Moment-based formulation of Navier–Maxwell slip boundary conditions for lattice Boltzmann simulations of rarefied flows in microchannels

We present an implementation of first-order Navier–Maxwell slip boundary conditions for simulating near-continuum rarefied flows in microchannels with the lattice Boltzmann method. Rather than imposing boundary conditions directly on the particle velocity distribution functions, following the existing discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory, we use a moment-based method to impose the Navier–Maxwell slip boundary conditions that relate the velocity and the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the\ud domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. The results are in excellent agreement with asymptotic solutions of the compressible Navier-Stokes equations for microchannel flows in the slip regime. Our moment formalism is also valuable for analysing the existing boundary conditions, and explains the origin of numerical slip in the bounce-back and other common boundary conditions that impose explicit conditions on the higher moments instead of on the local tangential velocity

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting

A locally adaptive time-stepping algorithm for\ud petroleum reservoir simulations

An algorithm for locally adapting the step-size for large scale finite volume simulations of multi-phase flow in petroleum reservoirs is suggested which allows for an “all-in-one” implicit calculation of behaviour over a very large time scale. Some numerical results for simple two-phase flow in one space dimension illustrate the promise of the algorithm, which has also been applied to very simple 3D cases. A description of the algorithm is presented here along with early results. Further development of the technique is hoped to facilitate useful scaling properties

Phase mixing vs. nonlinear advection in drift-kinetic plasma turbulence

A scaling theory of long-wavelength electrostatic turbulence in a magnetised, weakly collisional plasma (e.g., ITG turbulence) is proposed, with account taken both of the nonlinear advection of the perturbed particle distribution by fluctuating ExB flows and of its phase mixing, which is caused by the streaming of the particles along the mean magnetic field and, in a linear problem, would lead to Landau damping. It is found that it is possible to construct a consistent theory in which very little free energy leaks into high velocity moments of the distribution function, rendering the turbulent cascade in the energetically relevant part of the wave-number space essentially fluid-like. The velocity-space spectra of free energy expressed in terms of Hermite-moment orders are steep power laws and so the free-energy content of the phase space does not diverge at infinitesimal collisionality (while it does for a linear problem); collisional heating due to long-wavelength perturbations vanishes in this limit (also in contrast with the linear problem, in which it occurs at the finite rate equal to the Landau-damping rate). The ability of the free energy to stay in the low velocity moments of the distribution function is facilitated by the "anti-phase-mixing" effect, whose presence in the nonlinear system is due to the stochastic version of the plasma echo (the advecting velocity couples the phase-mixing and anti-phase-mixing perturbations). The partitioning of the wave-number space between the (energetically dominant) region where this is the case and the region where linear phase mixing wins its competition with nonlinear advection is governed by the "critical balance" between linear and nonlinear timescales (which for high Hermite moments splits into two thresholds, one demarcating the wave-number region where phase mixing predominates, the other where plasma echo does).Comment: 45 pages (single-column), 3 figures, replaced with version published in JP

Three-dimensional central-moments-based lattice Boltzmann method with external forcing: A consistent, concise and universal formulation

The cascaded or central-moments-based lattice Boltzmann method (CM-LBM) is a robust alternative to the more conventional BGK-LBM for the simulation of high-Reynolds number flows. Unfortunately, its original formulation makes its extension to a broader range of physics quite difficult. To tackle this issue, a recent work [A. De Rosis, Phys. Rev. E 95, 013310 (2017)] proposed a more generic way to derive concise and efficient three-dimensional CM-LBMs. Knowing the original model also relies on central moments that are derived in an adhoc manner, i.e., by mimicking those of the Maxwell-Boltzmann distribution to ensure their Galilean invariance a posteriori, a very recent effort [A. De Rosis and K. H. Luo, Phys. Rev. E 99, 013301 (2019)] was proposed to further generalize their derivation. The latter has shown that one could derive Galilean invariant CMs in a systematic and a priori manner by taking into account high-order Hermite polynomials in the derivation of the discrete equilibrium state. Combining these two approaches, a compact and mathematically sound formulation of the CM-LBM with external forcing is proposed. More specifically, the proposed formalism fully takes advantage of the D3Q27 discretization by relying on the corresponding set of 27 Hermite polynomials (up to the sixth order) for the derivation of both the discrete equilibrium state and the forcing term. The present methodology is more consistent than previous approaches, as it properly explains how to derive Galilean invariant CMs of the forcing term in an a priori manner. Furthermore, while keeping the numerical properties of the original CM-LBM, the present work leads to a compact and simple algorithm, representing a universal methodology based on CMs and external forcing within the lattice Boltzmann framework.Comment: Published in Phys. Fluids as Editor's Pic

Ground State Energy of the One-Component Charged Bose Gas

The model considered here is the `jellium' model in which there is a uniform, fixed background with charge density $-e\rho$ in a large volume $V$ and in which $N=\rho V$ particles of electric charge $+e$ and mass $m$ move --- the whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to investigate the ground state energy of this system for bosonic particles in the large $\rho$ limit. He found that the energy per particle is $-0.402 r_s^{-3/4} {me^4}/{\hbar^2}$ in this limit, where $r_s=(3/4\pi \rho)^{1/3}e^2m/\hbar^2$. Here we prove that this formula is correct, thereby validating, for the first time, at least one aspect of Bogolubov's pairing theory of the Bose gasComment: 38 pages latex. Typos corrected.Lemma 6.2 change

Lattice Boltzmann for Binary Fluids with Suspended Colloids

A new description of the binary fluid problem via the lattice Boltzmann method is presented which highlights the use of the moments in constructing two equilibrium distribution functions. This offers a number of benefits, including better isotropy, and a more natural route to the inclusion of multiple relaxation times for the binary fluid problem. In addition, the implementation of solid colloidal particles suspended in the binary mixture is addressed, which extends the solid-fluid boundary conditions for mass and momentum to include a single conserved compositional order parameter. A number of simple benchmark problems involving a single particle at or near a fluid-fluid interface are undertaken and show good agreement with available theoretical or numerical results.Comment: 10 pages, 4 figures, ICMMES 200

Brewing of filter coffee

We report progress on mathematical modelling of coffee grounds in a drip filter coffee machine. The report focuses on the evolution of the shape of the bed of coffee grounds during extraction with some work also carried out on the chemistry of extraction. This work was sponsored by Philips who are interested in understanding an observed correlation between the final shape of the coffee grounds and the quality of the coffee. We used experimental data gathered by Philips and ourselves to identify regimes in the coffee brewing process and relevant regions of parameter space. Our work makes it clear that a number of separate processes define the shape of the coffee bed depending on the values of the parameters involved e.g. the size of the grains and the speed of fluid flow during extraction. We began work on constructing mathematical models of the redistribution of the coffee grounds specialised to each region and on a model of extraction. A variety of analytic and numerical tools were used. Furthermore our research has progressed far enough to allow us to begin to exploit connections between this problem and other areas of science, in particular the areas of sedimentology and geomorphology, where the processes we have observed in coffee brewing have been studied