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Stability analysis and large-eddy simulation of rotating turbulence with organized eddies

Published online by Cambridge University Press:  26 April 2006

Claude Cambon ,
Jean-Pierre Benoit ,
Liang Shao  and
Laurent Jacquin
Claude Cambon
Affiliation:
Ecole Centrale de Lyon / Université Claude Bernard, Lyon 1, Laboratoire de Mécanique des Fluides et d'Acoustique, URA CNRS no. 263, BP 163, 69131 Ecully Cedex, France
Jean-Pierre Benoit
Affiliation:
Ecole Centrale de Lyon / Université Claude Bernard, Lyon 1, Laboratoire de Mécanique des Fluides et d'Acoustique, URA CNRS no. 263, BP 163, 69131 Ecully Cedex, France
Liang Shao
Affiliation:
Ecole Centrale de Lyon / Université Claude Bernard, Lyon 1, Laboratoire de Mécanique des Fluides et d'Acoustique, URA CNRS no. 263, BP 163, 69131 Ecully Cedex, France
Laurent Jacquin
Affiliation:
Office National d'Etudes et de Recherches Aérospatiales, BP 72, 92320 Châtillon Cedex, France
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Abstract

Rotation strongly affects the stability of turbulent flows in the presence of large eddies. In this paper, we examine the applicability of the classic Bradshaw-Richardson criterion to flows more general than a simple combination of rotation and pure shear. Two approaches are used. Firstly the linearized theory is applied to a class of rotating two-dimensional flows having arbitrary rates of strain and vorticity and streamfunctions that are quadratic. This class includes simple shear and elliptic flows as special cases. Secondly, we describe a large-eddy simulation of initially quasi-homogeneous three-dimensional turbulence superimposed on a periodic array of two-dimensional Taylor-Green vortices in a rotating frame.

The results of both approaches indicate that, for a large structure of vorticity W and subject to rotation Ω, maximum destabilization is obtained for zero tilting vorticity (½W + 2Ω = 0) whereas stability occurs for zero absolute vorticity (2Ω = 0) These results are consistent with the Bradshaw-Richardson criterion; however the numerical results show that in other cases the Bradshaw-Richardson number $B=2\Omega(W+2\Omega)/W^2$ is not always a good indicator of the flow stability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 278 , 10 November 1994 , pp. 175 - 200
Copyright
© 1994 Cambridge University Press

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References

Andersson, H. I. & Kristoffersen, R. 1992 Vizualisations and statistics of simulated turbulence in a rotating channel. Communication in Euromech 288, Ecole Centrale de Lyon (see Cambon 1994).
Bardina, J., Ferziger, J. M. & Reynolds, W. C. 1983 Technical Rep. TF-19, Stanford University.
Bartello, P., Métais, O. & Lesieur, M. 1994 Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. 273, 1–29.Google Scholar
Batchelor, G. K. & Proudman, I. 1954 The effect of rapid distortion on a fluid in turbulent motion Q. J. Mech. Appl. Maths, 7, 83–103.Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2171.Google Scholar
Benoit, J. P. 1992 Etude expérimentale et théorique d'une turbulence homogéne soumise à des effets couplés de rotation et déformation plane. Thése de doctorat, Ecole Centrale de Lyon.
Bertoglio, J. P. 1982 Turbulent shear flow within a rotating frame. AIAA J. 20, 1175.Google Scholar
Bidokhti, A.A. & Tritton, D. J. 1992 The structure of a turbulent free shear layer in a rotating fluid. J. Fluid Mech. 241, 469–502.Google Scholar
Bradshaw, P. 1969 The analogy between streamline curvature and buoyancy in turbulent shear flow. J. Fluid Mech. 36, 177–191.Google Scholar
Cambon, C. 1982 Etude spectrale d'un champ turbulent incompressible soumis à des effets couplés de déformation et de rotation imposés extérieurement. Thése de doctorat d’état, Université Claude Bernard-Lyon I.
Cambon, C. 1990 Contribution to single and double point modelling of homogeneous turbulence. Annual Research Briefs. Center for Turbulence Research, Stanford.
Cambon, C. 1994 Turbulent flows undergoing distortion and rotation: A report on EUROMECH 288. Fluid Dyn. Res. J., 13 (6), 281–298 (referred to herein as E288).Google Scholar
Cambon, C., Coleman, G. N. & Mansour, N. N. 1993 Rapid distortion analysis and direct simulation of compressible homogeneous turbulence at finite Mach number. J. Fluid Mech. 251, 641–665.Google Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.Google Scholar
Cambon, C., Teissédre, C. & Jeandel, D. 1985 Etude d’éffets couplés de déformation et de rotation sur une turbulence homogéne. J. Méc. Théo. Appl. 4, 629–657 (referred to herein as CTJ).Google Scholar
Chollet, J. P. & Lesieur, M. 1981 Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 2747–2757.Google Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765–777.Google Scholar
Craik, A. D. D. 1989 The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions. J. Fluid Mech. 198, 275–292.Google Scholar
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier-Stokes equations. Proc. R. Soc. Lond. A 406, 13–26.Google Scholar
Craya, A. 1958 Contribution à l'analyse de la turbulence associée a des vitesses moyennes. P. S. T. Ministére de l'Air, no 345.
Dang, K. & Roy, Ph. 1985 Direct and large eddy simulations of homogeneous turbulence submitted to solid body rotation. In Turbulent Shear Flow 5. Springer.
Eringen, A. C. 1967 Mechanics of Continua. Wiley.
Gence, J. N. & Mathieu, J. 1979 On the application of successive plane strains to grid generated turbulence. J. Fluid Mech. 93, 501–513.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859–872.Google Scholar
Jacquin, L. 1987 Etude théorique et expérimentale de la turbulence homogéne en rotation. Thése de doctorat d’état, Université Claude Bernard-Lyon I.
Jacquin, L., Leuchter, O., Cambon, C. & Mathieu, J. 1990 Homogeneous turbulence in the presence of rotation. J. Fluid Mech. 220, 1–52.Google Scholar
Jacquin, L., Salhi, A. & Benoit, J. P. 1992 On the stabilizing effects of rotation on swirling flows. Communication in Euromech 288, Ecole Centrale de Lyon (see Cambon 1994).
Johnston, J. P., Halleen, R. M. & Lezius, D. L. 1972 Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech. 56, 533–557.Google Scholar
Kraichnan, R. H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536.Google Scholar
Kristoffersen, R. & Andersson, H. I. 1993 Direct simulation of low-Reynolds-number turbulent flow in a rotating channel. J. Fluid Mech. 256, 163–197.Google Scholar
Lagnado, R. R., Phan-Thien, N. & Leal, L. G. 1984 The stability of two-dimensional linear flows. Phys. Fluids 27, 1094–1101.Google Scholar
Landman, M. J. & Saffman, P. G. 1987 The three-dimensional instability of strained vortices in a viscous fluid. Phys. Fluids 30, 2339–2342.Google Scholar
Lee, M. J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583.Google Scholar
Lesieur, M. 1990 Turbulence in Fluids, Kluwer. 2nd edn.
Leuchter, O., Benoit, J. P. & Cambon, C. 1992 Effects of plane strain and rotation on homogeneous turbulence. European Turbulence Conference 4, Delft.Google Scholar
Malkus, W. V. R. & Waleffe, F. 1991 The transition from order to disorder in elliptical flow: a direct path to shear flow turbulence. Advances in Turbulence 3 (ed. A. V. Johansson & P. H. Alfredsson) Springer.
Mansour, N. N., Cambon, C. & Speziale C. G. 1991a Theoretical and computational study of rotating isotropic turbulence. Studies in Turbulence (ed. T. B. Gatski, S. Sarkar & C. G. Speziale). Springer.
Mansour, N. N., Cambon, C. & Speziale C. G. 1991b Single point modelling of initially isotropic turbulence under uniform rotation. Annual Research Briefs. Center for Turbulence Research, Stanford University.
Métais, O., Yanase, S., Flores, C., Bartello, P. & Lesieur, M. 1991 Reorganisation of coherent vortices in shear layers under the action of solid-body rotation. Turbulent Shear Flows 8, Munich.
Michard, M., Mathieu, J., Morel, R., Alcaraz, E. & Bertoglio, J. P. 1986 Grid generated turbulence exhibiting a peak in the spectrum. European Turbulence Conference 1, Lyon.
Pedley, T. J. 1969 On the instability of viscous flow in a rapidly rotating pipe. J. Fluid Mech. 35, 97–115.Google Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.Google Scholar
Ayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. 93, 148–154.Google Scholar
Salhi, A. 1992 Homogeneous turbulent shear flow in a rotating frame: linear analysis (R. D. T.). Communication in Euromech 288, Ecole Centrale de Lyon (see Cambon 1994).
Speziale, C. G., Gatski, T. B. & Mac Giolla Mhuiris, N. 1990 A critical comparison of turbulence models for homogeneous shear flows in a rotating frame. Phys. Fluids A 2, 1678–1684.Google Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow revised version 1976 Cambridge University Press.
Tritton, D. J. 1992 Stabilization and destabilization of turbulent shear flow in a rotating fluid. J. Fluid Mech. 241, 503–523.Google Scholar
Waleffe, F. 1989 The three-dimensional instability of a strained vortex and its relation to turbulence. PhD thesis, MIT.
Waleffe, F. 1990 On the origin of the streak spacing in turbulent shear flows. Annual Research Briefs. Center for Turbulence Research, Stanford University.
Waleffe, F. 1992 Elliptical Flow Instabilities. Communication in Euromech 288, Ecole Centrale de Lyon (see Cambon 1994).
Waleffe, F. 1993 Inertial transfer in the helical decomposition. Phys. Fluids A 5, 677–685.Google Scholar
Yanase, S., Flores, C., Métais, O. & Riley, J. J. 1992 Rotating free shear flows I : linear stability analysis. Phys. Fluids A 5, 2725–2737.Google Scholar