BGK Approach Research Articles

A new collision operator for lattice Boltzmann shallow water model: a convergence and stability study

This work presents a new lattice Boltzmann model for steady and unsteady two-dimensional shallow water flows. Compared to previous lattice Boltzmann based shallow water models, the proposed method uses a consistent characteristic speed in the pressure term and in the viscosity. To preserve the isotropy of viscosity, the relaxation rates for the different cumulants have to be decoupled. This is only possible by using multiple relaxation rates. The recovery of the correct viscosity is investigated by a convergence study based on the decay of a Taylor Green Vortex. Results from shallow water models using the proposed collision operator are then compared to those derived from standard BGK approach and from a continuous model.

A BGK model for high temperature rarefied gas flows

High temperature gases, for instance in hypersonic reentry flows, show complex phenomena like excitation of rotational and vibrational energy modes, and even chemical reactions. For flows in the continuous regime, simulation codes use analytic or tabulated constitutive laws for pressure and temperature. In this paper, we propose a BGK model which is consistent with any arbitrary constitutive laws, and which is designed to make high temperature gas flow simulations in the rarefied regime. A Chapman-Enskog analysis gives the corresponding transport coefficients. Our approach is illustrated by a numerical comparison with a compressible Navier-Stokes solver with rotational and vibrational non equilibrium. The BGK approach gives a deterministic solver with a computational cost which is close to that of a simple monoatomic gas.

Implicit gas-kinetic BGK scheme with multigrid for 3D stationary transonic high-Reynolds number flows

Instead of solving the Euler and Navier–Stokes equations directly, the gas-kinetic BGK scheme based on the Boltzmann equation has been developed and attracted more and more attentions since the early 1990s. It shows high accuracy and robustness for a wide range of flow regimes. But an obvious disadvantage of the BGK scheme is the low computational efficiency, in particular for multidimensional problems. Till now it has not been widely used as a practical tool for science and engineering applications. To overcome this drawback, in this paper some acceleration techniques, including local time stepping, implicit LU-SGS method, and multigrid strategy, are implemented into the original BGK approach and the new scheme is applied to study 3D steady transonic viscous flows. Numerical results show the significant speed up of the scheme to capture the steady state solution.

Modeling of carbon dioxide condensation in the high pressure flows using the statistical BGK approach

In this work, we propose a new heat accommodation model to simulate freely expanding homogeneous condensation flows of gaseous carbon dioxide using a new approach, the statistical Bhatnagar-Gross-Krook method. The motivation for the present work comes from the earlier work of Li et al. [J. Phys. Chem. 114, 5276 (2010)] in which condensation models were proposed and used in the direct simulation Monte Carlo method to simulate the flow of carbon dioxide from supersonic expansions of small nozzles into near-vacuum conditions. Simulations conducted for stagnation pressures of one and three bar were compared with the measurements of gas and cluster number densities, cluster size, and carbon dioxide rotational temperature obtained by Ramos et al. [Phys. Rev. A 72, 3204 (2005)]. Due to the high computational cost of direct simulation Monte Carlo method, comparison between simulations and data could only be performed for these stagnation pressures, with good agreement obtained beyond the condensation onset point, in the farfield. As the stagnation pressure increases, the degree of condensation also increases; therefore, to improve the modeling of condensation onset, one must be able to simulate higher stagnation pressures. In simulations of an expanding flow of argon through a nozzle, Kumar et al. [AIAA J. 48, 1531 (2010)] found that the statistical Bhatnagar-Gross-Krook method provides the same accuracy as direct simulation Monte Carlo method, but, at one half of the computational cost. In this work, the statistical Bhatnagar-Gross-Krook method was modified to account for internal degrees of freedom for multi-species polyatomic gases. With the computational approach in hand, we developed and tested a new heat accommodation model for a polyatomic system to properly account for the heat release of condensation. We then developed condensation models in the framework of the statistical Bhatnagar-Gross-Krook method. Simulations were found to agree well with the experiment for all stagnation pressure cases (1-5 bar), validating the accuracy of the Bhatnagar-Gross-Krook based condensation model in capturing the physics of condensation.

A Bhatnagar–Gross–Krook kinetic approach to fast reactive mixtures: Relaxation problems

The paper deals with a consistent BGK-type approximation for the Boltzmann-like equations which govern the evolution of a gas undergoing bimolecular chemical reactions. In particular, model equations, specifically devised for physical situations in which chemical relaxation is as fast as mechanical relaxation, are discussed in comparison to previous models. This BGK approach preserves the main features of the reactive Boltzmann equations, including law of mass action and H -theorem. Numerical results illustrating the effects of the several varying parameters on the relaxation to equilibrium are presented and commented on.

A novel coupled lattice Boltzmann model for low Mach number combustion simulation

A novel coupled lattice Boltzmann model is developed for two- and three-dimensional low Mach number combustion simulations, in which the fluid density can bear sharp changes. Different to the hybrid lattice Boltzmann scheme [O. Filippova, D. Hanel, A novel lattice BGK approach for low Mach number combustion, J. Comput. Phys. 158 (2000) 139], this scheme is strictly pure lattice Boltzmann style (i.e., we solve the flow, temperature, and concentration fields using the lattice Boltzmann method only); different to the non-coupled lattice Boltzmann scheme [K. Yamamoto, X. He, G.D. Doolen, Simulation of combustion field with lattice Boltzmann method, J. Stat. Phys. 107 (2002) 367], the fluid density in this model is coupled directly with the temperature. In this model the time step and the fluid particle speed can be adjusted dynamically, depending on the “particle characteristic temperature”. And the algorithm is still a simple process of hopping from one grid point to the next, the same as the standard lattice Boltzmann method. Therefore the outstanding advantages of the standard lattice Boltzmann method are retained in this model besides better numerical stability. Excellent agreement between the present results and other numerical or experimental data shows that this scheme is an efficient numerical method for practical combustion simulations.

Boltzmann–BGK approach to simulating weakly compressible 3D turbulence: comparison between lattice Boltzmann and gas kinetic methods

Recently, the so-called gas-kinetic method (GKM) based on the Boltzmann-BGK equation has been successfully used in a variety of high Mach-number shock and laminar flow calculations. In this paper, we study the viability of extending the applicability of GKM to simulate turbulent flows. We evaluate the capability of GKM by making detailed comparisons against the lattice Boltzmann method and a Navier–Stokes solver. We perform three-dimensional direct numerical simulations (DNS) of decaying isotropic turbulence with the three methods and compare the evolution of kinetic energy, dissipation rate, and energy spectrum. Details of various macroscopic flow variables of interest are also examined. Further, the decay exponent calculated from the GKM is compared with the published results. The agreement in all categories considered is quite good. The results clearly demonstrate the potential promise of GKM for turbulence simulations over a broad range of Mach numbers.

Calculation of the velocity of a molecular gas slipping over a sphere with regard to accommodation coefficients

The velocity of molecular gas slipping over a spherical surface is calculated by exact analytical methods including the accommodation coefficients for the first two moments of the distribution function. An extension of the BGK approach to the Boltzmann kinetic model for the case of rotational degrees of freedom is used as the basic equation.