A second-order positivity-preserving finite volume upwind scheme for air-mixed droplet flow in atmospheric icing

A second-order positivity-preserving finite volume upwind scheme based on the approximate Riemann solver is developed for computing the Eulerian two-phase flow composed of air and small water droplets in atmospheric icing. In order to circumvent a numerical problem due to the non-strictly hyperbolic nature of the original Eulerian droplet equations, a simple technique based on splitting of the original system into the well-posed hyperbolic part and the source term is proposed. The positivity-preserving Harten-Lax-van Leer-Contact approximate Riemann solver is then applied to the well-posed hyperbolic part of the Eulerian droplet equations. It is demonstrated that the new scheme satisfies the positivity condition for the liquid water contents. The numerical results of one and two-dimensional test problems are also presented as the verification and validation of the new scheme. Lastly, the exact analytical Riemann solutions of the well-posed hyperbolic part of the droplet equations in wet and dry regions are given for the verification study.