Turbulent mass transfer through a flat shear-free surface

Mass transfer through the flat shear-free surface of a turbulent open-channel flow is investigated over a wide range of Schmidt number (1 $ \le $ Sc $ \le $ 200) by means of large-eddy simulations using a dynamic subgrid-scale model. In contrast with situations previously analysed using direct numerical simulation, the turbulent Reynolds number Re is high enough for the near-surface turbulence to be fairly close to isotropy and almost independent of the structure of the flow in the bottom region (the statistics of the velocity field are identical to those described by I. Calmet & J. Magnaudet J. Fluid Mech. vol. 474, 2003, p. 355). The main statistical features of the concentration field are analysed in connection with the structure of the turbulent motion below the free surface, characterized by a velocity macroscale $u$ and an integral length scale $L$. All near-surface statistical profiles are found to be Sc-independent when plotted vs. the dimensionless coordinate Sc$^{1 / 2}yu$/$\nu $ ($y$ is the distance to the surface and $\nu $ is the kinematic viscosity). Mean concentration profiles are observed to be linear throughout an inner diffusive sublayer whose thickness is about one Batchelor microscale, i.e. LSc$^{ - 1 / 2 }$Re$^{ - 3 / 4}$. In contrast, the concentration fluctuations are found to reach their maximum near the edge of the outer diffusive layer which scales as LSc$^{ - 1 / 2}$Re$^{ - 1 / 2}$. Instantaneous views of the near-surface isovalues of the concentration and vertical velocity are used to reveal the influence of the Schmidt number. In particular, it is observed that at high Schmidt number, the tiny concentration fluctuations that subsist in the diffusive sublayer just mirror the divergence of the two-component surface velocity field. Co-spectra of concentration and vertical velocity fluctuations indicate that the main contribution to the turbulent mass flux is provided by eddies whose horizontal size is close to $L$, which strongly supports the view that the mass transfer is governed by large-scale structures. The dimensionless mass transfer rate is observed to be proportional to Sc$^{ - 1 / 2}$ over the whole range of Schmidt number. Based on a frequency analysis of the concentration equation and on the Sc$^{ - 1 / 2}$Re$^{ - 3 / 4 }$scaling of the diffusive sublayer, it is shown that the mass transfer rate at a given Sc is proportional to $\langle {\beta ^2}\rangle ^{1 / 4}$, $\langle {\beta ^2}\rangle $ being the variance of the divergence of the surface velocity field. This yields dimensionless mass transfer rates of the form $\alpha$Sc$^{ - 1 / 2}$Re$^{ - 1 / 4}$, where the value of $\alpha$ is shown to result from both the kinematic blocking of the vertical velocity and the viscous damping of the horizontal vorticity components induced by the free surface.