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Breakdown regime of a shielded vortex interacting with a standing normal shock: a numerical study

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Abstract

Numerical simulation results of a convecting shielded vortex interacting with a normal shock using a compact scheme in the convecting upwind and split pressure framework are presented. We explore the parameter space spanned by vortex Mach number and incident Mach number to look for combinations of the parameters which lead to vortex breakdown. The incident and vortex Mach numbers covered are on the higher side, where relatively less information is available. It is well known that for a weak shock, the vortex retains its original shape and for stronger shocks it breaks down. In-between these two extremes, there is a region where the vortex neither retains its original shape nor does it break into small pieces. We determine the vortex breakdown and transition regions that have not so far been reported in shock–vortex interaction studies. A number of cases have been studied, and a vortex breakdown criterion for the cases considered is proposed.

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References

  1. Cutler, A.D., Levey, B.S.: Vortex breakdown in a supersonic jet. 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, HI, AIAA Paper 1991–1815 (1991). https://doi.org/10.2514/6.1991-1815

  2. Nelson, R.C., Pelletier, A.: The unsteady aerodynamics of slender wings and aircraft undergoing large amplitude maneuvers. Prog. Aerosp. Sci. 39, 185–248 (2003). https://doi.org/10.1016/S0376-0421(02)00088-X

    Article  Google Scholar 

  3. Rault, A., Chiavassa, G., Donat, R.: Shock-vortex interactions at high Mach numbers. J. Sci. Comput. 19(1–3), 347–371 (2003). https://doi.org/10.1023/A:1025316311633

    Article  MathSciNet  Google Scholar 

  4. Inoue, O., Hattori, Y.: Sound generation by shock–vortex interactions. J. Fluid Mech. 380, 81–116 (1999). https://doi.org/10.1017/S0022112098003565

    Article  MathSciNet  Google Scholar 

  5. Meadows, K.R., Kumar, A., Hussain, M.Y.: Computational study on the interaction between a vortex and a shock wave. AIAA J. 29(2), 174–179 (1991). https://doi.org/10.2514/3.59916

    Article  Google Scholar 

  6. Smart, M.K., Kalkhoran, I.M., Popovic, S.: Some aspects of streamwise vortex behavior during oblique shock wave/vortex interaction. Shock Waves 8, 243–255 (1998). https://doi.org/10.1007/s001930050117

    Article  Google Scholar 

  7. Hollingsworth, M.A., Richards, E.J.: A Schlieren study of the interaction between a vortex and a shock wave in a shock tube. Br. Aeronaut. Res. Counc. 17, 985 (1955). http://cir.nii.ac.jp/crid/1573387448929137920

  8. Dosanjh, D.S., Weeks, T.M.: Interaction of a starting vortex as well as a vortex street with a traveling shock wave. AIAA J. 3, 216–223 (1965). https://doi.org/10.2514/3.2833

    Article  Google Scholar 

  9. Naumann, A., Hermanns, E.: On the interaction between a shock wave and a vortex field. AGARD Conference Proceedings (1973)

  10. Ribner, H.S.: The sound generated by interaction of a single vortex with a shock wave. University of Toronto, Institute of Aerospace Studies (UTIA) Report No. 61 (1959)

  11. Ribner, H.S.: Cylindrical sound wave generated by shock–vortex interaction. AIAA J. 23, 1708–1715 (1985). https://doi.org/10.2514/3.9155

    Article  Google Scholar 

  12. Dosanjh, D.S., Weeks, T.M.: Sound generation by shock–vortex interaction. AIAA J. 5, 660–669 (1967). https://doi.org/10.2514/3.4045

    Article  Google Scholar 

  13. Ting, L.: Transmission of singularities through a shock wave and the sound generation. Phys. Fluids 17(1518), 1–21 (1974). https://doi.org/10.1063/1.1694928

    Article  Google Scholar 

  14. Zhang, S., Jiang, S., Zhang, Y.T., Shu, C.W.: The mechanism of sound generation in the interaction between a shock wave and two counter-rotating vortices. Phys. Fluids 21(076101), 1–9 (2009). https://doi.org/10.1063/1.3176473

    Article  Google Scholar 

  15. Pirozzoli, S.: Numerical methods for high-speed flows. Annu. Rev. Fluid Mech. 43, 163–194 (2010). https://doi.org/10.1146/annurev-fluid-122109-160718

    Article  MathSciNet  Google Scholar 

  16. Tatsumi, S., Martinelli, L., Jameson, A.: Flux-limited schemes for the compressible Navier–Stokes equations. AIAA J. 33(2), 252–261 (1995). https://doi.org/10.2514/3.12422

    Article  Google Scholar 

  17. Kundu, A., De, S.: Application of compact scheme in the CUSP framework for strong shock–vortex interaction. Comput. Fluids 126, 192–204 (2016). https://doi.org/10.1016/j.compfluid.2015.11.018

    Article  MathSciNet  Google Scholar 

  18. Guo-Hua, T., Xiang-Jiang, Y.: A characteristic-based shock-capturing scheme for hyperbolic problems. J. Comput. Phys. 225, 2083–2097 (2007). https://doi.org/10.1016/j.jcp.2007.03.007

    Article  MathSciNet  Google Scholar 

  19. Kundu, A., De, S.: Navier–Stokes simulation of shock–heavy bubble interaction: comparison of upwind and WENO schemes. Comput. Fluids 157, 131–145 (2017). https://doi.org/10.1016/j.compfluid.2017.08.025

    Article  MathSciNet  Google Scholar 

  20. Chang, K.S., Barik, H., Chang, S.M.: The shock–vortex interaction patterns affected by vortex flow regime and vortex models. Shock Waves 19, 349–360 (2009). https://doi.org/10.1007/s00193-009-0210-1

    Article  Google Scholar 

  21. Pirozzoli, S.: Conservative hybrid compact-WENO schemes for shock–turbulence interaction. J. Comput. Phys. 178, 81–117 (2002). https://doi.org/10.1006/jcph.2002.7021

    Article  MathSciNet  Google Scholar 

  22. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992). https://doi.org/10.1016/0021-9991(92)90324-R

    Article  MathSciNet  Google Scholar 

  23. Carpenter, M.H., Kennedy, C.A.: Fourth-order 2N-storage Runge–Kutta schemes. NASA TM 109112 (1994). https://ntrs.nasa.gov/citations/19940028444

  24. Richardson, L.F.: The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philos. Trans. R. Soc. Lond. 210, 307–357 (1911). https://doi.org/10.1098/rsta.1911.0009

    Article  Google Scholar 

  25. Roache, P.J.: Perspective: a method for uniform reporting of grid refinement studies. J. Fluids Eng. 116, 405–413 (1994). https://doi.org/10.1115/1.2910291

    Article  Google Scholar 

  26. Zhou, G., Xu, K., Liu, F.: Grid-converged solution and analysis of the unsteady viscous flow in a two-dimensional shock tube. Phys. Fluids 30(016102), 1–21 (2018). https://doi.org/10.1063/1.4998300

    Article  Google Scholar 

  27. Kundu, A., De, S., Thangadurai, M., Dora, C.L., Das, D.: Numerical visualization of shock tube-generated vortex–wall interaction using a fifth-order upwind scheme. J. Vis. 19, 667–678 (2016). https://doi.org/10.1007/s12650-016-0362-x

    Article  Google Scholar 

  28. Samtaney, R., Pullin, D.I.: On initial-value and self-similar solutions of the compressible Euler equations. Phys. Fluids 8, 2650 (1996). https://doi.org/10.1063/1.869050

    Article  MathSciNet  Google Scholar 

  29. Ellzey, J.L., Henneke, M.R., Picone, J.M., Oran, E.S.: The interaction of a shock with a vortex: shock distortion and the production of acoustic waves. Phys. Fluids 7, 172–184 (1995). https://doi.org/10.1063/1.868738

    Article  MathSciNet  Google Scholar 

  30. Fürst, J., Angot, P., Debieve, J. F., Kozel, K.: Two and three-dimensional applications of TVD and ENO schemes. Numerical Modelling in Continuum Mechanics, 3rd Summer Conference in Praha, MATFIZPRESS Charles Univ., 1, 103–111 (1997)

  31. Orlandi, P., van Heijst, G.F.: Numerical simulation of tripolar vortices in 2D flow. Fluid Dyn. Res. 9, 179–206 (1992). https://doi.org/10.1016/0169-5983(92)90004-G

    Article  Google Scholar 

  32. Paireau, O., Tabeling, P., Legras, B.: A vortex subjected to a shear: an experimental study. J. Fluid Mech. 351, 1–16 (1997). https://doi.org/10.1017/S0022112097006915

    Article  Google Scholar 

  33. Sun, M., Takayama, K.: A note on numerical simulation of vortical structures in shock diffraction. Shock Waves 13, 25–32 (2003). https://doi.org/10.1007/s00193-003-0195-0

    Article  Google Scholar 

  34. Rubidge, S., Skews, B.: Shear-layer instability in the Mach reflection of shock waves. Shock Waves 24, 479–488 (2014). https://doi.org/10.1007/s00193-014-0515-6

    Article  Google Scholar 

  35. Chen, Z., Yi, S.H., Tian, L.F., He, L., Zhu, Y.Z.: Flow visualization of supersonic laminar flow over a backward-facing step via NPLS. Shock Waves 23, 299–306 (2013). https://doi.org/10.1007/s00193-012-0378-7

    Article  Google Scholar 

  36. Johnsen, E., Larson, J., Bhagatwala, A.V., Cabot, W.H., Moin, P., Olson, B.J., Rawat, P.S., Shankar, S.K., Sjögreenc, B., Yee, H.C., Zhong, X., Lele, S.K.: Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229, 1213–1237 (2010). https://doi.org/10.1016/j.jcp.2009.10.028

    Article  MathSciNet  Google Scholar 

  37. Kundu, A., Thangadurai, M., Biswas, G.: Investigation on shear layer instabilities and generation of vortices during shock wave and boundary layer interaction. Comput. Fluids 224, 104966 (2021). https://doi.org/10.1016/j.compfluid.2021.104966

    Article  MathSciNet  Google Scholar 

  38. Kevlahan, N.K.R.: The vorticity jump across a shock in a non-uniform flow. J. Fluid Mech. 341, 371–384 (1997). https://doi.org/10.1017/S0022112097005752

  39. Thakare, P., Nair, V., Sinha, K.: A weakly nonlinear framework to study shock–vorticity interaction. J. Fluid Mech. 933, A48 (2022). https://doi.org/10.1017/jfm.2021.1076

    Article  MathSciNet  Google Scholar 

  40. Kevlahan, N.K.R.: The propagation of weak shocks in non-uniform flows. J. Fluid Mech. 327, 161–197 (1996). https://doi.org/10.1017/S0022112096008506

    Article  MathSciNet  Google Scholar 

  41. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995). https://doi.org/10.1017/S0022112095000462

    Article  MathSciNet  Google Scholar 

  42. Halder, P., De, S., Sinhamahapatra, K.P., Singh, N.: Numerical simulation of shock–vortex interaction in Schardin’s problem. Shock Waves 23, 495–504 (2013). https://doi.org/10.1007/s00193-013-0448-5

    Article  Google Scholar 

  43. Colonius, T., Lele, S.K.: Computational aeroacoustics: progress on nonlinear problems of sound generation. Prog. Aerosp. Sci. 40, 345–416 (2004). https://doi.org/10.1016/j.paerosci.2004.09.001

    Article  Google Scholar 

  44. Sinha, K.: Enstrophy in shock/homogeneous turbulence interaction. J. Fluid Mech. 707, 74–110 (2012). https://doi.org/10.1017/jfm.2012.265

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author thanks Sudipta De, former Principal Scientist at CSIR-CMERI, for his enormous contribution in developing the inhouse compressible-flow solver. The author gratefully acknowledges help and support from Anupam Sinha for providing the High-Performance Computing Facility of CSIR-CMERI.

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  1. Department of Applied Mechanics, Motilal Nehru National Institute of Technology, Allahabad, Uttar Pradesh, 211004, India

    A. Kundu

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Kundu, A. Breakdown regime of a shielded vortex interacting with a standing normal shock: a numerical study. Shock Waves 34, 21–36 (2024). https://doi.org/10.1007/s00193-024-01163-8

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