- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Onsager and the theory of hydrodynamic turbulence

Gregory L. Eyink and Katepalli R. Sreenivasan

Gregory L. Eyink

  • Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, Maryland 21218, USA

Katepalli R. Sreenivasan

  • International Center for Theoretical Physics, Trieste, Italy and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA

PDF

Rev. Mod. Phys. 78, 87 – Published 17 January, 2006

DOI: https://doi.org/10.1103/RevModPhys.78.87

Abstract

Lars Onsager, a giant of twentieth-century science and the 1968 Nobel Laureate in Chemistry, made deep contributions to several areas of physics and chemistry. Perhaps less well known is his ground-breaking work and lifelong interest in the subject of hydrodynamic turbulence. He wrote two papers on the subject in the 1940s, one of them just a short abstract. Unbeknownst to Onsager, one of his major results was derived a few years earlier by A. N. Kolmogorov, but Onsager’s work contains many gems and shows characteristic originality and deep understanding. His only full-length article on the subject in 1949 introduced two novel ideas—negative-temperature equilibria for two-dimensional ideal fluids and an energy-dissipation anomaly for singular Euler solutions—that stimulated much later work. However, a study of Onsager’s letters to his peers around that time, as well as his private papers of that period and the early 1970s, shows that he had much more to say about the problem than he published. Remarkably, his private notes of the 1940s contain the essential elements of at least four major results that appeared decades later in the literature: (1) a mean-field Poisson-Boltzmann equation and other thermodynamic relations for point vortices; (2) a relation similar to Kolmogorov’s 4∕5 law connecting singularities and dissipation; (3) the modern physical picture of spatial intermittency of velocity increments, explaining anomalous scaling of the spectrum; and (4) a spectral turbulence closure quite similar to the modern eddy-damped quasinormal Markovian equations. This paper is a summary of Onsager’s published and unpublished contributions to hydrodynamic turbulence and an account of their place in the field as the subject has evolved through the years. A discussion is also given of the historical context of the work, especially of Onsager’s interactions with his contemporaries who were acknowledged experts in the subject at the time. Finally, a brief speculation is offered as to why Onsager may have chosen not to publish several of his significant results.

View figure in article

Article Text

References (209)

  1. Adler, S. L., 1969, Phys. Rev. 177, 2426.
  2. Anselmet, F., Y. Gagne, E. J. Hopfinger, and R. A. Antonia, 1984, J. Fluid Mech. 140, 63.
  3. Arad, I., B. Dhruva, S. Kurien, V. S. L’vov, I. Procaccia, and K. R. Sreenivasan, 1998, Phys. Rev. Lett. 81, 5330.
  4. Barenghi, C. F., R. J. Donnelly, and W. F. Vinen, 2001, Quantized Vortex Dynamics and Superfluid Turbulence (Springer-Verlag, Berlin).
  5. Batchelor, G. K., 1946, Nature (London) 158, 883.
  6. Batchelor, G. K., 1947, Proc. Cambridge Philos. Soc. 43, 533.
  7. Batchelor, G. K., 1953, The Theory of Homogeneous Turbulence (Cambridge University Press, Cambridge).
  8. Batchelor, G. K., 1969, Phys. Fluids 11, 233.
  9. Bell, J. S., and R. Jackiw, 1969, Nuovo Cimento A 60, 47.
  10. Bouchet, F., and J. Sommeria, 2002, J. Fluid Mech. 464, 165.
  11. Bray, R. W., 1966, Ph.D. thesis (University of Cambridge, Pembroke College).
  12. Burgers, J. M., 1929a, Proc. K. Ned. Akad. Wet. 32, 414.
  13. Burgers, J. M., 1929b, Proc. K. Ned. Akad. Wet. 32, 643.
  14. Burgers, J. M., 1929c, Proc. K. Ned. Akad. Wet. 32, 818.
  15. Burgers, J. M., 1933a, Proc. K. Ned. Akad. Wet. 36, 276.
  16. Burgers, J. M., 1933b, Proc. K. Ned. Akad. Wet. 36, 390.
  17. Burgers, J. M., 1933c, Proc. K. Ned. Akad. Wet. 36, 487.
  18. Burgers, J. M., 1933d, Proc. K. Ned. Akad. Wet. 36, 620.
  19. Burgers, J. M., 1948, Adv. Appl. Mech. 1, 171.
  20. Cadot, O., Y. Couder, A. Daerr, S. Douady, and A. Tsinober, 1997, Phys. Rev. E 56, 427.
  21. Caglioti, E., P. L. Lions, C. Marchioro, and M. Pulvirenti, 1992, Commun. Math. Phys. 143, 501.
  22. Caglioti, E., P. L. Lions, C. Marchioro, and M. Pulvirenti, 1995, Commun. Math. Phys. 174, 229.
  23. Campbell, L. J., and K. O’Neil, 1991, J. Stat. Phys. 65, 495.
  24. Careri, G., 2000, Phys. Perspect. 2, 204.
  25. Chandrasekhar, S., 1939, An Introduction to the Theory of Stellar Structure (University of Chicago Press, Chicago).
  26. Chen, M.-S., J. Bonner, J. Nagle, and L. Onsager, 1974, J. Chem. Phys. 60, 405.
  27. Chen, S., B. Dhruva, S. Kurien, K. Sreenivasan, and M. Taylor, 2005, J. Fluid Mech. (in press).
  28. Chorin, A. J., 1996, in The Collected Works of Lars Onsager (with commentary), edited by P. C. Hemmer, H. Holden, and S. K. Ratkje (World Scientific, Singapore), pp. 743–745.
  29. Chow, K. W., N. W. M. Ko, R. C. K. Leung, and S. K. Tang, 1998, Phys. Fluids 10, 1111.
  30. Constantin, P., W. E, and E. S. Titi, 1994, Commun. Math. Phys. 165, 207.
  31. DeGroot, S. R., 1945, L’Effet Soret (North-Holland, Amsterdam).
  32. Domaradzki, J. A. and R. S. Rogallo, 1990, Phys. Fluids A 2, 413.
  33. Dryden, H. L., 1943, Q. Appl. Math. 1, 7.
  34. Duchon, J., and R. Robert, 2000, Nonlinearity 13, 249.
  35. Edwards, S. F., 1964, J. Fluid Mech. 18, 239.
  36. Edwards, S. F. and J. B. Taylor, 1974, Proc. R. Soc. London, Ser. A 336, 257.
  37. Emden, R., 1907, Gaskugeln (Teubner, Leipzig).
  38. Eyink, G. L., 1994, Physica D 78, 222.
  39. Eyink, G. L., 1995a, J. Stat. Phys. 78, 353.
  40. Eyink, G. L., 1995b, J. Stat. Phys. 78, 335.
  41. Eyink, G. L., 2001, Nonlinearity 14, 787.
  42. Eyink, G. L., 2003, Nonlinearity 16, 137.
  43. Eyink, G. L. and H. Spohn, 1993, J. Stat. Phys. 79, 833.
  44. Fairbank, W. M., M. J. Buckingham, and C. F. Kellers, 1958, in Proceedings of the Fifth International Conference on Low Temperature Physics, edited by J. R. Dillinger (University of Wisconsin Press, Madison, WI), p. 50.
  45. Falkovich, G., K. Gawȩdzki, and M. Vergassola, 2001, Rev. Mod. Phys. 73, 913.
  46. Fetter, A. L., 1966, Phys. Rev. 151, 100.
  47. Feynman, R. P., 1955, in Progress in Low Temperature Physics, edited by C. Gorter (North-Holland, Amsterdam), Vol. 1, pp. 17–51.
  48. Fine, K. S., A. C. Cass, W. G. Flynn, and C. F. Driscoll, 1995, Phys. Rev. Lett. 75, 3277.
  49. Fjørtoft, R., 1953, Tellus 22, 243.
  50. Fournier, J.-D., and U. Frisch, 1978, Phys. Rev. A 17, 747.
  51. Frisch, U., 1968, in Probabilistic Methods in Applied Mathematics, edited by A. Barucha-Reid (Academic, New York), Vol. 1, pp. 75–198.
  52. Frisch, U., 1995, Turbulence. The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, England).
  53. Frisch, U., and J. Bec, 2001, in Les Houches 2000: New Trends in Turbulence, edited by M. Lesieur, A. Yaglom, and F. David (Springer EDP-Sciences, New York), pp. 341–383.
  54. Fröhlich, J., and D. Ruelle, 1982, Commun. Math. Phys. 87, 1.
  55. Fuoss, R. M., and L. Onsager, 1955, Proc. Natl. Acad. Sci. U.S.A. 41, 274.
  56. Ginzburg, V. L., 1960, Sov. Phys. Solid State 2, 2031.
  57. Gosting, L. J., and L. Onsager, 1952, J. Am. Chem. Soc. 74, 6066.
  58. Heisenberg, W., 1948, Z. Phys. 124, 628.
  59. Helmholtz, H., 1867, Philos. Mag. 33, 4.
  60. Hemmer, P. C., H. Holden, and S. K. Ratkje, 1996, Eds., The Collected Works of Lars Onsager (with commentary) (World Scientific, Singapore).
  61. Hubbard, J., and L. Onsager, 1977, J. Chem. Phys. 67, 4850.
  62. Itzykson, C., and J. B. Zuber, 1980, Quantum Field Theory (McGraw-Hill, New York).
  63. Jackiw, R., and C. Rebbi, 1977, Phys. Rev. D 16, 1052.
  64. Jaffard, S., 2001, in Twentieth Century Harmonic Analysis: A Celebration, Vol. 33 of NATO Science Series. Series II, Mathematics, Physics, and Chemistry, edited by J. Byrnes (Kluwer, Dordrecht).
  65. Jin, D. Z., and D. H. E. Dubin, 1998, Phys. Rev. Lett. 80, 4434.
  66. Jin, D. Z., and D. H. E. Dubin, 2000, Phys. Plasmas 7, 1719.
  67. Joyce, G., and D. Montgomery, 1973, J. Plasma Phys. 10, 107.
  68. Kaneda, Y., T. Ishihara, M. Yokokawa, K. Itakura, and A. Uno, 2003, Phys. Fluids 15, 21.
  69. Kaufman, B., and L. Onsager, 1949, Phys. Rev. 76, 1244.
  70. Khanin, K. M., 1982, Physica D 2, 261.
  71. Kiessling, M. K.-H., 1993, Commun. Pure Appl. Math. 47, 27.
  72. Kiessling, M. K.-H., and J. L. Lebowitz, 1997, Lett. Math. Phys. 42, 43.
  73. Kirchhoff, G., 1883, Vorlesungen über Mathematische Physik, Vol. 1 (Teubner, Leipzig).
  74. Kolmogorov, A. N., 1941a, Dokl. Akad. Nauk SSSR 32, 16.
  75. Kolmogorov, A. N., 1941b, Dokl. Akad. Nauk SSSR 30, 9.
  76. Kolmogorov, A. N., 1941c, Dokl. Akad. Nauk SSSR 31, 538.
  77. Kolmogorov, A. N., 1962, J. Fluid Mech. 13, 82.
  78. Kraichnan, R. H., 1959, J. Fluid Mech. 5, 497.
  79. Kraichnan, R. H., 1967, Phys. Fluids 10, 1417.
  80. Kraichnan, R. H., 1971, J. Fluid Mech. 47, 513.
  81. Kuvshinov, B. N., and T. J. Shep, 2000, Phys. Fluids 12, 3282.
  82. Lamb, H., 1932, Hydrodynamics (Cambridge University Press, Cambridge, England).
  83. Landau, L. D., 1937a, Zh. Eksp. Teor. Fiz. 7, 19.
  84. Landau, L. D., 1937b, Zh. Eksp. Teor. Fiz. 7, 627.
  85. Landau, L. D., and E. M. Lifschitz, 1980, Statistical Physics, Part 1, 3rd ed. (Pergamon, Oxford, England).
  86. Landau, L. D., and E. M. Lifschitz, 1987, Fluid Mechanics, 2nd ed. (Pergamon, Oxford, England).
  87. Lax, P. D., 1972, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Vol. 11 of CBMS Regional Conference Series in Applied Math (Society for Industrial and Applied Mathematics, Philadelphia).
  88. Lebowitz, J. L., 1995, Ed., J. Stat. Phys. 78, 1, special issue in honor of the 90th birthday of Lars Onsager.
  89. Lee, T. D., 1951, J. Appl. Phys. 22, 524.
  90. Lee, T. D., 1952, Q. Appl. Math. 10, 69.
  91. Levanyuk, A. I., 1959, Zh. Eksp. Teor. Fiz. 36, 810.
  92. Lin, C.-C., 1941, Proc. Natl. Acad. Sci. U.S.A. 27, 570.
  93. Lin, C.-C., 1943, On the Motion of Vortices in Two Dimensions (Toronto University Press, Toronto).
  94. Lin, C.-C., 1945a, letter to T. von Kármán, 4 September. PTvK, CIT, 18.22. *
  95. Lin, C.-C., 1945b, letter to L. Onsager, 26 June. PTvK, CIT, 18.22. *
  96. Lin, C.-C., 1945c, letter to L. Onsager, 4 September. PTvK, CIT, 18.22. *
  97. Lin, F. H., and J. X. Xin, 1999, Commun. Math. Phys. 200, 249.
  98. London, F., 1948, Phys. Rev. 74, 562.
  99. Longuet-Higgins, M., and M. Fisher, 1995, J. Stat. Phys. 78, 605.
  100. Lundgren, T. S., and Y. B. Pointin, 1977, J. Stat. Phys. 17, 323.
  101. Lynden-Bell, D., 1967, Mon. Not. R. Astron. Soc. 136, 101.
  102. Machlup, S., and L. Onsager, 1953, Phys. Rev. 91, 1512.
  103. MacPhail, D. C., 1940, J. Aeronaut. Sci. 8, 73.
  104. Mandelbrot, B. B., 1969, in Turbulence of Fluids and Plasmas, edited by E. Weber (Interscience, New York).
  105. Mandelbrot, B. B., 1972, in Statistical Models and Turbulence, Vol. 12 of Lecture Notes in Physics, edited by M. Rosenblatt and C. V. Atta (Springer, New York), pp. 333–351.
  106. Mandelbrot, B. B., 1974, J. Fluid Mech. 62, 331.
  107. Marchioro, C., and M. Pulvirenti, 1994, Mathematical Theory of Incompressible Nonviscous Fluids (Springer, New York).
  108. Mazur, P., 1996, in The Collected Works of Lars Onsager (with commentary), edited by P. C. Hemmer, H. Holden, and S. K. Ratkje (World Scientific, Singapore), pp. 61–70.
  109. McCauley, J. L., Jr., 1977, J. Phys. C 10, 689.
  110. McCauley, J. L., Jr., 1995, J. Stat. Phys. 78, 531.
  111. Meixner, J., 1943, Z. Phys. Chem. Abt. B 53, 235.
  112. Meneveau, C., and K. R. Sreenivasan, 1991, J. Fluid Mech. 224, 429.
  113. Messer, J., and H. Spohn, 1982, J. Stat. Phys. 29, 561.
  114. Miller, D. G., 1995, J. Stat. Phys. 78, 563.
  115. Miller, J., 1990, Phys. Rev. Lett. 65, 2137.
  116. Miller, J., P. B. Weichman, and M. C. Cross, 1992, Phys. Rev. A 45, 2328.
  117. Montgomery, D., and G. Joyce, 1974, Phys. Fluids 17, 1139.
  118. Montgomery, D., W. H. Matthaeus, W. T. Stribling, D. Martínez, and S. Oughton, 1992, Phys. Fluids A 4, 3.
  119. Montgomery, D., X. Shan, and W. H. Matthaeus, 1993, Phys. Fluids A 5, 2207.
  120. Nielsen, N. K., and B. Schroer, 1977, Nucl. Phys. B 127, 493.
  121. Novikov, E. A., 1975, Sov. Phys. JETP 41, 937.
  122. Obukhov, A. M., 1941a, Dokl. Akad. Nauk SSSR 32, 22.
  123. Obukhov, A. M., 1941b, Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 5, 453.
  124. O’Neil, K. A., and R. A. Redner, 1991, J. Stat. Phys. 62, 399.
  125. Onsager, L., 1926, Phys. Z. 27, 388.
  126. Onsager, L., 1931a, Phys. Rev. 37, 405.
  127. Onsager, L., 1931b, Phys. Rev. 38, 2265.
  128. Onsager, L., 1932, Phys. Rev. 40, 1024.
  129. Onsager, L., 1933, Chem. Rev. (Washington, D.C.) 13, 73.
  130. Onsager, L., 1936, J. Am. Chem. Soc. 58, 1486.
  131. Onsager, L., 1939a, J. Chem. Phys. 43, 189.
  132. Onsager, L., 1939b, Phys. Rev. 55, 1109.
  133. Onsager, L., 1944, Phys. Rev. 65, 117.
  134. Onsager, L., circa 1945, private notes. LOA, NTNU, 11.129. †
  135. Onsager, L., 1945a, letter to L. Pauling, 15 March. PTvK, CIT, 18.22. *
  136. Onsager, L., 1945b, letter to C. Lin, June. PTvK, CIT, 18.22. *
  137. Onsager, L., 1945c, Phys. Rev. 68, 281.
  138. Onsager, L., 1945d, Ann. N.Y. Acad. Sci. 46, 241.
  139. Onsager, L., 1949a, Nuovo Cimento, Suppl. 6, 249.
  140. Onsager, L., 1949b, Nuovo Cimento, Suppl. 6, 261.
  141. Onsager, L., 1949c, Ann. N.Y. Acad. Sci. 51, 627.
  142. Onsager, L., 1949d, Nuovo Cimento, Suppl. 6, 279.
  143. Onsager, L., 1952, Philos. Mag. 43, 1006.
  144. Onsager, L., 1953a, in Proceedings of the International Conference on Theoretical Physics, Kyoto and Tokyo, Sept. 1953 (Science Council of Japan, Tokyo), pp. 669–675.
  145. Onsager, L., 1953b, in Proceedings of the International Conference on Theoretical Physics, Kyoto and Tokyo, Sept. 1953 (Science Council of Japan, Tokyo), pp. 877–880.
  146. Onsager, L., 1964, J. Am. Chem. Soc. 86, 3421.
  147. Onsager, L., 1973, in Physics and Chemistry of Ice, edited by E. Whalley, S. Jones, and L. Gold (Royal Society of Canada, Ottawa), pp. 7–12.
  148. Onsager, L., 1974a, in Structure of Water and Aqueous Solutions, Proceedings of the International Symposium, Marburg, July 1973, edited by W. Luck (Verlag Chemie, Weinheim), pp. 1–7.
  149. Onsager, L., 1974b, in Quantum Statistical Mechanics in the Natural Sciences, Coral Gables Conference, 1973, edited by S. Mintz and S. Widmayer (Plenum, New York), pp. 1–14.
  150. Onsager, L., circa 1975, private notes. LOA, NTNU, 11.132. †
  151. Onsager, L., and M.-S. Chen, 1977, J. Phys. Chem. 81, 2017.
  152. Onsager, L., and M. Dupuis, 1960, in Rend. Sci. Int. Fis. “Enrico Fermi,” Corso X, Varenna, 1959 (N. Zanichelli, Bologna), pp. 294–315.
  153. Onsager, L., W. H. Furry, and R. C. Jones, 1939, Phys. Rev. 55, 1083.
  154. Onsager, L., J. B. Hubbard, W. M. van Beek, and M. Mandel, 1977, Proc. Natl. Acad. Sci. U.S.A. 74, 401.
  155. Onsager, L., and B. Kaufman, 1947, in Rep. Intern. Conf. on Fund. Particles and Low Temp., Cambridge, July 1946 (The Physical Society, London), Vol. 2, pp. 137–144.
  156. Onsager, L., and S. Kim, 1977, J. Phys. Chem. 81, 1211.
  157. Onsager, L., and S. Machlup, 1953, Phys. Rev. 91, 1505.
  158. Onsager, L., and W. W. Watson, 1939, Phys. Rev. 56, 474.
  159. Orszag, S. A., 1970, J. Fluid Mech. 41, 363.
  160. Orszag, S. A., 1977, in Fluid Dynamics, edited by R. Balian and J.-L. Peube (Gordon and Breach, London), pp. 235–373.
  161. Parisi, G., and U. Frisch, 1985, in Turbulence and Predictability in Geophysical Fluid Dynamics, Proceedings of the International School of Physics “E. Fermi,” 1983, Varenna, Italy, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam), pp. 84–87.
  162. Pauling, L., 1945, letter to L. Onsager, 6 April. PTvK, CIT, 18.22. *
  163. Penrose, O., and L. Onsager, 1956, Phys. Rev. 104, 576.
  164. Poincaré, H., 1893, Théorie des Tourbillons (Georges Carré, Paris).
  165. Polyakov, A. M., 1993, Nucl. Phys. B 396, 367.
  166. Prigogine, I., 1947, Etude Thermodynamique des Processus Irréversibles (Desoer, Liége).
  167. Proudman, I., and W. D. Reid, 1954, Philos. Trans. R. Soc. London, Ser. A 247, 163.
  168. Purcell, E. M., and R. V. Pound, 1951, Phys. Rev. 81, 279.
  169. Robert, R., 1990, C. R. Acad. Sci., Ser. I: Math. 311, 575.
  170. Robert, R., 1991, J. Stat. Phys. 65, 531.
  171. Robert, R., 2003, in Handbook of Mathematical Fluid Dynamics, edited by S. Friedlander and D. Serre (Elsevier, Amsterdam), Vol. 2, pp. 1–54.
  172. Robert, R., and J. Sommeria, 1991, J. Fluid Mech. 229, 291.
  173. Rose, H. A., and P. L. Sulem, 1978, J. Phys. (Paris) 39, 441.
  174. Saffman, P. G., 1968, in Topics in Nonlinear Physics, edited by N. Zabusky (Springer-Verlag, New York), pp. 485–614.
  175. Schwinger, J., 1951, Phys. Rev. 82, 664.
  176. Smith, R. A., and T. M. O’Neil, 1990, Phys. Fluids B 2, 2961.
  177. Sommeria, J., C. Staquet, and R. Robert, 1991, J. Fluid Mech. 233, 661.
  178. Sreenivasan, K. R., 1984, Phys. Fluids 27, 1048.
  179. Sreenivasan, K. R., 1995, in Developments in Fluid Dynamics and Aerospace Engineering, edited by S. Deshpande, A. Prabhu, K. Sreenivasan, and P. Viswanath (Interline, Bangalore), pp. 159–190.
  180. Sreenivasan, K. R., 1998, Phys. Fluids 10, 528.
  181. Sreenivasan, K. R., and B. Dhruva, 1998, Prog. Theor. Phys. 130, 103.
  182. Staebler, D. L., S. Mascarenhas, and L. Onsager, 1978, J. Chem. Phys. 68, 3823.
  183. Stolovitzky, G., P. Kailasnath, and K. R. Sreenivasan, 1992, Phys. Rev. Lett. 69, 1178.
  184. Sulem, P.-L., and U. Frisch, 1975, J. Fluid Mech. 72, 417.
  185. Taylor, G. I., 1935, Proc. R. Soc. London, Ser. A 157, 421.
  186. Taylor, G. I., 1937, J. Aeronaut. Sci. 4, 311.
  187. Taylor, G. I., 1938, Proc. R. Soc. London, Ser. A 164, 476.
  188. Taylor, G. I., and A. E. Green, 1937, Proc. R. Soc. London, Ser. A 158, 499.
  189. Taylor, J. B., 1972, Phys. Lett. 40A, 1.
  190. Thouless, D., P. Ao, Q. Niu, M. Geller, and C. Wexler, 1999, Int. J. Mod. Phys. B 13, 675.
  191. Tollmien, W., 1933a, Z. Angew. Math. Mech. 13, 331.
  192. Tollmien, W., 1933b, Physics (N.Y.) 4, 289.
  193. Trubridge, G. F. P., 1934, Rep. Prog. Phys. 1, 43.
  194. Turkington, B., A. Majda, K. Haven, and M. DiBattista, 2001, Proc. Natl. Acad. Sci. U.S.A. 98, 12346.
  195. von Kármán, T., 1911, Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl. 12, 509.
  196. von Kármán, T., 1912, Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl. 13, 547.
  197. von Kármán, T., 1937, J. Aeronaut. Sci. 4, 131.
  198. von Kármán, T., C. Lin1945, letter to , 23 August. PTvK, CIT, 18.22. *
  199. von Kármán, T., and L. Howarth, 1938, Proc. R. Soc. London, Ser. A 164, 192.
  200. von Neumann, J., 1949, recent theories of turbulence, reproduced in Collected Works of J. von Neumann (1949–1963), Vol. 6, edited by A. H. Taub (Pergamon, New York, 1963), pp. 437–472.
  201. von Weizsäcker, C. F., 1948, Z. Phys. 124, 614.
  202. Watson, W. W., L. Onsager, and A. Zucker, 1949, Rev. Sci. Instrum. 20, 924.
  203. Wiener, N., 1938, Am. J. Math. 60, 897.
  204. Yakhot, V., and K. R. Sreenivasan, 2004, Physica A 343, 147.
  205. Yang, C. N., 1952, Phys. Rev. 85, 808.
  206. Yang, C. N., 1996, HFPN—Number 06, March 15. Available online at http://chris.kias.re.kr/yang.htm
  207. Yeung, P. K., and J. G. Brasseur, 1991, Phys. Fluids A 3, 884.
  208. Yin, Z., D. C. Montgomery, and H. J. H. Clercx, 2003, Phys. Fluids 15, 1937.
  209. Zygmund, A., 2002, Trigonometric Series, 3rd ed. (Cambridge University Press, Cambridge, England).

Outline

Information

Rev. Mod. Phys. 78, 87– Published 17 January, 2006

    © 2006 American Physical Society

    Sign In to Your Journals Account

    Filter

    Filter

    Article Lookup

    Enter a citation