Inverse Interscale Transport of the Reynolds Shear Stress in Plane Couette Turbulence

Interscale interaction between small-scale structures near the wall and large-scale structures away from the wall plays an increasingly important role with increasing Reynolds number in wall-bounded turbulence. While the top-down influence from the large- to small-scale structures is well known, it has been unclear whether the small scales near the wall also affect the large scales away from the wall. In this Letter we show that the small-scale near-wall structures indeed play a role to maintain the large-scale structures away from the wall, by showing that the Reynolds shear stress is transferred from small to large scales throughout the channel. This is in contrast to the turbulent kinetic energy transport which is from large to small scales. Such an “inverse” interscale transport of the Reynolds shear stress eventually supports the turbulent energy production at large scales.

Figure 1

Space-wavelength ($y$- or $\eta \text{−}{\lambda }_z$) diagrams of interscale transport of (a)–(c) the turbulent kinetic energy ${\mathrm{Tr}}_{k_t}$ and (d)–(f) the Reynolds shear stress ${\mathrm{Tr}}_{-uv}$ scaled by $u_{\tau }^3/h$ at (a), (d) $\mathrm{Re}=500$, (b), (e) 1000, and (c), (f) 2000. The black dashed line in each panel indicates the channel center $y/h=0$.

Figure 2

Profiles at the channel center $y/h=0$ of the premultiplied (a1) spectra of the turbulent kinetic energy $k_z{E}_{k_t}$, (b1) cospectra of the Reynolds shear stress $k_z{E}_{-uv}$, and (a2), (b2) their scale-by-scale production and interscale transport for four different Reynolds numbers. The values are scaled by $u_{\tau }^2$ in (a1) and (b1) while scaled by $u_{\tau }^3/h$ in (a2) and (b2). The colors represent different Reynolds number cases: blue, $\mathrm{Re}=500$; red, $\mathrm{Re}=1000$; yellow, $\mathrm{Re}=1500$; purple, $\mathrm{Re}=2000$; and the dashed and solid lines in (a2) and (b2) present the premultiplied production and interscale transport, respectively. The black dash-dotted line in (b2) represents $k_z(p{r}_{-uv}+t{r}_{-uv}+d_{-uv}^t)$ for $\mathrm{Re}=2000$.

Figure 3

Space-wavelength ($y$- or $\eta -{\lambda }_z$) diagrams of the premultiplied (a) Reynolds shear stress cospectra $k_z{E}_{-uv}$ and its scale-by-scale (b) production $k_{z}p{r}_{-uv}$, (c) interscale transport $k_{z}t{r}_{-uv}$, and (d) turbulent spatial transport $k_z{d}_{-uv}^t$ at $\mathrm{Re}=2000$. The values are scaled by $u_{\tau }^2$ for $k_z{E}_{-uv}$ and by $u_{\tau }^3/h$ for the other quantities, and the black dashed line in each panel indicates the channel center $y/h=0$.