Fluctuating hydrodynamics in periodic domains and heterogeneous adjacent multidomains: Thermal equilibrium.

We first study fluctuating hydrodynamics (FH) at equilibrium in periodic domains by use of the smoothed dissipative particle dynamics (SDPD) method. We examine the performance of SDPD by comparing it with the theory of FH. We find that the spatial correlation of particle velocity is always the Dirac δ function, irrespective of numerical resolution, in agreement with the theory. However, the spatial correlation of particle density has a finite range of r(c), which is due to the kernel smoothing procedure for the density. Nevertheless, this finite range of correlation can be reduced to an arbitrarily small value by increasing the resolution, that is, reducing r(c), similarly to how the smoothing kernel converges to the Dirac δ function. Moreover, we consider temporal correlation functions (CFs) of random field variables in Fourier space. For sufficient resolution, the CFs of SDPD simulations agree very well with analytical solutions of the linearized FH equations. This confirms that both the shear and sound modes are modeled accurately and that fluctuations are generated, transported, and dissipated in both thermodynamically and hydrodynamically consistent ways in SDPD. We also show that the CFs of the classical dissipative particle dynamics (DPD) method with proper parameters can recover very well the linearized solutions. As a reverse implication, the measurement of CFs provides an effective means of extracting viscosities and sound speed of a DPD system with a new set of input parameters. Subsequently, we study the FH in truncated domains in the context of multiscale coupling via the domain decomposition method, where a SDPD simulation in one subdomain is coupled with a Navier-Stokes (NS) solver in an adjacent subdomain with an overlapping region. At equilibrium, the mean values of the NS solution are known a priori and do not need to be extracted from actual simulations. To this end, we model a buffer region as an equilibrium boundary condition (EBC) at the truncated side of the SDPD simulation. In the EBC buffer, the velocity of particles is drawn from a known Gaussian distribution, that is, the Maxwell-Boltzmann distribution. Due to the finite range of spatial correlation, the density of particles in the EBC buffer must be drawn from a conditional Gaussian distribution, which takes into account the available density distribution of neighboring interior particles. We introduce a Kriging method to provide such a conditional distribution and hence preserve the spatial correlation of density. Spatial and temporal correlations of SDPD simulations in the truncated domain are compared to that in a single complete domain. We find that a gap region between the buffer and interior is important to reduce the extra dissipation generated by the artificial buffer at equilibrium, rendering more investigations necessary for thermal fluctuations in the multiscale coupling of nonequilibrium flows.