Droplet impact and freezing dynamics on ultra-cold surfaces: A scaling analysis of central-concave pattern

Understanding the freezing of impact water droplets on cold surfaces is of both fundamental and practical significance. Using the VOF (Volume of Fluid) method with the dynamic contact angle and Solidification/Melting models embedded, we simulate the impact and freezing processes of water droplets on ultra-cold surfaces, which extends the surface temperature from about -50 degrees C previously to -100 degrees C. Two distinct freezing morphologies of central-pointy or central-concave patterns are obtained and mapped at various surface temperatures and Weber numbers. An inner valley and an outer rim typically form the central-concave pattern. The influence of the surface temperature on the spreading of the contact line is greater than that of the valley. The maximum spreading factors of the contact line and the valley are related to the effective Reynolds number by a 1/3-power law and the Weber number by a 1/4-power one. Furthermore, a theoretical model incorporating the freezing dynamics and the shrinking of the valley is proposed to predict the spreading factors and freezing times of the valley and rim for the central-concave pattern, yielding a global maximum deviation of 20 %. Our results and analyses provide an insight into the coupling mechanism of the droplet impact and freezing dynamics on ultra-cold surfaces.