Inertial coupling method for particles in an incompressible fluctuating fluid

We develop an inertial coupling method for modeling the dynamics of point-like "blob" particles immersed in an incompressible fluid, generalizing previous work for compressible fluids (Balboa Usabiaga et al., 2013 [42]). The coupling consistently includes excess (positive or negative) inertia of the particles relative to the displaced fluid, and accounts for thermal fluctuations in the fluid momentum equation. The coupling between the fluid and the blob is based on a no-slip constraint equating the particle velocity with the local average of the fluid velocity, and conserves momentum and energy. We demonstrate that the formulation obeys a fluctuation-dissipation balance, owing to the non-dissipative nature of the no-slip coupling. We develop a spatio-temporal discretization that preserves, as best as possible, these properties of the continuum formulation. In the spatial discretization, the local averaging and spreading operations are accomplished using compact kernels commonly used in immersed boundary methods. We find that the special properties of these kernels allow the blob to provide an effective model of a particle; specifically, the volume, mass, and hydrodynamic properties of the blob are remarkably grid-independent. We develop a second-order semi-implicit temporal integrator that maintains discrete fluctuation-dissipation balance, and is not limited in stability by viscosity. Furthermore, the temporal scheme requires only constant-coefficient Poisson and Helmholtz linear solvers, enabling a very efficient and simple FFT-based implementation on GPUs. We numerically investigate the performance of the method on several standard test problems. In the deterministic setting, we find the blob to be a remarkably robust approximation to a rigid sphere, at both low and high Reynolds numbers. In the stochastic setting, we study in detail the short and long-time behavior of the velocity autocorrelation function and observe agreement with all of the known behavior for rigid sphere immersed in a fluctuating fluid. The proposed inertial coupling method provides a low-cost coarse-grained (minimal resolution) model of particulate flows over a wide range of time-scales ranging from Brownian to convection-driven motion. (C) 2013 Elsevier B.V. All rights reserved.