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On the algebraic stretching dynamics of variable-density mixing in shock–bubble interaction

Published online by Cambridge University Press:  04 February 2026

Xu Han ,
Bin Yu Open the ORCID record for Bin Yu [Opens in a new window]  and
Hong Liu Open the ORCID record for Hong Liu [Opens in a new window]
Xu Han
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, PR China
Bin Yu*
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, PR China Sichuan Research Institute, Shanghai Jiao Tong University, Sichuan 100190, PR China J.C. Wu Center for Aerodynamics, School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, PR China
Hong Liu*
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, PR China J.C. Wu Center for Aerodynamics, School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, PR China
*
Corresponding authors: Bin Yu, kianyu@sjtu.edu.cn; Hong Liu, hongliu@sjtu.edu.cn
Corresponding authors: Bin Yu, kianyu@sjtu.edu.cn; Hong Liu, hongliu@sjtu.edu.cn
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Abstract

The mixing mechanism within a single vortex has been a theoretical focus for decades, while it remains unclear especially under the variable-density (VD) scenario. This study investigates canonical single-vortex VD mixing in shock–bubble interactions (SBI) through high-resolution numerical simulations. Special attention is paid to examining the stretching dynamics and its impact on VD mixing within a single vortex, and this problem is investigated by quantitatively characterising the scalar dissipation rate (SDR), namely the mixing rate, and its time integral, referred to as mixedness. To study VD mixing, we first examine single-vortex passive-scalar (PS) mixing with the absence of a density difference. Mixing originates from diffusion and is further enhanced by the stretching dynamics. Under the axisymmetry and zero diffusion assumptions, the single-vortex stretching rate illustrates an algebraic growth of the length of scalar strips over time. By incorporating the diffusion process through the solution of the advection–diffusion equation along these stretched scalar strips, a PS mixing model for SDR is proposed based on the single-vortex algebraic stretching characteristic. Within this framework, density-gradient effects from two perspectives of the stretching dynamics and diffusion process are discovered to challenge the extension of the PS mixing model to VD mixing. First, the secondary baroclinic effect increases the VD stretching rate by the additional secondary baroclinic principal strain, while the algebraic stretching characteristic is still retained. Second, the density source effect, originating from the intrinsic nature of the density difference in the multi-component transport equation, suppresses the diffusion process. By accounting for both the secondary baroclinic effect on stretching and the density source effect on diffusion, a VD mixing model for SBI is further modified. This model establishes a quantitative relationship between the stretching dynamics and the evolution of the mixing rate and mixedness for single-vortex VD mixing over a broad range of Mach numbers. Furthermore, the essential role of the stretching dynamics on the mixing rate is demonstrated by the derived dependence of the time-averaged mixing rate $\overline {\langle \chi \rangle }$ on the Péclet number ${\textit{Pe}}$, which scales as $\overline {\langle \chi \rangle } \sim {\textit{Pe}}^{{2}/{3}}$.

JFM classification

Flow Control: Mixing enhancement Vortex Flows: Vortex dynamics Compressible Flows: Shock waves

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JFM Papers
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Journal of Fluid Mechanics , Volume 1028 , 10 February 2026 , A27
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© The Author(s), 2026. Published by Cambridge University Press

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Supplementary material: File

Han et al. supplementary movie 1

Time evolution of the density contour (top) and the vorticity contour (bottom) for VD and PS SBI cases with shock Mach number 2.4. Dashed-dot line indicates the isoline of Y = 0.01Ymax (Ymax = 1.0 for VD SBI and Ymax = 0.0001 for PS SBI). MV, main vortex; SBV, secondary baroclinic vorticity.
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Supplementary material: File

Han et al. supplementary movie 2

Time evolution of the mixedness contour (top) and the SDR contour (bottom) for VD and PS SBI cases with shock Mach number 2.4. Red dashed-dot line indicates the isoline of Y = 0.01Ymax (Ymax = 1.0 for VD SBI and Ymax = 0.0001 for PS SBI). MV, main vortex; Br, bridge; TB, trailing bubble.
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Han et al. supplementary movie 3

The result of the Lagrangian particle movement for a compressed semicircle composed of a series of scalar strips, each undergoing deformation due to the continuous stretching effect of the single vortex.
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Supplementary material: File

Han et al. supplementary movie 4

The results of the Lagrangian particle movement for the VD SBI case with shock Mach number 1.22.
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Supplementary material: File

Han et al. supplementary material

Han et al. supplementary material
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