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Receptivity of inviscid modes in supersonic boundary layers due to scattering of free-stream sound by localised wall roughness

Published online by Cambridge University Press:  04 June 2020

Ming Dong Open the ORCID record for Ming Dong [Opens in a new window] ,
Yinhui Liu Open the ORCID record for Yinhui Liu [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Ming Dong*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China
Yinhui Liu
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China
Xuesong Wu
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China Department of Mathematics, Imperial College London, 180, Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The present paper investigates the receptivity of inviscid first and second modes in super/hypersonic boundary layers due to the interaction between a weak free-stream acoustic wave and a small isolated surface roughness element. The large-Reynolds-number asymptotic analysis reveals the detailed processes of the excitation. The distortion of the acoustic signature by the curved wall contributes to the leading-order receptivity, producing an eigenmode of $O({\mathcal{E}}h)$ amplitude, where ${\mathcal{E}}\ll 1$ is the magnitude of the acoustic wave and $h\ll 1$ the roughness height normalised by the local boundary-layer thickness $\unicode[STIX]{x1D6FF}$. The interactions between the roughness-induced mean-flow distortion and the acoustic signature contribute to the second-order receptivity, which is of $O({\mathcal{E}}hR^{-1/3})$ with $R\gg 1$ being the Reynolds number based on $\unicode[STIX]{x1D6FF}$. Interestingly, the leading-order receptivity is equivalent to a canonic receptivity problem, the excitation by time-periodic blowing and suction through a local slot on the wall, and the effective periodic outflux velocity forced from the underneath Stokes layer can be determined explicitly in terms of the roughness shape function. This equivalence holds when $h=O(R^{-1/3})$, for which the roughness-induced mean-flow distortion becomes nonlinear. A systematic parametric study is carried out for the excitation of the first and second modes by both fast and slow free-stream acoustic waves, and the dependence of the receptivity efficiency on the relevant parameters is provided. Interestingly, the second-order receptivity could become dominant (e.g. in the case of slow acoustic waves with low frequencies and small incident angles), but the present mathematical theory remains valid. In order to check the accuracy of the asymptotic predictions, we have carried out direct numerical simulations (DNS) and also extended the existing finite-Reynolds-number theory to the supersonic regime. The asymptotic solutions agree with the results given by the finite-Reynolds-number calculations and DNS when $R$ is sufficiently large (typically $R=O(10^{5})$). An improved large-Reynolds-number approach is developed by replacing the non-penetration boundary condition by an unsteady outflux, which accounts for the $O(R^{-1/2})$ viscous correction by the Stokes layer. With this modification, the accuracy of the receptivity calculation at moderate Reynolds numbers (approximately a few thousands) is improved remarkably.

JFM classification

Boundary Layers: Boundary layer receptivity Compressible Flows: Compressible boundary layers Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 896 , 10 August 2020 , A23
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards.Google Scholar
Balakumar, P. & Kegerise, M. A. 2015 Receptivity of hypersonic boundary layers over straight and flared cones. AIAA J. 53, 2097–2109.CrossRefGoogle Scholar
Balakumar, P. & Malik, M. R. 1992 Discrete modes and continuous spectra in supersonic boundary layers. J. Fluid Mech. 239, 631–656.CrossRefGoogle Scholar
Blackaby, N., Cowley, S. & Hall, P. 1993 On the instability of hypersonic flow past a flat plate. J. Fluid Mech. 247, 369–416.CrossRefGoogle Scholar
Bodonyi, R. J. & Duck, P. W. 1992 Boundary-layer receptivity due to a wall suction and control of Tollmien–Schlichting waves. Phys. Fluids A 4, 1206–1214.CrossRefGoogle Scholar
Choudhari, M. 1993 Boundary-layer receptivity due to distributed surface imperfections of a deterministic or random nature. Theor. Comput. Fluid Dyn. 4, 101–117.CrossRefGoogle Scholar
Choudhari, M. & Duck, P. W.1996 Nonlinear excitation of inviscid stationary vortex in a boundary-layer flow. NASA Report 96-27.CrossRefGoogle Scholar
Choudhari, M. & Streett, C.1990 Boundary layer receptivity phenomena in three-dimensional and high-speed boundary layers. AIAA Paper 1990-5258.CrossRefGoogle Scholar
Choudhari, M. & Streett, C. L. 1992 A finite Reynolds-number approach for the prediction of boundary-layer receptivity in localized regions. Phys. Fluids A 4, 2495–2514.CrossRefGoogle Scholar
Cowley, S. & Hall, P. 1990 On the instability of hypersonic flow past a wedge. J. Fluid Mech. 214, 17–42.CrossRefGoogle Scholar
Crouch, J. D. 1992 Localized receptivity of boundary layers. Phys. Fluids A 4, 1408.CrossRefGoogle Scholar
Dietz, A. J. 1999 Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech. 378, 291–317.CrossRefGoogle Scholar
Dong, M. & Wu, X. 2013 On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. J. Fluid Mech. 732, 616–659.CrossRefGoogle Scholar
Dong, M. & Zhou, H. 2013 A simulation on bypass transition and its key mechanism. Sci. China Phys. Mech. Astron. 56, 775–784.CrossRefGoogle Scholar
Duck, P. W., Ruban, A. I. & Zhikarev, C. N. 1996 Generation of Tollmien–Schlichting waves by free-stream turbulence. J. Fluid Mech. 312, 341–371.CrossRefGoogle Scholar
Fedorov, A. V. 2003a Receptivity of a high-speed boundary layer to acoustic disturbances. J. Fluid Mech. 491, 101–129.CrossRefGoogle Scholar
Fedorov, A. V.2003b Receptivity of hypersonic boundary layer to acoustic disturbances scattered by surface roughness. AIAA Paper 2003-3731.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. V. & Khokhlov, A. P. 1991 Excitation of unstable modes in a supersonic boundary layer by acoustic waves. Fluid Dyn. 9, 456–467.Google Scholar
Fedorov, A. V. & Khokhlov, A. P. 2001 Prehistory of instability in a hypersonic boundary layer. Theor. Comput. Fluid Dyn. 14, 359–375.CrossRefGoogle Scholar
Fedorov, A. V. & Khokhlov, A. P. 2002 Receptivity of hypersonic boundary layer to wall disturbances. Theor. Comput. Fluid Dyn. 15, 231–254.CrossRefGoogle Scholar
Fong, K., Wang, X. & Zhong, X. 2014 Numerical simulation of roughness effect on the stability of a hypersonic boundary layer. Comput. Fluids 96, 350–367.CrossRefGoogle Scholar
Fong, K., Wang, X. & Zhong, X.2015 Parametric study on stabilization of hypersonic boundray-layer waves using 2-D surface roughness. AIAA Paper 2015-0837.CrossRefGoogle Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.CrossRefGoogle Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Schlichting 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, 509–530.CrossRefGoogle Scholar
Goldstein, M. E. & Ricco, P. 2018 Non-localized boundary layer instabilities resulting from leading edge receptivity at moderate supersonic Mach numbers. J. Fluid Mech. 838, 435–477.CrossRefGoogle Scholar
Guschin, V. A. & Fedorov, A. V. 1990 Excitation and development of unstable disturbances in supersonic boundary layer. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 3, 21–29.Google Scholar
Hall, P. & Smith, F. 1982 A suggested mechanism for nonlinear wall roughness effects on high Reynolds number flow stability. Stud. Appl. Maths 66, 241–265.CrossRefGoogle Scholar
Hernández, C. G. & Wu, X. 2019 Receptivity of supersonic boundary layers over smooth and wavy surfaces to impinging slow acoustic waves. J. Fluid Mech. 872, 849–888.CrossRefGoogle Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411–482.CrossRefGoogle Scholar
Liu, Y., Dong, M. & Wu, X. 2020 Generation of first mack modes in supersonic boundary layers by slow acoustic waves interacting with streamwise isolated wall roughness. J. Fluid Mech. 888, A10.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2003a Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions. J. Fluid Mech. 488, 31–78.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2003b Receptivity of a supersonic boundary layer over a flat plate. Part 2. Receptivity to free-stream sound. J. Fluid Mech. 488, 79–121.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2005 Receptivity of a supersonic boundary layer over a flat plate. Part 3. Effects of different types of free-stream disturbances. J. Fluid Mech. 532, 63–109.CrossRefGoogle Scholar
Mack, L. 1987 Review of linear compressible stability theory. In Stability of Time Dependent and Spatially Varying Flows (ed. Dwoyer, D. L. & Hussaini, M. Y.), pp. 164–187. Springer.CrossRefGoogle Scholar
Malik, M.1990 Finite difference solution of the compressible stability eigenvalue problem. NASA Tech. Rep. 16572 86, 376–413.Google Scholar
Maslov, A. A., Shiplyuk, A. N., Sidorenko, A. A. & Arnal, D. 2001 Leading-edge receptivity of a hypersonic boundary layer on a flat plate. J. Fluid Mech. 426, 73–94.CrossRefGoogle Scholar
Messiter, A. F. 1970 Boundary layer flow near the trailing edge on a flat plate. SIAM J. Appl. Maths 18, 241–257.CrossRefGoogle Scholar
Morkovin, M. V.1969 Critical evaluation of transition from laminar to turbulent shear layers with emphasis on hypersonically traveling bodies. Tech. Rep. AFFDL-TR, 68-149. US Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, Ohio.Google Scholar
Qin, F. & Wu, X. 2016 Response and receptivity of the hypersonic boundary layer past a wedge to free-stream acoustic, vortical and entropy disturbances. J. Fluid Mech. 797, 874–915.CrossRefGoogle Scholar
Qin, H. & Dong, M. 2016 Boundary-layer disturbances subjected to free-stream turbulence and simulation on bypass transition. Z. Angew. Math. Mech. 37, 967–986.Google Scholar
Ruban, A. I. 1984 On Tollimien–Schlichting wave generation by sound (in Russian). Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5, 44–52 (translation in Fluid Dyn. 19, 709–716).Google Scholar
Saric, W. S., Hoos, J. A. & Radeztsky, R. H. 1991 Boundary-layer receptivity of sound with roughness. In Boundary Layer Stability and Transition to Turbulence, vol. 114, pp. 17–22. ASME.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Smith, F. T. 1973 Laminar flow over a small hump on flat plate. J. Fluid Mech. 57, 803–824.CrossRefGoogle Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. 366, 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
Smith, F. T. & Brown, S. N. 1990 The inviscid instability of a Blasius boundary layer at large values of the Mach number. J. Fluid Mech. 219, 499–518.CrossRefGoogle Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate. Mathematika 16, 106–121.CrossRefGoogle Scholar
Terent’ev, E. D. 1984 The linear problem of a vibrator performing harmonic oscillations at supercritical frequencies in a subsonic boundary layer. PMM USSR 48, 184–191.Google Scholar
Tumin, A. 2007 Three-dimensional spatial normal modes in compressible boundary layers. J. Fluid Mech. 586, 295–322.CrossRefGoogle Scholar
Tumin, A. M. & Fedorov, A. V. 1984 Instability wave excitation by a localized vibrator in the boundary layer. J. Appl. Mech. Tech. Phys. 25, 867–873.CrossRefGoogle Scholar
Wu, X. 1999 Generation of Tollmien–Schlichting waves by convecting gusts interacting with sound. J. Fluid Mech. 397, 285–316.CrossRefGoogle Scholar
Wu, X. 2001a On local boundary-layer receptivity to vortical disturbances in the free stream. J. Fluid Mech. 449, 373–393.CrossRefGoogle Scholar
Wu, X. 2001b Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances: a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91–133.CrossRefGoogle Scholar
Wu, X. & Dong, M. 2016a Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers. J. Fluid Mech. 797, 683–782.CrossRefGoogle Scholar
Wu, X. & Dong, M. 2016b 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
Zhong, X. & Ma, Y. 2006 Boundary-layer receptivity of Mach 7.99 flow over a blunt cone to free-stream acoustic waves. J. Fluid Mech. 556, 55–103.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