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## Method of accelerating convergence for gas kinetic algorithm based on digital constitutive relation of macroscopic equations

Pi Xing-Cai, Zhu Lian-Hua, Li Zhi-Hui, Peng Ao-Ping, Zhang Yong-Hao
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• #### 摘要

在跨流域复杂流动问题的模拟中, 基于求解速度分布函数演化方程的气体动理论方法的效率问题一直受到工程应用领域关注. 研究提升气体动理论方法在定常流动模拟中的计算效率具有重要意义. 为了提升定常流动计算收敛速度, 本文提出了一种耦合宏观方程数值本构关系的气体动理论加速收敛方法. 通过求解Boltzmann模型方程, 将应力、热流高阶项的数值解与宏观方程耦合, 实现了宏观方程的封闭; 另一方面, 宏观方程的计算结果被用来更新Boltzmann模型方程的当地平衡态速度分布函数中的宏观物理量, 以此构造求解Boltzmann模型方程的全隐式数值格式. 通过跨流域方腔流动、超声速圆柱绕流及双圆柱干扰绕流案例的数值模拟, 对方法进行了广泛考核. 计算结果与常规气体动理论统一算法、直接模拟蒙特卡罗法符合良好, 证明该方法很好地描述了稀薄流动中的非线性本构关系, 以及激波、强壁面剪切、流动分离等强非平衡特征. 进一步, 对于低努森数Kn的流动, 方法能显著加速收敛过程, 提升计算效率; 随着努森数Kn增加, 气体对流输运效应减弱, 方法的加速收敛效果降低. 与此同时, 如何减少内迭代耗时, 进一步提升效率有待更多研究.

#### Abstract

In the simulation of complex multi-scale flows covering various flow regimes, the computational efficiency of gas kinetic method by which the evolution equation of velocity distribution function is solved directly is the key to engineering applications. In order to accelerate simulation for steady flows, a gas kinetic algorithm accelerated by utilizing the macroscopic conservative equations with a digital constitutive relation is developed. In this algorithm, the contribution of the high-order terms of stress and heat flux in macroscopic conservative equations is determined by the gas kinetic solution. Meanwhile, the solution of the macroscopic conservative equations provides the macroscopic quantities for the equilibrium distribution function in the Boltzmann model equation, where a fully implicit scheme to solve the Boltzmann model equation is developed. Extensive validations are performed for the cavity flow, the supersonic flow around the cylinder, and the interactive rarefied flow around two side-by-side cylinders. The results from the above method are in good agreement with the results from the conventional gas kinetic unified algorithm and the direct simulation Monte Carlo method. It can be concluded that the nonlinear constitutive relation of rarefied flow can be well captured by the present method. And the ability of this method to simulate complex flows such as shock wave, strong wall shear and flow separation is demonstrated. Furthermore, the present method has shown to be much faster than the conventional gas kinetic unified algorithm, especially for the low-Kn flows. As the value of Kn increases, the acceleration rate decreases, because the effect of flow convection becomes weak. Meanwhile, more effort is needed to reduce inner loop iterations to improve its efficiency.

#### 作者及机构信息

###### 通信作者: 李志辉, zhli0097@x263.net
• 基金项目: 国家自然科学基金青年科学基金(批准号: 11902339)资助的课题

#### Authors and contacts

###### Corresponding author: Li Zhi-Hui, zhli0097@x263.net
• Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11902339)

#### 施引文献

• 图 1  方腔流温度分布计算结果　(a) Kn = 1 ; (b) Kn = 0.075; (c) Kn = 0.01 (GKUA: 彩色背景及黑色实线; Coupled: 红色虚线)

Fig. 1.  Temperature distribution in cavity flow: (a) Kn = 1; (b) Kn = 0.075; (c) Kn = 0.01 (GKUA: coloured background and black solid lines; Coupled: red dashed lines).

图 2  方腔中心线上的速度场　(a) Kn = 1; (b) Kn = 0.075; (c) Kn = 0.01

Fig. 2.  Velocity profiles at the central lines of the cavity: (a) Kn = 1; (b) Kn = 0.075; (c) Kn = 0.01.

图 3  耦合加速收敛方法与常规GKUA的计算收敛历史

Fig. 3.  The convergence history between coupled acceleration method and the conventional GKUA.

图 4  Kn = 0.01圆柱绕流流场对比　(a) 压力; (b) 温度; (c) 马赫数 (GKUA: 彩色背景及白实线; Coupled: 红色虚线)

Fig. 4.  (a) Pressure, (b) temperature, (c) Mach number distribution around cylinder for Kn = 0.01 (GKUA: coloured background and white solid lines; Coupled: red dash lines).

图 5  Kn = 0.1圆柱绕流流场对比　(a) 压力; (b) 温度; (c) 马赫数(GKUA: 彩色背景及白实线; Coupled: 红色虚线)

Fig. 5.  (a) Pressure, (b) temperature, (c) Mach number distribution around cylinder for Kn = 0.1 (GKUA: coloured background and white solid lines; Coupled: red dash lines).

图 6  圆柱壁面的　(a) 压力, (b) 热流, (c) 剪切应力

Fig. 6.  (a) Pressure, (b) heat flux, and (c) shear stress profile along the wall surface of cylinder.

图 7  耦合加速收敛方法与常规GKUA的超声速圆柱绕流计算收敛情况对比

Fig. 7.  Comparison of the convergence history of supersonic flow around the cylinder between the coupled acceleration method and the conventional GKUA.

图 8  并列双圆柱多块网格布局

Fig. 8.  The multi-blocks mesh layout for two side-by-side cylinders.

图 9  并列双圆柱绕流流场对比　(a) 压力; (b) 温度; (c) 马赫数(GKUA: 彩色背景及白实线; Coupled: 红色虚线)

Fig. 9.  (a) Pressure, (b) temperature, (c) Mach number field for two side-by-side cylinders (GKUA: coloured background and white solid lines; Coupled: red dash lines).

图 10  上圆柱壁面的　(a) 压力, (b) 热流, (c) 剪切应力

Fig. 10.  (a) Pressure, (b) heat flux, and (c) shear stress profile along the surface of upper cylinder.

图 11  耦合加速收敛方法与常规GKUA的并列双圆柱超声速绕流计算收敛情况对比

Fig. 11.  Comparison of the convergence history of supersonic flow around two side-by-side cylinders between the coupled acceleration method and the conventional GKUA.