Low Mach number fluctuating hydrodynamics of binary liquid mixtures

Abstract

Continuing on our previous work (A. Donev, A. Nonaka, Y. Sun, T. G. Fai, A. L. Garcia and J. B. Bell, Comm. App. Math. and Comp. Sci. 9 (2014), no. 1, 47–105), we develop semi-implicit numerical methods for solving low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different densities and transport coefficients. We treat viscous dissipation implicitly using a recently developed variable-coefficient Stokes solver (M. Cai, A. J. Nonaka, J. B. Bell, B. E. Griffith and A. Donev, Commun. Comput. Phys. 16 (2014), no. 5, 1263–1297). This allows us to increase the time step size significantly for low Reynolds number flows with large Schmidt numbers compared to our earlier explicit temporal integrator. Also, unlike most existing deterministic methods for low Mach number equations, our methods do not use a fractional time-step approach in the spirit of projection methods, thus avoiding splitting errors and giving full second-order deterministic accuracy even in the presence of boundaries for a broad range of Reynolds numbers including steady Stokes flow. We incorporate the Stokes solver into two time-advancement schemes, where the first is suitable for inertial flows and the second is suitable for the overdamped limit (viscous-dominated flows), in which inertia vanishes and the fluid motion can be described by a steady Stokes equation. We also describe how to incorporate advanced higher-order Godunov advection schemes in the numerical method, allowing for the treatment of (very) large Péclet number flows with a vanishing mass diffusion coefficient. We incorporate thermal fluctuations in the description in both the inertial and overdamped regimes. We validate our algorithm with a series of stochastic and deterministic tests. Finally, we apply our algorithms to model the development of giant concentration fluctuations during the diffusive mixing of water and glycerol, and compare numerical results with experimental measurements. We find good agreement between the two, and observe propagative (nondiffusive) modes at small wavenumbers (large spatial scales), not reported in published experimental measurements of concentration fluctuations in fluid mixtures. Our work forms the foundation for developing low Mach number fluctuating hydrodynamics methods for miscible multispecies mixtures of chemically reacting fluids.