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Register a new userCascaded Lattice Boltzmann Method based on Central Moments for Axisymmetric Thermal Flows Including Swirling Effects
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A cascaded lattice Boltzmann (LB) approach based on central moments and multiple relaxation times to simulate thermal convective flows, which are driven by buoyancy forces and/or swirling effects, in the cylindrical coordinate system with axial symmetry is presented. In this regard, the dynamics of the axial and radial momentum components along with the pressure are represented by means of the 2D Navier-Stokes equations with geometric mass and momentum source terms in the pseudo Cartesian form, while the evolutions of the azimuthal momentum and the temperature field are each modeled by an advection-diffusion type equation with appropriate local source terms. Based on these, cascaded LB schemes involving three distribution functions are formulated to solve for the fluid motion in the meridian plane using a D2Q9 lattice, and to solve for the azimuthal momentum and the temperature field each using a D2Q5 lattice. The geometric mass and momentum source terms for the flow fields and the energy source term for the temperature field are included using a new symmetric operator splitting technique, via pre-collision and post-collision source steps around the cascaded collision step for each distribution function. These result in a particularly simple and compact formulation to directly represent the effect of various geometric source terms consistently in terms of changes in the appropriate zeroth and first order moments. Simulations of several complex buoyancy-driven thermal flows and including rotational effects in cylindrical geometries using the new axisymmetric cascaded LB schemes show good agreement with prior benchmark results for the structures of the velocity and thermal fields as well as the heat transfer rates given in terms of the Nusselt numbers.
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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.
In this paper, we develop and characterize the fully dissipative Lattice Boltzmann method for ultra-relativistic fluids in two dimensions using three equilibrium distribution functions: Maxwell-Juttner, Fermi-Dirac and Bose-Einstein. Our results stem from the expansion of these distribution functions up to fifth order in relativistic polynomials. We also obtain new Gaussian quadratures for square lattices that preserve the spatial resolution. Our models are validated with the Riemann problem and the limitations of lower order expansions to calculate higher order moments are shown. The kinematic viscosity and the thermal conductivity are numerically obtained using the Taylor-Green vortex and the Fourier flow respectively and these transport coefficients are compared with the theoretical prediction from Grads theory. In order to compare different expansion orders, we analyze the temperature and heat flux fields on the time evolution of a hot spot.
Operator split forcing schemes exploiting a symmetrization principle, i.e. Strang splitting, for cascaded lattice Boltzmann (LB) methods in two- and three-dimensions for fluid flows with impressed local forces are presented. Analogous scheme for the passive scalar transport represented by a convection-diffusion equation with a source term in a novel cascaded LB formulation is also derived. They are based on symmetric applications of the split solutions of the changes on the scalar field/fluid momentum due to the sources/forces over half time steps before and after the collision step. The latter step is effectively represented in terms of the post-collision change of moments at zeroth and first orders, respectively, to represent the effect of the sources on the scalar transport and forces on the fluid flow. Such symmetrized operator split cascaded LB schemes are consistent with the second-order Strang splitting and naturally avoid any discrete effects due to forces/sources by appropriately projecting their effects for higher order moments. All the force/source implementation steps are performed only in the moment space and they do not require formulations as extra terms and their additional transformations to the velocity space. These result in particularly simpler and efficient schemes to incorporate forces/sources in the cascaded LB methods unlike those considered previously. Numerical study for various benchmark problems in 2D and 3D for fluid flow problems with body forces and scalar transport with sources demonstrate the validity and accuracy, as well as the second-order convergence rate of the symmetrized operator split forcing/source schemes for the cascaded LB methods.
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.
In this brief report, a thermal lattice-Boltzmann (LB) model is presented for axisymmetric thermal flows in the incompressible limit. The model is based on the double-distribution-function LB method, which has attracted much attention since its emergence for its excellent numerical stability. Compared with the existing axisymmetric thermal LB models, the present model is simpler and retains the inherent features of the standard LB method. Numerical simulations are carried out for the thermally developing laminar flows in circular ducts and the natural convection in an annulus between two coaxial vertical cylinders. The Nusselt number obtained from the simulations agrees well with the analytical solutions and/or the results reported in previous studies.
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