Generalized Langevin dynamics in multiphase direct numerical simulations using hydrodynamically optimized memory kernels

We propose a novel methodology for performing continuum-based simulations of Brownian motion in systems of arbitrary geometric complexity at thermal equilibrium. The methodology is valid for a wide range of particle-to-fluid density ratios, ρp/ρf=[1,1000]. It is implemented in a multiphase direct numerical simulation framework, in which the complete hydrodynamic force acting on a particle can be obtained with high accuracy using the immersed-boundary method. The hydrodynamic force is then used with the particle velocity history in an optimization procedure, through which the hydrodynamic memory kernel can be established from a convolution integral without any a priori assumption about its functional form or scaling. The memory kernel is thereafter used to generate a colored Brownian force in agreement with the fluctuation–dissipation theorem. Finally, the hydrodynamic and Brownian forces are used to determine the particle acceleration, needed to evolve the particle trajectory, using the generalized Langevin equation. We show that the developed methodology correctly predicts the particle statistics in both unhindered and wall-adjacent Brownian motion, in good agreement with theoretical and experimental results. The current work, thus, lays the foundation for simulations of geometrically complex Brownian systems, where state-of-the-art multiphase techniques such as interface-capturing, turbulence modeling, heat and mass transfer, and chemical reactions can be accounted for. Furthermore, we discuss how the memory kernel, obtained on-the-fly as an integral part of the methodology, can potentially be used to correlate particle mobility with particle reactivity.