Abstract
The spatial development of disturbances with small and moderate amplitudes in a two-dimensional (2-D) supersonic flat-plate boundary layer at Mach 4.8 is investigated using direct numerical simulations based on the compressible 3-D Navier-Stokes equations. Disturbances are introduced into the boundary layer by blowing and suction within a narrow disturbance strip at the wall. In response to the timewise periodic forcing, two types of disturbance waves are generated, a “first-mode” wave and a “multiple-viscous-solution.” The “multiple-viscous-solution” was described by Mack (1969, 1984) but was not seen before in a direct numerical simulation. The results of the simulations are compared with results of linear stability theory, and the agreement is very good. In simulations for larger amplitudes, fundamental resonance is observed, where both types of 3-D waves are nonlinearly amplified and synchronize their phase velocities with the 2-D disturbance waves. Subharmonic resonance is found for 3-D waves with large wave numbers, where the phase velocities of the linear 2-D and 3-D waves are nearly the same.
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Adams, N.A., and Kleiser, L. (1993). Numerical Simulation of Transition in a Compressible Flat Plate Boundary Layer. In Transitional and Turbulent Compressible Flows — 1993 (L.D. Kral and T.A. Zang, eds.) FED, Vol. 151. ASME, New York, pp. 101–110.
Adams, N.A., Sandham, N.D., and Kleiser, L. (1992). A Method for Direct Numerical Simulation of Compressible Boundary-Layer Transition. Notes on Numerical Fluid Mechanics, Vol. 35 (J.B. Vos, A. Rizzi, and I. Ryhming, eds.). Vieweg Verlag, Braunschweig, pp. 523–532.
Balakumar, P., and Malik, M.R. (1992). Discrete Modes and Continuous Spectra in Supersonic Boundary Layers. J. Fluid Mech., 239, 631–656.
Eißler, W., Wolz, W., and Bestek, H. (1991). Numerische Untersuchungen des laminar-turbulenten Strömungsumschlags in Überschall-Plattengrenzschichten. Jahrbuch der DGLR 1991/I, Deutsche Gesellschaft für Luft- und Raumfahrt, Bonn, pp. 151–160.
El-Hady, N.M. (1989). Secondary Instability of Compressible Boundary Layers to Subharmonic Three-Dimensional Disturbances. AIAA Paper No. 89-0035.
Erlebacher, G., and Hussaini, M.Y. (1990). Numerical Experiments in Supersonic Boundary-Layer Stability. Phys. Fluids A, 2, 94–103.
Fasel, H., Rist, U., and Konzelmann, U. (1990). Numerical Investigation of the Three-Dimensional Development in Boundary-Layer Transition. AIAA J., 28, 29–37.
Fasel, H., Thumm, A., and Bestek, H. (1993). Direct Numerical Simulation of Transition in Supersonic Boundary Layers: Oblique Breakdown. In Transitional and Turbulent Compressible Flows — 1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, pp. 77–92.
Fedoroy, A.V., and Khokhlov, A.P. (1993). Excitation and Evolution of Unstable Disturbances in Supersonic Boundary Layer. In Transitional and Turbulent Compressible Flows — 1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, pp. 1–13.
Gottlieb, D., and Turkel, E. (1976). Dissipative Two-Four Methods for Time-Dependent Problems. Math. Comp., 30, 703–723.
Kachanov, Y.S., and Levchenko, V.Y. (1984). The Resonant Interaction of Disturbances at Laminar-Turbulent Transition in a Boundary Layer. J. Fluid Mech., 138, 209–247.
Kendall, J.M. (1967). Supersonic Boundary Layer Stability Experiments. Proc. Boundary-Layer Transition Study Group Meeting (W.D. McCauley, ed.), Vol. II. Aerospace Corporation, San Bernardino, CA, pp. 10–11.
Kosinov, A.D., Maslov, A.A., and Shevelkov, S.G. (1990). Experiments on the Stability of Supersonic Boundary Layers. J. Fluid Mech., 219, 621–633.
Mack, L.M. (1965). Computation of the Stability of the Laminar Boundary Layer. In Methods of Computational Physics (B. Alder, S. Fernbach, and M. Rotenberg, eds.), Vol. 4. Academic Press, New York, pp. 247–299.
Mack, L.M. (1969). Boundary Layer Stability Theory. Document No. 900-277, Rev. A, Jet Propulsion Laboratory, Pasadena, CA.
Mack, L.M. (1984). Boundary-Layer Linear Stability Theory. AGARD Rep., 709, 1–81.
Mack, L.M. (1987). Review of Linear Compressible Stability Theory. In Stability of Time Dependent and Spatially Varying Flows, ICASE Workshop, Hampton, VA, 1987 (D.L. Dwoyer and M.Y. Hussaini, eds.). Springer-Verlag, New York, pp. 164–187.
Mack, L.M. (1993). Private communication.
Maestrello, L., Bayliss, A., and Krishnan, R. (1991). On the Interaction between First- and Second-Mode Waves in a Supersonic Boundary Layer. Phys. Fluids A, 3, 3014–3020.
Masad, J.A., and Nayfeh, A.H. (1990). Subharmonic Instability of Compressible Boundary Layers. Phys. Fluids A, 2, 1380–1392.
Masad, J.A., and Nayfeh, A.H. (1991). Effect of Heat Transfer on the Subharmonic Instability of Compressible Boundary Layers. Phys. Fluids A, 3, 2148–2163.
Ng, L., and Erlebacher, G. (1992). Secondary Instabilities in Compressible Boundary Layers. Phys. Fluids A, 4, 710–726.
Pruett, C.D., and Chang, C.L. (1993). A Comparison of PSE and DNS for High-Speed Boundary-Layer Flows. In Transitional and Turbulent Compressible Flows — 1993 (L.D. Kral and T.A. Zang, eds.). FED, Vol. 151. ASME, New York, pp. 57–67.
Pruett, C.D., and Zang, T.A. (1992). Direct Numerical Simulation of Laminar Breakdown in High-Speed Axisymmetric Boundary Layers. Theoret. Comput. Fluid Dynamics, 3, 345–367.
Pruett, C.D., Ng, L., and Erlebacher, G. (1991). On the Nonlinear Stability of a High-Speed, Axisymmetric Boundary Layer. Phys. Fluids A, 3, 2910–2926.
Stetson, K.F., and Kimmel, R.L. (1992). On Hypersonic Boundary-Layer Stability. AIAA Paper No. 92-0737.
Thumm, A. (1991). Numerische Untersuchungen zum laminar-turbulenten Strömungsumschlag in transsonischen Grenzschicht-strömungen. Dissertation, Universität Stuttgart, Stuttgart.
Thumm, A., Wolz, W., and Fasel, H. (1990). Numerical Simulation of Spatially Growing Three-Dimensional Disturbance Waves in Compressible Boundary Layers. In Laminar-Turbulent Transition. IUTAM Symposium, Toulouse, France, 1989 (D. Arnal and R. Michel, eds.). Springer-Verlag. Berlin, pp. 303–308.
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Communicated by M.Y. Hussaini
This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, as part of SFB 259.
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Eißler, W., Bestek, H. Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers. Theoret. Comput. Fluid Dynamics 8, 219–235 (1996). https://doi.org/10.1007/BF00418059
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DOI: https://doi.org/10.1007/BF00418059