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Numerical investigation of the development of three-dimensional wavepackets in a sharp cone boundary layer at Mach 6

Published online by Cambridge University Press:  03 September 2014

Jayahar Sivasubramanian  and
Hermann F. Fasel
Jayahar Sivasubramanian*
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
Hermann F. Fasel
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
*
†Email address for correspondence: [email protected]
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Abstract

Direct numerical simulations were performed to investigate wavepackets in a sharp cone boundary layer at Mach 6. In order to understand the natural transition process in hypersonic cone boundary layers, the flow was forced by a short-duration (localized) pulse. The pulse disturbance developed into a three-dimensional wavepacket, which consisted of a wide range of disturbance frequencies and wavenumbers. First, the linear development of the wavepacket was studied by forcing the flow with a low-amplitude pulse (0.001 % of the free-stream velocity). The dominant waves within the resulting wavepacket were identified as the second-mode axisymmetric disturbance waves. In addition, weaker first-mode oblique disturbance waves were also observed on the lateral sides of the wavepacket. In order to investigate the nonlinear transition regime, large-amplitude pulse disturbances (0.5 % of the free-stream velocity) were introduced. The response of the flow to the large-amplitude pulse disturbances indicated the presence of a fundamental resonance mechanism. Lower secondary peaks in the disturbance wave spectrum were identified at approximately half the frequency of the high-amplitude frequency band, suggesting the possibility of a subharmonic resonance mechanism. However, the spectrum also indicated that the fundamental resonance was much stronger than the subharmonic resonance. A secondary stability investigation using controlled disturbances confirmed that fundamental resonance is indeed a dominant mechanism compared to subharmonic resonance. Furthermore, strong peaks in the disturbance wave spectrum were also observed for low-azimuthal-wavenumber second-mode oblique waves, hinting at a possible oblique breakdown mechanism. Thus, the wavepacket simulations indicate that the second-mode fundamental resonance and oblique breakdown mechanisms are the strongest for the investigated flow. Hence, both mechanisms are likely to be relevant in the natural transition process for a cone boundary layer at Mach 6.

JFM classification

Boundary Layers: Boundary layer stability Compressible Flows: Compressible boundary layers Aerodynamics: High-speed flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 756 , 10 October 2014 , pp. 600 - 649
Copyright
© 2014 Cambridge University Press 

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References

Adams, N. A. & Kleiser, L. 1993 Numerical simulation of transition in a compressible flat plate boundary layer. In Transitional and Turbulent Compressible Flows (ed. Kral, L. D. & Zang, T. A.), Proc. ASME FED Conf., vol. 151, pp. 101–110. ASME.Google Scholar
Alba, C. R., Casper, K. M., Beresh, S. J. & Schneider, S. P.2010 Comparison of experimentally measured and computed second-mode disturbances in hypersonic boundary-layers. AIAA Paper 2010-0897.CrossRefGoogle Scholar
Balakumar, P. & Malik, M. R. 1992 Discrete modes and continuous spectra in supersonic boundary layers. J. Fluid Mech. 239, 631–656.CrossRefGoogle Scholar
Balsara, D. S. & Shu, C.-W. 2000 Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452.CrossRefGoogle Scholar
Bountin, D., Shiplyuk, A. & Maslov, A. 2008 Evolution of nonlinear processes in a hypersonic boundary layer on a sharp cone. J. Fluid Mech. 611, 427–442.CrossRefGoogle Scholar
Casper, K. M., Beresh, S. J., Henfling, J. F., Spillers, R. W., Pruett, B. & Schneider, S. P.2009 Hypersonic wind-tunnel measurements of boundary-layer pressure fluctuations. AIAA Paper 2009-4054.CrossRefGoogle Scholar
Chang, C. L. & Malik, M. R.1993 Non-parallel stability of compressible boundary layers. AIAA Paper 1993-2912.CrossRefGoogle Scholar
Chang, C. L. & Malik, M. R. 1994 Oblique-mode breakdown and secondary instability in superonic boundary layers. J. Fluid Mech. 273, 323–360.CrossRefGoogle Scholar
Chokani, N. 1999 Nonlinear spectral dynamics of hypersonic laminar boundary layer flow. Phys. Fluids 11 (12), 3846–3851.CrossRefGoogle Scholar
Chokani, N. 2005 Nonlinear evolution of Mack modes in a hypersonic boundary layer. Phys. Fluids 17, 014102.CrossRefGoogle Scholar
Cohen, J. 1994 The initial evolution of a wavepacket in a laminar boundary layer. Phys. Fluids 6, 1133–1143.CrossRefGoogle Scholar
Cohen, J., Breuer, K. S. & Haritonidis, J. H. 1991 On the evolution of a wavepacket in a laminar boundary layer. J. Fluid Mech. 225, 595–621.CrossRefGoogle Scholar
Corke, T. C., Cavalieri, D. A. & Matlis, E. 2002 Boundary-layer instability on sharp cone at Mach 3.5 with controlled input. AIAA J. 40 (5), 1015–1018.CrossRefGoogle Scholar
Demetriades, A. 1960 An experiment on the stability of hypersonic laminar boundary layers. J. Fluid Mech. 7, 385–396.CrossRefGoogle Scholar
Demetriades, A. 1977 Boundary layer instability observations at Mach number 7. Trans. ASME: J. Appl. Mech. 99, 7–10.CrossRefGoogle Scholar
Eissler, W. & Bestek, H. 1996 Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers. Theor. Comput. Fluid Dyn. 8, 219–235.CrossRefGoogle Scholar
Ermolaev, Yu. G., Kosinov, A. D. & Semionov, N. V. 1996 Experimental investigation of laminar–turbulent transition process in supersonic boundary layer using controlled disturbances. In Nonlinear Instability and Transition in Three-Dimensional Boundary Layers (ed. Duck, P. W. & Hall, P.), pp. 17–26. Kluwer Academic.Google Scholar
Fasel, H. F. & Konzelmann, U. 1990 Non-parallel stability of a flat-plate boundary layer using the complete Navier–Stokes equations. J. Fluid Mech. 221, 311–347.CrossRefGoogle Scholar
Fasel, H., Thumm, A. & Bestek, H. 1993 Direct numerical simulation of transition in supersonic boundary layer: oblique breakdown. In Transitional and Turbulent Compressible Flows (ed. Kral, L. D. & Zang, T. A.), Proc. ASME FED Conf., vol. 151, pp. 77–92. ASME.Google Scholar
Fedorov, A. V. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43, 79–95.CrossRefGoogle Scholar
Ferziger, J. H. 1998 Numerical Methods for Engineering Application, 2nd edn. Wiley-Interscience.Google Scholar
Fezer, A. & Kloker, M. 1999 Spatial direct numerical simulation of transition phenomena in supersonic flat-plate boundary layers. In Laminar–Turbulent Transition (ed. Fasel, H. F. & Saric, W. S.), pp. 415–420. Springer.Google Scholar
Fezer, A. & Kloker, M.2001 Grenzschichtumschlag bei Überschallströmung. Sonderforschungsbericht 259. Deutsche Forschungsgemeinschaft.Google Scholar
Forgoston, E. & Tumin, A. 2005 Initial-value problem for three-dimensional disturbances in a compressible boundary layer. Phys. Fluids 17 (8), 084106.CrossRefGoogle Scholar
Forgoston, E. & Tumin, A. 2006 Three-dimensional wavepackets in a compressible boundary layer. Phys. Fluids 18 (10), 104103.CrossRefGoogle Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480.CrossRefGoogle Scholar
Gaster, M. 1975 A theoretical model of a wavepacket in the boundary layer on a flat plate. Proc. R. Soc. Lond. 347 (1649), 271–289.Google Scholar
Gaster, M. & Grant, I. 1975 An experimental investigation of the formation and development of a wavepacket in a laminar boundary layer. Proc. R. Soc. Lond. 347 (1649), 253–269.Google Scholar
Gross, A. & Fasel, H. F. 2008 High-order accurate numerical method for complex flows. AIAA J. 46, 204–214.CrossRefGoogle Scholar
Gross, A. & Fasel, H. F. 2010 Numerical investigation of supersonic flow for axisymmetric cones. Maths Comput. Simul. 81, 133–142.CrossRefGoogle Scholar
Harris, P. J.1997 Numerical investigation of transitional compressible plane wakes. PhD thesis, University of Arizona.Google Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20, 487–526.CrossRefGoogle Scholar
Husmeier, F.2008 Numerical investigations of transition in hypersonic flows over circular cones. PhD thesis, University of Arizona.CrossRefGoogle Scholar
Husmeier, F. & Fasel, H. F.2007 Numerical investigations of hypersonic boundary layer transition for circular cones. AIAA Paper 2007-3843.CrossRefGoogle Scholar
Husmeier, F., Mayer, C. S. J. & Fasel, H. F.2005 Investigation of transition of supersonic boundary layers at Mach 3 using DNS. AIAA Paper 2005-0095.CrossRefGoogle Scholar
Juliano, T. J., Schneider, S. P., Aradag, S. & Knight, D. 2008 Quiet-flow Ludwieg tube for hypersonic transition research. AIAA J. 46 (7), 1757–1763.CrossRefGoogle Scholar
Kachanov, Yu. S. 1994 Physical mechanisms of laminar boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411–482.CrossRefGoogle Scholar
Kendall, J. M. 1975 Wind tunnel experiments relating to supersonic and hypersonic boundary-layer transition. AIAA J. 13, 290–299.CrossRefGoogle Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
Konzelmann, U.1990 Numerische Untersuchungen zur räumlichen Entwicklung dreidimensionaler Wellenpakete in einer Plattengrenzschichtströmung. PhD thesis, Universität Stuttgart.Google Scholar
Konzelmann, U. & Fasel, H. 1991 Numerical simulation of a three-dimensional wave packet in a growing flat plate boundary layer. In Boundary Layer Transition and Control, University of Cambridge, UK, 8 December 1991, pp. 24.1–24.11.Google Scholar
Kosinov, A. D., Maslov, A. A. & Semionov, N. V. 1997 An experimental study of generation of unstable disturbances on the leading edge of a plate at $M=2$ . J. Appl. Mech. Tech. Phys. 38 (1), 45–50.CrossRefGoogle Scholar
Kosinov, A. D., Semionov, N. V. & Shevelkov, S. G. 1994a Investigation of supersonic boundary layer stability and transition using controlled disturbances. In Methods of Aerophysical Research, Institute of Theoretical and Applied Mechanics (ITAM), Novosibirsk, Russia (ed. Kharitonov, A. M.), vol. 2, pp. 159–166.Google Scholar
Kosinov, A. D., Semionov, N. V., Shevelkov, S. G. & Zinin, O. I. 1994b Experiments on the nonlinear instability of supersonic boundary layers. In Nonlinear Instability of Nonparallel Flows (ed. Valentine, D. T., Lin, S. P. & Philips, W. R. C.), pp. 196–205. Springer.CrossRefGoogle Scholar
Laible, A. C.2011 Numerical investigation of boundary-layer transition for cones at Mach 3.5 and 6.0. PhD thesis, University of Arizona.Google Scholar
Laible, A. C., Mayer, C. S. J. & Fasel, H. F.2008 Numerical investigation of supersonic transition for a circular cone at Mach 3.5. AIAA Paper 2008-4397.CrossRefGoogle Scholar
Laible, A. C., Mayer, C. S. J. & Fasel, H. F.2009 Numerical investigation of transition for a cone at Mach 3.5: oblique breakdown. AIAA Paper 2009-3557.CrossRefGoogle Scholar
van Leer, B. 1982 Flux-vector splitting for the Euler equations. In Eighth International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, vol. 170, pp. 507–512. Springer.CrossRefGoogle Scholar
Lin, T. C., Grabowsky, W. R. & Yelmgren, K. E. 1984 The search for optimum configurations for re-entry vehicles. J. Spacecr. Rockets 21 (2), 142–149.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2003 Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions. J. Fluid Mech. 488, 37–78.Google Scholar
Mack, L. M. 1965 Computation of the stability of the laminar compressible boundary layer. In Methods of Computational Physics (ed. Alder, B., Fernbach, S. & Rotenberg, M.), vol. 4, pp. 247–299. Academic Press.Google Scholar
Mack, L. M.1969 Boundary-layer stability theory. Internal Document 900-277. Jet Propulsion Laboratory, Pasadena, CA.Google Scholar
Mack, L. M.1987 Stability of axisymmetric boundary layers on sharp cones at hypersonic Mach numbers. AIAA Paper 1987-1413.CrossRefGoogle Scholar
Mangler, W. 1948 Zusammenhang zwischen ebenen und rotationssymmetrischen Grenzschichten in kompressiblen Flüssigkeiten. Z. Angew. Math. Mech. 28 (4), 97–103.CrossRefGoogle Scholar
Maslov, A. A., Shiplyuk, A. N., Sidorenko, A. A. & Arnal, D. 2001 Leading-edge receptivity of a hypersonic boundary layer on a flat plate. J. Fluid Mech. 426, 73–94.CrossRefGoogle Scholar
Matlis, E. H.2003 Controlled experiments on instabilities and transition to turbulence on sharp cones at supersonic Mach numbers. PhD thesis, University of Notre Dame.Google Scholar
Mayer, C. S. J. & Fasel, H. F.2008 Investigation of asymmetric subharmonic resonance in a supersonic boundary layer at Mach 2 using DNS. AIAA Paper 2008-0591.CrossRefGoogle Scholar
Mayer, C. S. J., von Terzi, D. A. & Fasel, H. F. 2011a Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3. J. Fluid Mech. 674, 5–42.CrossRefGoogle Scholar
Mayer, C. S. J., Wernz, S. & Fasel, H. F.2007 Investigation of oblique breakdown in a supersonic boundary layer at Mach 2 using DNS. AIAA Paper 2007-0949.CrossRefGoogle Scholar
Mayer, C. S. J., Wernz, S. & Fasel, H. F. 2011b Numerical investigation of the nonlinear transition regime in a Mach 2 boundary layer. J. Fluid Mech. 668, 113–149.CrossRefGoogle Scholar
Medeiros, M. A. F. & Gaster, M. 1999a The influence of phase on the nonlinear evolution of wavepackets in boundary layers. J. Fluid Mech. 397, 259–283.CrossRefGoogle Scholar
Medeiros, M. A. F. & Gaster, M. 1999b The production of subharmonic waves in the nonlinear evolution of wavepackets in boundary layers. J. Fluid Mech. 399, 301–318.CrossRefGoogle Scholar
Meitz, H. & Fasel, H. F. 2000 A compact-difference scheme for the Navier–Stokes equations in vorticity–velocity formulation. J. Comput. Phys. 157, 371–403.CrossRefGoogle Scholar
Poinsot, T. J. & Lele, S. K. 1992 Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101, 104–129.CrossRefGoogle Scholar
Pruett, C. D. & Chang, C.-L. 1995 Spatial direct numerical simulation of high-speed boundary-layer flows. Part II: transition on a cone in Mach 8 flow. Theor. Comput. Fluid Dyn. 7, 397–424.CrossRefGoogle Scholar
Rufer, S. J. & Schneider, S. P.2006 Hot-wire measurements of instability waves on cones at Mach 6. AIAA Paper 2006-3054.CrossRefGoogle Scholar
Sandham, N. D., Adams, N. A. & Kleiser, L. 1995 Direct simulation of breakdown to turbulence following oblique instability waves in a supersonic boundary layer. Appl. Sci. Res. 54, 223–234.CrossRefGoogle Scholar
Saric, W. S. & Nayfeh, A. H. 1975 Non-parallel stability of boundary layer flows. Phys. Fluids 18, 945–950.CrossRefGoogle Scholar
Schneider, S. P. 2001 Effects of high-speed tunnel noise on laminar–turbulent transition. J. Spacecr. Rockets 38 (3), 323–333.CrossRefGoogle Scholar
Schneider, S. P. 2004 Hypersonic laminar–turbulent transition on circular cones and scramjet forebodies. Prog. Aerosp. Sci. 40, 1–50.CrossRefGoogle Scholar
Schneider, S. P. 2006 Laminar–turbulent transition on reentry capsules and planetary probes. J. Spacecr. Rockets 43 (6), 1153–1173.CrossRefGoogle Scholar
Shiplyuk, A. N., Bountin, D. A., Maslov, A. A. & Chokani, N. 2003 Nonlinear mechanisms of the initial stage of laminar–turbulent transition at hypersonic velocities. J. Appl. Mech. Tech. Phys. 44 (5), 654–659.CrossRefGoogle Scholar
Sivasubramanian, J. & Fasel, H. F.2012 Growth and breakdown of a wavepacket into a turbulent spot in a cone boundary layer at Mach 6. AIAA Paper 2012-0085.CrossRefGoogle Scholar
Stetson, K. F., Thompson, E. R., Donaldson, J. C. & Siler, L. G.1983 Laminar boundary layer stability experiments on a cone at Mach 8. Part I: sharp cone. AIAA Paper 1983-1761.CrossRefGoogle Scholar
Thumm, A.1991 Numerische Untersuchungen zum laminar–turbulenten Strömungsumschlag in transsonischen Grenzschichtströmungen. PhD thesis, Universität Stuttgart.Google Scholar
Tumin, A. 2007 Three-dimensional spatial normal modes in compressible boundary layers. J. Fluid Mech. 586, 295–322.CrossRefGoogle Scholar
Zhong, X. 1998 High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition. J. Comput. Phys. 144, 662–709.CrossRefGoogle Scholar
Zhong, X. 2001 Leading-edge receptivity to free-stream disturbance waves for hypersonic flow over a parabola. J. Fluid Mech. 441, 315–367.CrossRefGoogle Scholar
Zhong, X. & Ma, Y. 2006 Boundary-layer receptivity of Mach 7.99 flow over a blunt cone to free-stream acoustic waves. J. Fluid Mech. 556, 55–103.CrossRefGoogle Scholar
Zhong, X. & Tatineni, M. 2003 High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition. J. Comput. Phys. 190, 419–458.CrossRefGoogle Scholar