Energy equations for gas-liquid compressible turbulent flows with phase change

We have derived the equations of turbulence kinetic energy (TKE), mean-flow kinetic energy (MFKE), and mean internal energy (MIE) for gas-liquid compressible flows with phase change. These equations allow us to explain the pathways of energy exchange within the gas or liquid phase, and between the gas and liquid phases. We explain the role of interfacial surface energy through the power of surface tension, and the role of phase change on the energy exchange through the phase change free energy. We relate the rate of the latter to the powers of mass flux, momentum flux, and heat flux due to phase change. This link explains the effect of evaporation or condensation on the modulation of TKE, MFKE and MIE in gas-liquid compressible turbulent flows. Furthermore, we rewrite these equations for the specific incompressible cases of droplet-laden homogeneous shear turbulence with phase change (DLHST-PC) and droplet-laden decaying homogeneous isotropic turbulence with phase change (DLHIT-PC).