Shock Waves

pp 1–16 | Cite as

Gaussian models for late-time evolution of two-dimensional shock–light cylindrical bubble interaction

  • Y. Li
  • Z. Wang
  • B. Yu
  • B. ZhangEmail author
  • H. Liu
Original Article


Two-dimensional shock–bubble interaction is an analogy of the steady three-dimensional jet flow in a scramjet. On the basis of Navier–Stokes simulations, a cylindrical bubble embedded with hydrogen surrounded by air was accelerated by a shock. The evolution can be divided into the lobe-emergence stage, the back-lobe suction stage, and the equilibrium stage. Based on the inhomogeneity between the hydrogen mass fraction and the vorticity field, a correlation coefficient is proposed to quantitatively determine the starting moment of the equilibrium stage. In the equilibrium stage, quasi-Gaussian distributions are modeled for the mass fraction and the vorticity. Surface integrals are performed to derive corresponding mixedness and circulation models, both controlled by two statistical parameters (standard deviation and peak value). Such Gaussian integrated models are universal for different cylindrical bubble aspect ratios (\(\mathrm {AR}=0.5\)–2) and shock Mach numbers (\(M=1.22\)–2). They provide a statistical perspective of late-time SBI evolution in addition to the description from certain physical quantities and help better understand the compressible mixing of scramjet combustors.


Shock–bubble interaction Equilibrium stage Quasi-Gaussian distribution Reduced-order model 



The authors thank the center of High Performance Computing of SJTU for its high-performance computer \(\pi \). This work is supported by the National Science Foundation for Young Scientists of China (Grant No. 51606120). The authors also thank the three reviewers and the editor for their constructive advice. The first author thanks Jun Li, Hanhan Zhu, Haochen Liu, Jun Cheng, Chengcheng Liu, and Mingyun Xie for their contributions to this article.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

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