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Register a new userNumerical Study of the Properties of the Central Moment Lattice Boltzmann Method
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Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Since the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as accuracy, grid convergence and stability for well defined canonical problems is lacking and the present work is intended to fulfill this need. We perform a quantitative study of the performance of the cascaded LBM for a set of benchmark problems of differing complexity, viz., Poiseuille flow, decaying Taylor-Green vortex flow and lid-driven cavity flow. We first establish its grid convergence and demonstrate second order accuracy under diffusive scaling for both the velocity field and its derivatives, i.e. components of the strain rate tensor, as well. The method is shown to quantitatively reproduce steady/unsteady analytical solutions or other numerical results with excellent accuracy. Numerical experiments further demonstrate that the central moment MRT LBM results in significant stability improvements when compared with certain existing collision models at moderate additional computational cost.
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Cascaded lattice-Boltzmann method (Cascaded-LBM) employs a new class of collision operators aiming to improve numerical stability. It achieves this and distinguishes from other collision operators, such as in the standard single or multiple relaxation time approaches, by performing relaxation process due to collisions in terms of moments shifted by the local hydrodynamic fluid velocity, i.e. central moments, in an ascending order-by-order at different relaxation rates. In this paper, we propose and derive source terms in the Cascaded-LBM to represent the effect of external or internal forces on the dynamics of fluid motion. This is essentially achieved by matching the continuous form of the central moments of the source or forcing terms with its discrete version. Different forms of continuous central moments of sources, including one that is obtained from a local Maxwellian, are considered in this regard. As a result, the forcing terms obtained in this new formulation are Galilean invariant by construction. The method of central moments along with the associated orthogonal properties of the moment basis completely determines the expressions for the source terms as a function of the force and macroscopic velocity fields. In contrast to the existing forcing schemes, it is found that they involve higher order terms in velocity space. It is shown that the proposed approach implies generalization of both local equilibrium and source terms in the usual lattice frame of reference, which depend on the ratio of the relaxation times of moments of different orders. An analysis by means of the Chapman-Enskog multiscale expansion shows that the Cascaded-LBM with forcing terms is consistent with the Navier-Stokes equations. Computational experiments with canonical problems involving different types of forces demonstrate its accuracy.
Fluid motion driven by thermal effects, such as that due to buoyancy in differentially heated three-dimensional (3D) enclosures, arise in several natural settings and engineering applications. It is represented by the solutions of the Navier-Stokes equations (NSE) in conjunction with the thermal energy transport equation represented as a convection-diffusion equation (CDE) for the temperature field. In this study, we develop new 3D lattice Boltzmann (LB) methods based on central moments and using multiple relaxation times for the three-dimensional, fifteen velocity (D3Q15) lattice, as well as it subset, i.e. the three-dimensional, seven velocity (D3Q7) lattice to solve the 3D CDE for the temperature field in a double distribution function framework. Their collision operators lead to a cascaded structure involving higher order terms resulting in improved stability. In this approach, the fluid motion is solved by another 3D cascaded LB model from prior work. Owing to the differences in the number of collision invariants to represent the dynamics of flow and the transport of the temperature field, the structure of the collision operator for the 3D cascaded LB formulation for the CDE is found to be markedly different from that for the NSE. The new 3D cascaded (LB) models for thermal convective flows are validated for natural convection of air driven thermally on two vertically opposite faces in a cubic cavity enclosure at different Rayleigh numbers against prior numerical benchmark solutions. Results show good quantitative agreement of the profiles of the flow and thermal fields, and the magnitudes of the peak convection velocities as well as the heat transfer rates given in terms of the Nusselt number.
We present a new 3D lattice Boltzmann (LB) algorithm based on central moments for the D3Q27 lattice using a cuboid grid, which is parameterized by two grid aspect ratios that are related to the ratios of the particle speeds with respect to that along a reference coordinate direction. The use of the cuboid lattice grid enables the method to compute flows having different characteristic length scales in different directions more efficiently. It is constructed to simulate the Navier-Stokes equations consistently via introducing counteracting corrections to the second order moment equilibria obtained via a Chapman-Enskog analysis that eliminate the errors associated with the grid anisotropy and the non-Galilean invariant terms. The implementation is shown to be compact and modular, with an interpretation based on special matrices, admitting ready extension of the standard algorithm for the cubic lattice to the cuboid lattice via appropriate scaling of moments based on grid aspect ratios before and after collision step and equilibria corrections. The resulting formulation is general in that the same grid corrections developed for the D3Q27 lattice for recovering the correct viscous stress tensor is applicable for other lattice subsets, and a variety of collision models, including those based on the relaxation of raw moments, central moments and cumulants, as well as their special case involving the distribution functions. The cuboid central moment LBM is validated against a variety of benchmark flows, and when used in lieu of the corresponding raw moment formulation for simulating shear flows, we show that it results in significant improvements in numerical stability. Finally, we demonstrate that our cuboid LB approach is efficient in simulating anisotropic shear flow problems with significant savings in computational cost and memory storage when compared to that based on the cubic lattice.
Simulating inhomogeneous flows with different characteristic scales in different coordinate directions using the collide-and-stream based lattice Boltzmann methods (LBM) can be accomplished efficiently using rectangular lattice grids. We develop and investigate a new rectangular central moment LBM based on non-orthogonal moment basis (referred to as RC-LBM). The equilibria to which the central moments relax under collision in this approach are obtained from matching with those corresponding to the continuous Maxwell distribution. A Chapman-Enskog analysis is performed to derive the correction terms to the second order moment equilibria involving the grid aspect ratio and velocity gradients that restores the isotropy of the viscous stress tensor and eliminates the non-Galilean invariant cubic velocity terms of the resulting hydrodynamical equations. A special case of this rectangular formulation involving the raw moments (referred to as the RNR-LBM) is also constructed. The resulting schemes represent a considerable simplification, especially for the transformation matrices and isotropy corrections, and improvement over the existing MRT-LB schemes on rectangular lattice grids that use orthogonal moment basis. Numerical validation study of both the RC-LBM and RNR-LBM for a variety of benchmark flow problems are performed that show good accuracy at various grid aspect ratios. The ability of our proposed schemes to simulate flows using relatively lower grid aspect ratios than considered in prior rectangular LB approaches is demonstrated. Furthermore, simulations reveal the superior stability characteristics of the RC-LBM over RNR-LBM in handling shear flows at lower viscosities and/or higher characteristic velocities. In addition, computational advantages of using our rectangular LB formulation in lieu of that based on the square lattice is shown.
An Onsager-like relation is proposed as a new criterion for constructing and analysing the lattice Boltzmann (LB) method. For LB models obeying the relation, we analyse their linearized stability, establish their diffusive limit, and find new constraints for those with free parameters. The new relation seems of fundamental importance for the LB method.
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