Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier-Maxwell slip boundary conditions

We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, and nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity and the pressure relative to asymptotic solutions of the compressible Navier–Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier–Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the 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. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back and other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself

A note on magnetic monopoles and the one dimensional MHD Riemann problem

INTRODUCTION The evolution in time of a magnetic field B is determined by an electric field E through the induction equation [4, 8, 12, 14] + r\ThetaE = 0; (1) one of Maxwell's equations. The magnetic field must also satisfy r\DeltaB = 0. This constraint expresses the absence of magnetic monopoles, which have never been observed experimentally. Since (1) implies @ t (r\DeltaB) = 0 this constraint is often treated as an initial condition, which will be preserved under subsequent evolution. The induction equation (1) is combined with the equations of gas dynamics to describe the behaviour of compressible electrically conducting fluids subject to magnetic fields. For non-relativistic fluids, where Maxwell's displacement current may be neglected, the combined system is referred to as the magnetohydrodynamic (MHD) equations. The compressible ideal (inviscid and perfectly conducting) MHD equations may be written as a hyperbolic system of conservation laws in the form [7, 9, 11] @ 6

Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

Quantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme for the Klein–Gordon equation

Ambipolar Diffusion: A Relaxation Technique for Constructing Force-Free Magnetic Fields, and Astrophysical Applications.

99> ffi away from the normal (` ! 60 ffi in the diagram) in order to launch a wind. This is because the angular velocity, rather than the angular momentum, of an orbiting fluid parcel is conserved in the presence of a sufficiently strong magnetic field, so Keplerian orbits may be destabilised by a sufficiently inclined field. Lubow et al (1994), and Reyes--Ruiz & Stepinski (1996) have considered whether a viscous disk could plausibly drag an external, vertical magnetic field strongly enough to satisfy BP's condition. Lubow et al considered a geometrically thin disk, whereas Reyes--Ruiz & Stepinski also solved for the magnetic field inside a disk of prescribed, but finite width. They concluded that the turbulent magnetic diffusivity j t must be orders of magnitude smaller than the turbulent viscosity t , even after taking the aspect ratio into account. It is difficult to envisage a tu

The role of the complete Coriolis force in cross-equatorial transport of abyssal ocean currents

In studies of the ocean it has become conventional to retain only the component of the Coriolis force associated with the radial component of the Earth’s rotation vector, the so-called “traditional approximation”. We investigate the role of the “non-traditional” component of the Coriolis force, corresponding to the non-radial component of the rotation vector, in transporting abyssal waters across the equator. We first derive a non-traditional generalisation of the multi-layer shallow water equations, which describe the flow of multiple superposed layers of inviscid, incompressible fluid with constant densities over prescribed topography in a rotating frame. We derive these equations both by averaging the three-dimensional governing equations over each layer, and via Hamilton’s principle. The latter derivation guarantees that conservation laws for mass, momentum, energy and potential vorticity are preserved. Within geophysically realistic parameters, including the complete Coriolis force modifies the domain of hyperbolicity of the multi-layer equations by no more than 5%. By contrast, long linear plane waves exhibit dramatic structural changes due to reconnection of the surface and internal wave modes in the long-wave limit. We use our non-traditional shallow water equations as an idealised model of an abyssal current flowing beneath a less dense upper ocean. We focus on the Antarctic Bottom Water, which crosses the equator in the western Atlantic ocean, where the bathymetry forms an almost-westward channel. Cross-equatorial flow is strongly constrained by potential vorticity conservation, which requires fluid to acquire a large relative vorticity in order to move between hemispheres. Including the complete Coriolis force accounts for the fact that fluid crossing the equator in an eastward/westward channel experiences a smaller change in angular momentum, and therefore acquires less relative vorticity. Our analytical and numerical solutions for shallow water flow over idealised channel topography show that the non-traditional component of the Coriolis force facilitates cross-equatorial flow through an almost-westward channel.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

The thermal shallow water equations, their quasi-geostrophic limit, and equatorial super-rotation in Jovian atmospheres

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.This thesis is not currently available in ORA