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Journal of ZheJiang University (Engineering Science)  2026, Vol. 60 Issue (10): 2099-2108    DOI: 10.3785/j.issn.1008-973X.2026.10.003
    
Interaction between horseshoe vortex and free surface at different Weber numbers
Weiyuan ZENG(),Shiying XIONG*()
School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China
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Abstract  

Direct numerical simulations were performed to investigate the interaction between a horseshoe vortex and a free surface at various Weber numbers. The incompressible Navier-Stokes equations were solved using the volume-of-fluid method coupled with the piecewise linear interface construction scheme. The continuum surface force model was incorporated to account for surface tension effects during interface deformation. The simulations covered a range of Weber numbers from 0.24 to 0.84. By quantitatively analyzing the evolution of total kinetic energy, dissipation rate, total helicity, and surface energy, the influence of Weber number on the vortex-interface interaction was elucidated. At low Weber numbers, surface tension dominated, maintaining the interface as a coherent ring-like structure and suppressing deformation and topological changes, while the vorticity field remained stable with negligible variations in helicity. Conversely, at high Weber numbers, inertial forces dominated over capillary effects, leading to significant interface deformation, breakup, and reconnection. This process resulted in vortex breakdown, accelerated kinetic energy decay, distinct peaks in energy dissipation, and a continuous decrease in helicity. These findings underscored the critical role of the Weber number in energy transfer and dissipation mechanisms, providing theoretical support for multiphase flow dynamics and guidelines for engineering applications such as the design of surface and trans-media vehicles.



Key wordsfree-surface horseshoe vortex      free surface      Weber number      vortex dynamics      surface tension     
Received: 31 December 2025      Published: 28 July 2026
CLC:  O 359  
Fund:  国家自然科学基金资助项目(12302294, 12432010).
Corresponding Authors: Shiying XIONG     E-mail: zengweiyuan@zju.edu.cn;shiying.xiong@zju.edu.cn
Cite this article:

Weiyuan ZENG,Shiying XIONG. Interaction between horseshoe vortex and free surface at different Weber numbers. Journal of ZheJiang University (Engineering Science), 2026, 60(10): 2099-2108.

URL:

https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2026.10.003     OR     https://www.zjujournals.com/eng/Y2026/V60/I10/2099


不同韦伯数下马蹄涡与自由界面的相互作用

通过直接数值模拟研究不同韦伯数下马蹄涡与自由界面之间的相互作用. 采用分段线性界面重构方法结合体积分数法,求解不可压Navier-Stokes方程,并引入连续表面力模型,对韦伯数0.24~0.84范围内的流动过程进行模拟. 通过对流场中总动能、耗散率、总螺旋度及表面能的演化进行定量分析,揭示了韦伯数对涡环与自由界面相互作用的影响. 在低韦伯数情况下,表面张力占主导地位,相界面保持紧凑的环状结构,涡量场稳定且螺旋度几乎不变;在高韦伯数下,惯性力主导,界面经历显著的拉伸、断裂和重联过程,涡结构发生破碎,动能衰减加速,出现明显的能量耗散峰值,螺旋度持续减小. 研究结果揭示了韦伯数在涡环与界面作用的能量传递与耗散机制中具有重要作用,可为多相流界面动力学特性提供理论支撑,并为跨界面航行器设计这类工程应用提供指导.


关键词: 自由表面马蹄涡,  自由界面,  韦伯数,  涡动力学,  表面张力 
Case$ \sigma $$ We $$ {N}^{3} $$ {\rho }_{{\mathrm{l}}}/{\rho }_{{\mathrm{g}}} $$ {\mu }_{{\mathrm{l}}}/{\mu }_{{\mathrm{g}}} $$ Re $
13500.24512310001001000
23000.28512310001001000
32500.34512310001001000
42000.42512310001001000
51500.56512310001001000
61000.84512310001001000
Tab.1 Parameters for free-surface horseshoe vortex case
Fig.1 Case setup schematic diagram
Fig.2 Evolution of normalized total kinetic energy with dimensionless time at different mesh resolutions
Fig.3 Comparison of evolution of low-viscosity liquid droplets driven by surface tension
Fig.4 Phase interface distribution at different Weber numbers
Fig.5 Evolution of phase interface and vorticity field at $ We=0.42 $
Fig.6 Evolution of total kinetic energy with dimensionless time at different Weber numbers
Fig.7 Temporal evolution of local kinetic energy density distribution at $ x=0.5L $ slice under different Weber numbers
Fig.8 Evolution of average dissipation rate with dimensionless time at different Weber numbers
Fig.9 Evolution of total helicity in left and right halves of domain as a function of dimensionless time for different Weber numbers
Fig.10 Temporal evolution of local helicity density distribution at $ y=0.5L $ slice under different Weber numbers
Fig.11 Evolution of normalized surface energy with dimensionless time at different Weber numbers
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