Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-28T20:20:15.728Z Has data issue: false hasContentIssue false

Convective instability and transient growth in flow over a backward-facing step

Published online by Cambridge University Press:  30 April 2008

H. M. BLACKBURN
Affiliation:
Department of Mechanical and Aerospace Engineering, Monash University, Victoria 3800, Australia
D. BARKLEY
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV47AL, UK, and Physique et Mécanique des Milieux Hétérogènes, Ecole Supérieure de Physique et Chimie Industrielles de Paris, (PMMH UMR 7636-CNRS-ESPCI-P6-P7), 10 rue Vauquelin, 75231 Paris, France
S. J. SHERWIN
Affiliation:
Department of Aeronautics, Imperial College London, SW72AZ, UK

Abstract

Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Barkley, D. 2005 Confined three-dimensional stability analysis of the cylinder wake. Phys. Rev. E 71, 017301.Google ScholarPubMed
Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.CrossRefGoogle Scholar
Barkley, D., Gomes, M. G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167190.CrossRefGoogle Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids (in press).CrossRefGoogle Scholar
Beaudoin, J. F., Cadot, O., Aider, J.-L. & Wesfreid, J. E. 2004 Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech. B/Fluids 23, 147155.CrossRefGoogle Scholar
Blackburn, H. M. 2002 Three-dimensional instability and state selection in an oscillatory axisymmetric swirling flow. Phys. Fluids 14 (11), 39833996.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 16371650.CrossRefGoogle Scholar
Chomaz, J., Huerre, P. & Redekopp, L. 1990 The effect of nonlinearity and forcing on global modes. In New Trends in Nonlinear Dynamics and Pattern-Forming Phenomena (ed. Coullet, P. & Huerre, P.), pp. 259274. Plenum.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.CrossRefGoogle Scholar
Chun, K. B. & Sung, H. J. 1996 Control of turbulent separated flow over a backward-facing step by local forcing. Exps. Fluids 21, 417426.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12 (1), 120130.CrossRefGoogle Scholar
Cossu, C. & Chomaz, J. M. 1997 Global measures of local convective instabilities. Phys. Rev. Lett. 78, 43874390.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, 209218.CrossRefGoogle Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31 (8), 20932102.CrossRefGoogle Scholar
Furuichi, N. & Kumada, M. 2002 An experimental study of a spanwise structure around a reattachment region of a two-dimensional backward-facing step. Exp. Fluids 32, 179187.CrossRefGoogle Scholar
Gresho, P. M., Gartling, D. K., Torczynski, J. R., Cliffe, K. A., Winters, K. H., Garratt, T. J., Spence, A. & Goodrich, J. W. 1993 Is the steady viscous incompressible two-dimensional flow over a backward-facing step at Re = 800 stable? Intl J. Numer. Meth. Fluids 17, 501541.CrossRefGoogle Scholar
Guermond, J. & Shen, J. 2003 Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41, 112134.CrossRefGoogle Scholar
Henderson, R. D. & Barkley, D. 1996 Secondary instability in the wake of a circular cylinder. Phys. Fluids 8, 16831685.CrossRefGoogle Scholar
Hœpffner, J., Brandt, L. & Henningson, D. S. 2005 Transient growth on boundary layer streaks. J. Fluid Mech. 537, 91100.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in shear layers. J. Fluid Mech. 159, 151168.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.CrossRefGoogle Scholar
Kaiktsis, L., Karniadakis, G. E. & Orszag, S. A. 1991 Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step. J. Fluid Mech. 231, 501528.CrossRefGoogle Scholar
Kaiktsis, L., Karniadakis, G. E. & Orszag, S. A. 1996 Unsteadiness and convective instabilities in two-dimensional flow over a backward-facing step. J. Fluid Mech. 321, 157187.CrossRefGoogle Scholar
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd edn. Oxford University Press.CrossRefGoogle Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97, 414443.CrossRefGoogle Scholar
Le, H., Moin, P. & Kim, J. 1997 Direct numerical simulation of turbulent flow over a backward-facing step. J. Fluid Mech. 330, 349374.CrossRefGoogle Scholar
Lee, I. & Sung, H. 2001 Characteristics of wall pressure fluctuations in separated flows over a backward-facing step. Part I: Time-mean statistics and cross-spectral analyses. Exps. Fluids 30, 262272.CrossRefGoogle Scholar
Lien, F. S. & Leschziner, M. A. 1994 Assessment of turbulence-transport models including non-linear RNG eddy-viscosity formulation and second-moment closure for flow over a backward-facing step. Computers Fluids 23, 9831004.CrossRefGoogle Scholar
Lindzen, R. S. 1988 Instability of plane parallel shear-flow (toward a mechanistic picture of how it works). Pure Appl. Geophys. 126, 103121.CrossRefGoogle Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289309.CrossRefGoogle Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2006 Global optimal perturbations in a separated flow over a backward-rounded-step. In 36th AIAA Fluid Dyn. Conf. and Exhibit. San Francisco, paper. 2006-2879.Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid. Part II: A viscous liquid. Proc. R. Irish Acad. A 27, 9138.Google Scholar
Schatz, M., Barkley, D. & Swinney, H. 1995 Instabilities in spatially periodic channel flow. Phys. Fluids 7, 344358.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sherwin, S. J. & Blackburn, H. M. 2005 Three-dimensional instabilities and transition of steady and pulsatile flows in an axisymmetric stenotic tube. J. Fluid Mech. 533, 297327.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578584.CrossRefGoogle ScholarPubMed
Tuckerman, L. S. & Barkley, D. 2000 Bifurcation analysis for timesteppers. In Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems (ed. Doedel, E. & Tuckerman, L. S.), pp. 453566. Springer.CrossRefGoogle Scholar