George E. Karniadakis
Nikolaos A. Gatsonis
David J. Olinger
Simon W. Evans
Mark W. Richman
This work presents the mathematical and computational aspects of a smooth dissipative particle dynamics with dynamic virtual particle allocation method (SDPD-DV) for modeling and simulation of mesoscopic fluids in wall-bounded domains. The SDPD-DV method is realized with fluid particles, boundary particles and dynamically allocated virtual particles near solid boundaries. The physical domain in SDPD-DV contains external and internal solid boundaries, periodic inlets and outlets, and the fluid region. The solid boundaries of the domain are represented with boundary particles which have an assigned position, wall velocity, and temperature upon initialization. The fluid domain is discretized with fluid particles placed in a global index. The algorithm for nearest neighbor particle search is based on a combination of the linked-cell and Verlet-list approaches and utilizes large rectangular cells for computational efficiency. The density model of a fluid particle in the proximity of a solid boundary includes the contribution from the virtual particles in its truncated support domain. The thermodynamic properties of a virtual particle are identical to those of the corresponding fluid particle. A periodic boundary particle allocation method is used at periodic inlets and outlets. Models for the conservative and dissipative forces on a fluid particle in the proximity of a solid boundary are presented and include the contributions of the virtual particles in its truncated support domain. The integration of the fluid particle position and momentum equations is accomplished with an implementation of the velocity-Verlet algorithm. The integration is supplemented by a bounce-forward algorithm in cases where the virtual particle force model is not able to prevent particle penetration. The integration of the entropy equation is based on the Runge-Kutta scheme. In isothermal simulations, the pressure of a fluid particle is obtained by an artificial compressibility formulation for liquids and the ideal gas law for compressible fluids. Sampling methods used for particle properties and transport coefficients in SDPD-DV are presented. The self-diffusion coefficient is obtained by an implementation of the generalized Einstein and the Green-Kubo relations. Field properties are obtained by sampling SDPD-DV outputs on a post-processing grid that allows harnessing the particle information on desired spatio-temporal scales. The isothermal (without the entropy equation) SDPD-DV method is verified and validated with simulations in bounded and periodic domains that cover the hydrodynamic and mesoscopic regimes. Verification is achieved with SDPD-DV simulations of transient, Poiseuille, body-force driven flow of liquid water between plates separated. The velocity profiles from the SDPD-DV simulations are in very good agreement with analytical estimates and the field density fluctuation near solid boundaries is shown to be below 5%. Additional verification involves SDPD-DV simulations of transient, planar, Couette liquid water flow. The top plate is moving and separated from the bottom stationary plate. The numerical results are in very good agreement with the analytical solutions. Additional SDPD-DV verification is accomplished with the simulation of a body-force driven, low-Reynolds number flow of water over a cylinder of radius R=0.02m. The SDPD-DV field velocity and pressure are compared with those obtained by FLUENT. An extensive set of SDPD-DV simulations of liquid water and gaseous nitrogen in mesoscopic periodic domains is presented. For the SDPD-DV simulations of liquid water the mass of the fluid particles is varied between 1.24 and 3.3e-7 real molecular masses and their corresponding size is between 1.08 and 323 physical length scales. For SDPD-DV simulations of gaseous nitrogen the mass of the fluid particles is varied between 6.37e3and 6.37e6 real molecular masses and their corresponding size is between 2.2e2 and 2.2e3 physical length scales. The equilibrium states are obtained and show that the particle speeds scale inversely with particle mass (or size) and that the translational temperature is scale-free. The self-diffusion coefficient for liquid water is obtained through the mean-square displacement and the velocity auto-correlation methods for the range of fluid particle masses (or sizes) considered. Various analytical expressions for the self-diffusivity of the SDPD fluid are developed in analogy to the real fluid. The numerical results are in very good agreement with the SDPD-fluid analytical expressions. The numerical self-diffusivity is shown to be scale dependent. For fluid particles approaching asymptotically the mass of the real particle the self-diffusivity is shown to approach the experimental value. The Schmidt numbers obtained from the SDPD-DV simulations are within the range expected for liquid water. The SDPD-DV method (with entropy) is verified and validated with simulations with an extensive set of simulations of gaseous nitrogen in mesoscopic, periodic domains in equilibrium. The simulations of N2(g) are performed in rectangular domains. The self-diffusion coefficient for N2(g) at equilibrium states is obtained through the mean-square displacement for the range of fluid particle masses (or sizes) considered. The numerical self-diffusion is shown to be scale dependent. The simulations show that self-diffusion decreases with increasing mass ratio. For a given mass ratio, increasing the smoothing length, increases the self-diffusion coefficient. The shear viscosity obtained from SDPD-DV is shown to be scale free and in good agreement with the real value. We examine also the effects of timestep in SDPD-DV simulations by examining thermodynamic parameters at equilibrium. These results show that the time step can lead to a significant error depending on the fluid particle mass and smoothing length. Fluctuations in thermodynamic variables obtained from SDPD-DV are compared with analytical estimates. Additional verification involves SDPD-DV simulations of steady planar thermal Couette flow of N2(g). The top plate at temperature T1 =330K is moving at Vxw =30m/s and is separated by 10-4 m from the bottom stationary plate at T2=300K. The SDPD-DV velocity and temperature fields are in excellent agreement with those obtained by FLUENT.
Worcester Polytechnic Institute
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Yang, J. (2013). A Smoothed Dissipative Particle Dynamics Methodology For Wall-Bounded Domains. Retrieved from https://digitalcommons.wpi.edu/etd-dissertations/225
Wall-Bounded, Mesoscopic, DPD, SPH, SDPD, Virtual Particle