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在传统中, 格子Boltzmann离散模型的数值表现影响因素分析主要关注于平衡分布的矩精度, 而模型的正值性(作为粒子分布描述, 分布函数需恒为正)则仅作为模型的附属属性, 用于计算时工况约束. 随着部分高斯-厄米特求积公式离散方案的提出, 模型正值性被发现是独立于矩精度的模型属性, 可以通过格子速度调整. 研究人员推测平衡分布正值性对格子Boltzmann方法的数值表现存在显著影响, 可以通过改善平衡分布正值性改善模型数值表现. 相比提升模型矩精度方案, 正值性改善方案具有计算量优势. 然而鉴于高阶模型边界处理的缺失, 相关推测并未得到具体数值计算证实. 本文采用周期边界的Taylor-Green涡算例, 回避了边界处理问题, 详细分析了正值性对数值表现的影响, 包括平衡分布正值区域内计算精度稳定性以及模型平衡分布正值区域大小对计算影响, 并与矩精度影响进行对比. 计算结果显示, 模型正值区域内计算精度并不恒定, 随着工况靠近正值区域上界, 计算精度下降, 但总体上均具备较好的精算精度. 模型数值表现同时受到矩精度与模型正值性影响, 矩精度影响主要体现在模型是否满足Galilean不变性上, 对于满足Galilean不变性模型, 其数值表现则取决于模型正值性. 基于此, 本文认为通过改善模型正值性提升格子Boltzmann方法数值表现是切实可行的方案, 并推荐基于满足Galilean不变性条件下选择具有最宽正值区域的模型, 而不必执着于模型矩精度. 另外从本文的数值结果来看, 高阶模型模型的数值表现均好于经典D2Q9模型, 特别是D2H3-2模型, 是文中涉及模型的最优者, 值得进一步深入研究. 总体而言, 通过数值分析首次系统性梳理了模型正值性对计算影响, 并与矩精度进行对比分析. 本文证实了正值性对计算的影响, 为离散模型选择和改进提供了新的方向.
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关键词:
- 格子Boltzmann/
- 离散模型/
- 矩精度/
- 正值性
Traditionally, the numerical performance of the lattice Boltzmann method is mainly determined by the moment degree of a discrete equilibrium distribution. The equilibrium distribution positivity is merely considered as an ancillary property which is used to constrict the numerical configuration. With the newly-developed partial Gaussian-Hermite quadrature scheme, the positivity of equilibrium distribution is validated as an independent property like moment degree which can be adjusted by discrete velocities. Researchers speculated that the positivity should also be significant for the numerical performance by the lattice Boltzmann method and can be used to improve the performance. Comparing with the classical improvement through moment degree, the positivity approach will not bring additional computation. However, due to the lack of boundary treatment, the speculation has not been validated by detailed numerical simulations. In this paper, through employing a periodic case, the Taylor-Green vortex, to avoid the boundary issue, we in depth analyze the numerical effect of the model positivity, including the numerical accuracy in the model positive range, the influence of positivity on the numerical performance, and the significance comparison between positivity and moment degree. The results show that for a given model, the numerical accuracy is not consistent in the whole positive range. As the configuration is close to the border of positive range, the accuracy will degrade though it is still acceptable. The numerical performance of a model depends on both moment degree and positivity. The role that the moment degree plays lies mainly in the qualification of a model on Galilean invariance. Once a model fulfills the Galilean invariance, its numerical performance is solely dependent on the positivity. Hence, the improvement approach through modifying the model positivity is a viable solution, and a Galilean invariant model with wider positive range does possess a better numerical performance regardless of its moment degree. Furthermore, based on the numerical results in this paper, all D nH mmodels with high moment degree are better than the classical D2Q9 model. Of the above models, the D2H3-2 model has the best performance and deserves to be further studied -
Keywords:
- lattice Boltzmann/
- discrete model/
- moment degree/
- positivity
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Hermite 多项式阶数
The degree of Hermite polynomial多项式表达式
Hermite polynomial0 ${ { H}_0}\left( x \right) = 1$ 1 ${ { H}_1}\left( x \right) = 2x$ 2 ${ { H}_2}\left( x \right) = 4x^2-2$ 3 ${ { H}_3}\left( x \right) = 8x^3-12x$ 4 ${ {H}_4}\left( x \right) = 16x^4-48x^2+12$ 
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Model name Discrete velocityset {$v_\alpha$} Lattice constantc Weights {${w}_\alpha$} D2H2 {$0, \pm 1$} $1.2247\times10^0$ {$6.6667\times10^{-1}$, 1.6667$\times10^{-1}$} D2H3-1 {$0, \pm 1, \pm 3$} $5.5343\times10^{-1}$ {$7.4464\times10^{-2}$, $4.1859\times10^{-1}$, $4.4182\times10^{-2}$} D2H3-2 {$0, \pm 2, \pm 5$} $3.4420\times10^{-1}$ {$3.1044\times10^{-1}$, $3.0997\times10^{-1}$, $3.4812\times10^{-2}$} D2H4 {$0, \pm 1, \pm 2, \pm 3$} $8.4639\times10^{-1}$ {$4.7667\times10^{-1}$, $2.3391\times10^{-1}$, $2.6938\times10^{-2}$, $8.1213\times10^{-4}$} D2H5 {$0, \pm 1, \pm 2, \pm 3, \pm 5$} $4.7940\times10^{-1}$ {$1.6724\times10^{-1}$, $3.0315\times10^{-1}$, $5.3303\times10^{-2}$, $5.7922\times10^{-2}$, $2.0013\times10^{-3}$} D2H6 {$0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5$} $6.8590\times10^{-1}$ {$3.8694\times10^{-1}$, $2.4178\times10^{-1}$, $5.8922\times10^{-2}$, $5.6153\times10^{-3}$, $2.0652\times10^{-4}$, $3.2745\times10^{-6}$} 下载:
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Case name Step time/s $u_{L{\rm B} ,0}$ a $1.570796\times10^{-2}$ 0.5 b $3.141593\times10^{-2}$ 1.0 c $4.712389\times10^{-2}$ 1.5 d $6.283185\times10^{-2}$ 2.0 下载:
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