Upscale energy transfer and flow topology in free-surface turbulence

Free-surface turbulence, albeit constrained onto a two-dimensional space, exhibits features that barely resemble predictions of simplified two-dimensional modeling. We demonstrate that, in a three-dimensional open channel flow, surface turbulence is characterized by upscale energy transfer, which controls the long-term evolution of the larger scales. We are able to associate downscale and upscale energy transfer at the surface with the two-dimensional divergence of velocity. We finally demonstrate that surface compressibility confirms the strongly three-dimensional nature of surface turbulence.

Figure 1

Time- and surface-averaged energy flux ${\Pi }^{\left(\Delta \right)}$ as a function of the Gaussian filter size $\left(\Delta \right)$ at the channel surface $\left(z_0=0\right)$ for ${\text{Re}}_{\tau }^L$ (solid line) and ${\text{Re}}_{\tau }^H$ (dashed line). The contribution of the positive energy flux (downscale energy transfer) and of the negative energy flux (upscale energy transfer) are also shown in the inset.

Figure 2

Contour maps of the energy flux ${\Pi }^{\left(\Delta \right)}$ (a) and of the two-dimensional surface divergence ${\nabla }_{2D}$ (b) computed at the free surface for ${\text{Re}}_{\tau }=171$.

Figure 3

Contour maps of the energy flux ${\Pi }^{\left(\Delta \right)}$ (a) and of the two-dimensional surface divergence ${\nabla }_{2D}$ (b) computed at the free surface for ${\text{Re}}_{\tau }=509$.

Figure 4

Correlation coefficient $\langle {\nabla }_{2D}^{+}{\Pi }_{+}^{\Delta }\rangle$ (solid line) and $\langle {\nabla }_{2D}^{-}{\Pi }_{-}^{\Delta }\rangle$ (dashed line) between the positive (negative) energy flux ${\Pi }_{+}^{\Delta } \left({\Pi }_{-}^{\Delta }\right)$ and the positive (negative) two-dimensional surface divergence ${\nabla }_{2D}^{+} \left({\nabla }_{2D}^{-}\right)$ computed along the streamwise direction $x$ and averaged in time for ${\text{Re}}_{\tau }^L=171$ (a). We also show the spatial distribution of ${\Pi }^{\left(\Delta \right)}$ (b) and ${\nabla }_{2D}$ (c) on the small square area depicted in Fig. 2. Flow streamlines have been superposed to each contour map to highlight regions of local flow expansion (sources) and of local flow compression (sinks).

Figure 5

Correlation coefficient $\langle {\nabla }_{2D}^{+}{\Pi }_{+}^{\Delta }\rangle$ (solid line) and $\langle {\nabla }_{2D}^{-}{\Pi }_{-}^{\Delta }\rangle$ (dashed line) between the positive (negative) energy flux ${\Pi }_{+}^{\Delta } \left({\Pi }_{-}^{\Delta }\right)$ and the positive (negative) two-dimensional surface divergence ${\nabla }_{2D}^{+} \left({\nabla }_{2D}^{-}\right)$ computed along the streamwise direction $x$ and averaged in time for ${\text{Re}}_{\tau }^H=509$ (a). We also show the spatial distribution of ${\Pi }^{\left(\Delta \right)}$ (b) and ${\nabla }_{2D}$ (c) on the small square area depicted in Fig. 3. Flow streamlines have been superposed to each contour map to highlight regions of local flow expansion (sources) and of local flow compression (sinks).

Figure 6

Third-order structure function $S_3\left(r\right)$ computed at the channel center (a) and at the free surface (b) for both ${\text{Re}}_{\tau }^L=171$ and ${\text{Re}}_{\tau }^H=509$. $S_3$ is shown as a function of $r/L_y$.

Figure 7

Second-order $[S_2\left(r\right)$, (a)] and fourth-order $[S_4\left(r\right)$, (b)] structure functions for ${\text{Re}}_{\tau }^L=171$ expressed as a function of $r/L_y$ and computed at the surface and at the center of the channel. The solid lines indicate the observed scaling in the inertial range. At the channel center (midchannel, $-\circ -$), $S_2\simeq r^{2/3}$ and $S_4\simeq r^{4/3}$.

Figure 8

Second-order $[S_2\left(r\right)$, (a)] and fourth-order $[S_4\left(r\right)$, (b)] structure functions for ${\text{Re}}_{\tau }^H=509$ expressed as a function of $r/L_y$ and computed at the surface and at the center of the channel. The solid lines indicate the observed scaling in the inertial range. At the channel center (midchannel, $-\circ -$), $S_2\simeq r^{2/3}$ and $S_4\simeq r^{4/3}$.

Figure 9

Structure function scaling exponent ${\xi }_p$ at the free surface for both ${\text{Re}}_{\tau }^L$ and ${\text{Re}}_{\tau }^H$. The dotted line indicates the classical Kolmogorov scaling $p/3$.

Figure 10

Behavior of the compressibility factor $C$ along the wall-normal direction $z/h$ for ${\text{Re}}_{\tau }^H$ and for ${\text{Re}}_{\tau }^L$ (inset). Note that $z=0$ represents the free surface. Values of the compressibility factor (i) for a two-dimensional cut of a three-dimensional homogeneous isotropic turbulent flow $\left(C\simeq 0.16\right)$ and (ii) for a compressible Kraichnan flow $\left(C=0.5\right)$ are also shown [10].