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Effect of compressibility on the global stability of axisymmetric wake flows

Published online by Cambridge University Press:  19 August 2010

P. MELIGA ,
D. SIPP  and
J.-M. CHOMAZ
P. MELIGA*
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
D. SIPP
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
J.-M. CHOMAZ
Affiliation:
LadHyX, CNRS-Ecole Polytechnique, 91128 Palaiseau, France
*
†Email address for correspondence: [email protected]
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Abstract

We study the linear dynamics of global eigenmodes in compressible axisymmetric wake flows, up to the high subsonic regime. We consider both an afterbody flow at zero angle of attack and a sphere, and find that the sequence of bifurcations destabilizing the axisymmetric steady flow is independent of the Mach number and reminiscent of that documented in the incompressible wake past a sphere and a disk (Natarajan & Acrivos, J. Fluid Mech., vol. 254, 1993, p. 323), hence suggesting that the onset of unsteadiness in this class of flows results from a global instability. We determine the boundary separating the stable and unstable domains in the (M, Re) plane, and show that an increase in the Mach number yields a stabilization of the afterbody flow, but a destabilization of the sphere flow. These compressible effects are further investigated by means of adjoint-based sensitivity analyses relying on the computation of gradients or sensitivity functions. Using this theoretical formalism, we show that they do not act through specific compressibility effects at the disturbance level but mainly through implicit base flow modifications, an effect that had not been taken into consideration by previous studies based on prescribed parallel base flow profiles. We propose a physical interpretation for the observed compressible effects, based on the competition between advection and production of disturbances, and provide evidence linking the stabilizing/destabilizing effect observed when varying the Mach number to a strengthening/weakening of the disturbance advection mechanism. We show, in particular, that the destabilizing effect of compressibility observed in the case of the sphere results from a significant increase of the backflow velocity in the whole recirculating bubble, which opposes the downstream advection of disturbances.

JFM classification

Compressible Flows: Compressible Flows Instability: Instability Wakes/Jets: Wakes/Jets
Type
Papers
Information
Journal of Fluid Mechanics , Volume 660 , 10 October 2010 , pp. 499 - 526
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Achenbach, E. 1972 Experiments on the flow past spheres at very high Reynolds numbers. J. Fluid Mech. 54, 565–575.CrossRefGoogle Scholar
Achenbach, E. 1974 Vortex shedding from spheres. J. Fluid Mech. 62, 209–221.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, 167–190.CrossRefGoogle Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.CrossRefGoogle Scholar
Bouhadji, A. & Braza, M. 2003 Physical analysis by numerical simulation of organised modes and shock-vortex interaction in transonic flows around an aerofoil. Part 1. Mach number effect. J. Comput. Fluids 32, 1233–1260.CrossRefGoogle Scholar
Brès, G. A. & Colonius, T. 2008 Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309–339.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
Colonius, T. 2004 Modeling artificial boundary conditions for compressible flow. Annu. Rev. Fluid Mech. 36, 315–345.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to streamwise pressure gradient. Phys. Fluids 12, 120–130.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2001 Optimal control of nonmodal disturbances in boundary layers. Theor. Comput. Fluid Dyn. 15, 65–81.CrossRefGoogle Scholar
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comput. Phys. 224, 924–940.CrossRefGoogle Scholar
Crouch, J. D., Garbaruk, A., Magidov, D. & Travin, A. 2009 Origin of transonic buffet on aerofoils. J. Fluid Mech. 628, 357–369.CrossRefGoogle Scholar
Davis, T. A. 2004 A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 (2), 165–195.CrossRefGoogle Scholar
Davis, T. A. & Duff, I. S. 1997 An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Appl. 18 (1), 140–158.CrossRefGoogle Scholar
Deprés, D., Reijasse, P. & Dussauge, J.-P. 2004 Analysis of unsteadiness in afterbody transonic flows. AIAA J. 42 (12), 2541–2550.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
Fabre, D., Auguste, F. & Magnaudet, J. 2008 Bifurcations and symmetry breaking in the wake of axisymmetric bodies. Phys. Fluids 20 (5), 051702.CrossRefGoogle Scholar
Fuchs, H. V., Mercker, E. & Michel, U. 1979 Large-scale coherent structures in the wake of axisymmetric bodies. J. Fluid Mech. 93, 185–207.CrossRefGoogle Scholar
Giannetti, F. & Luchini, P. 2003 Receptivity of the circular cylinder's first instability. In Proceedings of 5th European Fluid Mechanics Conference, Toulouse, France.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Gunzburger, M. D. 1999 Sensitivities, adjoints and flow optimization. Intl J. Numer. Math. Fluids 31 (1), 53–78.3.0.CO;2-Z>CrossRefGoogle Scholar
Hwang, Y. & Choi, H. 2006 Control of absolute instability by basic-flow modification in parallel wake at low Reynolds number. J. Fluid Mech. 560, 465–475.CrossRefGoogle Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.CrossRefGoogle Scholar
Jackson, T. L. & Grosch, C. E. 1990 Absolute/convective instabilities and the convective Mach number in a compressible mixing layer. Phys. Fluids A 2, 949–954.CrossRefGoogle Scholar
Jendoubi, S. & Strykowski, P. J. 1994 Absolute and convective instability of axisymmetric jets with external flow. Phys. Fluids 6 (9), 3000–3009.CrossRefGoogle Scholar
Johnson, T. A. & Patel, V. C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70.CrossRefGoogle Scholar
Lesshafft, L. & Huerre, P. 2007 Linear impulse response in hot round jets. Phys. Fluids 19 (2), 024102.CrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 1998 Görtler vortices: a backward-in-time approach to the receptivity problem. J. Fluid Mech. 363, 1–23.CrossRefGoogle Scholar
Mack, C. J., Schmid, P. J. & Sesterhenn, J. L. 2008 Global stability of swept flow around a parabolic body: connecting attachment-line and crossflow modes. J. Fluid Mech. 611, 205–214.CrossRefGoogle Scholar
Mair, W. A. 1965 The effect of a rear-mounted disc on the drag of a blunt-based body of revolution. Aeronaut. Q. 16, 350–360.CrossRefGoogle Scholar
Marquet, O., Lombardi, M., Chomaz, J.-M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21.CrossRefGoogle Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of the cylinder flow. J. Fluid Mech. 615, 221–252.CrossRefGoogle Scholar
Matsumoto, J. & Kawahara, M. 2000 Stable shape identification for fluid–structure interaction problem using MINI element. J. Appl. Mech. 3, 263–274.CrossRefGoogle Scholar
Meliga, P., Chomaz, J.-M. & Sipp, D. 2009 a Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. J. Fluid. Mech. 633, 159–189.CrossRefGoogle Scholar
Meliga, P., Chomaz, J.-M. & Sipp, D. 2009 b Unsteadiness in the wake of disks and spheres: instability, receptivity and control using direct and adjoint global stability analyses. J. Fluid. Struc. 25, 601–616.CrossRefGoogle Scholar
Meliga, P., Sipp, D. & Chomaz, J.-M. 2008 Absolute instability in axisymmetric wakes: compressible and density variation effects. J. Fluid Mech. 600, 373–401.CrossRefGoogle Scholar
Merz, R. A., Page, R. H. & Przirembel, C. E. G. 1978 Subsonic axisymmetric near-wake studies. AIAA J. 16, 656–662.CrossRefGoogle Scholar
Michalke, A. 1971 Instabilität eines kompressiblen runden Freistrahls unter Berücksichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Plugwiss. (19), 319–328 (English translation in NASA TM 75190, 1977).Google Scholar
Monkewitz, P. A. & Sohn, K. D. 1988 Absolute instability in hot jets. AIAA J. 26 (8), 911–916.CrossRefGoogle Scholar
Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.CrossRefGoogle Scholar
Nichols, J. W., Schmid, P. J. & Riley, J. J. 2007 Self-sustained oscillations in variable-density round jets. J. Fluid Mech. 582, 341–376.CrossRefGoogle Scholar
Ormières, D. & Provansal, M. 1999 Transition to turbulence in the wake of a sphere. Phys. Rev. Lett. 83, 80–83.CrossRefGoogle Scholar
Pavithran, S. & Redekopp, L. G. 1989 The absolute-convective transition in subsonic mixing layers. Phys. Fluids A 1 (10), 1736–1739.CrossRefGoogle Scholar
Robinet, J.-C. 2007 Bifurcations in shock-wave/laminar-boundary layer interaction: global instability approach. J. Fluid Mech. 579, 85–112.CrossRefGoogle Scholar
Schlichting, H. 1978 Boundary Layer Theory, 7th edn.McGraw-Hill.Google Scholar
Schmid, P. J. & Henningson, D. S 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Seidel, J., Siegel, S., Jeans, T., Aradag, S., Cohen, K. & McLaughlin, T. 2008 Analysis of an axisymmetric bluff body wake using Fourier transform and POD. AIAA Paper 2008-552.CrossRefGoogle Scholar
Sevilla, A. & Martínez-Bazàn, C. 2004 Vortex shedding in high Reynolds number axisymmetric bluff-body wakes: local linear instability and global bleed control. Phys. Fluids 16 (9), 3460–3469.CrossRefGoogle Scholar
Siegel, S. G. & Fasel, H. F. 2001 Effect of forcing on the wake drag of an axisymmetric bluff body. AIAA Paper 2001-0736.CrossRefGoogle Scholar
Siegel, S., Seidel, J., Cohen, K., Aradag, S. & McLaughlin, T. 2008 Open loop transient forcing of an axisymmetric bluff body wake. AIAA Paper 2008-595.CrossRefGoogle Scholar
Soteriou, M. C. & Ghoniem, A. F. 1995 Effects of the free-stream density ratio on free and forced spatially developing shear layers. Phys. Fluids 7 (8), 2036–2051.CrossRefGoogle Scholar
Taneda, S. 1978 Visual observations of the flow past a sphere at Reynolds numbers between 104 and 106. J. Fluid Mech. 85, 187–192.CrossRefGoogle Scholar
Tomboulides, A. G. & Orszag, S. A. 2000 Numerical investigation of transitional and weak turbulent flow past a sphere. J. Fluid Mech. 416, 45–73.CrossRefGoogle Scholar
Weickgenannt, A. & Monkewitz, P. A. 2000 Control of vortex shedding in an axisymmetric bluff body wake. Eur. J. Mech. B Fluids 19, 789–812.CrossRefGoogle Scholar