Lattice-Boltzmann (LB) models provide a systematic formulation of effective-field computational approaches to the calculation of multiphase flow by replacing the mathematical surface of separation between the vapor and liquid with a thin transition region, across which all magnitudes change continuously. Many existing multiphase models of this sort do not satisfy the rigorous hydrodynamic constitutive laws. Here, we extend the two-dimensional, seven-speed Swift et al. LB model1 to rectangular grids (nine speeds) by using symbolic manipulation (MathematicaTM) and compare the LB model predictions with benchmark problems, in order to evaluate its merits. Particular emphasis is placed on the stress tensor formulation. Comparison with the two-phase analogue of the Couette flow and with a flow involving shear and advection of a droplet surrounded by its vapor reveals that additional terms have to be introduced in the definition of the stress tensor in order to satisfy the Navier–Stokes equation in regions of high density gradients. The use of Mathematica obviates many of the difficulties with the calculations "by-hand," allowing at the same time more flexibility to the computational analyst to experiment with geometrical and physical parameters of the formulation.

No abstract

SUMMARYQuadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains. These quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries. Specifically, the integrands considered here consist of a quadratic polynomial multiplied by a Heaviside or Dirac delta function operating on a linear polynomial. This form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function. Exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values. Compared with methods involving explicit integration over subdomains, the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical Jacobian information describing the dependence of the integrals on the nodal level set values. The accuracy of the method is demonstrated for a simple conduction problem with the Neumann and Robin-type boundary conditions.

A numerical study of the isothermal migration of a two-dimensional bubble in Poiseuille flow is reported here for vapor–liquid density and dynamic viscosity ratios of 1/8, Red=1, and Ca=2. A lattice Boltzmann model with a van der Waals equation of state is employed to simulate the diffuse interface for three interface thickness to bubble diameter ratios, 1/5, 1/10, and 1/20. Point-by-point comparisons with the sharp-interface incompressible counterpart (reported in the literature) reveal velocity discrepancies which are more evident on the vapor side. These differences are a manifestation of a finite mass flux through the interface, associated with driven finite–thickness interfaces. An analytical study of the one-dimensional analog of the traveling diffuse interface problem explains this phenomenon and shows that this flux vanishes as a result of viscous dissipation as the interface thickness tends to zero. This trend is corroborated by the two-dimensional lattice Boltzmann results.

Motivated by systematic CO2 evaporation experiments which recently became available (J. Pettersen, “Flow vaporization of CO2 in microchannel tubes,” Doctor technicae thesis, Norwegian University of Science and Technology, 2002), the present work constitutes an exploratory investigation of isothermal flow of CO2 near its liquid–vapor critical point through a long 5 μm diameter microchannel. A modified van der Waals constitutive model—with properties closely approximating those of “real” near-critical CO2—is incorporated in a two-dimensional lattice Boltzmann hydrodynamics model by embedding a dimensionless parameter X, with X→1 denoting the “real” fluid. The hydrodynamic phenomena resulting by imposing a constant pressure gradient along a periodic channel are investigated by considering two regimes in tandem: (1) transition from bubbly to annular flow with a liquid film formed at the channel walls and (2) destabilization of the liquid film by the Kelvin–Helmholtz instability. Due to numerical constraints, intrinsic modeling errors are introduced and are shown to be associated with discrepancies in the relative vapor–liquid interfacial thickness, which is expressed by X. The effects of these errors are investigated both theoretically and numerically in the physical limit X→1. Numerically determined flow patterns compare qualitatively well with direct visualization results obtained by Pettersen. Overall, the characteristics of isothermal near-critical two-phase flow in microchannels can be reproduced by the appropriate modification of the thermophysical properties of CO2.