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.     

A unified gas kinetic scheme with moving mesh and velocity space adaptation

There is great difficulty for direct Boltzmann solvers to simulate high Knudsen number flow due to the severe steep slope and high concentration of the gas distribution function in a local particle velocity space. Local mesh adaptation becomes necessary in order to make the Boltzmann solver to be a practical tool in aerospace applications. The present research improves the unified gas-kinetic scheme (UGKS) in the following two aspects. First, the UGKS is extended in a physical space with moving mesh. This technique is important to study a freely flying object in a rarefied environment. Second, the adaptive quadtree method in the particle velocity space is implemented in the UGKS. Due to the new improvements in the discretization of a gas distribution function in the six dimensional phase space, the adaptive unified gas kinetic scheme (AUGKS) is able to deal with a wide range of flow problems under extreme flying conditions, such as the whole unsteady flying process of an object from a highly rarefied to a continuum flow regime. After validating the scheme, the capability of AUGKS is demonstrated in the following two challenge test cases. The first case is about the free movement of an ellipse flying at initial Mach number 5 in a rarefied flow at different Knudsen numbers. The force on the ellipse and the unsteady trajectory of the ellipse movement are fully captured. The gas distribution function around the ellipse is analyzed. The second case is about the study of unsteady flight of a nozzle under a bursting process of the compressed gas expanding into a rarefied environment. Due to the strong expansion wave and the huge density difference between interior and exterior regions around the nozzle, the particle distribution function changes dramatically in the particle velocity space. The use of an adaptive velocity space in the AUGKS becomes necessary to simulate such a flow and to control the computational cost to a tolerable level. The second test is a challenge problem for any existing rarefied flow solver.     

Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

In this paper, we develop a numerical method to solve Boltzmann like equations of kinetic theory which is able to capture the compressible Navier–Stokes dynamics at small Knudsen numbers. Our approach is based on the micro/macro decomposition technique, which applies to general collision operators. This decomposition is performed in all the phase space and leads to an equivalent formulation of the Boltzmann (or BGK) equation that couples a kinetic equation with macroscopic ones. This new formulation is then discretized with a semi-implicit time scheme combined with a staggered grid space discretization. Finally, several numerical tests are presented in order to illustrate the efficiency of our approach. Incidentally, we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible Navier–Stokes asymptotics in the case of the simplified BGK model. Up to our knowledge, this property is not known for general collision operators.     

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     

An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow

During the past decade gas-kinetic methods based on the BGK simplification of the Boltzmann equation have been employed to compute fluid flow in a finite-difference or finite-volume context. Among the most successful formulations is the finite-volume scheme proposed by Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (48) (2001) 289–335]. In this paper we build on this theoretical framework mainly with the aim to improve the efficiency and convergence of the scheme, and extend the range of application to three-dimensional complex geometries using general unstructured meshes. To that end we propose a modified BGK finite-volume scheme, which significantly reduces the computational cost, and improves the behavior on stretched unstructured meshes. Furthermore, a modified data reconstruction procedure is presented to remove the known problem that the Chapman–Enskog expansion of the BGK equation fixes the Prandtl number at unity. The new Prandtl number correction operates at the level of the partial differential equations and is also significantly cheaper for general formulations than previously published methods. We address the issue of convergence acceleration by applying multigrid techniques to the kinetic discretization. The proposed modifications and convergence acceleration help make large-scale computations feasible at a cost competitive with conventional discretization techniques, while still exploiting the advantages of the gas-kinetic discretization, such as computing full viscous fluxes for finite volume schemes on a simple two-point stencil.     

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...     

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows.In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms.Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen numbers.     

A high-order gas-kinetic Navier–Stokes flow solver

The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge–Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge–Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier–Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier–Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier–Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.     

稀薄气体动力学矩方法研究综述

临近空间位于航天器入轨与返回的必经区域, 也是临近空间高超声速飞行器长航时飞行空域, 空间环境的特殊性决定了飞行器在穿越时必须考虑稀薄大气环境对飞行器气动力防隔热通讯及控制的影响.Boltzmann方程作为描述气体分子速度分布函数演化规律的微分-积分形式, 在一定条件下能够描述从自由分子流到连续流全流域流动现象.作为Boltzmann方程的宏观表达形式, 矩方程这一经典流体力学方程形式涵盖了Euler方程N-S方程Burnett方程super-Burnett方程及近年来发展的广义流体力学方程——非线性本构关系模型等.由于成熟的CFD数值计算理论及有限矩方程较高的计算效率, 滑移过渡流矩方法相比粒子仿真与Boltzmann模型方程方法具有十分显著的优势和巨大的工程应用潜力.因此, 对近年来传统及新型矩方法研究所取得的进展进行归纳总结, 并针对关键科学问题开展理论与数值计算方法研究, 具有十分重要的理论与工程应用价值.      

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.     

A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (S-model) were constructed to correct the Prandtl number of the original BGK model through the modification of stress and heat flux. With the introduction of a new parameter to combine the ES-model and S-model, a generalized kinetic model can be developed. This new model can give the correct Navier-Stokes equations in the continuum flow regime. Through the adjustment of the new parameter, it provides abundant dynamic effect beyond the ES-model and S-model. Changing the free parameter, the physical performance of the new model has been tested numerically. The unified gas kinetic scheme (UGKS) is employed for the study of the new model. In transition flow regime, many physical problems, i.e., the shock structure and micro-flows, have been studied using the generalized model. With a careful choice of the free parameter, good results can be achieved for most test cases. Due to the property of the Boltzmann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the S-model predicts more accurate numerical solutions in most test cases presented in this paper than the ES-model, while ES-model performs better in the cases where the flow is mostly driven by temperature gradient, such as a channel flow with large boundary temperature variation at high Knudsen number.     

Variation character of stagnation point heat flux for hypersonic pointed bodies from continuum to rarefied flow states and its bridge function study

This paper is a research on the variation character of stagnation point heat flux for hypersonic pointed bodies from continuum to rarefied flow states by using theoretical analysis and numerical simulation methods. The newly developed near space hypersonic cruise vehicles have sharp noses and wingtips, which desires exact and relatively simple methods to estimate the stagnation point heat flux. With the decrease of the curvature radius of the leading edge, the flow becomes rarefied gradually, and viscous interaction effects and rarefied gas effects come forth successively, which results in that the classical Fay-Riddell equation under continuum hypothesis will become invalid and the variation of stagnation point heat flux is characterized by a new trend. The heat flux approaches the free molecular flow limit instead of an infinite value when the curvature radius of the leading edge tends to 0. The physical mechanism behind this phenomenon remains in need of theoretical study. Firstly, due to the fact that the whole flow regime can be described by Boltzmann equation, the continuum and rarefied flow are analyzed under a uniform framework. A relationship is established between the molecular collision insufficiency in rarefied flow and the failure of Fourier’s heat conduction law along with the increasing significance of the nonlinear heat flux. Then based on an inspiration drew from Burnett approximation, control factors are grasped and a specific heat flux expression containing the nonlinear term is designed in the stagnation region of hypersonic leading edge. Together with flow pattern analysis, the ratio of nonlinear to linear heat flux W _r is theoretically obtained as a parameter which reflects the influence of nonlinear factors, i.e. a criterion to classify the hypersonic rarefied flows. Ultimately, based on the characteristic parameter W _r , a bridge function with physical background is constructed, which predicts comparative reasonable results in coincidence well with DSMC and experimental data in the whole flow regime.     

反应流体的移动网格动理学格式

在移动网格上构造一种反应流的动理学格式.首先利用BGK模型推导含化学反应的流体力学方程组,并利用其积分形式构造移动网格上离散格式,再利用自适应移动网格方法得到网格速度,最后利用时间精确的动理学数值方法构造数值通量,得到移动网格单元上新的物理量.一维与二维的数值实验表明这种格式同时具有高精度、高分辨率的特点.     

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.     

Rayleigh-Bénard simulation using the gas-kinetic Bhatnagar-Gross-Krook scheme in the incompressible limit

In this paper, a gas-kinetic Bhatnagar-Gross-Krook (BGK) model is constructed for the Rayleigh-Bénard thermal convection in the incompressible flow limit, where the flow field and temperature field are described by two coupled BGK models. Since the collision times in the corresponding BGK models can be different, the Prandtl number can be changed to any value instead of a fixed Pr=1 in the original BGK model [P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954)]. The two-dimensional Rayleigh-Bénard thermal convection is studied and numerical results are compared with theoretical ones as well as other simulation results.     

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.     

Efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes

This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements.     

Implicit high-order method for calculating rarefied gas flow in a planar microchannel

An efficient numerical algorithm for calculating rarefied gas flows in planar microchannels on the basis of the Boltzmann kinetic equation with the linearized S-model collision integral is presented. The algorithm consists of a high-order spatial discretisation on unstructured meshes, conservative procedure to calculate macroscopic quantities and efficient one-step implicit time evolution method. It therefore works across all flow regimes from the free-molecular to nearly continuum ones and provides rapid convergence to steady state solutions. The parallel implementation is provided via MPI programming paradigm. The performance of the method is illustrated by computing the flow for length to the width ratios for up to 10³ in the wide range of Knudsen numbers. Numerical estimates of efficiency of implicit time marching and parallel performance as well as convergence studies in both physical and velocity spaces are provided. The end effects and other essentially two-dimensional features of the flow are analysed and a detailed comparison with the case of an infinitely long channel is carried out. The presented numerical data can be used as reference for future studies of nonlinear flows as well as for comparison with other numerical approaches and flow models.     

Kinetic theory of hydrodynamic flows. II. The drag on a sphere and on a cylinder

In this paper we continue the study of solutions of the extended Boltzmann equation started previously. In particular, we study an iterated solution of the equation that can be used to describe the flow of a rarefied gas around a macroscopic object. We discuss the rarefied flow and then show how the iterated solution can be extended into the hydrodynamic regime. The results for the drag force and for the distribution function of the gas molecules are shown to be identical to the results obtained in a previous paper by a generalization of the normal solution method. We also discuss the special properties of both rarefied and continuum flows around a cylinder and show that in both regions one must take into account Oseen-like terms which naturally appear in the extended Boltzmann equation. In the hydrodynamic regime we obtain Lamb's formula for the force on the cylinder. By relating the terms in the iterated expression to dynamical events taking place in the fluid, we are able to discuss the dynamical origin of the results obtained here.A preliminary report on the work described here and in Part I was given in Ref. 2.     

A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation

A high-order algorithm of the direct simulation Monte Carlo (DSMC) method, H-DSMC, has been developed to simulate rarefied flow regimes. The mth order Taylor series expansion has been employed to obtain a more generalized form of the time discretization for the collision part of Boltzmann equation. In the purposed algorithm, the higher order collision terms are introduced as well as higher order terms in the time step of the probabilistic coefficients. These newly implemented higher order terms improve the accuracy and efficiency of the solution and enhance the convergence rate quite significantly. Comparison between results of the classic DSMC method and the H-DSMC method shows the promising performance of the introduced technique.