Scaling Relation of the Scalar Diffusion in a Rotating Mixer

Scalar mixing is under the joint control of convection and diffusion. The ratio of the dissipative scale of velocity field to that of the scalar field depends on the Schmidt number. In the high Schmidt number limit, the scalar scale is much smaller than that of the momentum, which then requires either special treatment or ad hoc models for the scalar quantity in numerical simulations. In order to avoid model uncertainty or unnecessary numerical complexity, the direct numerical simulation is performed for studying the scalar mixing process in a confined rotating mixer tank. It has been found that in the range of negligible numerical diffusivity, the characteristic scalar mixing time is inversely proportional to the scalar diffusivity. Analysis based on the dimensional argument justifies such scaling relation as well, from which the unaccepted computational time of the mixing process in the high Schmidt number limit can be efficiently determined, without the use of ad hoc models. This scaling idea is also of practical meaningfulness for other similar problems.