Skip to main content
SpringerLink
Log in

Passive scalar mixing induced by the formation of compressible vortex rings

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

This study aims to determine the relationship between the physical features of a compressible vortex and the mixing process. Such relationship is of significant importance to design combustors that can achieve optimal or most effective mixing. The passive scalar mixing induced by the formation of a canonical compressible vortex ring (CVR) generated at the end of a shock tube is investigated by using numerical simulation. In addition, the method of finite-time Lyapunov exponent (FTLE) field are detected to identify the region of CVR, as well as to analyze the passive scalar mixing during the CVR formation. As the CVR rolls up, the ambient fluid outside the shock tube is entrained into the ring. The entrainment fraction (the mass of entrained fluid to the total mass of CVR) is found to strongly depend on two features of CVRs. One is the compressibility of CVRs, which is characterized by the Mach number of the incident shock denoted by Mach number (Ma). The other is pinch-off of CVRs, which happens at a certain timescale with narrow range of 2–4. As Ma increases, the entrainment fraction of the leading CVR decreases linearly due to smaller vortex core and weaker radial diffusion of vorticity generated by larger compressibility. After CVRs pinch off, trailing vortices appear and show less effective at entrainment than the leading CVRs do. Moreover, the tendency of the rate of entrainment is examined. The results indicate that increasing compressibility and total fluid flux are in favor of the rate of entrainment but restrain the entrainment fraction of total jet.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Shariff, K., Leonard, A.: Vortex rings. Annu. Rev. Fluid Mech. 24, 235–279 (1992)

    MathSciNet  MATH  Google Scholar 

  2. Buch, K.A., Dahm, W.J.A.: Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. Part 1 Sc ≫ 1. J. Fluid Mech. 317, 21–71 (1996)

    Google Scholar 

  3. de Sousa, P.J., Pereira, J.C.: Dynamics of passive scalars and tracers advected by a two-dimensional tripolar vortex. J. Fluid Mech. 634, 41–60 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Flohr, P., Vassilicos, J.C.: Accelerated scalar dissipation in a vortex. J. Fluid Mech. 348, 295–317 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Hermanson, J.C., Cetegen, B.M.: Shock-induced mixing of nonhomogeneous density turbulent jets. Phys. Fluids 12, 1210–1225 (2000)

    MATH  Google Scholar 

  6. Duplat, J., Villermaux, E.: Mixing by random stirring in confined mixtures. J. Fluid Mech. 617, 51–86 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Shadden, S.C., Katija, K., Rosenfeld, M., et al.: Transport and stirring induced by vortex formation. J. Fluid Mech. 593, 315–331 (2007)

    MATH  Google Scholar 

  8. Maxworthy, T.: The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32 (1972)

    Google Scholar 

  9. Muller, E.A., Didden, N.: Zur erzeugung der zirkulation bei der bildung eines ringwirbels an einer dusenmundung. Stroj. Casop 31, 363–372 (1980)

    Google Scholar 

  10. Dabiri, J.O., Gharib, M.: Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311–331 (2004)

    MATH  Google Scholar 

  11. Olcay, A.B., Krueger, P.S.: Measurement of ambient fluid entrainment during laminar vortex ring formation. Exp. Fluids 44, 235–247 (2007)

    Google Scholar 

  12. Gharib, M., Rambod, E., Shariff, K.: A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Sau, R., Mahesh, K.: Passive scalar mixing in vortex rings. J. Fluid Mech. 582, 449–461 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Liang, G., Yu, B., Zhang, B., et al.: Hidden flow structures in compressible mixing layer and a quantitative analysis of entrainment based on Lagrangian method. J. Hydrodyn. 31, 256–265 (2019)

    Google Scholar 

  15. Dimotakis, P.E.: Two-dimensional shear-layer entrainment. AIAA J. 24, 1791–1796 (1986)

    Google Scholar 

  16. Wonhas, A., Vassilicos, J.C.: Mixing in frozen and time-periodic two-dimensional vortical flows. J. Fluid Mech. 442, 359–385 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Cetegen, B.M., Hermanson, J.C.: Mixing characteristics of compressible vortex rings interacting with normal shock waves. Combust. Flame 100, 232–240 (1995)

    Google Scholar 

  18. Li, D., Guan, B., Wang, G.: Numerical study on shock-accelerated heavy gas cylinders with diffusive interfaces. Acta Mech. Sin. 35, 750–762 (2019)

    MathSciNet  Google Scholar 

  19. Chen, X., Dong, G., Li, B.: Numerical study of three-dimensional developments of premixed flame induced by multiple shock waves. Acta Mech. Sin. 34, 1035–1047 (2018)

    MathSciNet  Google Scholar 

  20. Zhang, S., Zhang, Y., Shu, C.W.: Multistage interaction of a shock wave and a strong vortex. Phys. Fluids 17, 116101 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Zhang, S., Zhang, H., Shu, C.W.: Topological structure of shock induced vortex breakdown. J. Fluid Mech. 639, 343–372 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Heister, S.D., Mcdonough, J.M., Karagozian, A.R., et al.: The compressible vortex pair. J. Fluid Mech. 220, 339–354 (1990)

    MATH  Google Scholar 

  23. Brouillette, M., Hebert, C.: Propagation and interaction of shock-generated vortices. Fluid Dyn. Res. 21, 159–169 (1997)

    Google Scholar 

  24. Dora, C.L., Murugan, T., De, S., et al.: Role of slipstream instability in formation of counter-rotating vortex rings ahead of a compressible vortex ring. J. Fluid Mech. 753, 29–48 (2014)

    Google Scholar 

  25. Qin, L., Xiang, Y., Lin, H., et al.: Formation and dynamics of compressible vortex rings generated by a shock tube. Exp. Fluids 61, 86 (2020)

    Google Scholar 

  26. Haller, G., Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence. Phys. D 147, 352–370 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Haller, G.: A variational theory of hyperbolic Lagrangian coherent structures. Phys. D 240, 574–598 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Haller, G.: Lagrangian coherent structures. Annu. Rev. Fluid Mech. 47, 137–162 (2014)

    MathSciNet  Google Scholar 

  29. Kontis, K., An, R., Edwards, J.A.: Compressible vortex-ring interaction studies with a number of generic body configurations. AIAA J. 44, 2962–2978 (2006)

    Google Scholar 

  30. Kruegera, P.S., Gharib, M.: The significance of vortex ring formation to the impulse and thrust of a starting jet. Phys. Fluids 15, 1271–1281 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Murugan, T., De, S., Dora, C.L., et al.: Numerical simulation and PIV study of compressible vortex ring evolution. Shock Waves 22, 69–83 (2011)

    Google Scholar 

  32. Xiang, Y., Lin, H., Zhang, B., et al.: Quantitative analysis of vortex added-mass and impulse generation during vortex ring formation based on elliptic Lagrangian coherent structures. Exp. Therm Fluid Sci. 94, 295–303 (2018)

    Google Scholar 

  33. Qin, S., Liu, H., Xiang, Y.: On the formation modes in vortex interaction for multiple co-axial and co-rotating vortex rings. Phys. Fluids 30, 011901 (2018)

    Google Scholar 

  34. Zhang, M., Wu, Q., Huang, B., et al.: Lagrangian-based numerical investigation of the aerodynamic performance for an oscillating foil. Acta. Mech. Sin. 34, 839–854 (2018)

    Google Scholar 

  35. Wu, Q., Huang, B., Wang, G.: Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil. Acta Mech. Sin. 32, 64–74 (2015)

    MathSciNet  MATH  Google Scholar 

  36. O’Farrell, C., Dabiri, J.O.: A Lagrangian approach to identifying vortex pinch-off. Chaos 20, 017513 (2010)

    Google Scholar 

  37. Wang, L., Feng, L.H., Wang, J.J., et al.: Characteristics and mechanism of mixing enhancement for noncircular synthetic jets at low Reynolds number. Exp. Therm. Fluid Sci. 98, 731–743 (2018)

    Google Scholar 

  38. Perez-Munuzuri, V.: Mixing and clustering in compressible chaotic stirred flows. Phys. Rev. E 89, 022917 (2014)

    Google Scholar 

  39. Perez-Munuzuri, V.: Clustering of inertial particles in compressible chaotic flows. Phys. Rev. E 91, 052906 (2015)

    MathSciNet  Google Scholar 

  40. Green, M.A., Rowley, C.W., Smits, A.J.: Using hyperbolic Lagrangian coherent structures to investigate vortices in bioinspired fluid flows. Chaos 20, 017510 (2010)

    MathSciNet  Google Scholar 

  41. Sun, M., Takayama, K.: Vorticity production in shock diffraction. J. Fluid Mech. 478, 237–256 (2003)

    MathSciNet  MATH  Google Scholar 

  42. Ishii, R., Fujimoto, H., Hatta, N., et al.: Experimental and numerical analysis of circular pulse jets. J. Fluid Mech. 392, 129–153 (1999)

    MATH  Google Scholar 

  43. Lawson, J.M., Dawson, J.R.: The formation of turbulent vortex rings by synthetic jets. Phys. Fluids 25, 105113 (2013)

    Google Scholar 

  44. Schlueter-Kuck, K., Dabiri, J.O.: Pressure evolution in the shear layer of forming vortex rings. Phys. Rev. Fluids 1, 012501 (2016)

    Google Scholar 

  45. Gao, L., Yu, S.C.M.: Development of the trailing shear layer in a starting jet during pinch-off. J. Fluid Mech. 700, 382–405 (2012)

    MATH  Google Scholar 

  46. Shusser, M., Rosenfeld, M., Dabiri, J.O., et al.: Effect of time-dependent piston velocity program on vortex ring formation in a piston/cylinder arrangement. Phys. Fluids 18, 033601 (2006)

    Google Scholar 

  47. Gao, L., Yu, S.C.M.: A model for the pinch-off process of the leading vortex ring in a starting jet. J. Fluid Mech. 656, 205–222 (2010)

    MATH  Google Scholar 

  48. Krueger, P.S., Dabiri, J.O., Gharib, M.: The formation number of vortex rings formed in uniform background co-flow. J. Fluid Mech. 556, 147–166 (2006)

    MATH  Google Scholar 

  49. Dabiri, J.O., Gharib, M.: Starting flow through nozzles with temporally variable exit diameter. J. Fluid Mech. 538, 111–136 (2005)

    MATH  Google Scholar 

  50. Thangadurai, M., Das, D.: Characteristics of counter-rotating formed ahead of a compressible vortex ring. Exp. Fluids 49, 1247–1261 (2010)

    Google Scholar 

  51. Saffman, P.G.: The velocity of viscous vortex rings. Stud. Appl. Math. 49, 371–380 (1970)

    MATH  Google Scholar 

  52. Weigand, A., Gharib, M.: On the evolution of laminar vortex rings. Exp. Fluids 22, 447–457 (1997)

    Google Scholar 

  53. Arakeri, J.H., Das, D., Krothapalli, A., et al.: Vortex ring formation at the open end of a shock tube: a particle image velocimetry study. Phys. Fluids 16, 1008–1019 (2004)

    MATH  Google Scholar 

Download references

Acknowledgements

We wish to acknowledge the support of the National Natural Science Foundation of China (NSFC) Project (Grant 91441205) and the National Science Foundation for Young Scientists of China (Grant 51606120). Besides, the authors would like to thank the Center for High Performance Computing of SJTU for providing the super computer \( \pi \) to support this research.

Author information

Authors and Affiliations

  1. J. C. Wu Center for Aerodynamics, School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Haiyan Lin, Yang Xiang, Hui Xu, Hong Liu & Bin Zhang

Authors
  1. Haiyan Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hui Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Xiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, H., Xiang, Y., Xu, H. et al. Passive scalar mixing induced by the formation of compressible vortex rings. Acta Mech. Sin. 36, 1258–1274 (2020). https://doi.org/10.1007/s10409-020-01006-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-020-01006-6

Keywords

Search

Navigation