Abstract
In a series of papers on the resistance experienced by a fluid in turbulent motion and on the application of statistical mechanics to the theory of turbulent fluid motion, it was attempted to obtain a picture of the relative probabilities of the various possible patterns of the secondary motion, in order to arrive at a calculation of the magnitude of the turbulent shearing stress and of the resistance experienced by the primary motion in consequence of this stress1). Serious difficulties, however, were encountered, which did not permit to bring the calculations to a satisfactory result, and which gave rise to doubts concerning the suitability of the method applied. In the last paper of the series it was suggested that it might be worth while to investigate the properties of certain systems of mathematical equations, which, although much simpler in structure than the equations of hydrodynamics, nevertheless show features which can be considered as the analogues of typical properties of the hydrodynamic equations.
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References
1) J. M. Burgers, On the resistance experienced by a fluid in turbulent motion, Proc. Acad. Amsterdam 26, p. 582, 1923
On the application of statistical mechanics to the theory of turbulent fluid motion, J. M. Burgers, On the resistance experienced by a fluid in turbulent motion, Proc. Acad. Amsterdam 32, pp. 414, 643, 818, 1929
J. M. Burgers, On the resistance experienced by a fluid in turbulent motion, Proc. Acad. Amsterdam 36, pp. 276, 390, 487, 620, 1933.
Compare e.g.: G. I. Taylor, Statistical theory of turbulence, Proc. Roy. Soc. (London) A 151, pp. 421–478, 1935
G. I. Taylor, Statistical theory of turbulence, Proc. Roy. Soc. (London) A 156, pp. 307–317, 1936
G. I. Taylor Journ. Aeron. Sciences 4, p. 311, 1937
Some recent developments in the study of turbulence, Proc. Vth Intern. Congr. for Applied Mechanics, Cambridge, Mass., 1938, p. 294.
Th. Von Karman, The fundamentals of the statistical theory of turbulence, Journ. Aeron. Sciences 4, p. 131, 1937
On the statistical theory of turbulence, Proc. Nat. Acad, of Sciences (Washington) 23, p. 98, 1937
Some remarks on the statistical theory of turbulence, Proc. Vth Intern. Congr. for Applied Mechanics, Cambridge, Mass., 1938, p. 347.
Th. De Karman and L. Howarth, On the statistical theory of isotropic turbulence, Proc. Roy. Soc. (London) A 164, pp. 192–215, 1938.
G. Darrieus, Contribution à l’analyse de la turbulence en tourbillons cellulaires, Proc. Vth Intern. Congr. for Applied Mechanics. Cambridge, Mass., 1938, p. 422.
These conclusions have been deduced by Von Karman with the aid of the “similarity hypothesis”. See: Th. Von Karman, Mechanische Ähnlichkeit und Turbulenz, Nachr. Gesellscih. d. Wiss. Göttingen (math.-physik. Kl.) 1930, p. 5&.
These features can already be deduced by means of elementary considerations,
G. I. Taylor and A. E. Green, Mechanism of the production of small eddies from large ones, Proc. Roy. Soc. (London) A 158, pp. 499–521, 1937.
G. I. taylor, Production and dissipation of vorticity in a turbulent field, Proc. Roy. Soc. (London) A 164, pp. 15–23, 1938
The spectrum of turbulence, G. I. taylor, Production and dissipation of vorticity in a turbulent field, Proc. Roy. Soc. (London) A 164, pp. 476–490, 1938.
G. Darrieus, l.c.. footnote 2) above.
O. Reynolds, Scientific Papers II, p. 535. — H. A. Lorentz, Abhandl. über theor. Phvsik I (Leipzig u. Berlin 1907), p. 43; Collected Papers IV (The Hague 1937), p. 15.
This has been pbserved already by O. Reynolds, l.c. footnote 6) above, p. 54L under (2) (c). -- See further the papers mentioned in footnote 5) above.
Compare B. Rlemann--H. Weber. Die part. Differentialgleich. d. mathem. Physik. Band I (Braunschweig 1900). p. 494 seq.
fig. 3 gradually will change into another type.
The limit appearing here coincides with the number M given by (10.4), which number determined the number of stationary solutions of eq. (7.2).
Compare E. T. Whittaker and G. N. Watson, Modem Analysis (3rd Ed., Cambridge 1920). p. 167, Art. 9.3.
B. Riemann—H. Weber, Die part. Differentialgleich. d. mathem. Physik, Band II (Braunschweig 1901), Abschnitt 22 u. 23.
Compare again E. T. Whittaker and G. N. Watson, Modern Analysis (3rd Ed., Cambridge 1920), p. 167, Art. 9.3, where also the conditions have been stated to which the course of the function f(y)is subjected.
Compare e.g.: B. Van Der Pol, Phil. Mag. (VII) 2. p. 978, 1926
B. Van Der Pol Jahrb. d. drahtl. Telegr. 28, p. 178, 1926
B. Van Der Pol Jahrb. d. drahtl. Telegr. 29, p. 114, 1927.
The importance of the mean values of certain products of the third degree comes forward also from the work of Von Karman (Compare Th. De Karman and L. Howarth, Proc. Roy. Soc. (London) A164, p. 201 “Triple correlations”, 1938. See also G. Darrieus, l.e. footnote 2), above).
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Burgers, J.M. (1995). Mathematical Examples Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion. In: Nieuwstadt, F.T.M., Steketee, J.A. (eds) Selected Papers of J. M. Burgers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0195-0_10
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