| 复杂流动间断问题的高效高精度算法及应用 | |
| Alternative Title | A Study of High-efficiency and High-accuracy Algorithms for Complex Discontinuous Problems |
李诗尧
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| Thesis Advisor | 申义庆 |
| 2023-05-23 | |
| Degree Grantor | 中国科学院大学 |
| Place of Conferral | 北京 |
| Subtype | 博士 |
| Degree Discipline | 流体力学 |
| Keyword | 复杂流动数值模拟 加权基本无振荡格式 共用权 自适应网格加密方法 凝聚炸药爆轰 |
| Abstract | 针对流动中存在的间断现象，发展高分辨率高精度的数值求解方法一直是计算流体力学研究的重要内容。加权基本无振荡（Weighted essentially non-oscillatory，简称WENO）格式对光滑解具有较高阶的精度而在间断附近具有基本无振荡的性质，因而在计算流体力学中得到较多的研究与应用，WENO算法也得到不断的发展和完善。本文针对当前WENO格式的一些不足，研究高效、高精度低耗散的WENO格式，并将所发展的方法应用于凝聚炸药爆轰的数值模拟研究。
（1）提出新的权值计算方法，构造了新的WENO-Z型格式（下文简称WENO-ZN格式）,极大提高了临界点的精度。临界点具有较强的数学物理意义，如一阶、二阶、三阶临界点可刻画解的极值、拐点、曲率变化等几何特征，高阶临界点可刻画更小尺度上解的变化。为了提高极值点精度，WENO-Z格式利用整体光滑因子来计算权值。本文通过分析已有的WENO-Z型格式的不足，提出利用整体模板上的最高阶差分的平方作为整体光滑因子，以此使得新格式在尽可能高阶的临界点获得期望的精度；进一步构造了自适应函数代替原权值计算公式的常数1，该函数在光滑区趋于一个远大于1的数，而在含间断的模板其值远小于1，从而获得了一类精度高、耗散低、鲁棒性好的WENO-ZN格式。利用数值算例验证了所发展的系列格式的优越性能。
（2）提出和发展了应用于Euler方程组的共用权WENO（Common-Weights WENO, Co-WENO）格式。当WENO格式应用于方程组时，传统的方法是对方程组中的每个分量分别计算权值及加权组合。本文提出的共用权WENO格式，是指方程组中的每个分量皆利用同一组权值进行加权，因此共用权WENO格式具有两个优点：其一是由于只需计算一组权值，因此该方法具有较高的计算效率；其二是共用权方法保证了同一个模板或者是同一个单元对方程组中的每一个分量具有相同的贡献。本文基于Euler方程组的通量分裂方法，发展了系列的共用权WENO格式。数值算例表明，共用权WENO格式具有效率高、耗散低等性能。
（3）发展了高效的自适应网格加密方法。自适应网格加密方法能够根据需要对流场局部进行加密计算，从而在初始粗网格上通过加密得到与对应均匀细网格几乎一致的计算结果，能显著提升计算效率。在本文的研究工作中，首先提出了一种新的流场加密判定方法，其需要的加密阈值具有较好的普适性；其次，在利用粗网格数值向加密网格的插值过程中，我们提出了共用权WENO插值方法，与通量构造的共用权WENO格式相似，共用权WENO插值极大地提高了网格自适应加密算法的计算效率及稳定性。
（4）凝聚态炸药爆轰研究在国防军事发展和国民经济建设中具有重要的应用价值。由于其复杂的状态方程，针对理想气体状态方程所发展的算法不能直接推广应用。本文发展了适用于凝聚炸药爆轰的Steger-Warming通量分裂方法和半隐式TVD Runge-Kutta时间迭代方法，并结合前文发展的数值格式，模拟了钝感高能炸药的起爆与爆轰波传播过程。数值试验表明本文发展的计算方法在爆轰模拟中也有较高的计算精度和分辨率以及良好的收敛性，能够准确模拟爆轰波衍射过程中的“死区”（Dead Zone）现象和爆轰波碰撞中的正规反射与马赫反射等流动细节。
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| Other Abstract | The development of high-precision and high-resolution numerical methods for solving discontinuity problems has been a significant topic in computational fluid dynamics (CFD). The weighted essentially non-oscillatory (WENO) schemes have been widely applied in CFD due to their ability to achieve high order accuracy on smooth solutions and maintain essential non-oscillatory property near discontinuities. However, even though the order of accuracy for WENO schemes can be designed to be arbitrarily high, there still is other performance needed to be improved. This paper aims to further improve the accuracy, resolution, and also the computational efficiency of the WENO schemes, and apply them to study some complex flow fields.
Firstly, the novel WENO-Z type (WENO-ZN) schemes are constructed to improve accuracy at critical points. The WENO-ZN scheme uses the square of the highest order difference on the global stencil as the global smoothness indicator, so that the scheme can obtain optimal accuracy at high order critical points. Furthermore, an adaptive function is constructed to replace the constant 1 used to calculate the un-normalized weights in the original WENO-Z schemes. The function can approach a small value if the global stencil contains a discontinuity or approach a large value if the solution is smooth enough. Numerical examples showed that the proposed WENO-ZN schemes have properties of high-precision, low-dissipation and good robustness.
Secondly, the common-Weights WENO (Co-WENO) scheme based on the flux vector splitting method is developed for the Euler equations. In the traditional WENO schemes, each component numerical flux calculates its weights separately. While the Co-WENO scheme utilizes the same set of weights for all component fluxes on one global stencil, it therefore improves the computational efficiency and keeps the same contribution of each sub-stencil to the numerical fluxes. Numerical experiments show the Co-WENO schemes have high computational efficiency and low numerical dissipation.
Thirdly, a high-efficiency adaptive mesh refinement (AMR) method with common-weights weighed essentially non-oscillatory (Co-WENO) difference scheme and interpolation algorithms is developed. (1) The Co-WENO difference scheme is implemented to calculate the numerical fluxes; (2) a common-weights WENO interpolation is proposed to prolong the data from the parent-cells to child-cells; (3) an effective method is suggested to detect the refinement mesh. Numerical experiments show that the proposed AMR method has the properties of low dissipation, high resolution and high computational efficiency.
Finally, we develop the Steger-Warming flux vector splitting method and semi-implicit TVD Runge-Kutta method for numerical simulation of the condensed explosive detonation. Combined with numerical methods developed in this paper, the initiation and propagation of detonation wave are simulated. The ``Dead Zone" phenomenon in the detonation diffraction and the regular reflection and Mach reflection in the detonation collision are clearly presented.
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| Language | 中文 |
| Document Type | 学位论文 |
| Identifier | http://dspace.imech.ac.cn/handle/311007/92343 |
| Collection | 高温气体动力学国家重点实验室 |
| Corresponding Author | 李诗尧 |
| Recommended Citation GB/T 7714 | 李诗尧. 复杂流动间断问题的高效高精度算法及应用[D]. 北京. 中国科学院大学,2023. |
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| 81927.pdf（16127KB） | 学位论文 | 开放获取 | CC BY-NC-SA | Application Full Text | ||
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