Low Mach number algorithm for droplet-laden turbulent channel flow including phase transition

In this study we propose a new numerical algorithm for droplet-laden turbulent channel flow with phase transitions at low Mach numbers. The carrier gas is treated as compressible flow. In order to avoid very small time steps at low Mach numbers that would arise from stability requirements associated with explicit time-stepping we propose a new semi-explicit time integration method, applied to the low Mach number compressible flow equations. We perform a perturbation analysis in powers of the Mach number of the system of governing equations. The obtained decomposition of pressure into a space-independent part and a hydrodynamic part permits to apply a special pressure-based time integration algorithm for compressible flows at low Mach numbers. An important feature of the new numerical approach is the independence of the maximum allowed time step on the Mach number. In this study we validate the new method by comparing it with a fully explicit code for compressible flow at general Mach numbers showing a good agreement in all quantities of interest. The differences between the results of the two codes are on the order of the square of the Mach number caused by the disregard of high-order terms in the Mach number in the new algorithm. The relative difference found for a specific low value of the Mach number of 0.05 is on the order of 1% for instantaneous and mean quantities of the two phases. We also quantify the efficiency of the new algorithm by comparing the computational time it takes to simulate one time unit with both codes