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Global stability and resolvent analyses of laminar boundary-layer flow interacting with viscoelastic patches

Published online by Cambridge University Press:  22 February 2022

J.-L. Pfister Open the ORCID record for J.-L. Pfister [Opens in a new window] ,
N. Fabbiane Open the ORCID record for N. Fabbiane [Opens in a new window]  and
O. Marquet Open the ORCID record for O. Marquet [Opens in a new window]
J.-L. Pfister*
Affiliation:
DAAA, ONERA, Paris Saclay University, F-92190 Meudon, France
N. Fabbiane
Affiliation:
DAAA, ONERA, Paris Saclay University, F-92190 Meudon, France
O. Marquet
Affiliation:
DAAA, ONERA, Paris Saclay University, F-92190 Meudon, France
*
†Email address for correspondence: [email protected]
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Abstract

The attenuation of two-dimensional boundary-layer instabilities by a finite-length, viscoelastic patch is investigated by means of global linear stability theory. First, the modal stability properties of the coupled problem are assessed, revealing unstable fluid-elastic travelling-wave flutter modes. Second, the Tollmien–Schlichting instabilities over a rigid wall are characterised via the analysis of the fluid resolvent operator in order to determine a baseline for the fluid-structural analysis. To investigate the effect of the elastic patch on the growth of these flow instabilities, we first consider the linear frequency response of the coupled fluid-elastic system to the dominant rigid-wall forcing modes. In the frequency range of Tollmien–Schlichting waves, the energetic flow amplification is clearly reduced. However, an amplification is observed for higher frequencies, associated with travelling-wave flutter. This increased complexity requires the analysis of the coupled fluid-structural resolvent operator; the optimal, coupled, resolvent modes confirm the attenuation of the Tollmien–Schlichting instabilities, while also being able to capture the amplification at the higher frequencies. Finally, a decomposition of the fluid-structural response is proposed to reveal the wave cancellation mechanism responsible for the attenuation of the Tollmien–Schlichting waves. The viscoelastic patch, excited by the incoming rigid-wall wave, provokes a fluid-elastic wave that is out-of-phase with the former, thus reducing its amplitude.

JFM classification

Aerodynamics: Flow-structure interactions Boundary Layers: Boundary layer stability Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A1
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D.S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids 27 (5), 501–513.CrossRefGoogle Scholar
Aleyev, Y.G. 1977 Nekton. Junk, The Hague.CrossRefGoogle Scholar
Amestoy, P., Buttari, A., Guermouche, A., L'Excellent, J.-Y. & Ucar, B. 2013 MUMPS: a multifrontal massively parallel sparse direct solver.Google Scholar
Benjamin, T.B. 1960 Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9 (4), 513–532.CrossRefGoogle Scholar
Benjamin, T.B. 1963 The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16 (3), 436–450.CrossRefGoogle Scholar
Brandt, L., Sipp, D., Pralits, J.O. & Marquet, O. 2011 Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.CrossRefGoogle Scholar
Carpenter, P.W., Davies, C. & Lucey, A.D. 2000 Hydrodynamics and compliant walls: does the dolphin have a secret? Curr. Sci. 79 (6), 758–765.Google Scholar
Carpenter, P.W. & Gajjar, J.S.B. 1990 A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theor. Comput. Fluid Dyn. 1, 349–378.CrossRefGoogle Scholar
Carpenter, P.W. & Garrad, A.D. 1985 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien–Schlichting instabilities. J. Fluid Mech. 155, 465–510.CrossRefGoogle Scholar
Carpenter, P.W. & Garrad, A.D. 1986 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199–232.CrossRefGoogle Scholar
Carpenter, P.W., Lucey, A.D. & Davies, C. 2001 Progress on the use of compliant walls for laminar-flow control. J. Aircr. 38 (3), 504–512.CrossRefGoogle Scholar
Christensen, R. 2012 Theory of Viscoelasticity: An Introduction. Elsevier.Google Scholar
Davies, C. & Carpenter, P.W. 1997 Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels. J. Fluid Mech. 335, 361–392.CrossRefGoogle Scholar
Donea, J., Huerta, A., Ponthot, J.-P. & Rodrigez-Ferran, A. 2004 Arbitrary Lagrangian-Eulerian methods. In Encyclopedia of Computational Mechanics.CrossRefGoogle Scholar
Dowell, E.H. 1971 Generalized aerodynamic forces on a flexible plate undergoing transient motion in a shear flow with an application to panel flutter. AIAA J. 9 (5), 834–841.CrossRefGoogle Scholar
Duncan, J.H. 1988 The dynamics of waves at the interface between a two-layer viscoelastic coating and a fluid flow. J. Fluids Struct. 2 (1), 35–51.CrossRefGoogle Scholar
Duncan, J.H., Waxman, A.M. & Tulin, M.P. 1985 The dynamics of waves at the interface between a viscoelastic coating and a fluid flow. J. Fluid Mech. 158, 177–197.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.CrossRefGoogle Scholar
Gad-El-Hak, M. 1996 Compliant coatings: a decade of progress. Appl. Mech. Rev. 49 (10S), S147–S157.CrossRefGoogle Scholar
Gad-El-Hak, M., Blackwelder, R.F. & Riley, J.J. 1984 On the interaction of compliant coatings with boundary-layer flows. J. Fluid Mech. 140, 257–280.CrossRefGoogle Scholar
Gaster, M. 1988 Is the dolphin a red herring? In Turbulence Management and Relaminarisation (ed. H.W. Liepmann & R. Narasimha), pp. 285–304. International Union of Theoretical and Applied Mechanics. Springer.CrossRefGoogle Scholar
Gray, J. 1936 Studies in animal locomotion: VI. The propulsive powers of the dolphin. J. Expl Biol. 13 (2), 192–199.CrossRefGoogle Scholar
Hecht, F. 2012 New development in FreeFem++. J. Numer. Maths 20 (3–4), 251–265.Google Scholar
Hughes, T.J.R, Liu, W.K. & Zimmermann, T.K 1981 Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Meth. Appl. Mech. Engng 29, 329–349.CrossRefGoogle Scholar
Kachanov, Y.S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26 (1), 411–482.CrossRefGoogle Scholar
Kramer, M.O. 1961 The dolphin's secret. Nav. Engrs J. 73 (1), 103–108.Google Scholar
Landahl, M.T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13 (4), 609–632.CrossRefGoogle Scholar
Lehoucq, R.B., Sorensen, D.C. & Yang, C. 1997 ARPACK users’ guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods.CrossRefGoogle Scholar
Lucey, A.D. & Carpenter, P.W. 1992 A numerical simulation of the interaction of a compliant wall and inviscid flow. J. Fluid Mech. 234, 121–146.CrossRefGoogle Scholar
Lucey, A.D. & Carpenter, P.W. 1993 The hydroelastic stability of three-dimensional disturbances of a finite compliant wall. J. Sound Vib. 165 (3), 527–552.CrossRefGoogle Scholar
Lucey, A.D. & Carpenter, P.W. 1995 Boundary layer instability over compliant walls: comparison between theory and experiment. Phys. Fluids 7 (10), 2355–2363.CrossRefGoogle Scholar
Luhar, M., Sharma, A.S. & McKeon, B.J. 2015 A framework for studying the effect of compliant surfaces on wall turbulence. J. Fluid Mech. 768, 415–441.CrossRefGoogle Scholar
Luhar, M., Sharma, A.S. & McKeon, B.J. 2016 On the design of optimal compliant walls for turbulence control. J. Turbul. 17 (8), 787–806.CrossRefGoogle Scholar
Nakajima, N. & Harrell, E.R. 2001 Rheology of PVC plastisol: formation of immobilized layer in pseudoplastic flow. J. Colloid Interface Sci. 238 (1), 116–124.CrossRefGoogle ScholarPubMed
Nakajima, N., Isner, J.D. & Harrell, E.R. 1981 Gelation and fusion mechanism of PVC plastisols observed by changes of morphology, viscoelastic properties, and ultimate mechanical properties. J. Macromol. Sci. B 20 (3), 349–364.CrossRefGoogle Scholar
Pfister, J.-L. 2019 Instabilities and optimization of elastic structures interacting with laminar flows. PhD thesis, Université Paris-Saclay.Google Scholar
Pfister, J.-L., Marquet, O. & Carini, M. 2019 Linear stability analysis of strongly coupled fluid–structure problems with the Arbitrary Lagrangian–Eulerian method. Comput. Meth. Appl. Mech. Engng 355, 663–689.CrossRefGoogle Scholar
Rayleigh, Lord 1885 On waves propagated along the plane surface of an elastic solid. Proc. Lond. Math. Soc. 1 (1), 4–11.CrossRefGoogle Scholar
Schlichting, H. 1933 Zur enststehung der turbulenz bei der platenstromung. Nachr. Ges. Wiss. Gottingen 182, 181–208.Google Scholar
Schlichting, H. 1979 Boundary-Layer Theory. McGraw-Hill.Google Scholar
Schmid, P.J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2012 Stability and Transition in Shear Flows, vol. 142. Springer Science & Business Media.Google Scholar
Schubauer, G.B. & Skramstad, H.K. 1947 Laminar boundary-layer oscillations and stability of laminar flow. J. Aeronaut. Sci. 14 (2), 69–78.CrossRefGoogle Scholar
Sen, P.K. & Arora, D.S. 1988 On the stability of laminar boundary-layer flow over a flat plate with a compliant surface. J. Fluid Mech. 197, 201–240.CrossRefGoogle Scholar
Sipp, D. & Marquet, O. 2013 a Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.CrossRefGoogle Scholar
Sipp, D. & Marquet, O. 2013 b Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.CrossRefGoogle Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.CrossRefGoogle Scholar
Stewart, P.S., Waters, S.L. & Jensen, O.E. 2009 Local and global instabilities of flow in a flexible-walled channel. Eur. J. Mech. B/Fluids 28 (4), 541–557.CrossRefGoogle Scholar
Tollmien, W. 1929 Über die entstehung der turbulenz. Nachr. Ges. Wiss. Göttingen 609, 21–24.Google Scholar
Trefethen, L.N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Tsigklifis, K. & Lucey, A.D. 2017 The interaction of Blasius boundary-layer flow with a compliant panel: global, local and transient analyses. J. Fluid Mech. 827, 155–193.CrossRefGoogle Scholar
Wiplier, O. & Ehrenstein, U. 2001 On the absolute instability in a boundary-layer flow with compliant coatings. Eur. J. Mech. B/Fluids 20 (1), 127–144.CrossRefGoogle Scholar
Yeo, K.S. 1988 The stability of boundary-layer flow over single-and multi-layer viscoelastic walls. J. Fluid Mech. 196, 359–408.CrossRefGoogle Scholar
Yeo, K.S. 1992 The three-dimensional stability of boundary-layer flow over compliant walls. J. Fluid Mech. 238, 537–577.CrossRefGoogle Scholar
Yeo, K.S., Khoo, B.C. & Chong, W.K. 1994 The linear stability of boundary-layer flow over compliant walls: effects of boundary-layer growth. J. Fluid Mech. 280, 199–225.CrossRefGoogle Scholar
Yeo, K.S., Khoo, B.C. & Zhao, H.Z. 1996 The absolute instability of boundary-layer flow over viscoelastic walls. Theor. Comput. Fluid Dyn. 8 (4), 237–252.CrossRefGoogle Scholar