Liangqi Zhang

Consistent lattice Boltzmann methods for incompressible axisymmetric flows.

In this work, consistent lattice Boltzmann (LB) methods for incompressible axisymmetric flows are developed based on two efficient axisymmetric LB models available in the literature. In accord with their respective original models, the proposed axisymmetric models evolve within the framework of the standard LB method and the source terms contain no gradient calculations. Moreover, the incompressibility conditions are realized with the Hermite expansion, thus the compressibility errors arising in the existing models are expected to be reduced by the proposed incompressible models. In addition, an extra relaxation parameter is added to the Bhatnagar-Gross-Krook collision operator to suppress the effect of the ghost variable and thus the numerical stability of the present models is significantly improved. Theoretical analyses, based on the Chapman-Enskog expansion and the equivalent moment system, are performed to derive the macroscopic equations from the LB models and the resulting truncation terms (i.e., the compressibility errors) are investigated. In addition, numerical validations are carried out based on four well-acknowledged benchmark tests and the accuracy and applicability of the proposed incompressible axisymmetric LB models are verified.

A comparative study of lattice Boltzmann models for incompressible flow

Abstract For incompressible flow, a comparative study on the four lattice Boltzmann (LB) models, the standard model, the He–Luo model, Guo’s model, and the present model, is performed. Theoretically, the macroscopic equations derived from the involved LB models are compared by the Chapman–Enskog analysis. Then, the analytical framework proposed in M. Junk’s work is applied to investigate the finite difference stencils and the equivalent moment systems pertaining to the concerned LB models. Conclusions are drawn from the theoretical derivations that the truncated error terms, which differ among the concerned LB models, have effects on the accuracy of the modeled deviatoric stress. Moreover, the cavity flow in two dimensions is adopted as a benchmark test to confirm the theoretical demonstrations. The resulting velocity fields from the present model are more in line with the reference solutions in the region of high deviatoric stress than other three LB models, which is consistent with the theoretical expectations and is further confirmed by the comparisons of the truncation error terms. In addition, we also conclude from the numerical tests that the present model has the advantage of better convergence efficiency but suffers from the worse stability.

Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit

Because the standard lattice Boltzmann (LB) method is proposed for Cartesian Navier-Stokes (NS) equations, additional source terms are necessary in the axisymmetric LB method for representing the axisymmetric effects. Therefore, the accuracy and applicability of the axisymmetric LB models depend on the forcing schemes adopted for discretization of the source terms. In this study, three forcing schemes, namely, the trapezium rule based scheme, the direct forcing scheme, and the semi-implicit centered scheme, are analyzed theoretically by investigating their derived macroscopic equations in the diffusive scale. Particularly, the finite difference interpretation of the standard LB method is extended to the LB equations with source terms, and then the accuracy of different forcing schemes is evaluated for the axisymmetric LB method. Theoretical analysis indicates that the discrete lattice effects arising from the direct forcing scheme are part of the truncation error terms and thus would not affect the overall accuracy of the standard LB method with general force term (i.e., only the source terms in the momentum equation are considered), but lead to incorrect macroscopic equations for the axisymmetric LB models. On the other hand, the trapezium rule based scheme and the semi-implicit centered scheme both have the advantage of avoiding the discrete lattice effects and recovering the correct macroscopic equations. Numerical tests applied for validating the theoretical analysis show that both the numerical stability and the accuracy of the axisymmetric LB simulations are affected by the direct forcing scheme, which indicate that forcing schemes free of the discrete lattice effects are necessary for the axisymmetric LB method.

A Fourier-Legendre spectral element method in polar coordinates

In this paper, a new Fourier-Legendre spectral element method based on the Galerkin formulation is proposed to solve the Poisson-type equations in polar coordinates. The 1/r singularity at r=0 is avoided by using Gauss-Radau type quadrature points. In order to break the time-step restriction in the time-dependent problems, the clustering of collocation points near the pole is prevented through the technique of domain decomposition in the radial direction. A number of Poisson-type equations subject to the Dirichlet or Neumann boundary condition are computed and compared with the results in literature, which reveals a desirable result.

Numerical study on the axial segregation dynamics of a binary-size granular mixture in a three-dimensional rotating drum

Granular materials are ubiquitous in our daily life and inherent in multitudinous industrial processes. Differences in the granular properties such as size and density inevitably induce segregation. By means of the discrete element method, a binary-size mixture in a three-dimensional rotating drum is numerically simulated to explore the segregation dynamics of the granular material along the axial direction. Snapshots of the distribution of the two particle types in the rotating drum are presented with respect to time to illustrate the spatial evolution of the size-induced segregation structure. The space-time plots of various axial characteristics indicate that (i) radial segregation does not affect the axial distribution of total mass and mass fraction, but axial segregation leads to the formation of axial bands; (ii) greater non-dimensionalized collision forces for both the large and small particles develop where the large particles dominate; and (iii) axial segregation gives rise to the variation of t...

DEM study of the size-induced segregation dynamics of a ternary-size granular mixture in the rolling-regime rotating drum

Segregation induced by size, shape, or density difference of the granular material is inevitable in both natural and industrial processes; unfortunately, the underlying mechanism is still not fully understood. In view of the ubiquitous continuous particle size distributions, this study builds on the considerable knowledge gained so far from binary-size mixtures and extends it to a ternary-size mixture to understand the impact of the presence of a third particle size in the three-dimensional rotating drum operating in the rolling flow regime. The discrete element method is employed. The evolution of segregation, the active-passive interface, and the dynamical response of the particle-scale characteristics of the different particle types in the two regions are investigated. The results reveal that the medium particles are spatially sandwiched in between the large and small particles in both the radial and axial directions and therefore exhibit behaviors intermediate to the other two particle types. Compared...

A comparative study of the axisymmetric lattice Boltzmann models under the incompressible limit

A comparative study on four axisymmetric lattice Boltzmann (LB) models, namely, the kinetic theory based model by Guo etal. (2009), the consistent model by Li etal. (2010), the centered scheme model by Zhou (2011), and our model (based on applying the centered scheme to the Guo etal. (2009) model), is conducted both theoretically and numerically. The finite difference interpretation of the LB method by Junk (2001) is applied to evaluate the accuracy of the models under the incompressible limit. Particularly, the finite difference stencils adopted for the spatial gradient terms in the macroscopic axisymmetric NavierStokes (NS) equations are compared. Besides, the numerical performance (i.e., the numerical accuracy, stability and the convergence efficiency) of the models is compared by two benchmark tests, namely, the unsteady-state Womersley flow and the cylindrical cavity flow. The numerical results accord well with the theoretical analysis. Additionally, it is also found that the numerical stability of the axisymmetric LB models is effectively improved by removing the effects from the non-hydrodynamic variables.

Three-dimensional multi-relaxation-time lattice Boltzmann front-tracking method for two-phase flow*

We developed a three-dimensional multi-relaxation-time lattice Boltzmann method for incompressible and immiscible two-phase flow by coupling with a front-tracking technique. The flow field was simulated by using an Eulerian grid, an adaptive unstructured triangular Lagrangian grid was applied to track explicitly the motion of the two-fluid interface, and an indicator function was introduced to update accurately the fluid properties. The surface tension was computed directly on a triangular Lagrangian grid, and then the surface tension was distributed to the background Eulerian grid. Three benchmarks of two-phase flow, including the Laplace law for a stationary drop, the oscillation of a three-dimensional ellipsoidal drop, and the drop deformation in a shear flow, were simulated to validate the present model.

An alternative lattice Boltzmann model for three-dimensional incompressible flow

a b s t r a c t In this work, an alternative lattice Boltzmann (LB) model for three-dimensional (3D) in- compressible flow is proposed. The equilibrium distribution function (EDF) of the present model is directly derived in accordance with the incompressibility conditions by applying the Hermite expansion. Moreover, an alternative formula for pressure computation is de- signed from the second order moment of the distribution function. The present 3D LB model inherits the advantageous features of Guos LB model: the density is a constant, the fluid pressure is independent of density and the Navier-Stokes (N-S) equations for incompress- ible flow can be derived. Two benchmark tests, flow over a backward-facing step and the lid-driven cavity flow, are applied to validate the present model. Accurate results for these tests are obtained with the present model, and further comparisons with the previous LB models (the standard LB model, the He-Luo model and Guos LB model) demonstrate that the present model provides better accuracy in the region of high deviatoric stress and such advantage is further enhanced by using the D3Q27 lattice.