Abstract
Evolution of the discrete spectrum in the M a = 4.5 boundary layer is studied with LST and PSE approaches. Both two-dimensional (2-D) and three-dimensional (3-D) disturbances are considered with streamwise curvature effects. The concave curvature shows a destabilizing effect on the 2-D second/third mode when the fast mode (mode F(1), mode F(2)...) synchronizes with the slow mode (mode S). The spectrum branching in the synchronization between the mode F(2) and mode S is also observed. The increase in the spanwise wavenumber(3-D disturbances), on the other hand, suppresses the synchronization between mode F and mode S and reduces the growth rate of the unstable mode. With regard to the 3-D disturbances subjecting to the concave curvature, the mode S originating from the slow acoustic wave amounts to the unsteady Görtler mode while the quasi-steady Görtler mode emanates from the continuous spectrum of the vorticity/entropy wave.
Similar content being viewed by others
References
Mack, L.M.: Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13(3), 278–289 (1975)
Mack, L.M.: Boundary-layer linear stability theory. AGARD Report 709, 1984. Special Course on Stability and Transition of Laminar Flows
Federov, A., Tumin, A.: High-speed boundary-layer instability: Old terminology and a new framework. AIAA J. 49(8), 1647–1657 (2011)
Wang, X., Zhong, X.: The stabilization of a hypersonic boundary layer using local sections of porous coating. Phys. Fluids 24(3) (2012)
Theofilis, V.: Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerospace Sci. 39(4), 249–315 (2003)
Theofilis, V.: Global linear instability. Ann. Rev. Fluid Mech. 43, 319–352 (2011)
Reed, H.L., Saric, W.S., Arnal, D.: Linear stability theory applied to boundary layers. Ann. Rev. Fluid Mech. 28, 389–428 (1996)
Wang, L., Fu, S.: Development of an intermittency equation for the modeling of the supersonic/hypersonic boundary layer flow transition. Flow, Turbulence Combustion 87(1), 165–187 (2011)
Wang, L., Fu, S., Carnarius, A., Mockett, C., Thiele, F.: A modular rans approach for modelling laminar-turbulent transition in turbomachinery flows. Int. J. Heat Fluid Flow 34, 62–69 (2012)
Fu, S., Wang, L.: Rans modeling of high-speed aerodynamic flow transition with consideration of stability theory. Prog. Aerospace Sci. 58, 36–59 (2013)
Ma, Y., Zhong, X.: Receptivity of a supersonic boundary layer over a flat plate. part 2. receptivity to free-stream sound. J. Fluid Mech. 488(7), 79–121 (2003)
Ma, Y., Zhong, X.: Receptivity of a supersonic boundary layer over a flat plate. part 3. effects of different types of free-stream disturbances. J. Fluid Mech. 532 (6), 63–109 (2005)
Fedorov, A.V.: Receptivity of a high-speed boundary layer to acoustic disturbances. J. Fluid Mech. 491(9), 101–129 (2003)
Fedorov, A.V., Ryzhov, A.A., Soudakov, V.G., Utyuzhnikov, S.V.: Receptivity of a high-speed boundary layer to temperature spottiness. J. Fluid Mech. 722, 533–553 (2013)
Tempelmann, D., Schrader, L.-U., Hanifi, A., Brandt, L., Henningson, D.S.: Swept wing boundary-layer receptivity to localized surface roughness. J. Fluid Mech. 711, 516–544 (2012)
Fedorov, A.V.: Receptivity of a supersonic boundary layer to solid particulates. J. Fluid Mech. 737, 105–131 (2013)
Ruban, A.I., Bernots, T., Pryce, D.: Receptivity of the boundary layer to vibrations of the wing surface. J. Fluid Mech. 723, 480–528 (2013)
Fedorov, A.V.: Transition and stability of high-speed boundary layers. Ann. Rev. Fluid Mech. 43, 79–95 (2011)
Zhong, X., Wang, X.: Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annual Rev. Fluid Mech. 44, 527–561 (2012)
Kovasznay, L.S.G.: Turbulence in supersonic flow. J. Aeronautical Sci. 20(10), 657–674 (1953)
Ma, Y., Zhong, X.: Receptivity of a supersonic boundary layer over a flat plate. part 1. wave structures and interactions. J. Fluid Mech. 488(7), 31–78 (2003)
Fedorov, A.V., Khokhlov, A.P.: Prehistory of instability in a hypersonic boundary layer. Theor. Comput. Fluid Dyn. 14(6), 359–375 (2001)
Federov, A., Tumin, A.: Initial-value problem for hypersonic boundary-layer flows. AIAA J. 41(3), 379–389 (2003)
Gushchin, V.R., Fedorov, A.V.: Excitation and development of unstable disturbances in a supersonic boundary layer. Fluid Dyn. 25(3), 344–352 (1990)
Lifshitz, Y., Degani, D., Tumin, A.: Study of discrete modes branching in high-speed boundary layers. AIAA J. 50(10), 2202–2210 (2012)
Hall, P.: The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58 (1983)
Day, H.P., Herbert, T., Saric, W.S.: Comparing local and marching analysis of Görtler instability. AIAA J. 28(6), 1010–1015 (1990)
Bottaro, A., Luchini, P.: Görtler vortices: Are they amenable to local eigenvalue analysis. Eur. J. Mech. - B/Fluids 18(1), 47–65 (1999)
Wu, X., Zhao, D., Luo, J.: Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100 (2011)
Schrader, L.-U., Brandt, L., Zaki, T.A.: Receptivity, instability and breakdown of Görtler flow. J. Fluid Mech. 682(9), 362–396 (2011)
Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: On excitation of Görtler vortices due to scattering of free-stream vortices on surface non-uniformities. J. Phys. Conf. Ser. 318(3), 032029 (2011)
Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Boundary-layer receptivity to surface non-uniformities leading to generation of Görtler vortices. J. Phys.: Conf. Ser. 318(3), 032031 (2011)
Dando, A.H., Seddougui, S.O.: The compressible Görtler problem in two-dimensional boundary layers. IMA J. Appl. Math. 51(1), 27–67 (1993)
Ren, J., Fu. S.: Nonlinear development of the multiple Görtler modes in hypersonic boundary layer flows. 43rd AIAA Fluid Dynamics Conference and Exhibit, AIAA-2013-2467 (2013)
Ren, J., Fu, S.: Competition of the multiple Görtler modes in hypersonic boundary layer flows. Sci. China Phys. Mech & Astronomy 57(6), 1178–1193 (2014)
Volino, R.J., Simon, T.W.: Spectral measurements in transitional boundary layers on a concave wall under high and low free-stream turbulence conditions. J. Turbomachinery 122, 450–457 (2000)
Boiko, A.V., Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Quasi-steady and unsteady goertler vortices on concave wall: experiment and theory. In: J.M.L.M. Palma, A. Silva Lopes (eds.) Advances in Turbulence XI, volume 117 of Springer Proceedings Physics, pp 173–175. Springer Berlin, Heidelberg (2007)
Boiko, A.V., Ivanov, A.V., Kachanov, Y.S., Mischenko, D.A.: Steady and unsteady Görtler boundary-layer instability on concave wall. Eur. J. Mech. - B/Fluids 29(2), 61–83 (2010)
Wang, X., Zhong, X.: Receptivity of a hypersonic flat-plate boundary layer to three-dimensional surface roughness. J. Spacecraft rockets 45(6), 1165–1175 (2008)
Whang, C., Zhong, X.: Receptivity of Görtler vortices in hypersonic boundary layers. 40th Aerospace Sciences Meeting & Exhibit, AIAA-2002-0151. (2002)
Whang, C., Zhong, X.: Leading edge receptivity of Görtler vortices in a mach 15 flow over a blunt wedge. 41st Aerospace Sciences Meeting & Exhibit, AIAA-2002-0151 (2002)
Chang, C.-L., Malik, M.R.: Oblique-mode breakdown and secondary instability in supersonic boundary layers. J. Fluid Mech. 273, 323–360 (1994)
Floryan, J.M., Saric, W.S.: Stability of Görtler vortices in boundary layers. AIAA J. 20(3), 316–324 (1982)
Floryan, J.M., Saric, W.S.: Wavelength selection and growth of Görtler vortices. AIAA J. 22(11), 1529–1538 (1984)
Bertolotti, F.P., Herbert, Th., Spalart, P.R.: Linear and nonlinear stability of the blasius boundary layer. J. Fluid Mech. 9, 441–474 (1992)
Herbert, T.: Parabolized stability equations. Ann. Rev. Fluid Mech. 29, 245–283 (1997)
Chang, C.-L., Malik, M., Erlebacher, G., Hussaini, M.: Compressible stability of growing boundary layers using parabolized stability equations. 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, AIAA-1991-1636 (1991)
Li, F., Malik, M.R.: On the nature of pse approximation. Theor. Comput. Fluid Dyn. 8(4), 253–273 (1996)
Andersson, P., Henningson, D.S., Hanifi, A.: On a stabilization procedure for the parabolic stability equations. J. Eng. Math. 33(3), 311–332 (1998)
Saric, W.S.: Görtler vortices.Ann. Rev. Fluid Mech. 26, 379–409 (1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ren, J., Fu, S. Study of the Discrete Spectrum in a Mach 4.5 Görtler Flow. Flow Turbulence Combust 94, 339–357 (2015). https://doi.org/10.1007/s10494-014-9575-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10494-014-9575-z