We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

Lattice boltzmann method for fluid flows

Perturbed Free Shear Layers

Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

http://dsec.pku.edu.cn/~tieli/notes/numer_anal/MCQMC_Caflisch.pdf

Monte Carlo and quasi-Monte Carlo methods

Structural features resulting from the interaction of a turbulent jet issuing transversely into a uniform stream are described with the help of flow visualization and hot-wire anemometry. Jet-to-crossflow velocity ratios from 2 to 10 were investigated at crossflow Reynolds numbers from 3800 to 11400. In particular, the origin and formation of the vortices in the wake are described and shown to be fundamentally different from the well-known phenomenon of vortex shedding from solid bluff bodies. The flow around a transverse jet does not separate from the jet and does not shed vorticity into the wake. Instead, the wake vortices have their origins in the laminar boundary layer of the wall from which the jet issues. It is argued that the closed flow around the jet imposes an adverse pressure gradient on the wall, on the downstream lateral sides of the jet, provoking 'separation events’ in the wall boundary layer on each side. These result in eruptions of boundary-layer fluid and formation of wake vortices that are convected downstream. The measured wake Strouhal frequencies, which depend on the jet-crossflow velocity ratio, match the measured frequencies of the separation events. The wake structure is most orderly and the corresponding wake Strouhal number (0.13) is most sharply defined for velocity ratios near the value 4. Measured wake profiles show deficits of both momentum and total pressure.

/pdf/vortical-structure-in-the-wake-of-a-transverse-jet-46o027fnli.pdf

Vortical structure in the wake of a transverse jet

Turbulent combustion modeling

Arguments are presented to show that the concept of gradient diffusion is inapplicable to mixing in turbulent shear layers. A new model is proposed for treating molecular mixing and chemical reaction in such flows at high Reynolds number. It is based upon the experimental observations that revealed the presence of coherent structures and that showed that fluid elements from the two streams are distributed unmixed throughout the layer by large-scale inviscid motions. The model incorporates features of the strained flame model and makes use of the Kolmogorov cascade in scales. Several model predictions differ markedly from those of diffusion models and suggest experiments for testing the two approaches.

/pdf/a-simple-model-of-mixing-and-chemical-reaction-in-a-wbi0dud8ol.pdf

A Simple Model of Mixing and Chemical Reaction in a Turbulent Shear Layer

The flow field induced by a jet in incompressible cross-flow is analysed and the results compared with those obtained in a reacting water-jet experiment. It is argued that the axial vortex pair in the flow arises from the jet momentum normal to the free stream, the momentum flux being equivalent to a normal force, i.e. to a lift.

/pdf/structure-and-mixing-of-a-transverse-jet-in-incompressible-30ab1i1ygp.pdf

Structure and mixing of a transverse jet in incompressible flow

The structure of a shock wave in a simple discrete velocity gas—a gas in which the molecules move with a finite set of velocities—is discussed. The Boltzmann equation becomes, for this gas, a set of coupled differential equations which, in the present example, can be solved exactly. The solution describes an infinite Mach number shock in a gas consisting of hard elastic spheres. Although only six molecular velocities are considered, and the solution is easy to obtain, it compares remarkably well with those of other investigators.

Shock Structure in a Simple Discrete Velocity Gas

The application of a simple discrete velocity model to low Mach number Couette and Rayleigh flow is investigated. In the model, the molecular velocities are restricted to a finite set and in this study only eight equal speed velocities are allowed. The Boltzmann equation is reduced by this approximation to a set of coupled differential equations which can be solved in closed form. The fluid velocity and shear stress in Couette flow are in approximate accord with those of Wang Chang & Uhlenbeck (1954) and of Lees (1959) over the complete range of Knudsen number. Similarly, the Rayleigh flow solution is remarkably like those found by other investigators using moment methods.

Study of rarefied shear flow by the discrete velocity method

A description of the turbulent diffusion flame is proposed in which the flame structure is composed of a distribution of laminar diffusion flame elements, whose thickness is small in comparison with the large eddies. These elements retain their identity during the flame development; they are strained in their own plane by the gas motion, a process that not only extends their surface area, but also establishes the rate at which a flame element consumes the reactants. Where this flame stretching process has produced a high flame surface density, the flame area per unit volume, adjacent flame elements may consume the intervening reactant, thereby annihilating both flame segments. This is the flame shortening mechanism which, in balance with the flame stretching process, establishes the local level of the flame density. The consumption rate of reactant is then given simply by the product of the local flame density and the reactang consumption rate per unit area of flame surface. The proposed description permits a rather complete separation of the turbulent flow structure, on one hand, and the flame structure, on the other, and in this manner permits the treatment of reactions with complex chemistry with a minimum of added labor. The structure of the strained laminar diffusion flame may be determined by analysis, numerical computation, and by experiment without significant change to the model.

/pdf/the-coherent-flame-model-for-turbulent-chemical-reactions-3dbw9ieaxw.pdf

James E. Broadwell

Papers

A Simple Model of Mixing and Chemical Reaction in a Turbulent Shear Layer

Structure and mixing of a transverse jet in incompressible flow

Shock Structure in a Simple Discrete Velocity Gas

Study of rarefied shear flow by the discrete velocity method

The Coherent Flame Model for Turbulent Chemical Reactions