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On the temporal evolution of finite ensembles

Part of: Basic methods in fluid mechanics

Published online by Cambridge University Press:  26 February 2010

A. P. Rothmayer  and
D. W. Black
A. P. Rothmayer
Affiliation:
Department of Aerospace Engineering and Engineering Mechanics, Iowa State University, Ames, Iowa 50011, U.S.A.
D. W. Black
Affiliation:
Department of Aerospace Engineering and Engineering Mechanics, Iowa State University, Ames, Iowa 50011, U.S.A.
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Abstract

The temporal evolution of nonlinear, incompressible ensembles is examined first for the one-dimensional Burgers' equation and then for the incompressible, unsteady Navier-Stokes equations. It is shown that local closure of the averaged problem can be obtained for finite ensembles of Burgers' equation in the limit as the number of moments tends to infinity. This limit behaviour is verified via direct numerical computations for the onedimensional inviscid and viscous Burgers' equation. Closure is found to occur at reasonably low order. It is shown that this technique can be extended to obtain a local closure of the convective terms of the Navier-Stokes Reynoldsaveraged equations.

MSC classification

Secondary: 76M35: Stochastic analysis
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 291 - 318
Copyright
Copyright © University College London 1992

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References

Baldwin, B. S. and Lomax, H.. 1978. Thin-Layer Approximation and Algebraic Model for Separated Turbulent Flows. AIAA Paper, 78–257.Google Scholar
Berge, P.Pomeau, Y. and Vidal, C.. 1984. Order Within Chaos (John Wiley & Sons).Google Scholar
Boussinesq, J.. 1877. Théorie de l'écoulement tourbillant. Mem. prés. Acad. Sci., Paris, 23 (46).Google Scholar
Burgers, J. M.. 1948. A Mathematical Model Illustrating the Theory of Turbulence. Adv. Appl. Mech., 1, 171–199.CrossRefGoogle Scholar
Cantwell, B.. 1990. Future Directions in Turbulence Research and the Role of Organized Motion. In Whither Turbulence? Turbulence at the Crossroads. Proceedings of a workshop held at Cornell Univ., 1989 (Springer), 97–131.CrossRefGoogle Scholar
Cebeci, T. and Smith, A. M. O.. 1974. Analysis of Turbulent Boundary Layers (Academic Press).Google Scholar
Chou, P. Y.. 1945. On Velocity Correlations and the Solutions of the Equations of Turbulent Fluctuation. Q. appl. Math., 3, 38–54.CrossRefGoogle Scholar
Heslot, F.Castaing, B. and Libchaber, A.. 1987. Transition to Turbulence in Helium Gas. Phys. Rev. A, 36 (12), 5870–5873.CrossRefGoogle ScholarPubMed
Hinze, J. O.. 1959. Turbulence (McGraw-Hill).Google Scholar
Hopf, E.. 1952. Statistical Hydromechanics and Functional Calculus. J. ratl. mech. Anal., 1, 87–123.Google Scholar
Johnson, D. A. and King, L. S.. 1985. A Mathematically Simple Turbulence Closure Model for Attached and Separated Turbulent Boundary Layers. AIAA Journal, 23, Nov. 1985, 1684–1692.CrossRefGoogle Scholar
Kraichnan, R. H.. 1959. The Structure of Isotropic Turbulence at Very High Reynolds Number. J. Fluid Mech., 5, 497–543.CrossRefGoogle Scholar
Launder, B. E.. 1989. Second-Moment Closure: Present… and Future? Int. J. Heat and Fluid Flow, 10 (4), 282–300.CrossRefGoogle Scholar
Launder, B. E.Reece, G. J. and Rodi, W.. 1975. Progress in the Development of a Reynolds Stress Turbulence Closure. J. Fluid Mech., 68 (3), 537.CrossRefGoogle Scholar
Lewis, R. M. and Kraichnan, R. H.. 1962. A Space-Time Functional Formalism for Turbulence. Commun. pure appl. Math., 15, 397–411.CrossRefGoogle Scholar
Libchaber, A.. 1987. From Chaos to Turbulence in Bernard Convection. Proc. R. Soc. Lond., A413, 63–69.Google Scholar
Lumley, J. L.. 1978. Computational Modeling of Turbulent Flows. In Advances in Applied Mechanics 18, ed. Yih, C. S. (Academic Press), 123–176.Google Scholar
McComb, W. D.. 1990. The Physics of Fluid Turbulence. Oxford Engineering Science Series 25 (Oxford University Press).CrossRefGoogle Scholar
Narasimha, R.. 1990. The Utility and Drawbacks of Traditional Approaches. In Whither Turbulence? Turbulence at the Crossroads. Proceedings of a workshop held at Cornell Univ., 1989 (Springer), 13–48.CrossRefGoogle Scholar
Orszag, S. A.. 1970. Analytical Theories of Turbulence. J. Fluid Mech., 41, 363–386.CrossRefGoogle Scholar
Prandtl, L.. 1925. Über dei ausgebildete Turbulenz. ZAMM, 5, 136–139.CrossRefGoogle Scholar
Reynolds, O.. 1883. On the Experimental Investigation of the Circumstances which Determine Whether the Motion of Water Shall be Direct or Sinuous, and the Law of Resistance in Parallel Channels. Phil. Trans. R. Soc., A174, 935–982.Google Scholar
Reynolds, O.. 1895. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Phil Trans. R. Soc., A186, 123–164.Google Scholar
Roux, B., Editor, 1990. Numerical Simulation of Oscillatory Convection in Low-Pr Fluids. Proceedings of a GAMM workshop held in Marseille 1988, Vieweg.CrossRefGoogle Scholar
Smith, F. T.. 1982. On the High Reynolds Number Theory of Laminar Flows. IMA J. Appl. Math. 28, 207–281.CrossRefGoogle Scholar
Smith, F. T.. 1991. Theoretical Aspects of Transition and Turbulence in Boundary Layers. AIAA Paper 91-0331, presented at the 29th Aerospace Sciences meeting.CrossRefGoogle Scholar
Yakhot, V. and Orszag, S. A.. 1986. Renormalization Group Analysis of Turbulence. I. Basic Theory. J. Scientific Computing, 1, 3–51.CrossRefGoogle Scholar