Computational Eulerian Hydrodynamics and Galilean Invariance

Eulerian hydrodynamical simulations are a powerful and popular tool for modeling fluids in astrophysical systems. In this work, we critically examine recent claims that these methods violate Galilean invariance of the Euler equations. We demonstrate that Eulerian hydrodynamics methods do converge to a Galilean-invariant solution, provided a well-defined convergent solution exists. Specifically, we show that numerical diffusion, resulting from diffusion-like terms in the discretized hydrodynamical equations solved by Eulerian methods, accounts for the effects previously identified as evidence for the Galilean non-invariance of these methods. These velocity-dependent diffusive terms lead to different results for different bulk velocities when the spatial resolution of the simulation is kept fixed, but their effect becomes negligible as the resolution of the simulation is increased to obtain a converged solution. In particular, we find that Kelvin-Helmholtz instabilities develop properly in realistic Eulerian calculations regardless of the bulk velocity provided the problem is simulated with sufficient resolution (a factor of 2-4 increase compared to the case without bulk flows for realistic velocities). Our results reiterate that high-resolution Eulerian methods can perform well and obtain a convergent solution, even in the presence of highly supersonic bulk flows.Comment: Version accepted by MNRAS Oct 2, 2009. Figures degraded. For high-resolution color figures and movies of the numerical simulations, please visit http://www.astro.caltech.edu/~brant/Site/Computational_Eulerian_Hydrodynamics_and_Galilean_Invariance.htm

Inertial Frame Independent Forcing for Discrete Velocity Boltzmann Equation: Implications for Filtered Turbulence Simulation

We present a systematic derivation of a model based on the central moment lattice Boltzmann equation that rigorously maintains Galilean invariance of forces to simulate inertial frame independent flow fields. In this regard, the central moments, i.e. moments shifted by the local fluid velocity, of the discrete source terms of the lattice Boltzmann equation are obtained by matching those of the continuous full Boltzmann equation of various orders. This results in an exact hierarchical identity between the central moments of the source terms of a given order and the components of the central moments of the distribution functions and sources of lower orders. The corresponding source terms in velocity space are then obtained from an exact inverse transformation due to a suitable choice of orthogonal basis for moments. Furthermore, such a central moment based kinetic model is further extended by incorporating reduced compressibility effects to represent incompressible flow. Moreover, the description and simulation of fluid turbulence for full or any subset of scales or their averaged behavior should remain independent of any inertial frame of reference. Thus, based on the above formulation, a new approach in lattice Boltzmann framework to incorporate turbulence models for simulation of Galilean invariant statistical averaged or filtered turbulent fluid motion is discussed.Comment: 37 pages, 1 figur

Multi-particle-collision dynamics: Flow around a circular and a square cylinder

A particle-based model for mesoscopic fluid dynamics is used to simulate steady and unsteady flows around a circular and a square cylinder in a two-dimensional channel for a range of Reynolds number between 10 and 130. Numerical results for the recirculation length, the drag coefficient, and the Strouhal number are reported and compared with previous experimental measurements and computational fluid dynamics data. The good agreement demonstrates the potential of this method for the investigation of complex flows.Comment: 6 pages, separated figures in .jpg format, to be published in Europhysics Letter

Improved three-dimensional color-gradient lattice Boltzmann model for immiscible multiphase flows

In this paper, an improved three-dimensional color-gradient lattice Boltzmann (LB) model is proposed for simulating immiscible multiphase flows. Compared with the previous three-dimensional color-gradient LB models, which suffer from the lack of Galilean invariance and considerable numerical errors in many cases owing to the error terms in the recovered macroscopic equations, the present model eliminates the error terms and therefore improves the numerical accuracy and enhances the Galilean invariance. To validate the proposed model, numerical simulation are performed. First, the test of a moving droplet in a uniform flow field is employed to verify the Galilean invariance of the improved model. Subsequently, numerical simulations are carried out for the layered two-phase flow and three-dimensional Rayleigh-Taylor instability. It is shown that, using the improved model, the numerical accuracy can be significantly improved in comparison with the color-gradient LB model without the improvements. Finally, the capability of the improved color-gradient LB model for simulating dynamic multiphase flows at a relatively large density ratio is demonstrated via the simulation of droplet impact on a solid surface.Comment: 9 Figure

A Lattice Boltzmann Model of Binary Fluid Mixture

We introduce a lattice Boltzmann for simulating an immiscible binary fluid mixture. Our collision rules are derived from a macroscopic thermodynamic description of the fluid in a way motivated by the Cahn-Hilliard approach to non-equilibrium dynamics. This ensures that a thermodynamically consistent state is reached in equilibrium. The non-equilibrium dynamics is investigated numerically and found to agree with simple analytic predictions in both the one-phase and the two-phase region of the phase diagram.Comment: 12 pages + 4 eps figure

Galilean invariance of lattice Boltzmann models

It is well-known that the original lattice Boltzmann (LB) equation deviates from the Navier-Stokes equations due to an unphysical velocity dependent viscosity. This unphysical dependency violates the Galilean invariance and limits the validation domain of the LB method to near incompressible flows. As previously shown, recovery of correct transport phenomena in kinetic equations depends on the higher hydrodynamic moments. In this Letter, we give specific criteria for recovery of various transport coefficients. The Galilean invariance of a general class of LB models is demonstrated via numerical experiments

Vortex-antivortex proliferation from an obstacle in thin film ferromagnets

Magnetization dynamics in thin film ferromagnets can be studied using a dispersive hydrodynamic formulation. The equations describing the magnetodynamics map to a compressible fluid with broken Galilean invariance parametrized by the longitudinal spin density and a magnetic analog of the fluid velocity that define spin-density waves. A direct consequence of these equations is the determination of a magnetic Mach number. Micromagnetic simulations reveal nucleation of nonlinear structures from an impenetrable object realized by an applied magnetic field spot or a defect. In this work, micromagnetic simulations demonstrate vortex-antivortex pair nucleation from an obstacle. Their interaction establishes either ordered or irregular vortex-antivortex complexes. Furthermore, when the magnetic Mach number exceeds unity (supersonic flow), a Mach cone and periodic wavefronts are observed, which can be well-described by solutions of the steady, linearized equations. These results are reminiscent of theoretical and experimental observations in Bose-Einstein condensates, and further supports the analogy between the magnetodynamics of a thin film ferromagnet and compressible fluids. The nucleation of nonlinear structures and vortex-antivortex complexes using this approach enables the study of their interactions and effects on the stability of spin-density waves.Comment: 23 pages, 7 figure