(325i) Characteristic Length and Time Scales on Turbulent Mass Transfer across a Free Surface in Fully Developed Turbulence

This study discusses turbulent mass transfer into a fully developed turbulent liquid across a free surface. Our previous research results (R. Nagaosa and R. A. Handler, Phys. Fluids 15, 375, 2003) tell that the mass transfer process is enhanced by an interaction of tubular vortices, which represent nonlinear behavior of turbulence, with the free surface. It is widely accepted that turbulence underneath the free surface, dominated by nonlinear behavior of viscous fluid flow, plays a great role in determining turbulence scalar transfer mechanism. An understanding turbulence and associated turbulence scalar transfer has not been progressed even in the last decade. One of the obstacles to prevent understanding such mechanism is that measurements and mappings of instantaneous velocity and scalar fluctuations three-dimensionally in a turbulent boundary layer below the interface is difficult to realize in laboratory experiments. The previous laboratory experiments have been therefore concentrated on evaluations of macro-scale, and ensemble-averaged hydrodynamics in the near-interface region, and small-scale information on turbulence has not been targeted for their measurements. Some of the researchers have tried to choose the other approaches, namely a computational fluid mechanics (CFD), to satisfy their desires to understand detailed structures of turbulence and associated turbulence scalar transfer in the near-interface region. Whereas many reports on physics of turbulence with the gas-liquid interface have been presented especially in the last two decades, only a handful of studies discuss turbulence scalar transfer mechanism at the interface in the open-channel so far. The author has presented hydrodynamics of turbulence and associated turbulence scalar transfer at the gas-liquid interface without interfacial shear stress in the last several years. It is crucial in practical purposes to elucidate scaling of turbulence scalar transfer by several nondimensional quantities. Only the two nondimensional quantities, the Reynolds number and the Schmidt number, are expected relevant to turbulence scalar flux in a case of clean (no contamination), flat, and zero-shear interface. The effect of the Reynolds number on the scalar transfer coefficient has been discussed by Komori et al. (J. Fluid Mech 203, 103, 1989) based on their laboratory experiments. Their studies imply that the Sherwood number is approximately proportional to the bulk Reynolds number, which is defined by water depth and the bulk-mean velocity. Magnaudet and Calmet (J. Fluid Mech 553, 155, 2006) also argued scaling of the scalar transfer coefficient with the Reynolds number based on revisit of previous studies and their large-eddy simulation (LES) results. Their conclusion, on the other hand, states that the Sherwood number is scaled by 3/4 power of the Reynolds number, which is defined by the integral length scale and the corresponding velocity scale. It should be very cautious to discuss scaling of the scalar transfer coefficient based on the two previous results concerning several aspects. First, the Reynolds numbers to scale the scalar transfer coefficient are different in their individual studies. Next, their definitions of the Sherwood number are also different. While Komori et al. defined it across the whole of water depth, Magnaudet and Calmet employ the Sherwood number defined across the interface-influenced boundary layer. A careful examination on scaling analyses of the scalar transfer coefficient is necessary to activate their experimental and numerical results. The scope of this brief communication is to argue scaling analyses of the Sherwood number by introducing two different definitions of the Reynolds numbers. We employ our DNS database of turbulent open-channel flows to examine the relation between turbulence scalar flux and the two Reynolds numbers. In the first part of this report, we consider scaling of the Sherwood number by the bulk Reynolds number. Our present numerical predictions are compared with the experimental measurements by Komori et al. The introduction of the turbulent Reynolds number, which is based on the integral length scale and the corresponding velocity scale, is attempted in the next step to scale the Sherwood number across the interface-influenced boundary layer. Our numerical predictions by DNS are compared with the proposed relation between the turbulent Reynolds number and the scalar transfer coefficient by Magnaudet and Calmet. In addition, the characteristic length scale for turbulence in the interface is examined by the interfacial divergence. The results of this numerical study indicate clearly that the present DNS suggest that the Sherwood number based on the water depth, Sh, is scaled by 3/4 power of the bulk Reynolds number, Rem, while Komori et al's laboratory experiments justifies that the Sherwood number is proportional to the bulk Reynolds number. We also evaluate the turbulent Reynolds number based on the thickness of turbulent boundary layer below the free surface and the corresponding turbulence kinetic energy, ReL, as done by Magnaudet and Calmet. The Sherwood number based on the concentration difference across the turbulent boundary layer, Sh', is correlated by the turbulent Reynolds number, ReL. This correlation reveals that the Sherwood number is also proportional to 3/4 power of the turbulent Reynolds number. It is further elucidated that the results of LES by Magnaudet and Calmet is predictable by the present correlation between Sh'-ReL. It is interesting to point out that the exponent of 3/4 is found in both the Sh-Rem and Sh'-ReL correlations.