The transition to turbulence

AUG 01, 1978

Modern optical and computer techniques and new concepts in the theory of nonlinear systems are yielding insights into such hydrodynamic instabilities as Couette flow, vortex streets and the Rayleigh–Bénard instability.

DOI: 10.1063/1.2995142

Harry L. Swinney

Jerry P. Gollub

Fluid flows have been studied systematically for more than a century and their equations of motion are well known, yet the transition from laminar flow to turbulent flow remains an enigma. The difficulty lies in the intractability of the nonlinear hydrodynamic equations that express the conservation of mass, momentum and energy for a fluid continuum. Although these equations can be linearized and readily solved for a system near thermodynamic equilibrium, the solutions of the nonlinear equations—required to describe fluids far from equilibrium—are generally neither unique nor obtainable.

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References

1. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford U.P., London (1961);
D. D. Joseph, Stability of Fluid Motions I and II, Springer, New York (1976);
D. J. Tritton, Physical Fluid Dynamics, Van Nostrand Reinhold, New York (1977);
J. Whitehead, in Fluctuations, Instabilities and Phase Transitions (T. Riste, ed.).
Plenum, N.Y., (1975).
2. R. M. May, Nature 261, 459 (1976); https://doi.org/NATUAS
Synergetics (H. Haken, ed.) Springer, New York (1977);
G. Nicolis, I. Prigogine, Self‐Organization in Nonequilibrium Systems, Wiley, New York (1977).
3. G. I. Taylor, Phil. Trans. Roy. Soc. (London) A 223, 289 (1923).
4. Lord Rayleigh, Phil. Mag. 32, 529 (1916).https://doi.org/PHMAA4
5. T. S. Durrani, C. A. Greated, Laser Systems in Flow Measurement, Plenum, New York (1977).
6. G. Ahlers, Phys. Rev. Lett. 33, 1185 (1974); https://doi.org/PRLTAO
G. Ahlers, R. Behringer, Phys. Rev. Lett. 40, 712 (1978); https://doi.org/PRLTAO
P. Bergé, M. Dubois, Opt. Comm. 19, 129 (1976); https://doi.org/OPCOB8
J. P. Gollub, M. H. Freilich, Phys. Fluids 19, 618 (1976); https://doi.org/PFLDAS
R. W. Walden, R. J. Donnelly, Bull. Am. Phys. Soc. 23, 524 (1978).https://doi.org/BAPSA6
7. J. P. Gollub, H. L. Swinney, Phys. Rev. Lett. 35, 927 (1975); https://doi.org/PRLTAO
H. L. Swinney, P. R. Fenstermacher, J. P. Gollub, in Synergetics (H. Haken, ed.), Springer, New York (1977) page 60.
8. E. N. Lorenz, J. Atm. Sci. 20, 448 (1963).https://doi.org/JAHSAK
9. A. Davey, R. C. DiPrima, J. T. Stuart, J. Fluid Mech. 31, 17 (1968); https://doi.org/JFLSA7
R. M. Clever, F. H. Busse, J. Fluid Mech. 65, 625 (1974).https://doi.org/JFLSA7
10. L. Landau, C. R. (Dokl.) Acad. Sci. USSR 44, 311 (1944);
L. D. Landau, E. M. Lifshitz, Fluid Mechanics, Pergamon, London (1959).
11. T. Li, J. A. Yorke, Am. Math. Monthly 82, 985 (1975).https://doi.org/AMMYAE
12. Ya Sinai, Sov. Math.‐Dokl. 4, 1818 (1963).
13. D. Ruelle, F. Takens, Commun. Math. Phys. 20, 167 (1971).https://doi.org/CMPHAY
14. J. B. McLaughlin, P. C. Martin, Phys. Rev. A 12, 186 (1975).https://doi.org/PLRAAN
15. J. Kaplan, J. Yorke, E. Yorke, in Bifurcation Theory and Its Applications in Scientific Disciplines (O. Gurel, O. E. Rössler, eds.) N.Y. Acad. Sci., New York (1978).

More about the Authors

Harry L. Swinney. University of Texas, Austin.

Jerry P. Gollub. Haverford College, Haverford, Pennsylvania.