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Instabilities and transition in cooled wall hypersonic boundary layers

Published online by Cambridge University Press:  11 March 2021

S. Unnikrishnan*
Mechanical Engineering, Florida State University, Tallahassee, FL32310, USA
Datta V. Gaitonde
Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH43210, USA
Email address for correspondence:


Wall cooling has substantial qualitative and quantitative effects on the development of instabilities and subsequent transition processes in hypersonic boundary layers (HBLs). A sequence of linear stability theory, nonlinear two-dimensional and three-dimensional direct numerical simulations is used to analyse Mach 6 boundary layers, with wall temperatures ranging from near-adiabatic to highly cooled conditions, where the second-mode instability is accompanied by radiation of energy. Decomposition of linear stability modes into their fluid-thermodynamic (acoustic, vortical and thermal) components shows that this radiation comprises both acoustic as well as vortical waves. Furthermore, in these cases, two-dimensional simulations show that the conventional ‘trapped’ nature of second-mode instability is ruptured. A quantitative analysis indicates that although the energy efflux of both acoustic and vortical components increases with wall cooling, the destabilization effect is much stronger and no significant abatement of pressure perturbations is realized. The direct impact of these mechanisms on the transition process itself is examined with high-fidelity simulations of three-dimensional second-mode wavepacket propagation. In the near-adiabatic HBL, the wavepacket remains trapped within the boundary layer and attenuates outside the region of linear instability. However, wavepackets in the cooled wall HBLs amplify and display nonlinear distortion, and transition more rapidly. The structure of the wavepacket also displays different behaviour; moderately cooled walls show bifurcation into a leading turbulent head region and a trailing harmonic region, while highly cooled wall cases display lower convection speeds and significant wavepacket elongation, with intermittent spurts of turbulence in the wake of the head region. This elongation effect is associated with a weakening of the lateral jet mechanism due to the breakdown of spanwise coherent structures. These features have a direct impact on wall loading, including skin friction and heat transfer. In moderately cooled walls, the spatially localized wall loading is similar to those in near-adiabatic walls, with dominant impact due to coherent structures in the leading turbulent head region. In highly cooled walls, the elongated near-wall streaks in the wake region of the wavepacket result in more than twice as large levels of skin friction and heat transfer over a sustained period of time.

JFM Papers
© The Author(s), 2021. Published by Cambridge University Press

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Anderson, J.D. 2000 Hypersonic and High Temperature Gas Dynamics. AIAA.Google 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 (2), 405452.CrossRefGoogle Scholar
Beam, R. & Warming, R. 1978 An implicit factored scheme for the compressible Navier–Stokes equations. AIAA J. 16 (4), 393402.CrossRefGoogle Scholar
Bhagatwala, A. & Lele, S.K. 2009 A modified artificial viscosity approach for compressible turbulence simulations. J. Comput. Phys. 228 (14), 49654969.CrossRefGoogle Scholar
Bitter, N.P. & Shepherd, J.E. 2015 Stability of highly cooled hypervelocity boundary layers. J. Fluid Mech. 778, 586620.CrossRefGoogle Scholar
Casper, K.M., Beresh, S.J. & Schneider, S.P. 2014 Pressure fluctuations beneath instability wavepackets and turbulent spots in a hypersonic boundary layer. J. Fluid Mech. 756, 10581091.CrossRefGoogle Scholar
Chang, C.-L., Kline, H.L. & Li, F. 2019 Effects of wall cooling on supersonic modes in high-enthalpy hypersonic boundary layers over a cone. AIAA Aviation 2019 Forum. AIAA Paper 2019-2852.CrossRefGoogle Scholar
Chang, C.-L., Vinh, H. & Malik, M.R. 1997 Hypersonic boundary-layer stability with chemical reactions using PSE. 28th Fluid Dynamics Conference. AIAA Paper 1997-2012.CrossRefGoogle Scholar
Chuvakhov, P.V. & Fedorov, A.V. 2016 Spontaneous radiation of sound by instability of a highly cooled hypersonic boundary layer. J. Fluid Mech. 805, 188206.CrossRefGoogle Scholar
Chynoweth, B.C., Schneider, S.P., Hader, C., Fasel, H., Batista, A., Kuehl, J., Juliano, T.J. & Wheaton, B.M. 2019 History and progress of boundary-layer transition on a mach-6 flared cone. J. Spacecr. Rockets 56 (2), 333346.CrossRefGoogle Scholar
Doak, P.E. 1989 Momentum potential theory of energy flux carried by momentum fluctuations. J. Sound Vib. 131 (1), 6790.CrossRefGoogle Scholar
Egorov, I.V., Fedorov, A.V. & Soudakov, V.G. 2006 Direct numerical simulation of disturbances generated by periodic suction-blowing in a hypersonic boundary layer. Theor. Comput. Fluid Dyn. 20 (1), 4154.CrossRefGoogle Scholar
Egorov, I.V. & Novikov, A.V. 2016 Direct numerical simulation of laminar–turbulent flow over a flat plate at hypersonic flow speeds. Comput. Maths Math. Phys. 56 (6), 10481064.CrossRefGoogle Scholar
Fedorov, A.V. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43, 7995.CrossRefGoogle Scholar
Fedorov, A.V. & Khokhlov, A.P. 2001 Prehistory of instability in a hypersonic boundary layer. Theor. Comput. Fluid Dyn. 14 (6), 359375.CrossRefGoogle Scholar
Fedorov, A.V. & Tumin, A. 2003 Initial-value problem for hypersonic boundary-layer flows. AIAA J. 41 (3), 379389.CrossRefGoogle Scholar
Fedorov, A. & Tumin, A. 2011 High-speed boundary-layer instability: old terminology and a new framework. AIAA J. 49 (8), 16471657.CrossRefGoogle Scholar
Fischer, M.C. 1972 Spreading of a turbulent disturbance. AIAA J. 10 (7), 957959.CrossRefGoogle Scholar
Glezer, A., Katz, Y. & Wygnanski, I. 1989 On the breakdown of the wave packet trailing a turbulent spot in a laminar boundary layer. J. Fluid Mech. 198, 126.CrossRefGoogle Scholar
Hader, C. & Fasel, H.F. 2019 Direct numerical simulations of hypersonic boundary-layer transition for a flared cone: fundamental breakdown. J. Fluid Mech. 869, 341384.CrossRefGoogle Scholar
Huang, J., Bretzke, J.-V. & Duan, L. 2019 Assessment of turbulence models in a hypersonic cold-wall turbulent boundary layer. Fluids 4 (1), 37.CrossRefGoogle Scholar
Jenvey, P.L. 1989 The sound power from turbulence: a theory of the exchange of energy between the acoustic and non-acoustic fields. J. Sound Vib. 131 (1), 3766.CrossRefGoogle Scholar
Jocksch, A. & Kleiser, L. 2008 Growth of turbulent spots in high-speed boundary layers on a flat plate. Intl J. Heat Fluid Flow 29 (6), 15431557.CrossRefGoogle Scholar
Knisely, C.P. & Zhong, X. 2019 a Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. I. Linear stability theory. Phys. Fluids 31 (2), 024103.CrossRefGoogle Scholar
Knisely, C.P. & Zhong, X. 2019 b Sound radiation by supersonic unstable modes in hypersonic blunt cone boundary layers. II. Direct numerical simulation. Phys. Fluids 31 (2), 024104.CrossRefGoogle Scholar
Krishnan, L. & Sandham, N.D. 2006 Effect of Mach number on the structure of turbulent spots. J. Fluid Mech. 566, 225234.CrossRefGoogle Scholar
van Leer, B. 1979 Towards the ultimate conservation difference scheme V, a second-order sequel to Godunov's method. J. Comput. Phys. 32, 101136.CrossRefGoogle Scholar
Lees, L. & Lin, C.C. 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA Tech note No. 1115.Google Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166178. Nauka.Google 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, 3178.CrossRefGoogle Scholar
Mack, L.M. 1975 Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13 (3), 278289.CrossRefGoogle Scholar
Mack, L.M. 1984 Boundary-layer linear stability theory. AGARD Rep. 709.Google Scholar
Mack, L.M. 1990 On the inviscid acoustic-mode instability of supersonic shear flows. Theor. Comput. Fluid Dyn. 2 (2), 97123.Google Scholar
Novikov, A.V. 2017 Direct numerical simulation of transitional boundary layer with local separation in hypersonic flight. 7th European Conference for Aeronautics and Space Sciences (EUCASS). DOI: 10.13009/EUCASS2017-656.CrossRefGoogle Scholar
Novikov, A., Egorov, I. & Fedorov, A. 2016 Direct numerical simulation of wave packets in hypersonic compression-corner flow. AIAA J. 54 (7), 20342050.CrossRefGoogle Scholar
Park, S. & Lauchle, G.C. 2009 Wall pressure fluctuation spectra due to boundary-layer transition. J. Sound Vib. 319 (3-5), 10671082.CrossRefGoogle Scholar
Pulliam, T.H. & Chaussee, D.S. 1981 A diagonal form of an implicit approximate-factorization algorithm. J. Comput. Phys. 39 (2), 347363.CrossRefGoogle Scholar
Redford, J.A., Sandham, N.D. & Roberts, G.T. 2012 Numerical simulations of turbulent spots in supersonic boundary layers: effects of Mach number and wall temperature. Prog. Aerosp. Sci. 52, 6779.CrossRefGoogle Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Annu. Rev. Fluid Mech. 8 (1), 311349.CrossRefGoogle Scholar
Roe, P.L. 1981 Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357372.CrossRefGoogle Scholar
Salemi, L.C. & Fasel, H.F. 2018 Synchronization of second-mode instability waves for high-enthalpy hypersonic boundary layers. J. Fluid Mech. 838, R2.CrossRefGoogle Scholar
Sayadi, T., Hamman, C.W. & Moin, P. 2013 Direct numerical simulation of complete h-type and k-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480509.CrossRefGoogle Scholar
Shadloo, M.S. & Hadjadj, A. 2017 Laminar-turbulent transition in supersonic boundary layers with surface heat transfer: a numerical study. Numer. Heat Trans. A Appl. 72 (1), 4053.CrossRefGoogle Scholar
Shu, C.W. & Osher, S. 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (2), 439471.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. I. Coherent structures. Q. Appl. Math. 45 (3), 561571.CrossRefGoogle Scholar
Sivasubramanian, J. & Fasel, H. 2010 Numerical investigation of boundary-layer transition initiated by a wave packet for a cone at mach 6. 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. AIAA Paper 2010-0900.CrossRefGoogle Scholar
Soudakov, V.G., Egorov, I.V., Fedorov, A.V. & Novikov, A.V. 2016 Numerical simulation of receptivity and stability of a supersonic boundary layer. In 27th International Congress of the Aeronautical Sciences.Google Scholar
Stetson, K.F. & Kimmel, R. 1992 On hypersonic boundary-layer stability. 30th Aerospace Sciences Meeting and Exhibit, Reno, NV. AIAA Paper 1992-737.Google Scholar
Stetson, K.F., Thompson, E.R., Donaldson, J.C. & Siler, L.G. 1989 Laminar boundary layer stability experiments on a cone at mach 8, Part 5: tests with a cooled model. 20th Fluid Dynamics, Plasma Dynamics and Lasers Conference, Buffalo, NY. AIAA Paper 1989-1895.Google Scholar
Tumin, A. 2020 a Lst and the eigenfunction expansion method for linearized navier-stokes equations–a summary. AIAA Scitech 2020 Forum. AIAA Paper 2020-0105.CrossRefGoogle Scholar
Tumin, A. 2020 b Wave packets and supersonic second modes in a high-speed boundary layer. AIAA Scitech 2020 Forum. AIAA Paper 2020-0106.CrossRefGoogle Scholar
Unnikrishnan, S. & Gaitonde, D.V. 2016 Acoustic, hydrodynamic and thermal modes in a supersonic cold jet. J. Fluid Mech. 800, 387432.CrossRefGoogle Scholar
Unnikrishnan, S. & Gaitonde, D.V. 2018 Transfer mechanisms from stochastic turbulence to organized acoustic radiation in a supersonic jet. Eur. J. Mech. (B/Fluids) 72, 3856.CrossRefGoogle Scholar
Unnikrishnan, S. & Gaitonde, D.V. 2019 a First-mode-induced nonlinear breakdown in a hypersonic boundary layer. Comput. Fluids 191, 104249.CrossRefGoogle Scholar
Unnikrishnan, S. & Gaitonde, D.V. 2019 b Interactions between vortical, acoustic and thermal components during hypersonic transition. J. Fluid Mech. 868, 611647.CrossRefGoogle Scholar
Wright, R. & Zoby, E. 1977 Flight boundary layer transition measurements on a slender cone at Mach 20. 10th Fluid and Plasmadynamics Conference. AIAA Paper 1977-719.CrossRefGoogle Scholar
Wygnanski, I., Sokolov, M. & Friedman, D. 1976 On a turbulent ‘spot’ in a laminar boundary layer. J. Fluid Mech. 78 (4), 785819.CrossRefGoogle Scholar
Wygnanski, I., Zilberman, M. & Haritonidis, J.H. 1982 On the spreading of a turbulent spot in the absence of a pressure gradient. J. Fluid Mech. 123, 6990.CrossRefGoogle Scholar
Yates, H.B., Tufts, M.W. & Juliano, T.J. 2020 Analysis of the hypersonic cross-flow instability with experimental wavenumber distributions. J. Fluid Mech. 883, A50.CrossRefGoogle Scholar
Zhang, S., Liu, J. & Luo, J. 2016 Effect of wall-cooling on mack-mode instability in high speed flat-plate boundary layers. Z. Angew. Math. Mech. 37 (9), 12191230.CrossRefGoogle Scholar
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