Article

Higher-order Galilean-invariant lattice Boltzmann model for microflows: Single-component gas

Division of Chemical and Biomolecular Engineering, School of Chemical and Biomedical Engineering, Nanyang Technological University, 637459 Singapore, Singapore.
Physical Review E (Impact Factor: 2.29). 10/2010; 82(4 Pt 2):046701. DOI: 10.1103/PhysRevE.82.046701
Source: PubMed

ABSTRACT

We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation (without any cubic error). In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.

Download full-text

Full-text

Available from: Santosh Ansumali, Apr 11, 2014
  • Source
    • "The implementation of the diffuse Maxwell boundary condition using Gauss–Laguerre off-lattice quadrature models in Ref. [41] shows good results for Couette flow up to Kn = 0.5. By using an alternative framework, a high-order LB model with only 27 discrete velocities has been developed by Yudistiawan et al. [33] and it was shown that the corresponding moment system is quite similar to Grad's 26-moment system. This off-lattice D3Q27 model is able to represent both, Knudsen layer effects and the Knudsen minimum for Poiseuille flow. "
    [Show abstract] [Hide abstract]
    ABSTRACT: We analyze a large number of high-order discrete velocity models for solving the Boltzmann-BGK equation for finite Knudsen number flows. Using the Chapman-Enskog formalism, we prove for isothermal flows a relation identifying the resolved flow regimes for low Mach numbers. Although high-order lattice Boltzmann models recover flow regimes beyond the Navier-Stokes level we observe for several models significant deviations from reference results. We found this to be caused by their inability to recover the Maxwell boundary condition exactly. By using supplementary conditions for the gas-surface interaction it is shown how to systematically generate discrete velocity models of any order with the inherent ability to fulfill the diffuse Maxwell boundary condition accurately. Both high-order quadratures and an exact representation of the boundary condition turn out to be crucial for achieving reliable results. For Poiseuille flow, we can reproduce the mass flow and slip velocity up to the Knudsen number of 1. Moreover, for small Knudsen numbers, the Knudsen layer behavior is recovered.
    Full-text · Article · Oct 2015
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A general analysis of the hydrodynamic limit of multi-relaxation time lattice Boltzmann models is presented. We examine multi-relaxation time BGK collision operators that are constructed similarly to those for the MRT case, however, without explicitly moving into a moment space representation. The corresponding 'moments' are derived as left eigenvectors of said collision operator in velocity space. Consequently we can, in a representation independent of the chosen base velocity set, generate the conservation equations. We find a significant degree of freedom in the choice of the collision matrix and the associated basis which leaves the collision operator invariant. Therefore we can explain why MRT implementations in the literature reproduce identical hydrodynamics despite being based on different orthogonalization relations. Comment: 9 Pages
    Full-text · Article · Nov 2010 · Communications in Computational Physics
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Recently, kinetic theory-based lattice Boltzmann (LB) models have been developed to model nonequilibrium gas flows. Depending on the order of quadratures, a hierarchy of LB models can be constructed which we have previously shown to capture rarefaction effects in the standing-shear wave problems. Here, we further examine the capability of high-order LB models in modeling nonequilibrium flows considering gas and surface interactions and their effect on the bulk flow. The Maxwellian gas and surface interaction model, which has been commonly used in other kinetic methods including the direct simulation Monte Carlo method, is used in the LB simulations. In general, the LB models with high-order Gauss-Hermite quadratures can capture flow characteristics in the Knudsen layer and higher order quadratures give more accurate prediction. However, for the Gauss-Hermite quadratures, the present simulation results show that the LB models with the quadratures obtained from the even-order Hermite polynomials perform significantly better than those from the odd-order polynomials. This may be attributed to the zero-velocity component in the odd-order discrete set, which does not participate in wall and gas collisions, and thus underestimates the wall effect.
    Full-text · Article · Mar 2011 · Physical Review E
Show more