Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T02:34:32.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum-entropy closure for a Galerkin model of an incompressible periodic wake

Published online by Cambridge University Press:  24 April 2012

Bernd R. Noack  and
Robert K. Niven
Bernd R. Noack*
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Robert K. Niven
Affiliation:
School of Engineering and Information Technology, The University of New South Wales at ADFA, Canberra, Australian Capital Territory, 2600, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A statistical closure is proposed for a Galerkin model of an incompressible periodic cylinder wake. This closure employs Jaynes’ maximum entropy principle to infer the probability distribution for mode amplitudes using exact statistical balance equations as side constraints. The analysis predicts mean amplitude values and modal energy levels in good agreement with direct Navier–Stokes simulation. In addition, it provides an analytical equation for the modal energy distribution.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Mathematical Foundations: Variational methods Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 700 , 10 June 2012 , pp. 187 - 213
Copyright
Copyright © Cambridge University Press 2012

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

1. Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
2. Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750–756.CrossRefGoogle Scholar
3. Boltzmann, L. 1877 Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respektive den Sätzen über das Wärmegleichgewicht (translation: On the relationship between the second main theorem of mechanical heat theory and the probability theory, respectively with the theorems about the heat equilibrium). In Sitzungsbericht der Kaiserlichen Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Classe, vol. LXXVI, Abt. II, pp. 373–435.Google Scholar
4. Boussinesq, J. 1877 Essai sur la théorie des eaux courantes. Mémoires présentés par divers savants à l’Académie des Sciences Paris 23 (1), 1–680, 24(2), pp. 1–60.Google Scholar
5. Busse, F. H. 1970 Bounds for turbulent shear flow. J. Fluid Mech. 41, 219–240.CrossRefGoogle Scholar
6. Cheung, S., Oliver, T., Prudencio, E., Prudhomme, S. & Moser, R. 2011 Bayesian uncertainty analysis with applications to turbulence modeling. Reliab. Engng & System Safety 96 (9), 1137–1149.CrossRefGoogle Scholar
7. Craig, N. C. 1988 Entropy analysis of four familiar processes. J. Chem. Ed. 65 (9), 760–764.CrossRefGoogle Scholar
8. Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Phys. Fluids A 3, 2337–2354.Google Scholar
9. Dewar, R. C. 2005 Maximum entropy production and the fluctuation theorem. J. Phys. A: Math. Gen. 38, L371–L381.CrossRefGoogle Scholar
10. Dewar, R. C. & Porté, A. 2008 Statistical mechanics unifies different ecological patterns. J. Theor. Biol. 251, 389–403.CrossRefGoogle ScholarPubMed
11. Ebeling, W. & Klimontovich, Y. 1984 Selforganization and Turbulence in Liquids, 1st edn. BSB B.G. Teubner Verlagsgesellschaft.Google Scholar
12. Eyink, G. L. & Sreenivasan, K. R. 2006 Onsager and the theory of turbulence. Rev. Mod. Phys. 78, 87–135.CrossRefGoogle Scholar
13. Farazmand, M. M., Kevlahan, N. K.-R. & Protas, B. 2011 Controlling the dual cascade of two-dimensional turbulence. J. Fluid. Mech. 668, 1–21.CrossRefGoogle Scholar
14. Frisch, U. 1995 Turbulence, 1st edn. Cambridge University Press.CrossRefGoogle Scholar
15. Gorban, A. N. & Karlin, I. V. 2005 Invariant Manifolds for Physical and Chemical Kinetics, Lecture Notes in Physics , vol. 660. Springer.Google Scholar
16. Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer.Google Scholar
17. Haken, H. 1983 Synergetics, An Introduction. Nonequilibrium Phase Transitions and Self-Organizations in Physics, Chemistry, and Biology, 3rd edn. Springer.Google Scholar
18. Holmes, P., Lumley, J. L. & Berkooz, G. 1998 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 1st edn. Cambridge University Press.Google Scholar
19. Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
20. Hopf, E. 1951 Statistical hydromechanics and functional analysis. J. Rat. Mech. Anal. 1, 87–123.Google Scholar
21. Howard, L. N. 1963 Heat transport by turbulent convection. J. Fluid Mech. 17, 405–432.Google Scholar
22. Jaynes, E. T. 1957 Information theory and statistical mechanics. Phys. Rev. 106, 620–630.CrossRefGoogle Scholar
23. Jaynes, E. T. 1979 Where do we stand on maximum entropy? In The Maximum Entropy Formalism (ed. Levine, R. D. & Tribus, M. ). pp. 1–104. MIT.Google Scholar
24. Jaynes, E. T. 2003 Probability Theory. The Logic of Science, 1st edn. Cambridge University Press.CrossRefGoogle Scholar
25. Kaneda, Y. & Ishihara, T. 2006 High-resolution direct numerical simulation of turbulence. J. Turbul. 7 (20), 1–17.CrossRefGoogle Scholar
26. Kapur, J. N. & Kevasan, H. K. 1992 Entropy Optimization Principles with Applications, 1st edn. Academic.CrossRefGoogle Scholar
27. Kraichnan, R. H. & Chen, S. 1989 Is there a statistical mechanics of turbulence?. Phys. D 37, 160–172.CrossRefGoogle Scholar
28. Kullback, S. & Leibler, R. A. 1951 On information and sufficiency. Ann. Math. Stat. 22, 79–86.CrossRefGoogle Scholar
29. Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn. Course of Theoretical Physics , vol. 6. Pergamon.Google Scholar
30. Launder, B. & Spalding, D. 1972 Lectures in Mathematical Models of Turbulence. Academic.Google Scholar
31. Lee, T. 1952 On some statistical properties of hydrodynamical and magneto-hydrodynamical fields. Quart. Appl. Math. X, 69–74.Google Scholar
32. Lesieur, M. 1993 Turbulence in Fluids, 2nd edn. Kluwer.Google Scholar
33. Luchtenburg, D. M., Günter, B., Noack, B. R., King, R. & Tadmor, G. 2009 A generalized mean-field model of the natural and actuated flows around a high-lift configuration. J. Fluid Mech. 623, 283–316.CrossRefGoogle Scholar
34. Luchtenburg, D. M., Schlegel, M., Noack, B. R., Aleksić, K., King, R., Tadmor, G. & Günther, B. 2010 Turbulence control based on reduced-order models and nonlinear control design. In Active Flow Control II (ed. King, R. ). Notes on Numerical Fluid Mechanics and Multidisciplinary Design , vol. 108. pp. 341–356. Springer.CrossRefGoogle Scholar
35. Majda, A. & Timofeyev, I. 2000 Remarkable statistical behaviour for truncated Burger–Hopf dynamics. Proc. Natl Acad. Sci USA 97 (23), 12413–12417.Google Scholar
36. Malkus, W. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.CrossRefGoogle Scholar
37. Manneville, P. 2004 Instabilities, Chaos and Turbulence, 1st edn. Imperial.CrossRefGoogle Scholar
38. Maxwell, J. C. 1866 On the viscosity or internal friction of air and other gases. Phil. Trans. R. Soc. Lond. 156, 249–268.Google Scholar
39. McComb, D. 1991 The Physics of Fluid Turbulence, 1st edn. Clarendon.Google Scholar
40. Millionshtchikov, M. 1941 On the theory of homogeneous isotropic turbulence. C. R. Acad. Sci. U.S.S.R. 32, 615.Google Scholar
41. Moehlis, J., Smith, T. R., Holmes, P. & Faisst, H. 2002 Models for turbulent plane Couette flow using the proper orthogonal decomposition. Phys. Fluids 14, 2493–2507.CrossRefGoogle Scholar
42. Niven, R. K. 2009a Jaynes’ MaxEnt, steady state flow systems and the maximum entropy production principle. In Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 29th International Workshop, Oxford, MS, 5–10 July 2009 (ed. Goggans, C.-Y. & Chan, P. M. ). pp. 397–404. AIP Conference Proceedings, 1193.Google Scholar
43. Niven, R. K. 2009b Steady state of a dissipative flow-controlled system and the maximum entropy production principle. Phys. Rev. E 80, 021113.CrossRefGoogle ScholarPubMed
44. Niven, R. K. 2010a Minimisation of a free-energy-like potential for non-equilibrium systems at steady state. Phil. Trans. B 365, 1323–1331.CrossRefGoogle ScholarPubMed
45. Niven, R. K. 2010b Simultaneous extrema in the entropy production for steady-state fluid flow in parallel pipes. J. Non-Equilib. Thermodyn. 35, 347–378.CrossRefGoogle Scholar
46. Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.Google Scholar
47. Noack, B. R., Morzyński, M. & Tadmor, G. (ed.) 2011 Reduced-Order Modelling for Flow Control, CISM Courses and Lectures , vol. 528. Springer.CrossRefGoogle Scholar
48. Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
49. Noack, B. R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P. & Tadmor, G. 2008 A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilib. Thermodyn. 33, 103–148.Google Scholar
50. Noack, B. R., Schlegel, M., Morzyński, M. & Tadmor, G. 2010 System reduction strategy for Galerkin models of fluid flows. Int. J. Numer. Meth. Fluids 63 (2), 231–248.CrossRefGoogle Scholar
51. Onsager, L. & Machlup, S. 1953 Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1515.Google Scholar
52. Ozawa, A., Ohmura, R. D., Lorenz, T. & Pujol, T. 2003 The second law of thermodynamics and the global climate system: A review of the maximum entropy production principle. Rev. Geophys. 41, article 4.CrossRefGoogle Scholar
53. Paltridge, G. W. 1975 Global dynamics and climate – a system of minimum entropy exchange. Q. J. R. Meteorol. Soc. 101, 475–484.Google Scholar
54. Planck, M. 1932 Introduction to Theoretical Physics, Vol. V: Theory of Heat, 3rd edn. Macmillan.Google Scholar
55. Planck, M. 1945 Treatise on Thermodynamics, 3rd edn. Dover.Google Scholar
56. Podvin, B. 2009 A proper-orthogonal-decomposition based model for the wall layer of a turbulent channel flow. Phys. Fluids 21, 015111.CrossRefGoogle Scholar
57. Prandtl, L. 1927 Über die ausgebildete Turbulenz (translation: On developed turbulence). In Verhandlungen des 2. internationalen Kongresses für technische Mechanik, pp. 1–14. Orell Füßli.Google Scholar
58. Prigogine, I. 1967 Introduction to Thermodynamics of Irreversible Processes, 3rd edn. Interscience Publ.Google Scholar
59. Rempfer, D. 1994 On the structure of dynamical systems describing the evolution of coherent structures in a convective boundary layer. Phys. Fluids 6 (3), 1402–1404.CrossRefGoogle Scholar
60. Rowley, C., Colonius, T. & Murray, R. 2004 Model reduction for compressible flows using POD and Galerkin projection. Phys. D 189, 115–129.CrossRefGoogle Scholar
61. Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 645, 115–127.CrossRefGoogle Scholar
62. Schewe, G. 1986 Sensitivity of transition phenomena to small perturbations in flow round a circular cylinder. J. Fluid Mech. 172, 33–46.CrossRefGoogle Scholar
63. Schmid, P. J. 2010 Dynamic mode decomposition for numerical and experimental data. J. Fluid. Mech. 656, 5–28.CrossRefGoogle Scholar
64. Schrödinger, E. 1989 Statistical Thermodynamics, 1st edn. Dover.Google Scholar
65. Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Rev. Mec. 63, 251–276.Google Scholar
66. Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I The basic experiment. Mon. Weather Rev. 3, 99–165.2.3.CO;2>CrossRefGoogle Scholar
67. Sparrow, C. 1982 The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, 1st edn. Applied Mathematical Sciences , vol. 41. Springer.CrossRefGoogle Scholar
68. Strong, L. E. & Halliwell, H. F. 1970 An alternative to free energy for undergraduate instruction. J. Chem. Ed. 47 (5), 347–352.Google Scholar
69. Stuart, J. 1971 Nonlinear stability theory. Ann. Rev. Fluid Mech. 3, 347–370.CrossRefGoogle Scholar
70. Tadmor, G., Lehmann, O., Noack, B. R. & Morzyński, M. 2010 Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22 (3)034102.CrossRefGoogle Scholar
71. Tribus, M. 1961 Thermodynamics and Thermostatics: An Introduction to Energy, Information and States of Matter, with Engineering Applications. D. Van Nostrand Company Inc.Google Scholar
72. Watson, J. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. J. Fluid Mech. 9, 371–389.CrossRefGoogle Scholar
73. Williamson, C. 1996 Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539.CrossRefGoogle Scholar