Gas-kinetic description of shock wave structures by solving Boltzmann model equation

On the basis of the mesoscopic theory of Boltzmann-type velocity distribution function, the modified Boltzmann model equation describing the one-dimensional gas flows from various flow regimes is presented by incorporating the molecular interaction models relating to the viscosity and diffusion cross-sections, density, temperature and the dependent exponent of viscosity into the molecular collision frequency. The gas-kinetic numerical method for directly solving the molecular velocity distribution function is studied by introducing the reduced distribution functions and the discrete velocity ordinate method, in which the unsteady time-splitting method and the NND finite difference scheme are applied. To study the inner flows of non-equilibrium shock wave structures, the one-dimensional unsteady shock-tube problems with various Knudsen numbers and the steady shock wave problems at different Mach numbers are numerically simulated. The computed results are found to give good agreement with the theoretical, DSMC and experimental results. The computing practice has confirmed the good precision and reliability of the gas-kinetic numerical algorithm in solving the highly nonequilibrium shock wave disturbances from various flow regimes.     

Gas-kinetic numerical schemes for one- and two-dimensional inner flows

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.     

The cumulant method for the space-homogeneous Boltzmann equation

In this work we give a comparison of the exact Bobylev/Krook-Wu solution to the space-homogeneous Boltzmann equation and numerical results obtained by a implementation of the cumulant method for the space-homogeneous case. We find excellent agreement of the numerical solution to the cumulant equations with the exact solution of the space-homogeneous Boltzmann equation as long as the exact, non-linear production terms are used. If a linearized variant of the production terms is used, relaxation rates may be underestimated due to convergence to the solution of the linearized equations.Received: 3 April 2004, Accepted: 3 September 2004, Published online: 22 February 2005PACS: 51.10. + y, 51.30. + i, 47.11 + j, 47.45.-n Correspondence to: K.H. Hoffmann     

Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation

Based on the Bhatnagar–Gross–Krook (BGK) Boltzmann model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non‐linear source terms. Based on the unsteady time‐splitting technique and the non‐oscillatory, containing no free parameters, and dissipative (NND) finite‐difference method, the gas kinetic finite‐difference second‐order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm for the gas dynamical problems from various flow regimes is developed. To test the reliability of the present numerical method, the one‐dimensional shock‐tube problems and the flows past two‐dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related flows indicate that both high resolution of the flow fields and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

On numerical diffusion of simplified lattice Boltzmann method

In this article, we analyze the numerical diffusion in the recently developed simplified lattice Boltzmann method (SLBM) and propose amending strategies towards lower numerical diffusion. It is noted that, in the original SLBM, the intermediate flow properties are utilized to evaluate the nonequilibrium distribution function, which may bring in excessive numerical diffusion. In the revised scheme, this evaluation strategy is nurtured by using the corrected flow properties to calculate the nonequilibrium distribution function. In the meantime, the numerically evaluated nonequilibrium distribution function only approximately fulfills the conservation relationship in the second order of accuracy. Although such approximation does not violate the global order of accuracy, offsetting the extra error would contribute to reducing the numerical diffusion. After implementing the proposed amending strategies, the revised SLBM (RSLBM) is validated through three numerical examples. The results indicate that RSLBM bears comparable order of accuracy as the original SLBM but shows lower numerical error on the same mesh size. And the reduced numerical error facilitates recovery of delicate flow structures. The proposed RSLBM can be flexibly implemented on nonuniform or body-fitted meshes, and in three-dimensional simulations.     

Gas-kinetic unified algorithm for plane external force-driven flows covering all flow regimes by modeling of Boltzmann equation

The nonequilibrium steady gas flows under the external forces are essentially associated with some extremely complicated nonlinear dynamics, due to the acceleration or deceleration effects of the external forces on the gas molecules by the velocity distribution function. In this article, the gas-kinetic unified algorithm (GKUA) for rarefied transition to continuum flows under external forces is developed by solving the unified Boltzmann model equation. The computable modeling of the Boltzmann equation with the external force terms is presented at the first time by introducing the gas molecular collision relaxing parameter and the local equilibrium distribution function integrated in the unified expression with the flow state controlling parameter, including the macroscopic flow variables, the gas viscosity transport coefficient, the thermodynamic effect, the molecular power law, and molecular models, covering a full spectrum of flow regimes. The conservative discrete velocity ordinate (DVO) method is utilized to transform the governing equation into the hyperbolic conservation forms at each of the DVO points. The corresponding numerical schemes are constructed, especially the forward-backward MacCormack predictor-corrector method for the convection term in the molecular velocity space, which is unlike the original type. Some typical numerical examples are conducted to test the present new algorithm. The results obtained by the relevant direct simulation Monte Carlo method, Euler/Navier-Stokes solver, unified gas-kinetic scheme, and moment methods are compared with the numerical analysis solutions of the present GKUA, which are in good agreement, demonstrating the high accuracy of the present algorithm. Besides, some anomalous features in these flows are observed and analyzed in detail. The numerical experience indicates that the present GKUA can provide potential applications for the simulations of the nonequilibrium external-force driven flows, such as the gravity, the electric force, and the Lorentz force fields covering all flow regimes.     

Immersed boundary method for unsteady kinetic model equations

Predicting unsteady flows and aerodynamic forces for large displacement motion of microstructures requires transient solution of Boltzmann equation with moving boundaries. For the inclusion of moving complex boundaries for these problems, three immersed boundary method flux formulations (interpolation, relaxation, and interrelaxation) are presented. These formulations are implemented in a 2‐D finite volume method solver for ellipsoidal‐statistical (ES)‐Bhatnagar‐Gross‐Krook (BGK) equations using unstructured meshes. For the verification, a transient analytical solution for free molecular 1‐D flow is derived, and results are compared with the immersed boundary (IB)‐ES‐BGK methods. In 2‐D, methods are verified with the conformal, non‐moving finite volume method, and it is shown that the interrelaxation flux formulation gives an error less than the interpolation and relaxation methods for a given mesh size. Furthermore, formulations applied to a thermally induced flow for a heated beam near a cold substrate show that interrelaxation formulation gives more accurate solution in terms of heat flux. As a 2‐D unsteady application, IB/ES‐BGK methods are used to determine flow properties and damping forces for impulsive motion of microbeam due to high inertial forces. IB/ES‐BGK methods are compared with Navier–Stokes solution at low Knudsen numbers, and it is shown that velocity slip in the transitional rarefied regime reduces the unsteady damping force. Copyright © 2015 John Wiley & Sons, Ltd.     

高浓度固-液两相流紊流的动理学模型

采用分子动理学方法,基于固-液两相流液相分子或颗粒相颗粒的Boltzmann方程,对Boltzmann方程分别取零矩和一次矩,则得到高浓度固-液两相流紊流的连续方程和动量方程,再和较成熟的低浓度两相流连续方程和动量方程比较,取低浓度两相流控制方程中较成熟合理的有关项和高浓度时由动理学方法推导出的颗粒间碰撞项,则得到高浓度固-液两相流紊流的最终控制方程:连续方程和动量方程.     

Variance‐reduced Monte Carlo solutions of the Boltzmann equation for low‐speed gas flows: A discontinuous Galerkin formulation

We present and discuss an efficient, high‐order numerical solution method for solving the Boltzmann equation for low‐speed dilute gas flows. The method's major ingredient is a new Monte Carlo technique for evaluating the weak form of the collision integral necessary for the discontinuous Galerkin formulation used here. The Monte Carlo technique extends the variance reduction ideas first presented in Baker and Hadjiconstantinou (Phys. Fluids 2005; 17 , art. no. 051703) and makes evaluation of the weak form of the collision integral not only tractable but also very efficient. The variance reduction, achieved by evaluating only the deviation from equilibrium, results in very low statistical uncertainty and the ability to capture arbitrarily small deviations from equilibrium (e.g. low‐flow speed) at a computational cost that is independent of the magnitude of this deviation. As a result, for low‐signal flows the proposed method holds a significant computational advantage compared with traditional particle methods such as direct simulation Monte Carlo (DSMC). Copyright © 2008 John Wiley & Sons, Ltd.     

A method for solving the factorized vorticity-stream function equations by finite elements

A new finite element method for solving the time-dependent incompressible Navier-Stokes equations with general boundary conditions is presented. The two second-order partial differential equations for the vorticity and the stream function are factorized, apart from the non-linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no-slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4-noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.     

High-order hybridisable discontinuous Galerkin method for the gas kinetic equation

ABSTRACT

The high-order hybridisable discontinuous Galerkin (HDG) method is used to find steady-state solution of gas kinetic equations on two-dimensional geometry. The velocity distribution function and its traces are approximated in piecewise polynomial space on triangular mesh and mesh skeleton, respectively. By employing a numerical flux derived from the upwind scheme and imposing its continuity on mesh skeleton, the global system for unknown traces is obtained with fewer coupled degrees of freedom, compared to the original DG method. The solutions of model equation for the Poiseuille flow through square channel show the higher order solver is faster than the lower order one. Moreover, the HDG scheme is more efficient than the original DG method when the degree of approximating polynomial is larger than 2. Finally, the developed scheme is extended to solve the Boltzmann equation with full collision operator, which can produce accurate results for shear-driven and thermally induced flows.     

Direct numerical test of the B-G-K model equation by the DSMC method

In the present paper the rarefied gas flow caused by the sudden change of the wall temperature and the Rayleigh problem are simulated by the DSMC method which has been validated by experiments both in global flow field and velocity distribution function level. The comparison of the simulated results with the accurate numerical solution of the B-G-K model equation shows that near equilibrium the B-G-K equation with corrected collision frequency can give accurate result but as farther away from equilibrium the B-G-K equation is not accurate. This is for the first time that the error caused by the B-G-K model equation has been revealed. The project supported by the National Natural Science Foundation of China (19772059, 19889209)     

气体运动论数值算法在微槽道流中的应用研究

简要介绍基于Boltzmann模型方程的气体运动论数值算法基本思想及其对二维微槽道流动问题数值计算的推广,并阐述适用于微尺度流动问题的气体运动论边界条件数值处理方法。通过对压力驱动的二维微槽道流动问题进行数值模拟,将不同Knudsen数下的微槽道流计算结果分别与有关DSMC模拟值和经滑移流理论修正的N—S方程解进行比较分析,表明基于Boltzmann模型方程的气体运动论数值算法对微槽道气体流动问题具有很好的模拟能力。     

A numerical method for KdV equation using collocation and radial basis functions

Recently, there has been an increasing interest in the study of initial boundary value problems for Korteweg–de Vries (KdV) equations. In this paper, we propose a numerical scheme to solve the third-order nonlinear KdV equation using collocation points and approximating the solution using multiquadric (MQ) radial basis function (RBF). The scheme works in a similar fashion as finite-difference methods. Numerical examples are given to confirm the good accuracy of the presented scheme.     

High order parallelisation of an unstructured grid,discontinuous-Galerkin finite element solver for the Boltzmann–BGK equation

ABSTRACT

This paper outlines the implementation and performance of a parallelisation approach involving partitioning of both physical space and velocity space domains for finite element solution of the Boltzmann-BGK equation. The numerical solver is based on a discontinuous Taylor–Galerkin approach. To the authors' knowledge this is the first time a ‘high order’ parallelisation, or `phase space parallelisation', approach has been attempted in conjunction with a numerical solver of this type. Restrictions on scalability have been overcome with the implementation detailed in this paper. The developed algorithm has major advantages over continuum solvers in applications where strong discontinuities prevail and/or in rarefied flow applications where the Knudsen number is large. Previous work by the authors has outlined the range of applications that this solver is capable of tackling. The paper demonstrates that the high order parallelisation implemented is significantly more effective than previous implementations at exploiting High Performance Computing architectures.     

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1].The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients C_m and K_TS, respectively.In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10].In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients C_m and K_TS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 176–182. Original Russian Text Copyright © 2004 by Poddoskin.