Skip to main content Accessibility help
×
Home

Nonlinear instability of a supersonic boundary layer with two-dimensional roughness

  • Olaf Marxen (a1), Gianluca Iaccarino (a1) and Eric S. G. Shaqfeh (a1)

Abstract

Nonlinear instability in a supersonic boundary layer at Mach 4.8 with two-dimensional roughness is investigated by means of spatial direct numerical simulations (DNS). It was previously found that an important effect of a two-dimensional roughness is to increase significantly the amplitude of two-dimensional waves downstream of the roughness in a certain frequency band through enhanced instability and transient growth, while waves outside this band are damped. Here, we investigate the nonlinear secondary instability induced by a large-amplitude two-dimensional wave, which has received a significant boost in amplitude from this additional roughness-induced amplification. Both subharmonic and fundamental secondary excitation of the oblique secondary waves are considered. We found that even though the growth rate of the secondary perturbations increases compared to their linear amplification, only in some of the cases was a fully resonant state attained by the streamwise end of the domain. A parametric investigation of the amplitude of the primary wave, the phase difference between the primary and the secondary waves, and the spanwise wavenumber has also been performed. The transient growth experienced by the primary wave was found to not influence the secondary instability for most parameter combinations. For unfavourable phase relations between the primary and the secondary waves, the phase speed of the secondary wave decreases significantly, and this hampers its growth. Finally, we also investigated the strongly nonlinear stage, for which both the primary and the subharmonic secondary waves had a comparable, finite amplitude. In this case, the growth of the primary waves was found to vanish downstream of the transient growth region, resulting in a lower amplitude than in the absence of the large-amplitude secondary wave. This feedback also decreases the amplification rate of the secondary wave.

Copyright

Corresponding author

Present address: Department of Mechanical Engineering, Imperial College London, Exhibition Road, South Kensington, London SW7 2AZ, UK. Email address for correspondence: o.marxen@imperial.ac.uk

References

Hide All
Adams, N. A. & Kleiser, L. 1996 Subharmonic transition to turbulence in a flat-plate boundary layer at Mach number 4.5. J. Fluid Mech. 317, 301335.
Bountin, D., Shiplyuk, A. & Maslov, A. 2008 Evolution of nonlinear processes in a hypersonic boundary layer on a sharp cone. J. Fluid Mech. 611, 427442.
Duan, L., Wang, X. & Zhong, X. 2013 Stabilization of a Mach 5.92 boundary layer by two-dimensional finite-height roughness. AIAA J. 51 (1), 266270.
Eissler, W. & Bestek, H. 1996 Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers. Theor. Comput. Fluid Dyn. 8 (3), 219235.
Fedorov, A. V. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43, 7995.
Fujii, K. 2006 Experiment of the two-dimensional roughness effect on hypersonic boundary-layer transition. J. Spacecr. Rockets 43 (4), 731738.
Heitmann, D. & Radespiel, R. 2013 Simulations of boundary-layer response to laser-generated disturbances at Mach 6. J. Spacecr. Rockets 50 (2), 305316.
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20, 487526.
Kendall, J. M. 1975 Wind tunnel experiments relating to supersonic and hypersonic boundary-layer transition. AIAA J. 13 (3), 290299.
Kimmel, R. L. & Poggie, J.1997 Disturbance evolution and breakdown to turbulence in a hypersonic boundary layer – ensemble-averaged structure. AIAA Paper 1997-555.
Lifshitz, Y., Degani, D. & Tumin, A. 2012 Study of discrete modes branching in high-speed boundary layers. AIAA J. 50 (10), 22022210.
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, 3178.
Mack, L. M.1969 Boundary layer stability theory. Tech. Rep. JPL-900-277-REV-A; NASA-CR-131501. Jet Propulsion Laboratory, NASA.
Marxen, O., Iaccarino, G. & Shaqfeh, E.S.G. 2010 Disturbance evolution in a Mach 4.8 boundary layer with two-dimensional roughness-induced separation and shock. J. Fluid Mech. 648, 435469.
Marxen, O., Magin, T., Iaccarino, G. & Shaqfeh, E.S.G. 2011 A high-order numerical method to study hypersonic boundary-layer instability including high-temperature gas effects. Phys. Fluids 23, 084108.
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, 542.
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, 113149.
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2003 A robust high-order method for large eddy simulation. J. Comput. Phys. 191, 392419.
Ng, L. L. & Erlebacher, G. 1992 Secondary instabilities in compressible boundary layers. Phys. Fluids A 4 (4), 710726.
Pagella, A., Rist, U. & Wagner, S. 2002 Numerical investigations of small-amplitude disturbances in a boundary layer with impinging shock wave at $Ma=4.8$ . Phys. Fluids 14 (7), 20882101.
Reshotko, E. & Tumin, A. 2004 Role of transient growth in roughness-induced transition. AIAA J. 42 (4), 766770.
Tumin, A. 2007 Three-dimensional spatial normal modes in compressible boundary layers. J. Fluid Mech. 586, 295322.
Van Driest, E. R. & Blumer, C. B. 1968 Boundary-layer transition at supersonic speeds – roughness effects with heat transfer. AIAA J. 6 (4), 603607.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed