爆炸冲击波强间断问题的高阶伪弧长算法研究

马天宝; 王晨涛; 赵金庆; 宁建国

doi:10.11883/bzycj-2020-0366

爆炸冲击波强间断问题的高阶伪弧长算法研究

doi: 10.11883/bzycj-2020-0366

北京理工大学爆炸科学与技术国家重点实验室, 北京 100081

基金项目: 国家自然科学基金（11822203, 11532012）

详细信息

作者简介:
马天宝（1981－　），男，博士，教授，madabal@bit.edu.cn

通讯作者:
宁建国（1963－　），男，博士，教授，jgning@bit.edu.cn

中图分类号: 0381
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- 文章访问数: 447
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- 被引次数: 0
出版历程
- 收稿日期: 2020-09-30
- 修回日期: 2020-12-07
- 网络出版日期: 2021-09-28
- 刊出日期: 2021-11-23

High order pseudo arc-length method for strong discontinuity of detonation wave

State Key Laboratory of Explosion Science and technology, Beijing Institute of Technology, Beijing 100081, China

摘要

摘要: 为了提高对冲击波强间断处的分辨率，通过引入弧长参数，使网格自适应地朝着间断处移动，并结合高精度WENO数值格式，进而达到了对大梯度物理量的高分辨率捕捉。针对网格移动造成的非均匀和非正交现象，通过坐标变换，使得计算过程在均匀正交的计算空间中进行。通过和有限体积下的数值结果对比，结合数值误差分析，可以看到高阶伪弧长数值算法不仅保证了高精度而且对间断的捕捉更加明显，在间断附近解的整体光滑性较好，网格的自适应移动使得解的奇异性得到了削弱，因此可以削弱高阶格式容易引起数值振荡这个缺点。最后采用高阶伪弧长算法计算了化学反应流问题，结果表明高阶伪弧长算法有着较快的收敛率，对于解决爆炸与冲击强间断问题有着较为明显的优势。
- 伪弧长算法 /
- 高精度 /
- 坐标变换 /
- 冲击波 /
- 强间断
Abstract: In this paper, in order to improve the resolution of capturing discontinuities, we introduce the pseudo arc-length parameter to make the mesh move to the discontinuities adaptively. By combining the high-precision WENO scheme with the pseudo arc-length algorithm, the advantages of both schemes can be shown, on the one hand, the solution has a higher convergence rate, on the other hand, it has a higher resolution for the region solution with larger physical variation . Because the traditional high-order scheme is based on Cartesian grid, and the grid in the pseudo arc-length numerical calculation is deformed. In view of the non-uniform grid and non-orthogonal deformation grid caused by the grid moving, the original deformed physical space is mapped to the uniform orthogonal arc-length calculation space by introducing coordinate transformation, and then the classical higher order scheme is used to solve the governing equations in the computational coordinate system. Through the comparison of some numerical examples and the analysis of numerical errors, it can be found that the pseudo arc-length algorithm is better than the finite volume method with fixed mesh. The high-order pseudo arc-length algorithm has a very high resolution to capture discontinuities, and the density of the grid near the discontinuities is very high. The adaptive grid movement weakens the singularity of the governing equation near the discontinuity, so the whole solution is smooth and the numerical oscillation is not obvious. This shows that the pseudo arc-length algorithm can overcome the shortcomings of high-order schemes which easily cause numerical oscillations. Finally, the chemical reaction flow problem is calculated. The results show that the high-order pseudo arc-length numerical algorithm with less mesh number has faster convergence rate and higher discontinuous resolution. Therefore, the high-order pseudo arc-length algorithm has obvious advantages in dealing with the strong discontinuity problem of explosion and shock.
- pseudo arc-length algorithm /
- high resolution /
- coordinate transformation /
- shock wave /
- strong discontinuity

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图 1 新旧网格转化示意图

Figure 1. Diagram of old grid transforming to new grid

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图 2 一维Euler方程激波管图

Figure 2. Results of shock tube for one dimensional Euler equation

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图 3 一维爆炸算例计算结果

Figure 3. Results of one dimensional explosion example

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图 4 密度云图（$ T = 2{\text{π}} $，网格$ 40 \times 40 $）

Figure 4. Density contours ($ T = 2{\text{π}} $, grid $ 40 \times 40 $)

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图 5 一维化学反应流密度

Figure 5. Density of one dimensional chemical reaction flow

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图 7 一维化学反应流的网格轨迹

Figure 7. Mesh trajectory of one dimensional chemical reaction flow

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图 6 一维化学反应流压力

Figure 6. Pressure of one dimensional chemical reaction flow

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图 8 二维化学反应流图

Figure 8. Results of two dimensional chemical reaction flow

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表 1 有限体积法与伪弧长算法在不同网格数下的误差和精度

Table 1. Numerical errors and precision of FVM and PALM changing with grid numbers (example 3)

网格数	L₁				Order
网格数	FG-2	PALM-2	FG-5	PALM-5	FG-2	PALM-2	FG-5	PALM-5
40	3.197×10⁻²	3.185×10⁻²	4.050×10⁻⁵	6.079×10⁻⁵
80	9.173×10⁻³	9.181×10⁻³	1.021×10⁻⁶	1.047×10⁻⁶	1.801	1.794	5.310	4.892
160	2.502×10⁻³	2.500×10⁻³	3.042×10⁻⁸	3.265×10⁻⁸	1.874	1.877	5.068	5.003
320	6.804×10⁻⁴	6.712×10⁻⁴	1.365×10⁻⁹	1.675×10⁻⁹	1.879	1.897	4.478	4.285

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表 2 有限体积法与伪弧长算法在不同网格数下的误差和精度

Table 2. Numerical errors and precision of FVM and PALM changing with grid numbers (Example 4)

Mesh	L₁				Order
Mesh	FG-2	PALM-2	FG-5	PALM-5	FG-2	PALM-2	FG-5	PALM-5
20×20	1.756	1.626	6.889×10⁻²	7.224×10⁻²
40×40	0.597	0.558	2.116×10⁻³	2.009×10⁻³	1.558	1.541	5.025	5.169
80×80	0.157	0.140	7.577×10⁻⁵	7.564×10⁻⁵	1.924	1.997	4.803	4.731
160×160	0.042	0.033	2.365×10⁻⁶	2.672×10⁻⁶	1.890	2.079	5.002	4.823

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参考文献(13)

[1]	GAO Z, FANG L L, WANG B S, et al. Seventh and ninth orders characteristic-wise alternative WENO finite difference schemes for hyperbolic conservation laws [J]. Computers & Fluids, 2020, 202: 104519. DOI: 10.1016/j.compfluid.2020.104519.
[2]	CASTRO M, COSTA B, DON W S. High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws [J]. Journal of Computational Physics, 2011, 230(5): 1766–1792. DOI: 10.1016/j.jcp.2010.11.028.
[3]	GANDE N R, BHISE A A. Third-order WENO schemes with kinetic flux vector splitting [J]. Applied Mathematics and Computation, 2020, 378: 125203. DOI: 10.1016/j.amc.2020.125203.
[4]	LIANG D, ZHOU Z G. The conservative splitting domain decomposition method for multicomponent contamination flows in porous media [J]. Journal of Computational Physics, 2020, 400: 108974. DOI: 10.1016/j.jcp.2019.108974.
[5]	TANG H Z, TANG T. Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws [J]. SIAM Journal on Numerical Analysis, 2003, 41(2): 487–515. DOI: 10.1137/S003614290138437X.
[6]	TANG H Z. A moving mesh method for the Euler flow calculations using a directional monitor function [J]. Communications in Computational Physics, 2006, 1(4): 656–676.
[7]	HE P, TANG H Z. An adaptive moving mesh method for two-dimensional relativistic hydrodynamics [J]. Communications in Computational Physics, 2012, 11(1): 114–146. DOI: 10.4208/cicp.291010.180311a.
[8]	马天宝, 陈建良, 宁建国, 等. 爆轰波强间断问题的伪弧长算法及其人为解验证 [J]. 爆炸与冲击, 2018, 38(2): 271–278. DOI: 10.11883/bzycj-2016-0216. MA T B, CHEN J L, NING J G, et al. A pseudo arc-length method for strong discontinuity of detonation wave and its man manufactured solution verification [J]. Explosion and Shock Waves, 2018, 38(2): 271–278. DOI: 10.11883/bzycj-2016-0216.
[9]	WANG C, ZHANG X X, SHU C W, et al. Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations [J]. Journal of Computational Physics, 2012, 231(2): 653–665. DOI: 10.1016/j.jcp.2011.10.002.
[10]	NING J G, YUAN X P, MA T B, et al. Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions [J]. Computers & Fluids, 2015, 123: 72–86. DOI: 10.1016/j.compfluid.2015.09.011.
[11]	WANG R, FENG H, SPITERI R J. Observations on the fifth-order WENO method with non-uniform meshes [J]. Applied Mathematics and Computation, 2008, 196(1): 433–447. DOI: 10.1016/j.amc.2007.06.024.
[12]	SMIT J, VAN SINT ANNALAND M, KUIPERS J A M. Grid adaptation with WENO schemes for non-uniform grids to solve convection-dominated partial differential equations [J]. Chemical Engineering Science, 2005, 60(10): 2609–2619. DOI: 10.1016/j.ces.2004.12.017.
[13]	HUANG W F, REN Y X, JIANG X. A simple algorithm to improve the performance of the WENO scheme on non-uniform grids [J]. Acta Mechanica Sinica, 2018, 34(1): 37–47. DOI: 10.1007/s10409-017-0715-2.