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Upper bounds on functions of the dissipation rate in turbulent shear flow

Published online by Cambridge University Press:  26 April 2006

W. V. R. Malkus  and
L. M. Smith
W. V. R. Malkus
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139, USA
L. M. Smith
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA 02139, USA
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Abstract

The search for a statistical stability criterion characterizing steady-state turbulent shear flows ‘has led us to the study of a function proportional to the ratio of the fluctuation dissipation rate and the dissipation rate of the mean. The first Euler–Lagrange equations for an upper bound of this function have optimal solutions with the observed scaling laws and an asymptotic velocity defect for turbulent channel flow. As in the case of maximum transport, the optimal solution for the model equations is a discrete spectrum of streamwise vortices. It is shown how these solutions can be brought closer to the realized flow with additional constraints on the smallest vortex.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 208 , November 1989 , pp. 479 - 507
Copyright
© 1989 Cambridge University Press

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References

Alfredsson, P. H. & Johansson, A. V. 1984 On the detection of turbulence-generating events. J. Fluid Mech. 139, 325.Google Scholar
Arnol'd, V. I. 1965 Conditions for non-linear stability of the plane stationary curvilinear flow of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975.Google Scholar
Bayly, B. 1986 Three dimensional instability of elliptic flow. Phys. Rev. Lett. 57, 2160.Google Scholar
Bayly, B., Orszag, S. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359.Google Scholar
Bergé, P., Pomeau, Y. & Vidal, C. 1987 Order within Chaos, Chapter IX. Wiley.
Busse, F. 1969 Bounds on the transport of mass and momentum by turbulent flow. Z. Angew. Math. Phys. 20, 1.Google Scholar
Busse, F. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41, 219.Google Scholar
Busse, F. 1978 The optimum theory of turbulence. Adv. Appl. Mech. 18, 77.Google Scholar
Chan, S. K. 1971 Infinite Prandtl number turbulent convection. Stud. Appl. Maths 50, 13.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. 406, 13.Google Scholar
Henningson, D. S., Spalart, P. & Kim, J. 1987 Numerical simulation of turbulent spots in plane Poiseuille and boundary-layer flows. Phys. Fluids 30, 2914.Google Scholar
Herbert, T. 1983 Secondary instability of plane channel flows to subharmonic three-dimensional disturbances. Phys. Fluids 26, 871.Google Scholar
Herbert, T. 1988 Secondary instability of boundary layers. Ann. Rev. Fluid Mech. 20, 487.Google Scholar
Herring, J. 1964 Investigation of problems in thermal convection. J. Atmos. Sci. 21, 277.Google Scholar
Holm, D. D., Marsden, J., Ratin, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405.Google Scholar
Howard, L. N. 1972 Bounds on flow quantities. Ann. Rev. Fluid Mech. 4, 473.Google Scholar
Ierley, G. R. & Malkus, W. V. R. 1988 Stability bounds on turbulent Poiseuille flow. J. Fluid Mech. 187, 435.Google Scholar
Johansson, A. V. & Alfredsson, P. H. 1983 Measurements of wall-bounded turbulent shear flows. J. Fluid Mech. 137, 408.Google Scholar
Joseph, D. D. 1968 Eigenvalue bounds for the Orr—Sommerfeld equation. J. Fluid Mech. 33, 617.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Molions, vols. 1 and 2. Springer.
Kraichnan, R. 1955 Direct-interaction approximation for isotropic homogeneous turbulence. Phys. Rev. 55, 18.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5, 497.Google Scholar
Krommes, J. A. & Smith, R. A. 1987 Rigorous upper bounds for transport due to passive advection by inhomogeneous turbulence. Annls Phys. 177, 246.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
MacCreadie, W. T. 1931 On the stability of the motion of a viscous fluid. Proc. Natl Acad. Sci. USA 17, 381.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521.Google Scholar
Malkus, W. V. R. 1963 Outline of a theory of turbulent convection. In Theory and Fundamental Research in Heat Transfer, p. 203. Pergamon.
Malkus, W. V. R. 1979 Turbulent velocity profiles from stability criteria. J. Fluid Mech. 90, 401.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225.Google Scholar
Millikan, C. B. 1938 A critical discussion of turbulent flow in channels and circular tubes. Proc. 5th Intl Congr. Appl. Mech., Cambridge, Ma., p. 386.Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. R. Irish Acad. 27, 9 and 69.Google Scholar
Orszag, S. & Patera, P. T. 1980 Subcritical transition to turbulence in plane channel flows. Phys. Rev. Lett. 45, 989.Google Scholar
Orszag, S. & Patera, P. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347.Google Scholar
Pierrehumbert, R. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Lett. 57, 2157.Google Scholar
Pierrehumbert, R. & Widnall, S. 1982 Two-and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59.Google Scholar
Squire, H. B. 1933 On the stability of three-dimensional disturbances. Proc. R. Soc. Lond. A 132, 621.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1.Google Scholar
Synge, J. 1938 Hydrodynamical stability. Semicent. Publ. Amer. Math. Soc., vol. 2. p. 227.Google Scholar
Yakhot, V. & Orszag, S. A. 1986 Renormalization-group approach to turbulence. J. Sci. Comput. 1, 3.Google Scholar
Zaff, D. 1987 Secondary instabilities in Ekman boundary flow. Ph.D. thesis, Dept. of Maths, MIT.