高分辨率捕捉强间断多介质流场的伪弧长方法

doi:10.12382/bgxb.2025.0420

摘要/Abstract

摘要：

可压缩多介质气液两相流中,介质热力学差异导致界面区域呈现强非线性波系干涉,显著增加数值模拟难度。尤其在双曲型守恒律方程框架下,流场易形成激波、接触间断和稀疏波等奇异结构,对算法精度、间断捕捉能力及稳定性提出更高要求。为此,构建了一种面向多介质气液两相流场的高精度伪弧长方法,通过引入弧长参数将方程映射至正交弧长空间,缓解强间断导致的数值奇异性,并引入高阶重构格式保证解的精度。为精准刻画界面动力学行为,提出基于弧长空间的符号距离函数演化机制,并结合真实虚拟流体方法严格定义边界条件,确保界面物理量连续性。数值结果表明,该方法具备对强间断及复杂波系结构的高分辨率解析能力。

关键词: 多介质, 双曲型守恒律方程, 伪弧长方法, 非线性, 强间断

Abstract:

In compressible multi-medium gas-liquid two-phase flow, the thermodynamic disparities between different materials lead to strongly nonlinear wave interactions in the interfacial region, significantly increasing the difficulty of numerical simulation. Particularly within the framework of hyperbolic conservation law equation, the flow field is prone to form singular structures such as shock waves, contact discontinuities (CD), and rarefaction waves, which impose stricter requirements on the accuracy, discontinuity-capturing capability, and stability of numerical algorithms. To address these challenges, this paper develops a high-order pseudo arc-length method (PALM) tailored for multi-medium gas-liquid two-phase flow. By introducing an arc-length parameter, the governing equations are transformed into an orthogonal arc-length space, thereby alleviating numerical singularities induced by strong discontinuities. A high-order reconstruction scheme is incorporated to ensure solution accuracy. For the precise description of interfacial dynamics, an evolution mechanism for the signed distance function based on the arc-length space is proposed, combined with the real ghost fluid method to rigorously define boundary conditions and maintain the continuity of physical quantities across the interface. Numerical results demonstrate that the proposed method achieves high-resolution resolution of strong discontinuities and complex wave structures.

Key words: multi-medium, hyperbolic conservation law equation, pseudo arc-length method (PALM), nonlinear, strong discontinuities

李坤, 马天宝, 王渊芃. 高分辨率捕捉强间断多介质流场的伪弧长方法[J]. 兵工学报, 2025, 46(10): 250420-.

LI Kun, MA Tianbao, WANG Yuanpeng. Pseudo Arc-Length Method for High-Resolution Capturing of Strong Discontinuities in Multi-Medium Flows[J]. Acta Armamentarii, 2025, 46(10): 250420-.

图/表 9

图1 一维PALM原理

Fig.1 Principle of one-dimensional (1D) PALM algorithm

图2 二维PALM原理

Fig.2 Principle of two-dimensional (2D) PALM algorithm

图3 物理量守恒

Fig.3 Conservation of physical quantities

图4 弧长空间中一维RGFM定义

Fig.4 Definition of the 1D RGFM in arc-length space

图5 弧长空间中二维RGFM定义

Fig.5 Definition of the 2D RGFM in arc-length space

图6 Hamilton-Jacobi方程计算结果

Fig.6 Results of the Hamilton-Jacobi equation computation

表1 数值精度验证

Table 1 Numerical accuracy verification

网格	WENO-Z		WENO-${Z}_{{\tau }_{8}^{*}}$		PALM
N×M	L₁误差	收敛阶	L₁误差	收敛阶	L₁误差	收敛阶
40×40	4.72×10^-3	-	4.58×10^-3	-	1.72×10^-3	-
80×80	2.22×10^-4	4.41	2.04×10^-4	4.49	6.48×10^-5	4.73
160×160	6.75×10^-6	5.04	6.33×10^-6	5.01	1.81×10^-6	5.16

图7 强激波冲击气-水界面问题

Fig.7 Strong shock wave impacts on the air-water interface

图8 水中激波诱导气泡变形坍塌问题

Fig.8 Shock-induced bubble deformation and collapse in water

参考文献 45

[11]	BORGES R, CARMONA M, COSTA B, et al. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. Journal of Computational Physics, 2008, 227(6): 3191-3211.
[12]	XU X Z, SU X, NING J G. A novel high resolution fifth-order weighted essentially non-oscillatory scheme for solving hyperbolic equations[J]. Physics of Fluids, 2023, 35(11): 116110.
[13]	LI K, MA T B, WANG C T. Positivity-preserving pseudo arc-length method for solving extreme explosive shock wave flow field problems in hyperbolic conservation law equations[J]. Physics of Fluids, 2025, 37(1): 016122.
[14]	HENRICK A K, ASLAM T D, POWERS J M. Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points[J]. Journal of Computational Physics, 2005, 207(2): 542-567.
[15]	朱君, 赵宁. 一种MWENO格式的构造和应用[J]. 空气动力学学报, 2005, 23(3): 330-333.
	ZHU J, ZHAO N. A kind of MWENO scheme and its applications[J]. Acta Aerodynamica Sinica, 2005, 23(3): 330-333. (in Chinese)
[16]	NING J G, SU X, XU X Z. Improved fifth-order weighted essentially non-oscillatory scheme with low dissipation and high resolution for compressible flows[J]. Physics of Fluids, 2022, 34(5): 056105.
[17]	LUO X, WU S P. An improved WENO-Z+ scheme for solving hyperbolic conservation laws[J]. Journal of Computational Physics, 2021, 445(2): 110608.
[18]	孙肖元, 邓枫, 刘学强. 某型双机身飞机水上迫降数值模拟[J]. 兵工学报, 2023, 44(7): 2066-2079.
	SUN X Y, DENG F, LIU X Q. Numerical simulation of ditching of a twin-fuselage aircraft[J]. Acta Armamentarii, 2023, 44(7): 2066-2079. (in Chinese)
[1]	张轶凡, 刘亮涛, 王金相, 等. 水下爆炸冲击波和气泡载荷对典型圆柱壳结构的毁伤特性[J]. 兵工学报, 2023, 44(2): 345-359.
	ZHANG Y F, LIU L T, WANG J X, et al. Damagecharacteristics of underwater explosion shock wave and bubble load on typical cylindrical shell structure[J]. Acta Armamentarii, 2023, 44(2): 345-359. (in Chinese)
[2]	BIGDELOU P, LIU C, TAREY P, et al. An efficient ghost fluid method to remove overheating from material interfaces in compressible multi-medium flows[J]. Computers & Fluids, 2022, 233:105250.
[3]	李旭, 岳松林, 邱艳宇, 等. 近场水下爆炸气泡与混凝土组合板相互作用的试验研究[J]. 兵工学报, 2023, 44(S1): 79-89.
	LI X, YUE S L, QIU Y Y, et al. Experimental study on interaction between bubble and concrete composite slab in near-field underwater explosion[J]. Acta Armamentarii, 2023, 44(S1): 79-89. (in Chinese)
[4]	何志伟, 田保林, 李理, 等. 可压缩多介质流动问题的高精度数值模拟方法[J]. 空气动力学学报, 2021, 39(1): 177-190.
	HE Z W, TIAN B L, LI L, et al. High-order numerical simulation method for compressible multi-material flow problems[J]. Acta Aerodynamica Sinica, 2021, 39(1): 177-190. (in Chinese)
[5]	姚成宝, 付梅艳, 韩峰, 等. 基于多介质Riemann问题的流体-固体耦合数值方法及其在爆炸与冲击问题中的应用[J]. 兵工学报, 2021, 42(2): 340-355.
	YAO C B, FU M Y, HAN F, et al. A numerical scheme for fluid-solid interactions based on multi-medium riemann problem and its application in explosion and impact problems[J]. Acta Armamentarii, 2021, 42(2): 340-355. (in Chinese)
[6]	HARTEN A. High resolution schemes for hyperbolic conservation laws[J]. Journal of Computational Physics, 1997, 135(2): 260-278.
[19]	HIRT C W, NICHOLS B D. volume of fluid (VOF) method for the dynamics of free boundaries[J]. Journal of Computational Physics, 1981, 39(1): 201-225.
[20]	DYADECHKO V, SHASHKOV M. Reconstruction of multi-material interfaces from moment data[J]. Journal of Computational Physics, 2008, 227(11): 5361-5384.
[21]	张莉, 吴开腾. 固定界面Level Set方法在爆轰波阵面传播中的应用[J]. 兵工学报, 2019, 40(10): 2080-2087.
	ZHANG L, WU K T. The Application of fixed-interface level set method in the calculation of detonation wavefront propagation[J]. Acta Armamentarii, 2019, 40(10): 2080-2087. (in Chinese)
[22]	BAI J S, LI P, ZHONG M M, et al. A program burn method based on detonation shock dynamics and level set (LS) methods[J]. Explosion & Shock Waves, 2008, 28(5): 401-406.
[23]	SUSSMAN M, SMEREKA P, OSHER S. A level set approach for computing solutions to incompressible two-phase flow[J]. Journal of Computational Physics, 1994, 114 (1): 146-159.
[24]	FEDKIW R P. The ghost fluid method for numerical treatment of discontinuities and interfaces[M]. New York: Springer, 2001.
[25]	许亮, 冯成亮, 刘铁钢. 弹性板水下爆炸冲击载荷的修正虚拟流体方法分析[J]. 计算物理, 2017, 34(1): 1-9.
	XU L, FENG C L, LIU T G. Analysis ofimpact load on elastic plate in underwater explosions with modified ghost fluid method[J]. Chinese Journal of Computational Physics, 2017, 34(1): 1-9. (in Chinese)
[26]	刘铁钢, 许亮. 模拟多介质界面问题的虚拟流体方法综述[J]. 气体物理, 2019, 4(2): 1-16.
	LIU T G, XU L. A review of ghost fluid methods for multi-medium interface simulation[J]. Physics of Gases, 2019, 4(2): 1-16. (in Chinese)
[27]	WANG C W, LIU T G, KHOO B C. A real ghost fluid method for the simulation of multimedium compressible flow[J]. SIAM Journal on Scientific Computing, 2006, 28 (1): 278-302.
[28]	RIKS E. An incremental approach to the solution of snapping and buckling problems[J]. International Journal of Solids and Structures, 1979, 15(7): 529-551.
[29]	CHAN T. Newton-like pseudo-arclength methods for computing simple turning points[J]. Siam Journal on Scientific and Statistical Computing, 1984, 5(1): 135-148.
[30]	MASARATI P. Robust static analysis using general-purpose multibody dynamics[J]. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2014, 229(2): 152-165.
[31]	武际可, 许为厚, 丁红丽. 求解微分方程初值问题的一种弧长法[J]. 应用数学和力学, 1999, 20(8):875-880.
	WU J K, XU W H, DING H L. Arc-length method for differential equations[J]. Applied Mathematics and Mechanics, 1999, 20(8): 875-880. (in Chinese)
[32]	宁建国, 原新鹏, 马天宝, 等. 计算动力学中的伪弧长方法研究[J]. 力学学报, 2017, 49(3): 703-715.
	NING J G, YUAN X P, MA T B, et al. Pseudo arc-length numerical algorithm for computational dynamics[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(3): 703-715. (in Chinese)
[33]	赵金庆, 马天宝. 基于伪弧长移动网格算法的爆炸与冲击多介质问题数值模拟[J]. 兵工学报, 2020, 41(S2): 200-210.
	ZHAO J Q, MA T B. Numerical simulation of explosion and shock problems of multi-material flows by pseudo-arc length moving mesh algorithm[J]. Acta Armamentarii, 2020, 41(S2): 200-210. (in Chinese)
[34]	MA T B, WANG C T, XU X Z. Conservative high precision pseudo arc-length method for strong discontinuity of detonation wave[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43 (3): 417-436.
[35]	王晨涛, 马天宝, 李坤. 强间断多介质流的高精度伪弧长方法[J]. 北京理工大学学报, 2022, 42(9): 900-908.
	WANG C T, MA T B, LI K. High-order pseudo arc-length method for strongly discontinuity of multi-medium flows[J]. Transactions of Beijing Institute of Technology, 2022, 42(9): 900-908. (in Chinese)
[36]	MA T B, LI K, WANG C T. A high-order pseudo arc-length method with positivity-preserving flux limiter for compressible multi-medium flows[J]. Computers & Fluids, 2024, 274(8): 106234.
[37]	WANG C W, TANG H Z, LIU T G. An adaptive ghost fluid finite volume method for compressible gas-water simulations[J]. Journal of Computational Physics, 2008, 227(12): 6385-6409.
[38]	LEE J M. Riemannian manifolds: An introduction to curvature[M]. New York: Springer, 1997.
[39]	毛枚良, 姜屹, 闵耀兵, 等. 高阶精度有限差分方法几何守恒律研究进展[J]. 空气动力学学报, 2021, 39(1): 157-167.
	MAO M L, JIANG Y, MIN Y B, et al. A survey of geometry conservation law for high order finite difference method[J]. Acta Aerodynamica Sinica, 2021, 39(1): 157-167. (in Chinese)
[40]	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.
[41]	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(1): 72-86.
[42]	TORO E F. Riemann solvers and numerical methods for fluid dynamics: a practical introduction[M]. Heidelberg: Springer, 2009.
[43]	OSHER S, SHU C W. High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations[J]. SIAM Journal on numerical analysis, 1991, 28(4): 907-922.
[7]	HARTEN A, OSHER S, ENGQUIST B, et al. Some results on uniformly high-order accurate essentially nonoscillatory schemes[J]. Applied Numerical Mathematics, 1986, 2(3-5): 347-377.
[8]	LIU X, OSHER S, CHAN T. Weighted essentially non-oscillatory schemes[J]. Journal of Computational Physics, 1994, 115(1): 200-212.
[9]	JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1): 202-228.
[10]	宁建国, 苏璇, 许香照. 冲击波强间断的高分辨率低耗散的精确捕捉数值计算方法[J]. 中国科学:技术科学, 2024, 54(5): 870-880.
	NING J G, SU X, XU X Z. A high-resolution and low-dissipation numerical method for accurate capture of shock waves[J]. Scientia Sinica Technologica, 2024, 54(5): 870-880. (in Chinese)
[44]	NOURGALIEV R R, DINH T N, THEOFANOUS T G. Adaptive characteristics-based matching for compressible multifluid dynamics[J]. Journal of Computational Physics, 2006, 213(2): 500-529.
[45]	BOURNE N K, FIELD J E. Shock-induced collapse of single cavities in liquids[J]. Journal of Fluid Mechanics, 1992, 244: 225-240.