Abstract In this article, two schemes have been presented for applying Neumann boundary condition in thermal lattice Boltzmann method. In Scheme 1, the heat flux is directly applied to the gradients of unknown distribution functions. In the following, using the results of this scheme, Scheme 2 is introduced having simpler structure than Scheme 1. Both schemes are independent of lattice arrangement. However, D2Q9 lattice arrangement is employed in this article. These two schemes are analyzed in several examples at different relaxation times, and second-order convergence accuracy is achieved for both schemes at the relaxation times. In contrast, other conventional methods, especially the ones which are based on passive scalar approach, give second-order convergence accuracy only at particular relaxation times. Moreover, it is shown that the two proposed schemes in this paper have better accuracy than the conventional method used for simulating adiabatic boundary condition, whereas they do not impose additional computational cost. In addition, the two schemes have also been applied on curved boundaries, and good agreement has been observed with exact solution.

中文翻译：

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在热晶格 Boltzmann 方法中应用 Neumann 边界条件的两种新方法

摘要 本文提出了两种在热晶格玻尔兹曼方法中应用诺依曼边界条件的方案。在方案 1 中，热通量直接应用于未知分布函数的梯度。下面利用该方案的结果，介绍结构比方案1简单的方案2。两种方案均独立于晶格排列。然而，本文采用 D2Q9 晶格排列。这两种方案在不同弛豫时间的几个例子中进行了分析，并且两种方案在弛豫时间都达到了二阶收敛精度。相比之下，其他传统方法，尤其是基于被动标量方法的方法，仅在特定弛豫时间给出二阶收敛精度。而且，结果表明，本文提出的两种方案比用于模拟绝热边界条件的传统方法具有更好的精度，而它们不增加额外的计算成本。此外，这两种方案也已应用于弯曲边界，并且与精确解具有良好的一致性。