Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T14:40:22.793Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear stability of hypersonic flow over a cone with passive porous walls

Published online by Cambridge University Press:  22 October 2012

Vipin Michael  and
Sharon O. Stephen
Vipin Michael*
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
Sharon O. Stephen
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This study investigates the nonlinear stability of hypersonic viscous flow over a sharp slender cone with passive porous walls. The attached shock and effect of curvature are taken into account. Asymptotic methods are used for large Reynolds number and large Mach number to examine the viscous modes of instability (first Mack mode), which may be described by a triple-deck structure. A weakly nonlinear stability analysis is carried out allowing an equation for the amplitude of disturbances to be derived. The coefficients of the terms in the amplitude equation are evaluated for axisymmetric and non-axisymmetric disturbances. The stabilizing or destabilizing effect of nonlinearity is found to depend on the cone radius. The presence of porous walls significantly influences the effect of nonlinearity, and results for three types of porous wall (regular, random and mesh microstructure) are compared.

JFM classification

Compressible Flows: Compressible Flows Compressible Flows: Compressible boundary layers Instability: Instability Instability: Nonlinear instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 713 , 25 December 2012 , pp. 528 - 563
Copyright
©2012 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bassom, A. P. 1989 Weakly nonlinear lower-branch stability of fully developed and developing free-surface flows. IMA J. Appl. Maths 42, 269–301.CrossRefGoogle Scholar
Bestek, H. & Eissler, W. 1996 Direct numerical simulation of transition in Mach 4.8 boundary layers at flight conditions. In Engineering Turbulence Modelling and Experiments (ed. Rodi, W. & Bergeles, G.). vol. 3. Elsevier.Google Scholar
Bountin, D., Maslov, A., Chimytov, C. & Shiplyuk, A. 2010 Bispectral analysis of nonlinear processes in the hypersonic boundary layer on a porous cone surface. Fluid Dyn. 45, 415–421.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
Bountin, D., Shiplyuk, A. & Sidorenko, A. 2000 Experimental investigations of disturbance development in the hypersonic boundary layer on conical models. In Fifth IUTAM Symposium on Laminar–Turbulent Transition (ed. Fasel, H. F. & Saric, W. S.). Springer.Google Scholar
Brown, S. N., Khorrami, A. M., Neish, A. & Smith, F. T. 1991 On hypersonic boundary-layer interactions and transition. Phil. Trans. R. Soc. Lond. A 335, 139–152.Google Scholar
Chang, C.-L., Malik, M. R. & Hussaini, M. Y. 1990 Effects of shock on the stability of hypersonic boundary layers. AIAA Paper 90-1448.CrossRefGoogle Scholar
Chokani, N. 1999 Nonlinear spectral dynamics of hypersonic laminar boundary layer flow. Phys. Fluids 11, 3846–3851.CrossRefGoogle Scholar
Chokani, N. 2005 Nonlinear evolution of Mack modes in a hypersonic boundary layer. Phys. Fluids 17, 014102.CrossRefGoogle Scholar
Chokani, N., Bountin, D. A., Shiplyuk, A. N. & Maslov, A. A. 2005 Nonlinear aspects of hypersonic boundary-layer stability on a porous surface. AIAA J. 43, 149–155.CrossRefGoogle Scholar
Cowley, S. J. & Hall, P. 1990 On the instability of hypersonic flow past a wedge. J. Fluid Mech. 214, 17–42.CrossRefGoogle Scholar
De Tullio, N. & Sandham, N. D. 2010 Direct numerical simulation of breakdown to turbulence in a Mach 6 boundary layer over a porous surface. Phys. Fluids 22, 094105–15.CrossRefGoogle Scholar
Duck, P. W. & Hall, P. 1989 On the interaction of Tollmien–Schlichting waves in axisymmetric supersonic flows. Q. J. Mech. Appl. Maths 42, 115–130.CrossRefGoogle Scholar
Duck, P. W. & Hall, P. 1990 Non-axisymmetric viscous lower-branch modes in axisymmetric supersonic flows. J. Fluid Mech. 213, 191–201.CrossRefGoogle Scholar
Egorov, I. V., Fedorov, A. V. & Soudakov, V. G. 2008 Receptivity of a hypersonic boundary layer over a flat plate with a porous coating. J. Fluid Mech. 601, 165–187.CrossRefGoogle Scholar
Fedorov, A. V. 2011 Transition and stability of high-speed boundary layers. Annu Rev. Fluid Mech. 43, 79–95.CrossRefGoogle Scholar
Fedorov, A., Kozlov, V., Shiplyuk, A., Maslov, A. & Malmuth, N. 2006 Stability of hypersonic boundary layer on porous wall with regular microstructure. AIAA J. 44, 1866–1871.CrossRefGoogle Scholar
Fedorov, A. V., Malmuth, N. D., Rasheed, A. & Hornung, H. G. 2001 Stabilization of hypersonic boundary layers by porous coatings. AIAA J. 39, 605–610.CrossRefGoogle Scholar
Fedorov, A. V., Shiplyuk, A., Maslov, A. A., Burov, E. & Malmuth, N. D. 2003a Stabilization of high speed boundary layer using a porous coating. AIAA Paper 2003-1970.CrossRefGoogle Scholar
Fedorov, A. V., Shiplyuk, A., Maslov, A. A., Burov, E. & Malmuth, N. D. 2003b Stabilization of a hypersonic boundary layer using an ultrasonically absorptive coating. J. Fluid Mech. 479, 99–124.CrossRefGoogle Scholar
Fedorov, A. & Tumin, A. 2010 Branching of discrete modes in high-speed boundary layers and terminology issues. AIAA Paper 2010-5003.CrossRefGoogle Scholar
Gaponov, S., Ermolaev, Y., Kosinov, A., Lysenko, V., Semenov, N. & Smorodsky, B. 2010 The influence of surface porosity on the stability and transition of supersonic boundary layer on a flat plate. Thermophys. Aeromech. 17, 259–268.CrossRefGoogle Scholar
Gaponov, S. & Terekhova, N. M. 2009 Three-wave interactions between disturbances in the hypersonic boundary layer on impermeable and porous surfaces. Fluid Dyn. 44, 362–371.CrossRefGoogle Scholar
Germain, P. D. & Hornung, H. G. 1997 Transition on a slender cone in hypervelocity flow. Exp. Fluids 22 (3), 183–190.CrossRefGoogle Scholar
Hall, P. & Smith, F. T. 1982 A suggested mechanism for nonlinear wall roughness effects on high Reynolds number flow stability. Stud. Appl. Maths 66, 241–265.CrossRefGoogle Scholar
Hall, P. & Smith, F. T. 1984 On the effects of nonparallelism, three-dimensionality, and mode interaction in nonlinear boundary-layer stability. Stud. Appl. Maths 70, 91–120.CrossRefGoogle Scholar
Hayes, W. D. & Probstein, R. F. 1966 Hypersonic Flow Theory, 2nd edn., vol. 1. Academic.Google Scholar
Husmeier, F. & Fasel, H. F. 2007 Numerical investigations of hypersonic boundary layer transition for circular cones. AIAA Paper 2007-3843.CrossRefGoogle Scholar
Kimmel, R. 2003 Aspects of hypersonic boundary layer transition control. AIAA Paper 2003-0772.CrossRefGoogle Scholar
Kimmel, R. L., Demetriades, A. & Donaldson, J. C. 1996 Space–time correlation measurements in a hypersonic transitional boundary layer. AIAA J. 34, 2484–2489.CrossRefGoogle Scholar
Kimmel, R. L. & Kendall, J. M. 1991 Nonlinear disturbance in a hypersonic boundary layer. AIAA Paper 91-0320.CrossRefGoogle Scholar
Kluwick, A., Gittler, P. & Bodonyi, R. J. 1984 Viscous-inviscid interactions on axisymmetric bodies of revolution in supersonic flow. J. Fluid Mech. 140, 280–301.CrossRefGoogle Scholar
Koevary, C., Laible, A., Mayer, C. & Fasel, H. F. 2010 Numerical simulations of controlled transition for a sharp circular cone at Mach 8. AIAA Paper 2010-4598.CrossRefGoogle Scholar
Kozlov, V. F., Fedorov, A. & Malmuth, N. 2005 Acoustic properties of rarefied gases inside pores of simple geometries. J. Acoust. Soc. Am. 117, 3402–3412.CrossRefGoogle ScholarPubMed
Lachowicz, J. T., Chokani, N. & Wilkinson, S. P. 1996 Boundary-layer stability measurements in a hypersonic quiet tunnel. AIAA J. 34 (12), 2496–2500.CrossRefGoogle Scholar
Laible, A. & Fasel, H. F. 2011 Numerical investigation of hypersonic transition for a flared and a straight cone at Mach 6. AIAA Paper 2011-3565.CrossRefGoogle Scholar
Leung, K. K. & Emanuel, G. 1995 Hypersonic inviscid and viscous flow over a wedge and cone. J. Aircraft 32, 385–391.CrossRefGoogle Scholar
Lukashevich, S. V., Maslov, A. A., Shiplyuk, A. N., Fedorov, A. V. & Soudakov, V. G. 2010 Stabilization of high-speed boundary layer using porous coatings of various thicknesses. AIAA Paper 2010-4720.CrossRefGoogle Scholar
Mack, L. M. 1984 Boundary layer stability theory. In Special Course on Stability and Transition of Laminar Flow, AGARD Rep. 709.Google Scholar
Malik, M. R. 1990 Numerical-methods for hypersonic boundary-layer stability. J. Comput. Phys. 86, 376–413.CrossRefGoogle Scholar
Maslov, A. 2003 Experimental and theoretical studies of hypersonic laminar flow control using ultrasonically absorptive coatings (U.A.C.). Tech. Rep. ISTC 2172-2001. International Science and Technology Centre.Google Scholar
Maslov, A. A., Poplavskaya, T. & Bountin, D. A. 2010 Hypersonic boundary layer transition and control. In Seventh IUTAM Symposium on Laminar–Turbulent Transition (ed. Schlatter, P. & Henningson, D.). pp. 19–26. Springer.CrossRefGoogle Scholar
Michael, V. 2012 Effects of passive porous walls on the first Mack mode instability of hypersonic boundary layers over a sharp cone. PhD thesis, University of Birmingham..Google Scholar
Neiland, V. Y. 1969 Theory of laminar boundary layer separation in supersonic flow. Izv. Akad. Nauk. SSSR Mech. Zhidk Gaza 53–57.Google 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. Theoret. Comput. Fluid Dyn 7, 397–424.CrossRefGoogle Scholar
Rasmussen, M. 1994 Hypersonic Flow. Wiley.Google Scholar
Reed, H. L., Kimmel, R., Schneider, S., Arnal, D. & Saric, W. 1997 Drag prediction and transition in hypersonic flow. AIAA Paper 97-1818.CrossRefGoogle Scholar
Schneider, S. P. 2004 Hypersonic laminar-turbulent transition on circular cones and scramjet forebodies. Prog. Aerosp. Sci. 40 (1-2), 1–50.CrossRefGoogle Scholar
Seddougui, S. O. 1994 Stability of hypersonic flow over a cone. In Transition, Turbulence and Combustion (ed. Hussaini, M. Y., Gatski, T. B. & Jackson, T. L.). pp. 50–59. Kluwer.Google Scholar
Seddougui, S. O. & Bassom, A. P. 1994 Nonlinear instability of viscous modes in hypersonic flow past a wedge. Q. J. Mech. App. Maths 47, 557–582.CrossRefGoogle Scholar
Seddougui, S. O. & Bassom, A. P. 1997 Instability of hypersonic flow over a cone. J. Fluid Mech. 345, 383–411.CrossRefGoogle Scholar
Shiplyuk, A. N., Bountin, D. A., Maslov, A. A. & Chokani, N. 2003 Nonlinear mechanisms of the initial stage of the laminar–turbulent transition at hypersonic velocities. J. Appl. Mech. Tech. Phy. 44, 654–659.CrossRefGoogle Scholar
Smith, F. T. 1979 Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573–589.Google Scholar
Smith, F. T. 1980 Corrections to ‘Nonlinear stability of boundary layers for disturbances of various sizes’. Proc. R. Soc. Lond. A 371, 439–440.Google Scholar
Smith, F. T. 1989 On the first-mode instability in subsonic, supersonic or hypersonic boundary layers. J. Fluid Mech. 198, 127–153.CrossRefGoogle Scholar
Stephen, S. O. 2006 Nonlinear instability of hypersonic flow over a cone. Q. J. Mech. App. Maths 59, 301–319.CrossRefGoogle Scholar
Stephen, S. O. & Michael, V. 2010a Effect of passive porous walls on hypersonic boundary layer. In Seventh IUTAM Symposium on Laminar-Turbulent Transition (ed. Schlatter, P. & Henningson, D.). pp. 581–584. Springer.CrossRefGoogle Scholar
Stephen, S. O. & Michael, V. 2010b Effects of porous walls on hypersonic boundary layer over a sharp cone. AIAA Paper 2010-4286.CrossRefGoogle Scholar
Stephen, S. O. & Michael, V. 2012 Effects of porous walls on hypersonic boundary layers over a sharp cone. School of Mathematics Preprint Series 2012/06, University of Birmingham.CrossRefGoogle Scholar
Stetson, K. F. 1988 On nonlinear aspects of hypersonic boundary-layer stability. AIAA J. 26 (7), 883–885.CrossRefGoogle Scholar
Stetson, K. & Kimmel, R. 1992 On hypersonic boundary-layer stability. AIAA Paper 92-0737.CrossRefGoogle Scholar
Stetson, K. F., Kimmel, R., Thompson, E. R., Donaldson, J. C. & Siler, L. G. 1991 A comparison of planar and conical boundary layer stability and transition at a Mach number of 8. AIAA Paper 91-1639.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 1: sharp cone. AIAA Paper 83-1761.CrossRefGoogle Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press.CrossRefGoogle Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181–206.Google Scholar
Stilla, J. 1994 Engineering transition prediction for a hypersonic axisymmetric boundary layer. J. Aircraft 31, 1358–1364.CrossRefGoogle Scholar
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. J. Fluid Mech. 9, 353–370.CrossRefGoogle Scholar
Stuckert, G. & Reed, H. L. 1994 Linear disturbances in hypersonic, chemically reacting shock layers. AIAA J. 32, 1384–1393.CrossRefGoogle Scholar
Taylor, G. & Maccoll, J. 1933 The air pressure on a cone moving at high speeds.- I. Proc. R. Soc. Lond. A 139, 278–297.Google Scholar
Wang, X. & Zhong, X. 2009 Numerical simulation and theoretical analysis on boundary-layer instability affected by porous coating. AIAA Paper 2009-3679.CrossRefGoogle Scholar
Wang, X. & Zhong, X. 2010 Effect of porous coating on boundary-layer instability. AIAA Paper 2010-1243.CrossRefGoogle Scholar
Wang, X. & Zhong, X. 2011a Phase angle of porous coating admittance and its effect on boundary-layer stabilization. AIAA Paper 2011-3080.CrossRefGoogle Scholar
Wang, X. & Zhong, X. 2011b Numerical simulations on mode S growth over feltmetal and regular porous coatings of a Mach 5.92 flow. AIAA Paper 2011-375.CrossRefGoogle Scholar
Whitehead, A. Jr. 1989 NASP aerodynamics. AIAA Paper 89-5013.CrossRefGoogle Scholar
Zhong, X. & Wang, X. 2012 Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annu. Rev. Fluid Mech. 44, 527–561.CrossRefGoogle Scholar