Session E29: CFD: Lattice Boltzmann Methods

How good is the Lattice Boltzmann method?

Conflicting opinions exist in literature regarding how efficient the lattice Boltzmann method is relative to high-order finite difference approximations of the Navier-Stokes equations on Cartesian meshes, especially at high Mach numbers. We address the question from the pragmatic viewpoint of a practitioner. Dispersion, dissipation and aliasing errors of various lattice Boltzmann models are systematically quantified. The number of floating point operations and memory required for a desired accuracy level are carefully compared for the two numerical methods. Turbulent kinetic energy budgets for several standard test cases such as the decaying Taylor-Green vortex problem are used to evaluate how effective the stabilization mechanisms necessary for lattice Boltzmann method at high Reynolds numbers are. Detailed comments regarding the cyclomatic complexity of the underlying software, scalability of the underlying algorithm on state-of-the-art high-performance computing platforms and wall clock times and relative accuracy for selected simulations conducted using the two approaches are also made.   

Lattice Boltzmann Models for Flows with Axial Symmetry and Mass and Momentum Sources without Cubic Velocity Errors

Three-dimensional flows with axial symmetry arise in numerous applications, which can be solved more efficiently on a two-dimensional Cartesian coordinate system with appropriate source terms. Lattice Boltzmann (LB) method is a promising recent development in CFD. However, existing LB models are not Galilean invariant (GI) due to the degeneracy of the resulting third-order longitudinal moments, which leads to cubic velocity truncation errors. This can lead to anisotropic stress tensor with velocity-dependent viscosities and numerical instability under high shear even with finer grids. In this investigation, we develop a new radius-weighted LB model for axisymmetric flows using a non-orthogonal moment basis with an extended moment equilibria and restore GI on standard lattices. Also, as another related example, we consider flows with mass and momentum sources, which are important in various contexts, including acoustics, reacting flows and flows undergoing phase change. To handle such problems, we develop a new LB model by incorporating sources in its zeroth and first order moments, with extended moment equilibria to eliminate the cubic velocity errors. Both the resulting new models will be validated for benchmark problems.   

An Improved Lattice Boltzmann Model for Non-Newtonian Flows with Applications to Solid-Fluid Interactions in External Flows

Fluid mechanics of non-Newtonian fluids, which arise in numerous settings, are characterized by non-linear constitutive models that pose certain unique challenges for computational methods. Here, we consider the lattice Boltzmann method (LBM), which offers some computational advantages due to its kinetic basis and its simpler stream-and-collide procedure enabling efficient simulations. However, further improvements are necessary to improve its numerical stability and accuracy for computations involving broader parameter ranges. Hence, in this study, we extend the cascaded LBM formulation by modifying its moment equilibria and relaxation parameters to handle a variety of non-Newtonian constitutive equations, including power-law and Bingham fluids, with improved stability. In addition, we include corrections to the moment equilibria to obtain an inertial frame invariant scheme without cubic-velocity defects. After preforming its validation study for various benchmark flows, we study the physics of non-Newtonian flow over pairs of circular and square cylinders in a tandem arrangement, especially the wake structure interactions and their effects on resulting forces in each cylinder, and elucidate the effect of the various characteristic parameters.   

The flow around a flapping foil

The flow around a two-dimensional flapping foil immersed in a uniform stream is studied numerically using a Lattice-Boltzmann model, for Reynolds numbers between $100$ and $250$, and flapping Strouhal numbers between $0.01$ and $0.6$. The computation of the hydrodynamic force on the foil is related to the the wake structure. When the foil's is fixed in space, numerical results suggest a relation between drag coefficient behaviour and the flapping frequency which determines the transition from the von K\'arm\'an to the inverted von K\'arm\'an wake. When the foil is free of translational motion up-stream swimming at constant speed is observed at certain values of the flapping Strouhal.   

LBM-DSMC Hybrid Method for Complex Out-of-Equilibrium Flows

Many complex flows are characterized by the simultaneous presence of a range of non-equilibrium and rarefaction effects in different regions of the flow field. We recently developed a Direct Simulation Monte Carlo (DSMC)-Lattice Boltzmann Method (LBM) hybrid scheme, based on domain decomposition technique and on Grad’s moments method, able to accurately and efficiently simulate such flows. While DSMC is employed to compute the flow field only where large non-equilibrium effects are present, the more computationally efficient LBM is employed wherever the non-equilibrium effects can be dealt with perturbatively, i.e. according to Navier-Stokes hydrodynamics. Here we present the results on the application of the hybrid method to complex three-dimensional flows, in particular to the flow around a microsphere and through a disk-shaped expansion channel. The solutions provided by the hybrid method are compared against full DSMC simulations and the computational gain guaranteed by the application of the hybrid method over the full DSMC is also demonstrated.