Direct numerical simulation of droplet-laden homogeneous shear turbulence & a mass-conserving method for gas-liquid flows with phase change

Abstract

The interaction of dispersed droplets and turbulence is important in many natural and industrial processes, e.g., rain formation, liquid-liquid emulsion, spray cooling and spray atomization in combustors. In these flows, the droplet volume fraction is typically on the order of 1-10\% such that the turbulence is altered by droplet feedback on the surrounding fluid and by droplet-droplet interactions. The main goals of this work are to: (i) investigate the physical mechanisms of non-evaporating droplet-turbulence interaction for homogeneous shear turbulence (HST) by using direct numerical simulation (DNS), and (ii) imporve and verify our numerical method for DNS of evaporating droplets. To achieve the first objective, we have developed a new pressure-correction method, called FastRK3P*, and coupled it with the volume-of-fluid (VoF) method to simulate incompressible two-fluid flows subjected to a uniform shear flow. FastRK3P* has two main qualities: first, it does not use the solution from the previous time step to advance the solution in time, and second, it only requires one solution of the Poisson equation for pressure per time step. The first quality ensures stability for simulating HST, and the second makes the solver faster than the standard RK3 or Crank–Nicholson methods that require solving the Poisson equation multiple times per time step. Using FastRK3P*, we perform DNS of 1,258 non-evaporating droplets of initial diameter approximately equal to twice the Taylor length scale of turbulence, droplet-to-fluid density and viscosity ratios equal to ten, and a 5\% droplet volume fraction. We investigate the effects of shear number and Weber number on the modulation of turbulence kinetic energy (TKE). We derive the TKE equations for the two-fluid, carrier-fluid, and droplet-fluid flow in HST and the relationship between the power of surface tension and the rate of change of total droplet surface area, providing the pathways of TKE for two-fluid incompressible HST. Our DNS results show that for varying Weber numbers, the rate of change of TKE can increase or decrease with respect to single-phase cases. Such modulation is explained from the analysis of production and dissipation in the carrier-fluid and droplet-fluid flows, and power of surface tension in the two-fluid flow. Finally, we explain the effects of droplets on the production and dissipation rate of TKE through the droplet `catching-up' mechanism, and on the power of the surface tension through the droplet `shearing' mechanism. To achieve the second objective, we consider several 1D, and 2D test cases to verify FastP*PC, a coupled volume-of-fluid and pressure-correction flow solver for incompressible gas-liquid flows with phase change. Previous studies have reported spatial convergence rates for some flow variables, but no study presents a comprehensive analysis of convergence rates of all relevant flow variables. We present the results of our test cases in comparison with analytical solutions when available, and report spatial convergence rates for interface location, mass, velocity, temperature, vapor mass fraction, and mass flux.