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Investigation of Görtler vortices in high-speed boundary layers via an efficient numerical solution to the non-linear boundary region equations

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Abstract

Streamwise vortices and the associated streaks evolve in boundary layers over flat or concave surfaces due to disturbances initiated upstream or triggered by the wall surface. Following the transient growth phase, the fully developed vortex structures become susceptible to inviscid secondary instabilities resulting in early transition to turbulence via ‘bursting’ processes. In high-speed boundary layers, more complications arise due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this paper, we study Görtler vortices developing in high-speed boundary layers using the boundary region equations (BRE) formalism, which we solve using an efficient numerical algorithm. Streaks are excited using a small transpiration velocity at the wall. Our BRE-based algorithm is found to be superior to direct numerical simulation (DNS) and ad hoc nonlinear parabolized stability equation (PSE) models. BRE solutions are less computationally costly than a full DNS and have a more rigorous theoretical foundation than PSE-based models. For example, the full development of a Görtler vortex system in high-speed boundary layers can be predicted in a matter of minutes using a single processor via the BRE approach. This substantial reduction in calculation time is one of the major achievements of this work. We show, among other things, that it allows investigation into feedback control in reasonable total computational times. We investigate the development of the Görtler vortex system via the BRE solution with feedback control parametrically at various freestream Mach numbers \(M_\infty \) and spanwise separations \(\lambda \) of the inflow disturbances.

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References

  1. Bagheri, S., Hanifi, A.: The stabilizing effect of streaks on Tollmien–Schlichting and oblique waves: a parametric study. Phys. Fluids 19, 078103 (2007)

    Article  MATH  Google Scholar 

  2. Chen, X., Huang, G.L., Lee, C.B.: Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices. J. Fluid Mech. 865, 1–40 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ciolkosz, L.D., Spina, E.F.: An experimental study of Görtler vortices in compressible flow. AIAA Paper 4512 (2006)

  4. Dando, A.H., Seddougui, S.O.: The compressible Görtler problem in two-dimensional boundary layers. IMA J. Appl. Math. 51(1), 27–67 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. De Luca, L., Cardone, G., Aymer De La Chevalerie, D., Fonteneau, A.: Görtler instability of a hypersonic boundary layer. Exp. Fluids 16, 10–16 (1993)

    Article  Google Scholar 

  6. El-Hady, N.M., Verma, A.K.: Growth of Görtler vortices in compressible boundary layers along curved surfaces. J. Eng. Appl. Sci. 2(3), 213–238 (1983)

    Google Scholar 

  7. Elliot, J.W., Bassom, A.P.: The effect of wall cooling on compressible Görtler vortices. Eur. J. Mech. B Fluids 19, 37–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Es-Sahli, O., Sescu, A., Afsar, M., Hattori, Y., Hirota, M.: Effect of localized wall cooling or heating on streaks in high-speed boundary layers. AIAA Scitech 2021 Forum. AIAA, 2021-0853 (2021)

  9. Görtler, H.: Instabilita-umt laminarer Grenzchichten an Konkaven Wanden gegenber gewissen dreidimensionalen Storungen. ZAMM 21, 250–252; English version: NACA Report 1375 (1954) (1941)

  10. Hall, P.: Taylor–Görtler vortices in fully developed or boundary-layer flows: linear theory. J. Fluid Mech. 124, 475–494 (1982)

    Article  MATH  Google Scholar 

  11. Hall, P.: The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hall, P.: The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243–266 (1989)

    Article  MATH  Google Scholar 

  13. Hall, P., Horseman, N.: The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hall, P., Fu, Y.: On the Görtler vortex instability mechanism at hypersonic speeds. Theor. Comput. Fluid Dyn. 1(3), 125–134 (1989)

    Article  MATH  Google Scholar 

  15. Hall, P., Malik, M.: The growth of Görtler vortices in compressible boundary layers. J. Eng. Math. 23(3), 239–251 (1989)

    Article  MATH  Google Scholar 

  16. Kendall, J.M.: Experiments on boundary-layer receptivity to freestream turbulence. AIAA Paper, 2004-2335 (1998)

  17. Kobayashi, R., Kohama, Y.: Taylor–Görtler instability of compressible boundary layers. AIAA J. 15(12), 1723–1727 (1977)

    Article  Google Scholar 

  18. Landahl, M.T.: A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Leib, S.J., Wundrow, W., Goldstein, M.: Effect of freestream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, F., Malik, M.: Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, F., Choudhari, M., Chang, C.-L., Greene, P., Wu, M.: Development and breakdown of Gortler vortices in high speed boundary layers. In: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Aerospace Sciences Meetings (2010)

  22. Li, F., Choudhari, M., Paredes, P.: Nonlinear Görtler vortices and their secondary instability in a hypersonic boundary layer. AIAA Paper, 2019-3216 (2019)

  23. Malik, M.R., Hussaini, M.Y.: Numerical simulation of interactions between Görtler vortices and Tollmien–Schlichting waves. J. Fluid Mech. 210, 183–199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Marensi, E., Ricco, P.: Growth and wall-transpiration control of nonlinear unsteady Görtler vortices forced by free-stream vortical disturbances. Phys. Fluids 29, 114106 (2017)

    Article  Google Scholar 

  25. Marensi, E., Ricco, P., Wu, X.: Nonlinear unsteady streaks engendered by the interaction of freestream vorticity with a compressible boundary layer. J. Fluid Mech. 817, 80–121 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Matsubara, M., Alfredsson, P.H.: Disturbance growth in boundary layers subjected to free stream turbulence. J. Fluid Mech. 430, 149 (2001)

    Article  MATH  Google Scholar 

  27. Ren, J., Fu, S.: Secondary instabilities of Görtler vortices in high-speed boundary layer flows. J. Fluid Mech. 781, 388–421 (2017)

    Article  MATH  Google Scholar 

  28. Ricco, P.: Response of a compressible laminar boundary layer to freestream turbulence. Ph.D. thesis, University of London (2006)

  29. Ricco, P., Wu, X.: Response of a compressible laminar boundary layer to freestream vortical disturbances. J. Fluid Mech. 587, 97–138 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ricco, P., Shah, D., Hicks, P.D.: Compressible laminar streaks with wall suction. Phys. Fluids 25, 054110 (2013)

    Article  Google Scholar 

  31. Ricco, P., Tran, D.-L., Ye, G.: Wall heat transfer effects on Klebanoff modes and Tollmien–Schlichting waves in a compressible boundary layer. Phys. Fluids 21, 024106 (2009)

    Article  MATH  Google Scholar 

  32. Sescu, A., Pendyala, R., Thompson, D.: On the growth of Görtler vortices excited by distributed roughness elements. AIAA Paper, 2014-2885 (2014)

  33. Sescu, A., Thompson, D.: On the excitation of Görtler vortices by distributed roughness elements. Theor. Comput. Fluids Dyn. 57, 1159–1171 (2015)

    Google Scholar 

  34. Sescu, A., Alaziz, R., Afsar, M.: Effect of wall transpiration and heat transfer on nonlinear Görtler vortices in high-speed boundary layers. AIAA J. 9, 797–803 (2019)

    Google Scholar 

  35. Song, R., Zhao, L., Huang, Z.: Secondary instability of stationary Görtler vortices originating from first/second Mack mode. Phys. Fluids 32, 034109 (2020)

    Article  Google Scholar 

  36. Spall, R.E., Malik, M.R.: Görtler vortices in supersonic and hypersonic boundary layers. Phys. Fluids A 1, 1822 (1989)

    Article  MATH  Google Scholar 

  37. Swearingen, J.D., Blackwelder, R.F.: The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290 (1987)

    Article  Google Scholar 

  38. Viasro, S., Ricco, P.: Compressible unsteady Görtler vortices subject to freestream vortical disturbances. J. Fluid Mech. 867, 250–299 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wadey, P.D.: On the linear development of Görtler vortices in compressible boundary layers. Eur. J. Mech. (B/Fluids) 11, 705–717 (1992)

    MATH  Google Scholar 

  40. Wang, Q.-C., Wang, Z.-G., Zhao, Y.-X.: Visualization of Görtler vortices in supersonic concave boundary layer. J. Vis. 21(1), 57–62 (2018)

    Article  MathSciNet  Google Scholar 

  41. Wu, X., Zhao, D., Luo, J.: Excitation of steady and unsteady Görtler vortices by freestream vortical disturbances. J. Fluid Mech. 682, 66–100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu, D., Zhang, Y., Wu, X.: Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by freestream vortical disturbances. J. Fluid Mech. 829, 681–730 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Mississippi State University, Starkville, USA

    Omar Es-Sahli & Adrian Sescu

  2. Department of Mechanical and Aerospace Engineering, Strathclyde University, Glasgow, UK

    Mohammed Afsar

  3. Institute of Fluid Science, Tohoku University, Sendai, Japan

    Yuji Hattori

Authors
  1. Omar Es-Sahli
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  2. Adrian Sescu
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  3. Mohammed Afsar
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  4. Yuji Hattori
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Correspondence to Omar Es-Sahli.

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Communicated by Vassilios Theofilis.

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Es-Sahli, O., Sescu, A., Afsar, M. et al. Investigation of Görtler vortices in high-speed boundary layers via an efficient numerical solution to the non-linear boundary region equations. Theor. Comput. Fluid Dyn. 36, 237–249 (2022). https://doi.org/10.1007/s00162-021-00576-w

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  • DOI: https://doi.org/10.1007/s00162-021-00576-w

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