We construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.

A new boundary condition, aimed at inhibiting near-wall condensation effects in lattice Boltzmann simulations of capillary flows in micro-corrugated channels, is introduced. The new boundary condition is validated against analytical solutions for smooth channels and demonstrated for the case of three-dimensional microflows over randomly corrugated walls.

A quasi non-overlapping hybrid scheme that combines the finite-difference time-domain (FDTD) method and the finite-element time-domain (FETD) method with nonconforming meshes is developed for time-domain solutions of Maxwell’s equations. The FETD method uses mixed-order basis functions for electric and magnetic fields, while the FDTD method uses the traditional Yee’s grid; the two methods are joined by a buffer zone with the FETD method and the discontinuous Galerkin method is used for the domain decomposition in the FETD subdomains. The main features of this technique is that it allows non-conforming meshes and an arbitrary numbers of FETD and FDTD subdomains. The hybrid method is completely stable for the time steps up to the stability limit for the FDTD method and FETD method. Numerical results demonstrate the validity of this technique.

Recent studies on applications of the lattice Boltzmann method (LBM) and the finite-difference lattice Boltzmann method (FDLBM) to velocity slip simulations are mostly on one-dimensional (1D) problems such as a shear flow between parallel plates. Applications to a 2D problem may raise new issues. The author performed numerical simulations of rotational slip flow in coaxial cylinders as an example of 2D problem. Two types of 2D models were used. The first were multi-speed FDLBM models proposed by the author. The second was a standard LBM, the D2Q9 model. The simulations were performed applying a finite difference scheme to both the models. The study had two objectives. The first was to investigate the accuracies of LBM and FDLBM on applications to rotational slip flow. The second was to obtain an experience on application of the cylindrical coordinate system. The FDLBM model with 8 directions and the D2Q9 model showed an anisotropic flow pattern when the relaxation time constant or the Knudsen number was large. The FDLBM model with 24 directions showed accurate results even at large Knudsen numbers.

We analyze the Lattice Boltzmann method for the simulation of fluctuating hydrodynamics by Adhikari et al. [Europhys. Lett., 71 (2005), 473-479] and find that it shows excellent agreement with theory even for small wavelengths as long as a stationary system is considered. This is in contrast to other finite difference and older lattice Boltzmann implementations that show convergence only in the limit of large wavelengths. In particular cross correlators vanish to less than 0.5%. For larger mean velocities, however, Galilean invariance violations manifest themselves.

In order to model the dispersal of volcanic particles in the atmosphere and their deposition on the ground, one has to simulate an advection-diffusion-sedimentation process on a large spatial area. Here we compare a Lattice Boltzmann and a Cellular Automata approach. Our results show that for high Peclet regimes, the cellular automata model produce results that are as accurate as the lattice Boltzmann model and is computationally more effective.

The lattice Boltzmann method is applied to the investigations of the diffusivity and the permeability in the gas diffusion layer (GDL) of the polymer electrolyte fuel cell (PEFC). The effects of the configuration of water droplets, the porosity of the GDL, the viscosity ratio of water to air, and the surface wettability of the GDL are investigated. From the simulations under the PEFC operating conditions, it is found that the heterogeneous water network and the high porosity improve the diffusivity and the permeability, and the hydrophobic surface decreases the permeability.

A thermal lattice Boltzmann method (LBM) for two-phase fluid flows in nucleate pool boiling process is proposed. In the present method, a new function for heat transfer is introduced to the isothermal LBM for two-phase immiscible fluids with large density differences. The calculated temperature is substituted into the pressure tensor, which is used for the calculation of an order parameter representing two phases so that bubbles can be formed by nucleate boiling. By using this method, two-dimensional simulations of nucleate pool boiling by a heat source on a solid wall are carried out with the boundary condition for a constant heat flux. The flow characteristics and temperature distribution in the nucleate pool boiling process are obtained. It is seen that a bubble nucleation is formed at first and then the bubble grows and leaves the wall, finally going up with deformation by the buoyant effect. In addition, the effects of the gravity and the surface wettability on the bubble diameter at departure are numerically investigated. The calculated results are in qualitative agreement with other theoretical predictions with available experimental data.

Multi-component flow with chemical reactions is a common problem in different industrial applications: the mixing chamber of a reaction injection molding (RIM) machine; the dynamics of diesel soot particles interacting with a porous-ceramic particulate filter; reactive transport in porous media; bio-chemical processes involving enzyme-catalyzed kinetics. In all these cases, mass diffusion/convection and wall or volume chemical interactions among components play an important role. In the present paper we underline the importance of diffusion/convection/reaction mechanisms in bio-chemical processes using the Lattice Boltzmann (LB) technique. The bio-application where we studied diffusion/convection/reaction mechanisms is the quorum-sensing pathway for the bio-synthesis of the AI-2, a molecule that allows the bacteria to launch a coordinated attack on a host immune system (see [9,10] for more details of the bio-application). The overall goal is to create a micro-device to screen potential drugs that inhibit AI-2 bio-synthesis. The Michaelis-Menten saturation kinetic model is implemented at the reactive surface and the results are shown in terms of two dimensionless numbers: Damkohler (Da) and Peclet (Pe) number. For high Pe number a small conversion of reactants into products is obtained at the reactive surface, but the overall flux of products is high; moreover, a fast saturation of the conversion of reactants to products is obtained for high Da numbers. The trade-off for setting the Pe and Da numbers depends on the specific application and the technologies used in the micro-device (e.g., sensitivity of the detector, cost of reactants).

Coupling the immersed boundary (IB) method and the lattice Boltzmann (LB) method might be a promising approach to simulate fluid-structure interaction (FSI) problems with flexible structures and complex boundaries, because the former is a general simulation method for FSIs in biological systems, the latter is an efficient scheme for fluid flow simulations, and both of them work on regular Cartesian grids. In this paper an IB-LB coupling scheme is proposed and its feasibility is verified. The scheme is suitable for FSI problems concerning rapid flexible boundary motion and a large pressure gradient across the boundary. We first analyze the respective concepts, formulae and advantages of the IB and LB methods, and then explain the coupling strategy and detailed implementation procedures. To verify the effectiveness and accuracy, FSI problems arising from the relaxation of a distorted balloon immersed in a viscous fluid, an unsteady wake flow caused by an impulsively started circular cylinder at Reynolds number 9500, and an unsteady vortex shedding flow past a suddenly started rotating circular cylinder at Reynolds number 1000 are simulated. The first example is a benchmark case for flexible boundary FSI with a large pressure gradient across the boundary, the second is a fixed complex boundary problem, and the third is a typical moving boundary example. The results are in good agreement with the analytical and existing numerical data. It is shown that the proposed scheme is capable of modeling flexible boundary and complex boundary problems at a second-order spatial convergence; the volume leakage defect of the conventional IB method has been remedied by using a new method of introducing the unsteady and non-uniform external force; and the LB method makes the IB method simulation simpler and more efficient.

The lattice Boltzmann method (LBM) with an elastic model is applied to the simulation of two-phase flows containing a deformable body with a viscoelastic membrane. The numerical method is based on the LBM for incompressible two-phase fluid flows with the same density. The body has an internal fluid covered by a viscoelastic membrane of a finite thickness. An elastic model is introduced to the LBM in order to determine the elastic forces acting on the viscoelastic membrane of the body. In the present method, we take account of changes in surface area of the membrane and in total volume of the body as well as shear deformation of the membrane. By using this method, we calculate two problems, the behavior of an initially spherical body under shear flow and the motion of a body with initially spherical or biconcave discoidal shape in square pipe flow. Calculated deformations of the body (the Taylor shape parameter) for various shear rates are in good agreement with other numerical results. Moreover, tank-treading motion, which is a characteristic motion of viscoelastic bodies in shear flows, is simulated by the present method.

Free energy lattice Boltzmann methods are well suited for the simulation of two phase flow problems. The model for the interface is based on well understood physical grounds. In most cases a numerical interface is used instead of the physical one because of lattice resolution limitations. In this paper we present a framework where we can both follow the droplet behavior in a coarse scale and solve the interface in a fine scale simultaneously. We apply the method for the simulation of a droplet using an interface to diameter size ratio of 1 to 280. In a second simulation, a small droplet coalesces with a 42 times larger droplet producing on it only a small capillary wave that propagates and dissipates.