Analytical solutions for transitional flow around hydrometeors: Implications for particle dynamics and atmospheric modeling

Accurate modeling of hydrometeor dynamics is critical for advancing weather and climate predictions. However, this remains a long-standing challenge in atmospheric science, particularly within the transitional Reynolds number regime, due to the lack of analytical solutions to the Navier–Stokes equations. This study presents two analytical solutions for two-dimensional, incompressible, and viscous flow around a sphere, addressing the gap between low and high Reynolds number regimes. The first framework, an extension of classical expansion methods using hypergeometric, provides a direct analytical relation that characterizes separation angle across the transitional regime. To address post-separation dynamics, a second formulation based on a novel variable transformation is introduced, yielding Bessel function solutions capable of reproducing closed streamline patterns and toroidal vortex rings. These solutions provide a computationally efficient and physically interpretable framework, enhancing the fidelity of hydrometeor fall velocity and drag force parameterizations in large-scale atmospheric models.