Turbulent Mass Transfer to a Wall at Large Schmidt Numbers (Convection Velocity, Simulation, Wave Number Spectra, Drag-Reducing Polimers, Spatial Correlations)

Abstract

The velocity-concentration relationship for turbulent mass trans- fer to a wall was investigated in this study.

Numerical solutions of the unaveraged mass balance equation were sought using a random input for the velocity field. The relative amount of energy of the high and low frequency velocity fluctuations, their median frequency and spatial variation were varied in order to explore their effect on mass transport.

Analytical solutions of a linearized form of the mass balance equations and experimental measurements have shown that the effect of streamwise convection on mass transport is negligible. The numerical solutions of the nonlinear mass balance equation have also shown that the mass transport is independent of the spatial scale in the spanwise direction.

The results of the nonlinear calculations have shown that the mass transfer process is controlled by velocity fluctuations normal to the wall. The dependency of the average and fluctuating properties of the concentration and mass transfer field on the properties of the velocity field and on Schmidt numbers, Sc, was found to vary between two limiting cases.

It was found that for the limiting case of low Schmidt numbers the whole spectrum of the velocity frequencies is effective in transporting mass. For a given shape of the velocity spectrum, the property of the velocity field which is of critical importance is the energy at the fluc- tuations, (beta)('2). The following relation for the average mass transfer rate was obtained for a Newtonian velocity input: (')K = 7.391 x 10('-3) (.) (beta)('2('1/6)) Sc('-2/3). As the Schmidt number increases, an increased dampening of the high frequency concentration fluctuation occurs. At the limit of very large Schmidt numbers only low frequencies velocity fluctua- tions are effective in transporting mass. For a Newtonian velocity input it is shown that (')K = .574 W(,(beta))(O)('1/4)Sc('-3/4), where W(,(beta))(O) is the spectral functions of the velocity fluctuations at low frequencies. In an intermediate range of Schmidt numbers it is shown that (')K = .339 W(,(beta))(O)('1/5)(omega)(,m)('1/10)Sc('-7/10) where (omega)(,m) is the median frequency of the fluctuations.

The effect of the shape of the velocity spectral function was explored by using a white noise input. These calculations provided a possible interpretation of experimental results obtained in drag-reducing flows.