多组分反应流离散化中的逐层离散证书

中文摘要 AI 辅助

多组分反应流离散化必须守恒总物种质量、保持正的质量分数，并耗散一个凸的混合势。然而，这些性质在求解器的不同阶段受到控制，在一个阶段建立的证书不一定在下一阶段仍然有效。我们针对混合物平均扩散、非正交有限体积重构和保正的Patankar时间积分研究了这一兼容性问题。该分析是等温的，使用给定的密度，并在化学测试中使用合成的细致平衡反应网络。在面层面，熵变量跳跃被证明等于对数平均吉布斯海森矩阵精确作用于质量分数跳跃。要求在单纯形切空间上质量修正项的守恒和熵中性，则产生唯一的分量正的面组成和精确的两点熵证书。算术权重面准则也由生产中性空气/碳数据和准中性双极性空气控制满足，但面级证书不会自动扩展到后续离散层。在测试的原型中，加权最小二乘非正交修正可以将耗散的两点算子转换为反耗散的多点算子。同样，Patankar重标定可以在增加自由能的同时保持正性和守恒；一个显式的有限步有理反例独立于系综统计确立了这一失败。一个完全离散的两单元例子进一步表明，正的面熵裕度并不能保证完全离散更新的正性。因此，守恒、正性和自由能耗散是层特定的性质，需要不同的离散证书。

英文摘要

Multicomponent reacting-flow discretizations must conserve total species mass, preserve positive mass fractions, and dissipate a convex mixing potential. These properties are controlled at different stages of a solver, however, and a certificate established at one stage need not survive the next. We study this compatibility problem for mixture-averaged diffusion, non-orthogonal finite-volume reconstruction, and positivity-preserving Patankar time integration. The analysis is isothermal, uses prescribed density, and uses synthetic detailed-balance reaction networks for the chemistry tests. At the face level, the entropy-variable jump is shown to equal a log-mean Gibbs Hessian applied exactly to the mass-fraction jump. Requiring conservation and entropy neutrality of the mass-correction term on the simplex tangent space then yields a unique componentwise-positive face composition and an exact two-point entropy certificate. The arithmetic-weight face criterion is also satisfied by production neutral-air/carbon data and a quasi-neutral ambipolar-air control, but a face-level certificate does not automatically extend to subsequent discretization layers. A weighted least-squares non-orthogonal correction can, in the tested prototype, convert a dissipative two-point operator into an anti-dissipative multipoint operator. Likewise, Patankar rescaling can preserve positivity and conservation while increasing the free energy; an explicit finite-step rational counterexample establishes this failure independently of ensemble statistics. A fully discrete two-cell example further shows that a positive face entropy margin does not guarantee positivity of the fully discrete update. Thus conservation, positivity, and free-energy dissipation are layer-specific properties and require distinct discrete certificates.

发表机构

• Hangzhou International Innovation Institute, Beihang University（北京航空航天大学杭州创新研究院）
• School of Aeronautic Science and Engineering, Beihang University（北京航空航天大学航空科学与工程学院）

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