Pseudo Arc-Length Method for High-Resolution Capturing of Strong Discontinuities in Multi-Medium Flows

doi:10.12382/bgxb.2025.0420

Abstract

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

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-.

Figures/Tables 9

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

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

Fig.3 Conservation of physical quantities

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

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

Fig.6 Results of the Hamilton-Jacobi equation computation

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

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

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

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