【作者】 任炯； 王刚；

【Author】 Jiong Ren;Gang Wang;Dept.of Fluid Dynamics,School of Aeronautics,Northwestern Polytechnical University;

【机构】 西北工业大学航空学院流体力学系；

【摘要】 采用通量限制器控制一维熵相容格式的耗散项可以提高该格式对激波等间断解捕捉的分辨率,由此得到的格式称为高分辨率熵相容格式(简记为ECL格式)。由于不同的通量限制器对格式中耗散项的限制程度不同,导致不同限制器的高分辨率熵相容格式对间断捕捉的分辨率也不同。在不引起数值振荡或者保证通量限制器满足TVD条件的前提下,显然空间格式的数值耗散越少,其对间断位置的分辨率就越高。基于此思想,本文以总熵的数值耗散为参考,将Sweby提出的一类通量限制器分为迎风和对称两种形式插入熵相容格式,并做了大量的数值研究,最终发展出一类更一般化且满足二阶TVD条件的通量限制器,称之为Gsbee类限制器。针对该类限制器,有规律地比较了其中某些限制器对应的高分辨率熵相容格式引起的数值熵耗散的多少。在此基础上发现了一个新的通量限制器——最小耗散通量限制器(简记为MDF限制器),该限制器引起的数值耗散相对其它常用的通量限制器更少,且不会引起不稳定的数值振荡。针对该限制器的最小耗散性,本文给了相关的数学证明,并用更多典型的数值算例验证了其稳定性,有效性和对间断解的捕捉能力。更多还原

【Abstract】 The high resolution entropy-consistent scheme(ECL scheme for short) for 1D Euler equations can be achieved by entropy-consistent scheme with some suitable flux-limiters and indeed highly improve the resolution in discontinuities, while it still has other room for improvement in resolution because not a truly minimized dissipation flux-limiter was used to sufficiently reduce the dissipation. In this spirit, this paper develop a more general kind of limiters which subject to the second-order TVD constraints, termed GSbee limiters, from a class of Sweby’s limiters based on the analysis of ECL flux function and comparing two kinds of commonly used flux-limiters(upwind and symmetry) by modeling the rarefaction with a stationary shock of 1D Burgers equation. Finally, a new flux-limiter, termed MDF limiter, is proposed to achieve the minimum of limited dissipative flux. The high resolution entropy-consistent scheme with MDF upwind limiter, termed ECL-MDFu scheme, for Euler equations not only possess simple form, satisfy entropy stable condition, completely numerical stability, but also has much higher resolution for shock and contact discontinuity than any other existing limiters, which will be proved mathematically and verified numerically by comparing the change of discrete total entropy. Several numerical simulations for 1D and 2D Euler test cases sufficiently illustrate the excellent capability of new MDF limiter in resolution.更多还原