Heat and water vapor transfer in the wake of a falling ice sphere and its implication for secondary ice formation in clouds

Abstract

We perform direct numerical simulations of the settling of an ice sphere in an ambient fluid accounting for heat and mass transfer with the aim of studying in a meteorological context the case of falling graupel in humid air. The study is motivated by the fact that falling graupels in clouds are heated by the latent heat released during the accretion of liquid water droplets. They may therefore be considerably warmer than their surrounding and evaporate water vapor, which mixes with the surrounding air in the wake of the graupel, thereby creating transient zones of supersaturation there. The problem of a falling graupel is modeled as that of a heated sphere falling in a quiescent ambient fluid under the action of gravity. The coupling between the temperature and velocity fields is accounted for by the Boussinesq approximation. This problem can be parameterized by four parameters: the particle/fluid density ratio ${\rho }_{p}/{\rho }_{\infty }$ , the Galileo number Ga = u _g D / ν (where D is the diameter of the sphere, ν the viscosity of the fluid, ${u}_{g}=\sqrt{\left|({\rho }_{p}/{\rho }_{\infty }-1)g\right|D}$ , and g the gravitational acceleration), the Prandtl number Pr = ν / D _T (where D _T stands for the thermal diffusivity), and the Richardson number ${{Ri}}_{T}=\beta ({T}_{p}-{T}_{\infty })/(\tfrac{{\rho }_{p}}{{\rho }_{\infty }}-1)$ , where T _p − T _∞ is the temperature difference between the sphere and the ambient fluid and β the thermal expansion coefficient of the fluid. A separate scalar transport equation accounts for the vapor transport. Typical cloud conditions involve small temperature differences between the sphere and the surrounding, yielding relatively small Richardson numbers for both heat and mass transport. We give a special emphasis to the Galileo numbers 150, 170, 200 and 300 in order to analyze the specificities of each settling regime. The questions addressed in this study are mainly methodological and concern the influence of the settling regime and the mobility of the sphere on the structure of the scalar fields, the possible influence of modest Richardson numbers on the structure of the wake, and the possible application of this simulation framework to the investigation of the saturation in the wake of a falling graupel. We observe that the body behaves similar to a body with infinitely large density. Buoyancy effects upon the wake at the values of the Richardson number corresponding to the atmospheric context are found to be negligible. We discuss the necessity to distinguish between the diffusivity of temperature and vapor content and for this the requirement to solve both scalar transport equations separately. The simulations reveal the structure of the saturation field which features zones of supersaturation that might indeed be the sites of secondary ice nucleation (formation of additional ice crystals). The potential error in not solving both fields separately is relatively low but affects the regions of the flow that feature the largest supersaturation, such that it could be preferable to separate both transport equations depending on the future questions addressed.

Keywords

• settling sphere
• heat and mass transfer
• numerical simulation
• wake
• saturation
• ice sphere