Simplified and Highly Stable Lattice Boltzmann Method

The following sections are included:

• Dedication
• Preface
• Contents
• List of Tables
• List of Figures

The following sections are included:

• Fluid mechanics: physical laws and mathematical models
• The kinetic theory and the Boltzmann equation
• From the Boltzmann equation to the lattice Boltzmann equation
• Lattice Boltzmann models
• From lattice Boltzmann equation to Navier-Stokes equations: Chapman-Enskog expansion analysis
• Boundary treatments in lattice Boltzmann method
  • Heuristic schemes
  • Hydrodynamic schemes
  • Extrapolation schemes
• Computational sequence of lattice Boltzmann method
• Advantages and limitations of lattice Boltzmann method
• Motivations of developing simplified and highly stable lattice Boltzmann method (SHSLBM)

In this chapter, a new approach of constructing models within the lattice Boltzmann framework, namely the simplified and highly stable lattice Boltzmann method (SHSLBM), will be presented, aiming to reduce the memory cost, facilitate the implementation of physical boundary conditions, and enhance the numerical stability while maintaining merits of the conventional lattice Boltzmann method (LBM) such as the kinetic nature, simplicity and explicitness. With the notion that two major limitations of the conventional LBM originate from the evolution of distribution functions, a promising direction of improvement is to replace the dependent variable in LBM with the macroscopic flow properties. The connection between the distribution functions and the macroscopic flow properties has been established in the Chapman-Enskog (C-E) expansion analysis, paving the way of such improvement. The resultant SHSLBM essentially reconstructs solutions to the macroscopic equations recovered from the lattice Boltzmann equation (LBE) through the C-E analysis and resolved in a predictor-corrector manner. The new method brings two immediate merits. Firstly, by directly evolving the macroscopic flow variables instead of the distribution functions, significant reduction in memory cost can be expected. In the meantime, the physical boundary conditions can be employed straightforwardly without tedious transformation into conditions for the distribution functions. It will also be shown in this chapter that, by incorporating proper interpolation algorithm, the SHSLBM can be flexibly implemented on non-uniform or body-fitted meshes.

In this chapter, the simplified and highly stable lattice Boltzmann method (SHSLBM) derived in Chapter 2 will be comprehensively evaluated in terms of accuracy, stability, memory cost and efficiency. The construction process of the SHSLBM suggests that it bears comparable accuracy as the conventional LBM. To validate its practical accuracy, numerical test of the Taylor-Green vortex flow will be carried out. In the meantime, the stability of the SHSLBM will be revealed both analytically and numerically. The von Neumann stability analysis demonstrates that the SHSLBM is stable at all wavenumbers in high Reynolds number scenario, which outperforms the conventional LBM. Numerical tests at high Reynolds numbers and on coarse meshes later validate the analytical proofs. It is also shown in numerical experiments that, in general cases, the SHSLBM consumes less virtual memory but is less efficient than the conventional LBM.

A sample application of the SHSLBM to 2D lid-driven cavity flow will be elaborated in this chapter. The source code written in Fortran 90 language as well as detailed comments will be attached to facilitate readers' implementation. The numerical results verify that the SHSLBM is comparably accurate as the conventional LBM but is much more stable at high Reynolds numbers.

In this chapter, the simplified and highly stable lattice Boltzmann method (SHSLBM) is extended to model fluid systems that exert non-Newtonian behaviors or present thermal effects. In the simulations of non-Newtonian fluid flows, the truncated power-law model is applied to establish the constitutional relationship which locally adjusts the physical viscosity with the shear rate. The relaxation parameter is thus no longer a global quantity and may experience significant variation over the flow field. Compared with the conventional LBM, the SHSLBM has better numerical stability at high Reynolds numbers, which is more adaptable to the wide variation range of the relaxation parameter in non-Newtonian simulations.

To cope with thermal simulations, the simplified and highly stable thermal lattice Boltzmann method (SHSTLBM) is developed with proper extension from the isothermal model. The SHSTLBM starts from the double-distribution-function thermal lattice Boltzmann method and reconstructs solutions to the governing equations recovered at the macroscopic scale. The new method removes intrinsic limitations of the thermal lattice Boltzmann method, such as high cost of virtual memory and inconvenient implementation of physical boundary conditions. It also inherits good numerical stability from its isothermal counterpart, which can be proven both theoretically and numerically in cases with high Reynolds or Rayleigh numbers. Sample applications to benchmark problems will also be presented in this chapter to illustrate the accuracy and robustness of the proposed solvers.

In this chapter, the simplified and highly stable lattice Boltzmann method (SHSLBM) is consolidated with the boundary condition-enforced immersed boundary method to enrich its applicability in problems with complex geometries or moving boundaries. The resultant immersed boundary – simplified lattice Boltzmann method (IB-SLBM) resolves the problem in two steps. In the predictor step, the SHSLBM is utilized to resolve the flow field without considering the immersed objects. Due to its direct evolution of macroscopic variables, the SHSLBM allows direct interpretations of the physical boundary conditions and thus is more compatible with the immersed boundary method than the conventional LBM. In the corrector step, the boundary condition – enforced immersed boundary method is implemented to impose the boundary effects of the immersed object on the flow field. The intermediate flow properties obtained in the predictor step are corrected based on the principle of fulfilling boundary conditions on the surface of the immersed objects. Such corrections represent the momentum and energy exchanges on the boundary, which also facilitates the evaluation of hydrodynamic forces and thermal parameters on the immersed object. Through some sample applications to isothermal or thermal problems with complex geometries or moving boundaries, the robustness of this solver will be comprehensively validated.

The axisymmetric assumption is an effective approach to downgrade the problem from three-dimensionality to two-dimensionality. In this chapter, the simplified and highly stable lattice Boltzmann method (SHSLBM) is applied to resolve flow problems that fulfill the axisymmetric assumption, which derives the simplified axisymmetric lattice Boltzmann method (SALBM). The procedure is still to reconstruct solutions to the macroscopic equations recovered from the axisymmetric lattice Boltzmann equation through the Chapman-Enskog (C-E) expansion analysis. However, given the large magnitudes of the forcing terms involved in the axisymmetric lattice Boltzmann equation, extra measures should be taken in the reconstruction process to interpret their effects on the solutions. These measures will be detailed in this chapter and provide instructions for the next chapter on the multiphase flow simulations in which extra forcing terms should also be considered in the reconstruction process.

In this chapter, the simplified and highly stable lattice Boltzmann method (SHSLBM) is extended to tackle flow problems with multiple components. The multiphase flow solver is derived from reconstructing solutions to macroscopic equations recovered from the multiphase lattice Boltzmann equation through the Chapman-Enskog (C-E) expansion analysis and resolved in a predictor-corrector manner. Similar to the axisymmetric flow simulations stated in the previous chapter, governing equations for the multiphase flows also involve large forcing terms whose effects on the non-equilibrium part of the distribution functions cannot be ignored. By delicately consolidating the effects of these forcing terms into the solution reconstruction, a simplified multiphase lattice Boltzmann method (SMLBM) is presented in this chapter for modelling incompressible multiphase flows with large density ratios and complex interfaces. The diffuse interface (DI) method, which resolves the Cahn-Hilliard (C-H) equation, is utilized as the interface tracking algorithm, whose solution is also reconstructed using the idea of the SHSLBM. The resultant formulations of the SMLBM directly update macroscopic variables instead of distribution functions, which remarkably reduces the cost of virtual memory and allows interpreting physical boundary conditions in a straightforward way. Sample applications will also be provided to evaluate the SMLBM in challenging multiphase simulations with large density ratios, complex interfaces, and high Reynolds numbers.

The simplified and highly stable lattice Boltzmann method (SHSLBM) and the various extensions presented in the previous chapters essentially reconstruct solutions to the macroscopic equations recovered from the lattice Boltzmann equation (LBE) and resolved in a predictor-corrector scheme. Apart from obvious merits like the low cost in virtual memory and the convenient implementation of physical boundary conditions, an intriguing advantage of the SHSLBM is its good numerical stability. Through the von Neumann stability analysis, the SHSLBM can be theoretically proven to be much more stable than the conventional LBM at high Reynolds or Rayleigh numbers. However, this advantage might be constrained by the second order accuracy. Although the SHSLBM can achieve numerical convergence on coarse meshes, the obtained results are inaccurate due to its second-order nature. Therefore, improving the accuracy of the SHSLBM while maintaining its merits could enrich its applicability in engineering problems…

The following sections are included:

• Appendix A Description of Sample Codes
  • Sample Code of the SHSLBM for 2D Lid-driven Cavity Flow
  • Sample Code of the SHSTLBM for 2D Natural Convection in A Square Cavity
  • Sample Code of the IB-SLBM for Flow Past A Uniformly Accelerated Plate
  • Sample Code of the SMLBM for Laplace Law
  • Sample Code of the HSLBM for Taylor-Green Vortex Flow
• Bibliography
• Index