Research News Direct Numerical Simulation on Single-Droplet Flow with Mass Transfer By Rho-Taek Jung* and Toru Sato 1 Introduction The sequestration of carbon dioxide (CO 2 ), one of the so- called greenhouse gases, in the ocean is a promising method to reduce CO 2 in the atmosphere [1,2]. The CO 2 extracted from thermal power plants on land is liquefied (i.e., LCO 2 ) and transported by ships, and injected in the form of droplets at the depth of 1000±2000 m in the deep ocean. Emitted LCO 2 forms a plume of rising droplets, which entrains surrounding seawater. The LCO 2 droplets are dissolved out during the rise and water of the large concentration of CO 2 is expected to peel out of the rising plume, to sink as a density current and to intrude into the surrounding stratified ocean whose density is equal to that of the CO 2 -rich water. A prior proposition on CO 2 ocean sequestration is how to realize the slow dissolution of CO 2 into seawater and, consequently, the large dilution of dissolved CO 2 near the injection site in order to minimize biological impacts in the deep ocean. To accomplish both the slow dissolution and large dilution of CO 2 , it is very important to reach laminar droplet formation and to control the initial droplet size, which determines its rise velocity and the dissolution rate of CO 2 . If there is a mass transfer, the flow is divided into three characteristic regions depending on the Schmidt number. In the regime where the Schmidt number is much greater than unity, mass diffusion proceeds at a slower speed than momentum diffusion, and the mass boundary layer becomes much thinner than the momentum boundary layer. There are some experimental studies of the high mass transfer over a sphere [3,4]. Experiments on mass transfer from droplets need enduring experimental techniques, while numerical simula- tion may become a more appropriate method for under- standing mechanisms, as the calculation speed and capacity of hardware increases continuously. The aim of this study is to develop three-dimensional CFD codes for two-phase flows with unidirectional dissolution from a dispersed phase to a continuous phase. Direct CFD codes of front-capturing mechanisms for bubble/droplet flows have been developed by a number of researchers[5±7]. The inter- face is expressed by a variety of scalar-functions, e.g., volume of fluid (VOF), marker-density function (MDF), VOF in micro-cells, or the level-set. On the other hand, Matsumoto et al. [8] applied a front-tracking method with boundary-fitted coordinates to single-bubble flows. Although it seems that the front-tracking type gives more accurate representation of the interface shape than the front-capturing type because it tracks the interface in a geometrically direct way, grid skew may cause numerical inaccuracy when the deformation of an interface becomes large. Tryggvason et al. [9] developed the explicit front-tracking mesh for bubble/droplet flows, in which the interface is expressed by moving mesh in orthogonal grids. This seems to concur the grid skew problem. Here we have selected the MDF method because the front- capturing method with volume fraction has more flexibility in coping with large curvature, coalescence, pinch-off, and so on, than the front-tracking type. In the numerical procedure of the MDF, steep shock surfaces are transferred, so that the present method adopts the total variation-diminishing (TVD) scheme to suppress artificial oscillations. The details are found in Ref. [10]. Since it is still not cheap to treat a number of droplets by direct simulations with current computing facilities, we only focus on the movement of a single droplet. For the same reason, the thickness of mass boundary layer is assumed to have the same order of magnitude of that of the momentum boundary layer in this study. Accordingly, the Schmidt number handled by our front-capturing method is at the order of unity. In order to solve the high Schmidt number problems about a droplet, we have been developing the second code by using the front-tracking method on unstructured grids. So far we have carried out a simulation on flow over a solid sphere with mass transfer. Very thin layer grids are generated to calculate the steep gradient of mass concentration at the high Schmidt number along the sphere. Moreover, the formation of CO 2 clathrate hydrate is left as a challenge for future investigations. The problems we tackle in this study are (1) the movement of the interface between a droplet and the liquid continuous phase, (2) the dissolution of mass from the droplet to the continuous phase through the interface for low Sc by the front capturing method, (3) the transfer of dissolved mass on the solid sphere for high Sc by the front-tracking method. This article gives an outline of the two CFD methods and leads to results from case studies carried out for their validation. 2 Front Capturing Method in Orthogonal Grids Since we are interested in liquid-liquid two-phase systems, in which the densities and viscosities of both phases do not differ very much compared with gas-liquid systems, a one-fluid Chem. Eng. Technol. 24 (2001) 10, Ó WILEY-VCH Verlag GmbH, D-69469 Weinheim, 2001 0930-7516/01/1010-01071 $ 17.50+.50/0 1071 ± [*] Rho-Taek Jung,Toru Sato, The University of Tokyo, Environmental and Ocean Engineering Department, Bunkyou-ku, Tokyo, 133-5868, Japan. 0930-7516/01/1010-1071 $ 17.50+.50/0