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Effects of spanwise-periodic surface heating on supersonic boundary-layer instability

Published online by Cambridge University Press:  08 April 2022

Kaixin Zhu Open the ORCID record for Kaixin Zhu [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Kaixin Zhu
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, PR China
Xuesong Wu*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, PR China Department of Mathematics, Imperial College London, 180, Queen's Gate, London SW7 2AZ, UK
*
†Email address for correspondence: x.wu@ic.ac.uk
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Abstract

The effects of streamwise-elongated, spanwise-periodic surface heating on a supersonic boundary-layer instability are investigated under the assumption of high Reynolds number. Our focus is on the lower-branch viscous instability and so the spanwise spacing of the elements is chosen to be of $O({\textit {Re}}^{-3/8}L)$, the wavelength of the latter, where ${\textit {Re}}$ is the Reynolds number based on $L$, the distance from the leading edge to the centre of the elements. The streamwise length is assumed to be much longer in order to simplify the mathematical description. Starting with classical triple-deck theory, the equations governing the heating-induced streaky flow are derived by appropriate rescaling. When Chapman's viscosity law is adopted, a similarity solution is found. The stability of the streaky flow, which is of a bi-global nature, is shown to be governed by a novel triple-deck structure characterised by fully compressible dynamics in the lower deck. Through asymptotic analysis, the bi-global stability is reduced to a one-dimensional eigenvalue problem, which involves only the spanwise-dependent wall temperature and wall shear. The instability modes may be viewed as a continuation of oncoming first Mack modes, but might also be considered as a new kind since they exhibit two distinctive features: strong temperature perturbation near the wall and spontaneous radiation of an acoustic wave to the far field, neither of which is shared by first Mack modes. Numerical calculations, performed for two simple patterns of spanwise-periodic heating elements, demonstrate their stabilising/destabiling effects on modes with different frequencies and spanwise wavelengths.

JFM classification

Boundary Layers: Boundary layer stability Compressible Flows: Supersonic flow Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 940 , 10 June 2022 , A20
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Aljohani, A.F. & Gajjar, J.S.B. 2017 a Subsonic flow past localised heating elements in boundary layers. J. Fluid Mech. 821, R2.CrossRefGoogle Scholar
Aljohani, A.F. & Gajjar, J.S.B. 2017 b Subsonic flow past three-dimensional localised heating elements in boundary layers. Fluid Dyn. Res. 49 (6), 065503.CrossRefGoogle Scholar
Aljohani, A.F. & Gajjar, J.S.B. 2018 Transonic flow over localised heating elements in boundary layers. J. Fluid Mech. 844, 746–765.CrossRefGoogle Scholar
Brennan, G.S., Gajjar, J.S.B. & Hewitt, R.E. 2021 Tollmien–Schlichting wave cancellation via localised heating elements in boundary layers. J. Fluid Mech. 909, A16.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, 188–206.CrossRefGoogle Scholar
Corke, T.C. & Mangano, R.A. 1989 Resonant growth of three-dimensional modes in transitioning Blasius boundary layers. J. Fluid Mech. 209, 93–150.CrossRefGoogle Scholar
Dovgal, A.V., Levchenko, V.Y. & Timopeev, V.A. 1990 Boundary layer control by a local heating of the wall. In Laminar-Turbulent Transition (ed. D. Arnal & R. Michel), pp. 113–121. Springer.CrossRefGoogle Scholar
Downs, R.S. & Fransson, J.H.M. 2014 Tollmien–schlichting wave growth over spanwise-periodic surface patterns. J. Fluid Mech. 754, 39–74.CrossRefGoogle Scholar
Fedorov, A., Soudakov, V., Egorov, I., Sidorenko, A., Gromyko, Y., Bountin, D., Polivanov, P. & Maslov, A. 2015 High-speed boundary-layer stability on a cone with localized wall heating or cooling. AIAA J. 53 (9), 1–13.CrossRefGoogle Scholar
Kátai, C.B. & Wu, X. 2020 Effects of streamwise-elongated and spanwise-periodic surface roughness elements on boundary-layer instability. J. Fluid Mech. 899, A34.CrossRefGoogle Scholar
Koroteev, M.V. & Lipatov, I.I. 2009 Supersonic boundary layer in regions with small temperature perturbations on the wall. SIAM. J. Appl. Math. 70 (4), 1139–1156.CrossRefGoogle Scholar
Koroteev, M.V. & Lipatov, I.I. 2012 Local temperature perturbations of the boundary layer in the regime of free viscous–inviscid interaction. J. Fluid Mech. 707, 595–605.CrossRefGoogle Scholar
Kral, L.D. & Fasel, H.F. 1991 Numerical investigation of three-dimensional active control of boundary-layer transition. AIAA J. 29 (9), 1407–1417.CrossRefGoogle Scholar
Kral, L.D., Wlezien, R.W., Smith, J.M. & Masad, J.A. 1994 Boundary-layer transition control by localized heating: DNS and experiment. In Transition, Turbulence and Combustion, pp. 355–367. Springer.CrossRefGoogle Scholar
Liepmann, H.W., Brown, G.L. & Nosenchuck, D.M. 1982 Control of laminar-instability waves using a new technique. J. Fluid Mech. 118, 187–200.CrossRefGoogle Scholar
Liepmann, H.W. & Nosenchuck, D.M. 1982 Active control of laminar-turbulent transition. J. Fluid Mech. 118, 201–204.CrossRefGoogle Scholar
Lin, C.C. 1946 On the stability of two-dimensional parallel flows. III. Stability in a viscous fluid. Q. Appl. Math. 3 (4), 277–301.CrossRefGoogle Scholar
Lipatov, I.I. 2006 Disturbed boundary layer flow with local time-dependent surface heating. Fluid Dyn. 41 (5), 725–735.CrossRefGoogle Scholar
Lysenko, V.I. & Maslov, A.A. 1984 The effect of cooling on supersonic boundary-layer stability. J. Fluid Mech. 147, 39–52.CrossRefGoogle Scholar
Mack, L.M. 1975 Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13 (3), 278–289.CrossRefGoogle Scholar
Mack, L.M. 1984 Boundary-layer linear stability theory. AGARD Report 709.Google Scholar
Masad, J.A. 1995 Transition in flow over heat-transfer strips. Phys. Fluids 7 (9), 2163–2174.CrossRefGoogle Scholar
Masad, J.A. & Nayfeh, A.H. 1992 Laminar flow control of subsonic boundary layers by suction and heat-transfer strips. Phys. Fluids 4 (6), 1259–1272.CrossRefGoogle Scholar
Méndez, F., Treviño, C. & Liñán, A. 1992 Boundary layer separation by a step in surface temperature. Int. J. Heat Mass Transf. 35 (10), 2725–2738.CrossRefGoogle Scholar
Muller, D.E. 1956 A method for solving algebraic equations using an automatic computer. Mathematical tables and other aids to computation 10 (56), 208–215.CrossRefGoogle Scholar
Neiland, V.Y. 1969 Theory of laminar boundary layer separation in supersonic flow. Fluid Dyn. 4 (4), 33–35.Google Scholar
Renardy, M. & Rogers, R.C. 2006 An Introduction to Partial Differential Equations, vol. 13. Springer Science & Business Media.Google Scholar
Smith, F.T. 1979 a Instability of flow through pipes of general cross-section, part 1. Mathematika 26 (2), 187–210.CrossRefGoogle Scholar
Smith, F.T. 1979 b On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. 366 (1724), 91–109.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
Stewartson, K. & Williams, P.G. 1969 Self-induced separation. Proc. R. Soc. Lond. 312 (1509), 181–206.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.CrossRefGoogle Scholar
Treviño, C. & Liñán, A. 1996 The effects of displacement induced by thermal perturbations on the structure and stability of boundary-layer flows. Theor. Comput. Fluid Dyn. 8 (1), 57–72.CrossRefGoogle Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar
Walton, A.G. & Patel, R.A. 1998 On the neutral stability of spanwise-periodic boundary-layer and triple-deck flows. Q. J. Mech. Appl. Math. 51 (2), 311–328.CrossRefGoogle Scholar
Wu, X. & Dong, M. 2016 A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue. J. Fluid Mech. 794, 68–108.CrossRefGoogle Scholar
Wu, X. & Hogg, L.W. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 307–347.CrossRefGoogle Scholar