Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-06-11T20:50:16.205Z Has data issue: false hasContentIssue false
  • English
  • Français

The interaction of Blasius boundary-layer flow with a compliant panel: global, local and transient analyses

Published online by Cambridge University Press:  22 August 2017

Konstantinos Tsigklifis Open the ORCID record for Konstantinos Tsigklifis [Opens in a new window]  and
Anthony D. Lucey
Konstantinos Tsigklifis*
Affiliation:
Fluid Dynamics Research Group, Department of Mechanical Engineering, Curtin University, Western Australia 6845, Australia
Anthony D. Lucey
Affiliation:
Fluid Dynamics Research Group, Department of Mechanical Engineering, Curtin University, Western Australia 6845, Australia
*
†Email address for correspondence: k.tsigklifis@curtin.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the fluid–structure interaction (FSI) of a compliant panel with developing Blasius boundary-layer flow. The linearised Navier–Stokes equations in velocity–vorticity form are solved using a Helmholtz decomposition coupled with the dynamics of a plate-spring compliant panel couched in finite-difference form. The FSI system is written as an eigenvalue problem and the various flow- and wall-based instabilities are analysed. It is shown that global temporal instability can occur through the interaction of travelling wave flutter (TWF) with a structural mode or as a resonance between Tollmien–Schlichting wave (TSW) instability and discrete structural modes of the compliant panel. The former is independent of compliant panel length and upstream inflow disturbances while the specific behaviour arising from the latter phenomenon is dependent upon the frequency of a disturbance introduced upstream of the compliant panel. The inclusion of axial displacements in the wall model does not lead to any further global instabilities. The dependence of instability-onset Reynolds numbers with structural stiffness and damping for the global modes is quantified. It is also shown that the TWF-based global instability is stabilised as the boundary layer progresses downstream while the TSW-based global instability exhibits discrete resonance-type behaviour as Reynolds number increases. At sufficiently high Reynolds numbers, a globally unstable divergence instability is identified when the wavelength of its wall-based mode is longer than that of the least stable TSW mode. Finally, a non-modal analysis reveals a high level of transient growth when the flow interacts with a compliant panel which has structural properties capable of reducing TSW growth but which is prone to global instability through wall-based modes.

JFM classification

Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 827 , 25 September 2017 , pp. 155 - 193
Check for updates
Copyright
© 2017 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.)

References

Alder, M. 2015 Development and validation of a fluid–structure solver for transonic panel flutter. AIAA J. 53 (12), 3509–3521.CrossRefGoogle Scholar
Alder, M. 2016 Nonlinear dynamics of prestressed panels in low supersonic turbulent flow. AIAA J. 54 (11), 3632–3646.CrossRefGoogle Scholar
Ashpis, D. E. & Reshotko, E. 1990 The vibrating ribbon problem revisited. J. Fluid Mech. 213, 531–547.CrossRefGoogle Scholar
Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2007 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. (B/Fluids) 27 (5), 501–513.CrossRefGoogle Scholar
Baltensperger, R. & Trummer, M. R. 2002 Spectral differencing with a twist. SIAM J. Sci. Comput. 24, 1465–1487.CrossRefGoogle Scholar
Benjamin, T. B. 1963 The three-fold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech. 16, 436–450.CrossRefGoogle Scholar
Bridges, T. J. & Morris, P. J. 1984 Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55 (3), 437–460.CrossRefGoogle Scholar
Bushnell, D. M. 1977 Effects of compliant wall motion on turbulent boundary layers. Phys. Fluids 20, S31–S48.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Methods in Fluid Dynamics. Springer.Google Scholar
Carpenter, P. W. 1990 Status of transition delay using compliant walls. In Viscous Drag Reduction in Boundary Layers (ed. Bushnell, D. M. & Heffner, J. N.), Progress in Astronautics and Aeronautics, vol. 123, pp. 79–113.Google Scholar
Carpenter, P. W.1991 The optimization of multiple-panel compliant walls for delay of laminar-turbulent transition. 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Fluid Dynamics and Co-located Conferences, Paper No. AlAA 91-1772.Google Scholar
Carpenter, P. W. 1993 Optimization of multiple-panel compliant walls for delay of laminar-turbulent transition. AIAA J. 31 (7), 1187–1188.CrossRefGoogle Scholar
Carpenter, P. W., Davies, C. & Lucey, A. D. 2001 Does the dolphin have a secret? Curr. Sci. 79, 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 (6), 349–378.CrossRefGoogle Scholar
Carpenter, P. W. & Garrad, A. D. 1985 The hydrodynamics 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 hydrodynamics 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. & Morris, P. J. 1990 The effects of anisotropic wall compliance on boundary-layer stability and transition. J. Fluid Mech. 218, 171–223.CrossRefGoogle Scholar
Choi, K. S., Yang, X., Clayton, B. R., Glover, E. J., Atlar, M., Semenov, B. N. & Kulik, V. M. 1997 Turbulent drag reduction using compliant surfaces. Proc. R. Soc. Lond. A 453 (1965), 2229–2240.CrossRefGoogle Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Coppola, G. & de Luca, L. 2010 Non-modal dynamics before flow-induced instability in fluid–structure interactions. J. Sound Vib. 329 (7), 848–865.CrossRefGoogle Scholar
Cossu, C. & Chomaz, J. 1997 Global measures of local convective instability. Phys. Rev. Lett. 77, 4387–4390.CrossRefGoogle Scholar
Crighton, D. G. & Oswell, J. E. 1991 Fluid loading with mean flow. I. Response of an elastic plate to localized excitation. Phil. Trans. R. Soc. Lond. A 335, 557–592.Google Scholar
Danabasoglu, G. & Biringen, S. 1990 A chebyshev matrix method for the spatial modes of the Orr–Sommerfeld equation. Intl J. Numer. Meth. Fluids 11 (7), 1033–1037.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997a Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 1997b Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels. J. Fluid Mech. 335, 361–392.CrossRefGoogle Scholar
Davies, C. & Carpenter, P. W. 2001 A novel velocity–vorticity formulation of the Navier–Stokes equations with applications to boundary layer disturbance evolution. J. Comput. Phys. 172, 119–165.CrossRefGoogle Scholar
Dixon, A. E., Lucey, A. D. & Carpenter, P. W. 1994 The optimization of viscoelastic walls for transition delay. AIAA J. 32, 256–267.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
Dowell, E. H. 1973 Aerodynamic boundary layer effects on flutter and damping of plates. J. Aircraft 10 (12), 734–738.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
Fukagata, K., Kern, S., Chatelain, P., Koumoutsakos, P. & Kasagi, N. 2008 Evolutionary optimization of an anisotropic compliant surface for turbulent friction drag reduction. J. Turbul. 9 (35), 1–17.Google Scholar
Gad-el-Hak, M. 1998 Compliant coatings: the simpler alternative. Exp. Therm. Fluid Sci. 16, 141–156.CrossRefGoogle Scholar
Garrad, A. D. & Carpenter, P. W. 1982 A theoretical investigation of flow-induced instabilities in compliant coatings. J. Sound Vib. 85 (4), 483–500.CrossRefGoogle Scholar
Gaster, M. 1988 Is the dolphin a red herring? In Turbulence Management and Relaminarisation (ed. Liepmann, H. W. & Narasimha, R.), pp. 285–304. Springer.CrossRefGoogle Scholar
Hashimoto, A., Aoyama, T. & Nakamura, Y. 2009 Effects of turbulent boundary layer on panel flutter. AIAA J. 47 (12), 2785–2791.CrossRefGoogle Scholar
Houghton, E. L. & Carpenter, P. W. 2003 Aerodynamics for Engineering Students, 5th edn. Butterworth-Heinemann.Google Scholar
Huerre, P. & Monkewitz, P. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.CrossRefGoogle Scholar
Joslin, R. D. & Morris, P. J. 1992 Effect of compliant walls on secondary instabilities in boundary transition. AIAA J. 30, 332–339.CrossRefGoogle Scholar
Joslin, R. D., Morris, P. J. & Carpenter, P. W. 1991 The role of three-dimensional instabilities in compliant wall boundary-layer transition. AIAA J. 29, 1603–1610.CrossRefGoogle Scholar
Katz, J. & Plotkin, A. 1991 Low Speed Aerodynamics: From Wing Theory to Panel Methods. McGraw-Hill.Google Scholar
Kempka, S. N., Strickland, J. H., Glass, M. W., Peery, J. S. & Ingber, M. S. 1995 Velocity boundary conditions for vorticity formulations of the incompressible Navier–Stokes equations. In Forum on Vortex Methods for Engineering Applications, Sponsored by Sandia National Labs. Sandia National Laboratories.Google Scholar
Kim, E. & Choi, H. 2014 Space-time characteristics of a compliant wall in a turbulent channel flow. J. Fluid Mech. 756, 30–53.CrossRefGoogle Scholar
Kramer, M. O. 1957 Boundary layer stabilization by distributed damping. J. Aero. Sci. 24, 459–460.Google Scholar
Kramer, M. O. 1960 Boundary layer stabilization by distributed damping. J. Am. Soc. Nav. Eng. 72, 25–34.Google Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.CrossRefGoogle Scholar
Lucey, A. 1998 The excitation of waves on a flexible panel in a uniform flow. Phil. Trans. R. Soc. Lond. A 356, 2999–3039.CrossRefGoogle Scholar
Lucey, A. & Carpenter, P. 1995 Boundary layer instability over compliant walls: comparison between theory and experiment. Phys. Fluids 7 (10), 2355–2363.CrossRefGoogle Scholar
Lucey, A. & Peake, N. 2003 Wave excitation on flexible walls in the presence of a fluid flow. In Flow Past Highly Compliant Boundaries and in Collapsible Tubes (ed. Carpenter, P. W. & Pedley, T. J.), vol. 72, chap. 6, pp. 119–145. Springer.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 On the difference between the hydroelastic instability of infinite and very long compliant panels. J. Sound Vib. 163 (1), 176–181.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
Moler, C. B. & Stewart, G. W. 1973 An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10 (2), 241–256.CrossRefGoogle Scholar
Orr, W. M. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part ii: a viscous liquid. Proc. R. Irish Acad. A 27, 69–138.Google Scholar
Pavlov, V. V. 2006 Dolphin skin as a natural anisotropic compliant wall. Bioinspir. Biomim. 1, 31–40.CrossRefGoogle ScholarPubMed
Peake, N. 2004 On the unsteady motion of a long fluid-loaded elastic plate with mean flow. J. Fluid Mech. 507, 335–366.CrossRefGoogle Scholar
Pitman, M. W. & Lucey, A. D. 2009 On the direct determination of the eigenmodes of finite flow–structure systems. Proc. R. Soc. Lond. A 465, 257–281.Google Scholar
Pitman, M. W. & Lucey, A. D. 2010 Stability of plane-Poiseuille flow interacting with a finite compliant panel. In 17th Australasian Fluid Mechanics Conference. University of Auckland.Google Scholar
Rempfer, D., Blossey, P., Parsons, L. & Lumley, J. 2001 Low-dimensional dynamical model of a turbulent boundary layer over a compliant surface: preliminary results. In Fluid Mechanics and the Environment: Dynamical Approachess, vol. 566, pp. 267–283. Springer.CrossRefGoogle Scholar
Schlichting, H. 1979 Boundary Layer Theory, 7th edn. McGraw-Hill.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.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
Shankar, V. & Kumaran, V. 2002 Stability of wall modes in fluid flow past a flexible surface. Phys. Fluids 14 (7), 2324–2338.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, 541–557.CrossRefGoogle Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.CrossRefGoogle Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.CrossRefGoogle Scholar
Tsigklifis, K. & Lucey, A. D. 2015 Global instabilities and transient growth in Blasius boundary-layer flow over a compliant panel. Sadhana 40, 945–960.CrossRefGoogle Scholar
Wiplier, O. & Ehrenstein, U. 2000 Numerical simulation of linear and nonlinear disturbance evolution in a boundary layer with compliant walls. J. Fluids Struct. 14, 157–182.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, 127–144.CrossRefGoogle Scholar
Wu, J. C. & Thompson, J. F. 1973 Numerical solutions of time-dependent incompressible Navier–Stokes equations using an integro-differential formulation. Comput. Fluids 1, 197–215.CrossRefGoogle Scholar
Xu, S., Rempfer, D. & Lumley, J. 2003 Turbulence over a compliant surface: numerical simulation and analysis. J. Fluid Mech. 478, 11–34.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. 1990 The hydrodynamic stability of boundary-layer flow over a class of anisotropic compliant walls. J. Fluid Mech. 220, 125–160.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. & Dowling, A. P. 1987 The stability of inviscid flow over passive compliant walls. J. Fluid Mech. 183, 265–292.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, 237–252.CrossRefGoogle Scholar
Zengl, M. & Rist, U. 2012 Linear-stability investigations for flow-control experiments related to flow over compliant walls. In Nature-Inspired Fluid Mechanics (ed. Tropea, C. & Bleckmann, H.), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 119, pp. 223–237. Springer.CrossRefGoogle Scholar