考虑转动能的一维/二维Boltzmann-Rykov模型方程数值算法

研究考虑转动能的Boltzmann-Rykov模型方程,基于转动自由度对气体分子速度分布函数矩积分,引入约化速度分布函数,应用离散速度坐标法与数值积分技术,将气体运动论模型方程化为在离散速度坐标点处关于三个约化速度分布函数的联立方程组.应用拓展计算流体力学有限差分方法,数值计算考虑转动自由度的双原子气体一维、二维Boltzmann模型方程,得到高、低Knudsen数一维激波管内流动和二维竖直平板绕流问题的流场,分析验证考虑转动能的Boltzmann-Rykov模型方程全流域统一算法求解一维/二维气体流动问题的可靠性.结果表明,气体稀薄程度与分子内自由度对流场具有较大影响,且Knudsen数较高的稀薄气体流动呈现严重的非平衡流动特点.     

耦合不可压流场输运方程的格子Boltzmann方法研究

基于格子Boltzmann方法,提出了求解耦合不可压缩流场输运方程的一种改进数值方法. 该方法使用格子Boltzmann方法求解流场方程,并根据流场格子模型的密度分布函数构建了输运方程的二阶离散格式. 通过二维平板通道流场输运系统验证了该方法的有效性.数值结果表明,该方法可以有效地减少计算过程中出现的非物理耗散, 并克服了传统模型所需巨大存储量的缺点.     

跨流域高超声速绕流环境Boltzmann模型方程统一算法研究

如何准确可靠地模拟从外层空间高稀薄流到近地面连续流的航天器高超声速绕流环境与复杂流动变化机理是流体物理的前沿基础科学问题. 基于对Boltzmann方程碰撞积分的物理分析与可计算建模, 确立了可描述自由分子流到连续流区各流域不同马赫数复杂流动输运现象统一的Boltzmann模型速度分布函数方程, 发展了适于高、低不同马赫数绕流问题的离散速度坐标法和直接求解分子速度分布函数演化更新的气体动理论数值格式, 建立了模拟复杂飞行器跨流域高超声速飞行热环境绕流问题的气体动理论统一算法. 对稀薄流到连续流不同Knudsen数0.002 ≤Kn_∞ ≤1.618、不同马赫数下可重复使用卫星体再入过程(110–70 km)中高超声速绕流问题进行算法验证分析, 计算结果与典型文献的Monte Carlo直接模拟值及相关理论分析符合得较好. 研究揭示了飞行器跨流域不同高度高超声速复杂流动机理、绕流现象与气动力/热变化规律, 提出了一个通过数值求解介观Boltzmann模型方程, 可靠模拟高稀薄自由分子流到连续流跨流域高超声速气动力/热绕流特性统一算法.     

基于气体动理论的二维Karman涡街数值模拟

基于从稀薄流到连续流的跨流域气体动理论统一算法（gas-kinetic unified algorithm,GKUA）,通过数值求解考虑转动自由度激发的Boltzmann-Rykov模型方程,得到了一种跨流域非定常流动数值模拟的方法.该求解方法以Boltzmann模型方程为控制方程,在常温状态下如果考虑转动能激发的情况则选用Rykov模型.文中数值求解Rykov模型时,首先基于转动能模对速度分布函数积分以消去分子转动能量这一自变量,在速度空间应用自适应离散速度坐标法与数值积分演化更新计算技术,在位置空间应用3阶WENO空间离散格式和3阶显式Runge-Kutta时间推进.针对经典的二维Karman涡街流动现象进行数值模拟,说明该跨流域非定常流动模拟算法对于连续流区低速流动的适应性.      

用格子Boltzmann模型模拟垂直平板间的热对流

引入一个新的能量分布函数，利用该能量分布函数与粒子速度分布函数耦合来求解一个热流场. 因而，这一能量分布函数与粒子速度分布函数和Boltzmann方程构成了一个新的格子Boltzmann模型. 这一模型满足质量、动量和能量守恒的准则. 用该模型对垂直平板间的狭缝热对流进行了数值模拟，数值结果表明，在Prandtl数为1，Grashof数在1.3×10²—1×10⁶之间时，流场将出现多个旋涡结构的流型. 得出了与Lee相一致的结论. 关键词： 能量分布函数 Boltzmann方程 热对流     

Burgers-Korteweg-de Vries复合方程的格子Boltzmann方法模拟

针对Burgers-Korteweg-de Vries(cBKdV)复合方程提出一种格子Boltzmann模型.通过恰当地处理色散项u_xxx并运用Chapman-Enskog展开从格子Boltzmann方程推导出宏观方程,从而得到联系微观量与宏观量的局部平衡分布函数.对不同微分方程进行数值实验,数值解与解析解非常吻合,相比于其它数值结果,该格子Boltzmann模型的数值结果更精确,说明该数值模型的高效性.     

Burgers-Korteweg-de Vries复合方程的格子Boltzmann方法模拟（英文）

针对Burgers-Korteweg-de Vries(cBKdV)复合方程提出一种格子Boltzmann模型.通过恰当地处理色散项uxxx并运用Chapman-Enskog展开从格子Boltzmann方程推导出宏观方程,从而得到联系微观量与宏观量的局部平衡分布函数.对不同微分方程进行数值实验,数值解与解析解非常吻合,相比于其它数值结果,该格子Boltzmann模型的数值结果更精确,说明该数值模型的高效性.     

离散速度方向模型的H定理及其充分条件

H定理直接联系着动力学方法的稳定性,一直是各种简化求解Boltzmann方程方法的重要研究方向之一。本文证明了离散速度方向模型在平衡态和非平衡态下以及全部流动领域内都存在H定理,表明离散速度方向模型具有内在的稳定性。为了提高离散速度方向模型的数值稳定性,本文找到了一组该模型满足H定理的充分条件,该条件具有清晰的物理意义和简单的数学形式,可以方便地应用在数值计算中。     

用格子Boltzmann模型模拟非等温流场

根据微观和宏观之间的质量、动量、能量守恒准则和在原格子Boltzmann模型基础上,建立了几个新的格子Boltzmann模型,使得在外力场中的格子Boltzmann模型得到进一步完善.通过还原宏观流体力学方程,捕捉到了浮力强迫系数与Grashof数之间的关系.所得动量方程和Navier Stokes方程相比,在黏性输运项上有明显的改进,说明黏性应力不但与流体的速度梯度和流体的压缩性有关,而且还与非定常的内能梯度和动量通量有关.该模型对非等温流场的数值结果证明了其具有很好的数值稳定性和适用性. 关键词： Boltzmann模型 平衡分布函数 流体力学方程     

过渡区微尺度流动的有效黏性多松弛系数格子Boltzmann模拟

为提高采用二维九速离散速度模型的格子Boltzmann方法 (LBM)模拟微尺度流动中非线性现象的精度和效率,引入Dongari等提出的有效平均分子自由程对黏性进行修正(Dongari N,Zhang Y H,Reese J M2011 J.Fluids Eng.133 071101);并针对以往研究微尺度流动时采用边界处理格式含有离散误差的问题,采用多松弛系数格子Boltzmann方法结合二阶滑移边界条件,对微尺度Couette流动和周期性Poiseuille流动进行模拟,并将速度分布以及质量流量等模拟结果与直接模拟蒙特卡罗方法模拟数据、线性Boltzmann方程的数值解以及现有的LBM模型模拟结果进行对比.结果表明,相对于现有的LBM模型,引入新的修正函数所建立的有效黏性多松弛系数LBM模型有效提高了LBM模拟过渡区的微尺度流动中的非线性现象的能力.     

The Δ- Method for the Boltzmann Equation

A new numerical method is proposed to solve the Boltzmann equation. A frame is set up by using a discrete velocity approximation in the infinite velocity space, but by considering only those distribution function points which are not too small. The distribution function points may occur anywhere in the infinite discrete velocity space and are not constrained to a pre-specified region. A fourth-order finite difference is used for the convection terms. A Monte Carlo-like method is applied to the discrete velocity model of the collision integral. The effort of the method is proportional to the number of discrete points. Numerical examples are given for the full Boltzmann equation and results for some benchmark problems are compared with analytical or prior solutions.     

Simulating thermohydrodynamics by finite difference solutions of the Boltzmann equation

The formulation of a consistent thermohydrodynamics with a discrete model of the Boltzmann equation requires the representation of the velocity moments up to the fourth order. Space-filling discrete sets of velocities with increasing accuracy were obtained using a systematic approach in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev. E, 73 (5), n. 056702, 2006). These sets of velocities are suitable for collision-propagation schemes, where the discrete velocity and physical spaces are coupled and the Courant number is unitary. The space-filling requirement leads to sets of discrete velocities which can be large in thermal models. In this work, although the discrete sets of velocities are also obtained with a quadrature method based on prescribed abscissas, the lattices are not required to be space-filling. This leads to a reduced number of discrete velocities for the same approximation order but requires the use of an alternative numerical scheme. The use of finite difference schemes for the advection term in the continuous Boltzmann equation has shown to have some advantages with respect to the collision-propagation LBM method by freeing the Courant number from its unitary value and reducing the discretization error. In this work, a second order Runge-Kutta method was used for the simulation of the Sod's shock tube problem, the Couette flow and the Lid-driven cavity flow. Boundary conditions without velocity slip and temperature jumps were written for these discrete Boltzmann equation by splitting the velocity distribution function into an equilibrium and a non-equilibrium part. The equilibrium part was set using the local velocity and temperature at the wall and the non-equilibrium part by extrapolating the non-equilibrium moments to the wall sites.     

Study on the unified algorithm for three-dimensional complex problems covering various flow regimes using Boltzmann model equation

The Boltzmann simplified velocity distribution function equation describing the gas transfer phenomena from various flow regimes will be explored and solved numerically in this study. The discrete velocity ordinate method of the gas kinetic theory is studied and applied to simulate the complex multi-scale flows. Based on the uncoupling technique on molecular movement and colliding in the DSMC method, the gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by extending and applying the unsteady time-splitting method from computational fluid dynamics. The Gauss-type discrete velocity numerical quadrature technique for different Mach number flows is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established to study the three-dimensional complex flows from rarefied transition to continuum regimes. The parallel strategy adapted to the gas-kinetic numerical algorithm is investigated by analyzing the inner parallel degree of the algorithm, and then the HPF parallel processing program is developed. To test the reliability of the present gas-kinetic numerical method, the three-dimensional complex flows around sphere and spacecraft shape with various Knudsen numbers are simulated by HPF parallel computing. The computational results are found in high resolution of the flow fields and good agreement with the theoretical and experimental data. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view of solving the Boltzmann model equation. Supported by the National Natural Science Foundation of China (Grant Nos. 90205009 and 10321002) and the National Parallel Computing Center     

Numerical Simulations of Unsteady Flows from Rarefied Transition to Continuum Using Gas-Kinetic Unified Algorithm

Numerical simulations of unsteady gas flows are studied on the basis of Gas-Kinetic Unified Algorithm (GKUA) from rarefied transition to continuum flow regimes. Several typical examples are adopted. An unsteady flow solver is developed by solving the Boltzmann model equations, including the Shakhov model and the Rykov model etc. The Rykov kinetic equation involving the effect of rotational energy can be transformed into two kinetic governing equations with inelastic and elastic collisions by integrating the molecular velocity distribution function with the weight factor on the energy of rotational motion. Then, the reduced velocity distribution functions are devised to further simplify the governing equation for one- and two-dimensional flows. The simultaneous equations are numerically solved by the discrete velocity ordinate (DVO) method in velocity space and the finite-difference schemes in physical space. The time-explicit operator-splitting scheme is constructed, and numerical stability conditions to ascertain the time step are discussed. As the application of the newly developed GKUA, several unsteady varying processes of one- and two-dimensional flows with different Knudsen number are simulated, and the unsteady transport phenomena and rarefied effects are revealed and analyzed. It is validated that the GKUA solver is competent for simulations of unsteady gas dynamics covering various flow regimes.     

The Incompressible Navier–Stokes for the Nonlinear Discrete Velocity Models

Abstract

We establish the incompressible Navier–Stokes limit for the discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which remain in a suitable small neighborhood of the global Maxwellian. Appropriately scaled families solutions of discrete Boltzmann equation are shown to have fluctuations that locally in time converge strongly to a limit governed by a solution of Incompressible Navier–Stokes provided that the initial fluctuation is smooth, and converges to appropriate initial data. As applications of our results, we study the Carleman model and the one-dimensional Broadwell model.     

Kinetic boundary condition at a vapor-liquid interface

By molecular dynamics simulations, the boundary condition for the Boltzmann equation at a vapor-liquid interface is found to be the product of three one-dimensional Maxwellian distributions for the three velocity components of vapor molecules and a factor including a well-defined condensation coefficient. The Maxwellian distribution for the velocity component normal to the interface is characterized by the liquid temperature, as in a conventional model boundary condition, while those for the tangential components are prescribed by a different temperature, which is a linear function of energy flux across the interface. The condensation coefficient is found to be constant and equal to the evaporation coefficient determined by the liquid temperature only.     

New Discrete Model Boltzmann Equations for Arbitrary Partitions of the Velocity Space

Modified discrete Boltzmann equations for arbitrary partitions of the velocity space are established. The new equations can be derived from the continuous Boltzmann equation and are a generalization of previous discrete-velocity models. They preserve mass, momentum, and energy, and an H-theorem holds. The new model equations are tested by comparing their solutions with the analytical ones of the continuous Boltzmann equation for the Krook–Wu and the very hard particle models.     

Approximation of the Boltzmann equation by discrete velocity models

Two convergence results related to the approximation of the Boltzmann equation by discrete velocity models are presented. First we construct a sequence of deterministic discrete velocity models and prove convergence (as the number of discrete velocities tends to infinity) of their solutions to the solution of a spatially homogeneous Boltzmann equation. Second we introduce a sequence of Markov jump processes (interpreted as random discrete velocity models) and prove convergence (as the intensity of jumps tends to infinity) of these processes to the solution of a deterministic discrete velocity model.     

含振动能激发Boltzmann模型方程气体动理论统一算法验证与分析

为模拟研究高温高马赫数下多原子气体内能激发对跨流域非平衡流动的影响,将转动能、振动能分别作为气体分子速度分布函数的自变量,把转动能和振动能处理为连续分布的能量模式,将Boltzmann方程的碰撞项分解成弹性碰撞项和非弹性碰撞项,同时将非弹性碰撞按一定松弛速率分解为平动-转动能松弛过程和平动-转动-振动能松弛过程,构造了一类考虑振动能激发的Boltzmann模型方程,并证明了其守恒性和H定理.基于内部能量变量对分布函数无穷积分,引入三个约化速度分布函数,得到一组考虑振动能激发的约化速度分布函数控制方程组,使用离散速度坐标法,基于LU-SGS隐式格式和有限体积法求解离散速度分布函数,建立含振动能激发的气体动理论统一算法.通过开展高稀薄流到连续流圆柱绕流问题统一算法与直接模拟蒙特卡罗法模拟结果对比分析,特别是过渡流区平动、转动、振动非平衡效应对绕流流场与物面力热特性的影响机制,证实了所建立的含振动能激发的Boltzmann模型方程及气体动理论统一算法的准确可靠性.     

Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations

High resolution finite difference schemes for solving the nonlinear model Boltzmann equations are presented for the computations of rarefied gas flows. The discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model Boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space. Then a high order essentially nonoscillatory method due to Harten et al. (J. Comput. Phys. 71, 231, 1987) is adapted and extended to solve them. Explicit methods using operator splitting and implicit methods using the lower-upper factorization are described to treat multidimensional problems. The methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use. The computed results using model Boltzmann equations are found to compare well both with those using the direct simulation Monte Carlo results in the transitional regime flows and those with the continuum Navier-Stokes calculations in near continuum regime flows.