inner and outer motions are very close to each other, imposing significant difficulties to separate them. Even more, we still do not know the critical Reynolds number for the emergence of the energetic outer motions. In the present study, we propose an assumption that the inner motions that are free of outer motions and transitional effects should be well scaled by wall units, for which we name “the turbulence with only inner motions” (TIM). By carrying out direct numerical simulation (DNS) at low Reynolds numbers, the critical Reynolds number is determined to be ${\text{Re}}_{\tau }\approx 110$, where ${\text{Re}}_{\tau }$ is the friction Reynolds number. Empirical fitting formulas for the broadband streamwise turbulence intensity and premultiplied spectra of TIM are proposed based on the DNS data. Furthermore, with publicly available DNS database of Reynolds number up to ${\text{Re}}_{\tau }=5200$, the turbulence energy contributions by the outer motions are studied by subtracting that of TIM from the total. The result implies possibly distinct modulation effects on motions with different scales.

Figure 1

Comparison of (a) the mean streamwise velocity profile $U^{+}\left(y^{+}\right)$ and (b) the velocity fluctuations $(u_{\text{rms}}^{+},v_{\text{rms}}^{+},u_{\text{rms}}^{+})$ and tangential Reynolds stress $\langle u^{+}{v}^{+}\rangle$ between the present DNS and [62] (indicated by LM15) at ${\text{Re}}_{\tau }=180$.

Figure 2

Comparisons of the streamwise and spanwise premultiplied spectra of streamwise velocity fluctuation ($k_x{E}_{uu}/u_{\tau }^2$ and $k_z{E}_{uu}/u_{\tau }^2$) between the present DNS and [62] at ${\text{Re}}_{\tau }=180$. The levels of contour lines are 0.2, 0.4, 0.6, and 0.8 times the maximum premultiplied spectral densities.

Figure 3

Contours of instantaneous streamwise velocity fluctuations $u^{+}$ in the $x-z$ plane at $y^{+}\approx 15$ for (a) ${\text{Re}}_{\tau }=180$, (b) ${\text{Re}}_{\tau }=150$, (c) ${\text{Re}}_{\tau }=130$, (d) ${\text{Re}}_{\tau }=110,$ and (e) ${\text{Re}}_{\tau }=90$.

Figure 4

Contours of instantaneous streamwise velocity fluctuations $u^{+}$ in the $x-z$ plane at $y/\delta \approx 0.5$ for (a) ${\text{Re}}_{\tau }=180$, (b) ${\text{Re}}_{\tau }=150$, (c) ${\text{Re}}_{\tau }=130$, (d) ${\text{Re}}_{\tau }=110,$ and (e) ${\text{Re}}_{\tau }=90$.

Figure 5

Comparison of (a) the mean streamwise velocity profile $U^{+}\left(y^{+}\right)$ and (b) the streamwise velocity fluctuations ${\langle u^2\rangle }^{+}$ at different Reynolds numbers.

Figure 6

Comparisons of streamwise premuliplied energy spectra of streamwise velocity fluctuation $k_x{E}_{uu}/u_{\tau }^2$ at low Reynolds numbers. The contour levels (lines) are from 0.6 to 1.8 with a constant increment of 0.4.

Figure 7

Comparisons of spanwise premuliplied energy spectra of streamwise velocity fluctuation $k_z{E}_{uu}/u_{\tau }^2$ at low Reynolds numbers. The contour levels (lines) are 1.0, 1.8, and 3.6, respectively.

Figure 8

Comparison of the empirical fittings of Eq. (3) for the streamwise turbulence intensities.

Figure 9

(a) Variations of the fitting coefficients ${\alpha }_1\sim {\alpha }_3$ in Eqs. (5) and (6) with $y^{+}$ and (b) comparison of the streamwise premultiplied spectra $k_x{E}_{uu}/u_{\tau }^2$ between DNS at ${\text{Re}}_{\tau }=110$ and the fitting formula Eq. (5).

Figure 10

(a) Variations of the fitting coefficients ${\beta }_1\sim {\beta }_3$ in Eqs. (7) and (8) with $y^{+}$ and (b) comparison of the spanwise premultiplied spectra $k_z{E}_{uu}/u_{\tau }^2$ between DNS at ${\text{Re}}_{\tau }=110$ and the fitting formula Eq. (6).

Figure 11

The contributions of streamwise turbulence intensities by outer motions.

Figure 12

Streamwise premultiplied spectra of outer motions by subtracting the contributions by TIM: (a) ${\text{Re}}_{\tau }=550$, (b) ${\text{Re}}_{\tau }=1000$, (c) ${\text{Re}}_{\tau }=2000,$ and (d) ${\text{Re}}_{\tau }=5200$.

Figure 13

Spanwise premultiplied energy spectra of outer motions by subtracting the contributions by TIM: (a) ${\text{Re}}_{\tau }=550$, (b) ${\text{Re}}_{\tau }=1000$, (c) ${\text{Re}}_{\tau }=2000,$ and (d) ${\text{Re}}_{\tau }=5200$.