Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T04:19:37.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Tollmien–Schlichting wave cancellation via localised heating elements in boundary layers

Published online by Cambridge University Press:  23 December 2020

G. S. Brennan Open the ORCID record for G. S. Brennan [Opens in a new window] ,
J. S. B. Gajjar Open the ORCID record for J. S. B. Gajjar [Opens in a new window]  and
R. E. Hewitt Open the ORCID record for R. E. Hewitt [Opens in a new window]
G. S. Brennan
Affiliation:
Department of Mathematics, University of Manchester, ManchesterM13 9PL, UK
J. S. B. Gajjar*
Affiliation:
Department of Mathematics, University of Manchester, ManchesterM13 9PL, UK
R. E. Hewitt
Affiliation:
Department of Mathematics, University of Manchester, ManchesterM13 9PL, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Instability to Tollmien–Schlichting waves is one of the primary routes to transition to turbulence for two-dimensional boundary layers in quiet disturbance environments. Cancellation of Tollmien–Schlichting waves using surface heating was first demonstrated in the experiments of Liepmann et al. (J. Fluid Mech., vol. 118, 1982, pp. 187–200) and Liepmann & Nosenchuck (J. Fluid Mech., vol. 118, 1982, pp. 201–204). Here we consider a similar theoretical formulation that includes the effects of localised (unsteady) wall heating/cooling. The resulting problem is closely related to that of Terent'ev (Prikl. Mat. Mekh., vol. 45, 1981, pp. 1049–1055; Prikl. Mat. Mekh., vol. 48, 1984, pp. 264–272) on the generation of Tollmien–Schlichting waves by a vibrating ribbon, but with thermal effects. The nonlinear receptivity problem based on triple-deck scales is formulated and the linearised version solved both analytically as well as numerically. The most significant result is that the wall heating/cooling function can be chosen such that there is no pressure response to the disturbance, meaning there is no generation of Tollmien–Schlichting waves. Numerical calculations substantiate this with an approximation based on the exact analytical result. Previous numerical studies of the unsteady triple-deck equations have shown difficulties in capturing the convective wave packet that develops in the initial-value problem and we show that these arise from the choice of time steps as well as the range of the Fourier modes taken.

JFM classification

Boundary Layers: Boundary layer stability Flow Control: Instability control Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 909 , 25 February 2021 , A16
Check for updates
Copyright
© The Author(s), 2020. Published by 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.)

Footnotes

‡

Present address: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK.

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. 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
Cassel, K. W., Ruban, A. I. & Walker, J. D. A. 1995 An instability in supersonic boundary-layer flow over a compression ramp. J. Fluid Mech. 300, 265–285.CrossRefGoogle Scholar
De Tullio, N. & Ruban, A. I. 2015 A numerical evaluation of the asymptotic theory for subsonic compressible boundary layers. J. Fluid Mech. 771, 520–546.CrossRefGoogle Scholar
van Dyke, M. 1952 Impulsive motion of an infinite plate in a viscous compressible fluid. Z. Angew. Math. Phys. 3 (5), 343–353.CrossRefGoogle Scholar
Fletcher, A. J. P., Ruban, A. I. & Walker, J. D. A. 2004 Instabilities in supersonic compression ramp flow. J. Fluid Mech. 517, 309–330.CrossRefGoogle Scholar
Gajjar, J. S. B. 1996 Nonlinear stability of non-stationary cross-flow vortices in compressible boundary layers. Stud. Appl. Maths 96, 53–84.CrossRefGoogle Scholar
Gaster, M. 1965 On the generation of spatially growing instability waves in the boundary layer. J. Fluid Mech. 22, 433–441.CrossRefGoogle Scholar
Gaster, M. & Grant, I. 1975 An experimental investigation of the formation and development of a wave packet in a laminar boundary layer. Proc. R. Soc. Lond. A 347, 253–269.Google Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Sclichting waves near a leading edge. J. Fluid Mech. 127, 59–81.CrossRefGoogle Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509–529.CrossRefGoogle Scholar
Koroteev, M. V. & Lipatov, I. I. 2009 Supersonic boundary layer in regions with small temperature perturbations on the wall. SIAM J. Appl. Maths 70, 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
Koroteev, M. V. & Lipatov, I. I. 2013 Steady subsonic boundary layer in domains of local surface heating. Appl. Maths Mech. 77, 486–493.CrossRefGoogle Scholar
Liepmann, H., Brown, G. & Nosenchuck, D. 1982 Control of laminar instability waves using a new technique. J. Fluid Mech. 118, 187–200.CrossRefGoogle Scholar
Liepmann, H. & Nosenchuck, D. 1982 Active control of laminar turbulent transition. J. Fluid Mech. 118, 201–204.CrossRefGoogle Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Lipatov, I. I. 2006 Disturbed boundary layer flow with local time-dependent surface heating. Fluid Dyn. 41, 55–65.CrossRefGoogle Scholar
Löfdahl, L. & Gad-el-Hak, M. 1999 MEMS applications in turbulence and flow control. Prog. Aerosp. Sci. 35 (2), 101–203.CrossRefGoogle Scholar
Logue, R. P. 2008 Numerical studies of bifurcation in flows governed by the triple-deck and other equations. PhD thesis, University of Manchester.Google Scholar
Logue, R. P., Gajjar, J. S. B. & Ruban, A. I. 2014 Instability of supersonic compression ramp flow. Phil. Trans. R. Soc. Lond. A 372 (2020), 20130342.Google ScholarPubMed
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18, 241–257.CrossRefGoogle Scholar
Neiland, V. Y. 1969 Theory of laminar boundary layer separation in supersonic flow. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza 4, 53–57.Google Scholar
Neiland, V. Y., Bogolepov, V., Dudin, G. & Lipatov, I. 2007 Asymptotic Theory of Supersonic Viscous Gas Flows. Elsevier, Butterworth-Heinemann.Google Scholar
Prandtl, L. 1904 Über flüssigkeitsbewegung bei sehr kleiner reibung. In Verh. III. Intern. Math. Kongr., Heidelberg, Teubner, Leipzig, 1905, pp. 484–491. English trans. 2001 in Early Developments of Modern Aerodynamics (ed. J. A. K. Ackroyd, B. P. Axcell & A. I. Ruban), p. 77. Butterworth-HeinemannGoogle Scholar
Ruban, A. I. 1984 On Tollmien–Schlichting wave generation by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5, 44–52.Google Scholar
Ruban, A. I., Bernots, T. & Kravtsova, M. A. 2016 Linear and nonlinear receptivity of the boundary layer in transonic flows. J. Fluid Mech. 786, 154–189.CrossRefGoogle Scholar
Schubauer, G. B. & Skramstad, H. K. 1948 Laminar boundary layer oscillations and transition on a flat plate. NACA Tech. Rep. 909.Google Scholar
Seddougui, S., Bowles, R. & Smith, F. 1991 Surface-cooling effects on boundary-layer instability and upstream influence. Eur. J. Mech. (B/Fluids) 10, 117–145.Google Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T. 1982 On the high Reynolds number theory of laminar flows. IMA J. Appl. Maths 28 (3), 207–281.CrossRefGoogle Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14, 145–239.CrossRefGoogle Scholar
Stewartson, K. 1981 D'Alembert's paradox. SIAM Rev. 23 (2), 308–343.CrossRefGoogle Scholar
Stewartson, K. & Williams, P. 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181–206.Google Scholar
Terent'ev, E. D. 1981 The linear problem of a vibrator in a subsonic boundary layer. Prikl. Mat. Mekh. 45, 1049–1055.Google Scholar
Terent'ev, E. D. 1984 Linear problem of a vibrator peforming harmonic oscillations at super-critical frequencies in a subsonic boundary layer. Prikl. Mat. Mekh. 48, 264–272.Google Scholar
Trevin̄o, C. & Lin̄án, A. 1996 The effects of displacement induced by thermal perturbations on the structure and stability of boundary-layer flows. Theor. Comput. Fluid Dyn. 8, 57–72.CrossRefGoogle Scholar
Tutty, O. R. & Cowley, S. J. 1986 On the stability and the numerical solution of the unsteady interactive boundary-layer equation. J. Fluid Mech. 168, 431–456.CrossRefGoogle Scholar
Walker, J. D. A., Fletcher, A. & Ruban, A. I. 2006 Instabilities of a flexible surface in supersonic flow. Q. J. Mech. Appl. Maths 59 (2), 253–276.CrossRefGoogle Scholar
Supplementary material: File

Brennan et al. supplementary material

Brennan et al. supplementary material

Download Brennan et al. supplementary material(File)
File 299.1 KB