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Interaction of Stationary Disturbances with Tollmien—Schlichting Waves in a Supersonic Boundary Layer

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Abstract

The possibility of controlling unsteady disturbances, traveling Tollmien—Schlichting waves, and stationary streamwise structures is studied. The investigation is performed for the flat-plate boundary layer at the freestream Mach number M = 2. The possible enhancement and suppression of the growth of these waves by stationary streaky structures of the stability eigenproblem of supersonic boundary layer is studied. The problem is solved in the local-parallel approximation within the framework of the three-wave resonance interaction. The pumping wave is a stationary, near-streaky formation. It is shown that even in the stability domain Tollmien—Schlichting waves grow under the influence of the streamwise structures. It is established that under certain conditions the effect of stationary disturbances on these waves can be considerable also in the instability domain and the Reynolds number ranges in which the steady disturbances suppress traveling waves are determined.

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REFERENCES

  1. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1968).

    MATH  Google Scholar 

  2. C.C. Lin, The Theory of Hydrodynamic Stability (Cambridge Univ. Press, 1955).

    MATH  Google Scholar 

  3. P.S. Klebanoff and K.D. Tidstrom, “Evolution of amplified waves leading to transition in a boundary layer with zero pressure gradient,” NASA TN D-195 (1959).

  4. W.S. Saric, H.L. Reed, and E.J. Kerschen, “Boundary-layer receptivity to free stream disturbances,” Annu. Rev. Fluid Mech. 34, 291–319 (2002).

    Article  ADS  Google Scholar 

  5. A.V. Boiko, G.R. Grek, A.V. Dovgal’, and V.V. Kozlov, Physical Mechanisms of Transition to Turbulence in Open Systems (Research Center “Regular and Chaotic Dynamics”, Izhevsk, 2006) [in Russian].

  6. S.C. Crow, “The spanwise perturbation of two-dimensional boundary layers,” J. Fluid Mech. 24, 153–104 (1966).

    Article  ADS  MathSciNet  Google Scholar 

  7. P. Bradshaw, “The effect of wind tunnel screens on ‘two-dimensional’ boundary layers,” Nat. Phys. Lab. Aero. Rep. No. 1085 (1963).

  8. P.A. Libby and H. Fox, “Some perturbation solutions in laminar boundary-layer theory. Part 1. The momentum equation,” J. Fluid Mech. 17, 433–449 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  9. F.P. Bertolotti, “Response of the Blasius boundary layer to free-stream vorticity,” Phys. Fluids9(8), 2286–2299 (1997).

    Article  ADS  Google Scholar 

  10. M.V. Ustinov, “Receptivity of the flat-plate boundary layer to free-stream turbulence,” FluidDynamics38(3), 397–408 (2003).

    Article  MathSciNet  Google Scholar 

  11. M.E. Goldstein, “Effect of free-stream turbulence on boundary layer transition,” Phil. Trans. Roy. Soc. A 372, 372 (2014).

    Article  MathSciNet  Google Scholar 

  12. S.A. Gaponov and A.V. Yudin, “Interaction of hydrodynamic external disturbances with the boundary layer,” J. Appl. Mech. Techn. Phys.43(1), 83—89 (2002).

    Article  ADS  Google Scholar 

  13. S.A. Gaponov, “Interaction of external vortical and thermal disturbances with boundary layer,” Intern. J. Mech. 1(1), 15–20 (2007).

    Google Scholar 

  14. S.A. Gaponov, “Interaction between a supersonic boundary layer and acoustic disturbances,” Fluid Dynamics12(6), 858–862 (1977).

    Article  ADS  Google Scholar 

  15. C.E. Grosch and H. Salwen, “The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions,” J. Fluid Mech.87, 33–54 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  16. C.E. Grosch and H. Salwen, “The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions,” J. Fluid Mech.104, 445–465 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  17. M. Dong and X. Wu, “On continuous spectra of the Orr-Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances,” J. Fluid Mech. 732, 616–659 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  18. P. Luchini, “Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations,” J. Fluid Mech. 404, 289–309 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  19. P. Andersson, M. Berggren, and D.S. Henningson, “Optimal disturbances in boundary layers,” in: Proc. AFOSR Workshop on Optimal Design and Control, Ed. by J.T. Borggaard, J. Burns, E. Cliff, and S. Schreck (Boston, 1998).

  20. P. Andersson, M. Berggren, and D.S. Henningson, “Optimal disturbances and bypass transition in boundary layers,” Phys. Fluids11, 134–150 (1999).

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  22. H. Hultgren and Gustavsson, “Algebraic growth of disturbances in a laminar boundary layer,” Phys. Fluids24, 1000–1004 (1981).

    Article  ADS  Google Scholar 

  23. D. S. Henningson, “An eigenfunction expansion of localized disturbances,” in: Advances in Turbulence 3, Ed. by A.V. Johansson and P.H. Alfredsson (1991), pp. 162–169.

    Google Scholar 

  24. S.A. Gaponov, “Quasi-resonance excitation of stationary disturbances in compressible boundary layer,” Intern. J. Mech. 11, 120–127 (2017).

    Google Scholar 

  25. S.A. Gaponov, G.V. Petrov, and B.V. Smorodskii, “Linear and nonlinear interaction of acoustic waves with a supersonic boundary layer,” Aeromekhanika Gazovaya Dinamika, No. 3, 21–30 (2002).

    Google Scholar 

  26. S.A. Gaponov, “Growth of disturbances in a supersonic boundary layer,” J. Appl. Mech. Techn. Phys. 32(6), 910–913 (1991).

    Article  ADS  Google Scholar 

  27. S.A. Gaponov and N.M. Terekhova, “Stationary disturbances in a a supersonic boundary layer,” AeromekhanikaGazovayaDinamika No. 4, 35–42 (2002).

    Google Scholar 

  28. A. Thumm, W. Wolz, and H. Fasel, “Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundary layers,” in: Proc. IUTAM Symp. Toulouse, France, 1990, pp. 303–308.

  29. P.J. Schmid and D.S. Henningson, “A new mechanism for rapid transition involving a pair of oblique waves,” Phys. Fluids A 4, 1986–1989 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  30. C.-L. Chang and M.R. Malik, “Oblique-mode breakdown and secondary instability in supersonic boundary layers,” J. Fluid Mech.273, 323–360 (1994).

    Article  ADS  Google Scholar 

  31. W.S. Saric, R.B. Carillo, and M.S. Reibert, “Leading edge roughness as a transition control mechanism,” AIAA Paper No. 0781 (1998).

  32. W.S. Saric and H.L. Reed, “Supersonic laminar flow control on swept wings using distributed roughness,” AIAA Paper No. 0147 (2002).

  33. N.V. Semionov and A.D. Kosinov, “Method of laminar-turbulent transition control of supersonic boundary layer on a swept wing,” ThermophysicsAeromechanics14(3), 337–341 (2007).

    Article  ADS  Google Scholar 

  34. A.D.D. Craik, “Non-linear resonant instability in boundary layers,” J. Fluid Mech. 50, 393–413 (1971).

    Article  ADS  Google Scholar 

  35. S.A. Gaponov and I.I. Maslennikova, “Subharmonic instability of supersonic boundary layer,” ThermophysicsAeromechanics4(1), 3–12 (1997).

    Google Scholar 

  36. S.A. Gaponov, I.I. Maslennikova, and V.Yu. Tyushin, Nonlinear effect of external low-frequency acoustics on eigen-oscillations in a supersonic boundary layer,” J. Appl. Mech. Techn. Phys.40(5), 865–870 (1999).

    Article  ADS  Google Scholar 

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Funding

The study was supported by the Program of Basic Scientific Investigations of the State Academies of Sciences for the years 2013–2020, project no. АААА-22.6.4. and the Russian Foundation for Basic Research, project no. 17-19-01289.

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Authors and Affiliations

  1. Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences, 630090, Novosibirsk, Russia

    S. A. Gaponov & N. M. Terekhova

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  1. S. A. Gaponov
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  2. N. M. Terekhova
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Correspondence to S. A. Gaponov or N. M. Terekhova.

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Translated by M. Lebedev

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Gaponov, S.A., Terekhova, N.M. Interaction of Stationary Disturbances with Tollmien—Schlichting Waves in a Supersonic Boundary Layer. Fluid Dyn 55, 433–440 (2020). https://doi.org/10.1134/S0015462820040059

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  • DOI: https://doi.org/10.1134/S0015462820040059

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