Thermodynamic consistency of liquid-gas lattice Boltzmann simulations

Lattice Boltzmann simulations have been very successful in simulating liquid-gas and other multi-phase fluid systems. However, the underlying second order analysis of the equation of motion has long been known to be insufficient to consistently derive the fourth order terms that are necessary to represent an extended interface. These same terms are also responsible for thermodynamic consistency, i.e. to obtain a true equilibrium solution with both a constant chemical potential and a constant pressure. In this article we present an equilibrium analysis of non-ideal lattice Boltzmann methods of sufficient order to identify those higher order terms that lead to a lack of thermodynamic consistency. We then introduce a thermodynamically consistent forcing method.Comment: 12 pages, 8 figure

Singular forces and point-like colloids in lattice Boltzmann hydrodynamics

We present a second-order accurate method to include arbitrary distributions of force densities in the lattice Boltzmann formulation of hydrodynamics. Our method may be used to represent singular force densities arising either from momentum-conserving internal forces or from external forces which do not conserve momentum. We validate our method with several examples involving point forces and find excellent agreement with analytical results. A minimal model for dilute sedimenting particles is presented using the method which promises a substantial gain in computational efficiency.Comment: 22 pages, 9 figures. Submitted to Phys. Rev.

Lattice Boltzmann simulations of soft matter systems

This article concerns numerical simulations of the dynamics of particles immersed in a continuum solvent. As prototypical systems, we consider colloidal dispersions of spherical particles and solutions of uncharged polymers. After a brief explanation of the concept of hydrodynamic interactions, we give a general overview over the various simulation methods that have been developed to cope with the resulting computational problems. We then focus on the approach we have developed, which couples a system of particles to a lattice Boltzmann model representing the solvent degrees of freedom. The standard D3Q19 lattice Boltzmann model is derived and explained in depth, followed by a detailed discussion of complementary methods for the coupling of solvent and solute. Colloidal dispersions are best described in terms of extended particles with appropriate boundary conditions at the surfaces, while particles with internal degrees of freedom are easier to simulate as an arrangement of mass points with frictional coupling to the solvent. In both cases, particular care has been taken to simulate thermal fluctuations in a consistent way. The usefulness of this methodology is illustrated by studies from our own research, where the dynamics of colloidal and polymeric systems has been investigated in both equilibrium and nonequilibrium situations.Comment: Review article, submitted to Advances in Polymer Science. 16 figures, 76 page

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. 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 make the discrete blob a particle with surprisingly physically-consistent volume, mass, and hydrodynamic properties. 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...Comment: Contains a number of corrections and an additional Figure 7 (and associated discussion) relative to published versio

Link-wise Artificial Compressibility Method

The Artificial Compressibility Method (ACM) for the incompressible Navier-Stokes equations is (link-wise) reformulated (referred to as LW-ACM) by a finite set of discrete directions (links) on a regular Cartesian mesh, in analogy with the Lattice Boltzmann Method (LBM). The main advantage is the possibility of exploiting well established technologies originally developed for LBM and classical computational fluid dynamics, with special emphasis on finite differences (at least in the present paper), at the cost of minor changes. For instance, wall boundaries not aligned with the background Cartesian mesh can be taken into account by tracing the intersections of each link with the wall (analogously to LBM technology). LW-ACM requires no high-order moments beyond hydrodynamics (often referred to as ghost moments) and no kinetic expansion. Like finite difference schemes, only standard Taylor expansion is needed for analyzing consistency. Preliminary efforts towards optimal implementations have shown that LW-ACM is capable of similar computational speed as optimized (BGK-) LBM. In addition, the memory demand is significantly smaller than (BGK-) LBM. Importantly, with an efficient implementation, this algorithm may be one of the few which is compute-bound and not memory-bound. Two- and three-dimensional benchmarks are investigated, and an extensive comparative study between the present approach and state of the art methods from the literature is carried out. Numerical evidences suggest that LW-ACM represents an excellent alternative in terms of simplicity, stability and accuracy.Comment: 62 pages, 20 figure

Steering in computational science: mesoscale modelling and simulation

This paper outlines the benefits of computational steering for high performance computing applications. Lattice-Boltzmann mesoscale fluid simulations of binary and ternary amphiphilic fluids in two and three dimensions are used to illustrate the substantial improvements which computational steering offers in terms of resource efficiency and time to discover new physics. We discuss details of our current steering implementations and describe their future outlook with the advent of computational grids.Comment: 40 pages, 11 figures. Accepted for publication in Contemporary Physic

Lattice Boltzmann methods for direct numerical simulation of turbulent fluid flows

We study the use of lattice Boltzmann (LB) methods for simulation of turbulent fluid flows motivated by their high computational throughput and amenability to highly parallel platforms such as graphics processing units (GPUs). Several algorithmic improvements are unearthed including work on non-unit Courant numbers, the force operator, use of alternative topologies based on face and body centered cubic lattices and a new formulation using a generalized eigendecomposition that allows a new freedom in tuning the eigenvectors of the linearised collision operator. Applications include a variable bulk viscosity and the use of a stretched grid, our implementation of which reduces errors present in previous efforts. We present details for numerous lattices including all required matrices, their moments the procedures and programs used to generate these and perform linear stability analysis. Small Mach number flows where density variations are negligible except in the buoyancy force term allow the use of a highly accurate finite volume solver to simulate the evolution of the buoyancy field which is coupled to the LB simulation as an external force. We use a multidimensional flux limited third order flux integral based advection scheme. The simplified algorithm we have devised is easier to implement, has higher performance and does not sacrifice any accuracy compared to the leading alternative. Our algorithm is particularly suited to an outflow based implementation which furthers the stated benefits. We present numerical experiments confirming the third order accuracy of our scheme when applied to multidimensional advection. The coupled solver is implemented in a new code that runs in parallel across multiple machines using GPUs. Our code achieves high computational throughput and accuracy and is used to simulate a range of turbulent flows. Details regarding turbulent channel flow and sheared convective boundary layer simulations are presented including some new insight into the scaling properties of the latter flow