Discretization of the velocity space in solution of the Boltzmann equation

We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to be equivalent to truncating the Hermite expansion of the distribution function to the corresponding order. The truncated part of the distribution has no contribution to the moments of low orders and is negligible at small Mach numbers. Higher order approximations to the Boltzmann equation can be achieved by using more velocities in the quadrature

Using the lid-driven cavity flow to validate moment-based boundary conditions for the Lattice Boltzmann Equation

The accuracy of the Moment Method for imposing no-slip boundary conditions in the lattice Boltzmann algorithm is investigated numerically using lid-driven cavity flow. Boundary conditions are imposed directly upon the hydrodynamic moments of the lattice Boltzmann equations, rather than the distribution functions, to ensure the constraints are satisfied precisely at grid points. Both single and multiple relaxation time models are applied. The results are in excellent agreement with data obtained from state-of-the-art numerical methods and are shown to converge with second order accuracy in grid spacing

Evaluation of recombinant adenovirus-mediated gene delivery for expression of tracer genes in catecholaminergic neurons

Selective labeling of small populations of neurons of a given phenotype for conventional neuronal tracing is difficult because tracers can be taken up by all neurons at the injection site, resulting in nonspecific labeling of unrelated pathways. To overcome these problems, genetic approaches have been developed that introduce tracer proteins as transgenes under the control of cell-type-specific promoter elements for visualization of specific neuronal pathways. The aim of this study was to explore the use of tracer gene expression for neuroanatomical tracing to chart the complex interconnections of the central nervous system. Genetic tracing methods allow for expression of tracer molecules using cell-type-specific promoters to facilitate neuronal tracing. In this study, the rat tyrosine hydroxylase (TH) promoter and an adenoviral delivery system were used to express tracers specifically in dopaminergic and noradrenergic neurons. Region-specific expression of the transgenes was then analyzed. Initially, we characterized cell-type-specific expression of GFP or RFP in cultured cell lines. We then injected an adenovirus carrying the tracer transgene into several brain regions using a stereotaxic apparatus. Three days after injection, strong GFP expression was observed in the injected site of the brain. RFP and WGA were expressed in a cell-type-specific manner in the cerebellum, locus coeruleus, and ventral tegmental regions. Our results demonstrate that selective tracing of catecholaminergic neuronal circuits is possible in the rat brain using the TH promoter and adenoviral expression

On the Three-dimensional Central Moment Lattice Boltzmann Method

A three-dimensional (3D) lattice Boltzmann method based on central moments is derived. Two main elements are the local attractors in the collision term and the source terms representing the effect of external and/or self-consistent internal forces. For suitable choices of the orthogonal moment basis for the three-dimensional, twenty seven velocity (D3Q27), and, its subset, fifteen velocity (D3Q15) lattice models, attractors are expressed in terms of factorization of lower order moments as suggested in an earlier work; the corresponding source terms are specified to correctly influence lower order hydrodynamic fields, while avoiding aliasing effects for higher order moments. These are achieved by successively matching the corresponding continuous and discrete central moments at various orders, with the final expressions written in terms of raw moments via a transformation based on the binomial theorem. Furthermore, to alleviate the discrete effects with the source terms, they are treated to be temporally semi-implicit and second-order, with the implicitness subsequently removed by means of a transformation. As a result, the approach is frame-invariant by construction and its emergent dynamics describing fully 3D fluid motion in the presence of force fields is Galilean invariant. Numerical experiments for a set of benchmark problems demonstrate its accuracy.Comment: 55 pages, 8 figure