Abstract
The study of the properties of the turbulent flow of a fluid through a cylindrical tube or between parallel walls forms one of the most interesting problems of hydrodynamics. The opinion has been expressed by various authors that this problem has to be attacked by the use of statistical methods. 1) When compared with the classical problem of the kinetic theory of gases, however, the hydrodynamical case presents the difficulty that the system is not a conservative one: energy is continually being dissipated in consequence of the viscosity of the fluid, so that work has to be supplied from without in order that the mean flow may present a stationary character. A second difficulty is that it is not immediately clear what elementary processes or types of motion can be used as “objects” to be counted in order to arrive at a definition of the probability of a given type of flow.
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References
Comp. f. i. Th. VON Karman, Zeitschr. f. ang. Math. u. Mech. 1, p. 250, 1921
Proc. Ist. Intern. Congress for Appl. Mech., Delft, 1924, p. 105
R. VON Mises, Zeitschr. f. ang. Math. u. Mech. 1, p. 428, 1921
also a paper by the present writer in these Proc. 26, p. 582, 1923.
O. Reynolds, Scientific Papers II, p. 535.
O. Reynolds, l.c.; H. A. Lorentz, Abhandl. über theoret. Physik, I, p. 43.
Compare O. Reynolds and H. A. Lorentz, l.l.c.c., and also J. M. Burgers, these Proc. 26, p. 585, 1923.
Comp. J. M. Burgers, l.c., equation (14).
Comp f.i.). J. H. Jeans, The dynamical theory of gases (Cambridge 1916), p. 73.
J. H. Jeans, l.c. p. 72.
E. Trefftz, Zu den Grundlagen der Schwingungstheorie, Mathem. Ann. 95, p. 307, 1925.
As might be inferred from the circumstance that the Reynolds’ number must be sufficiently high, the question will be connected with the problem of the stability of the turbulent motion, which has been considered by various authors (comp. f.i. F. Noether, Zeitschr. f. ang. Math. u. Mech. 1, p. 125, 1921), and about which opinions still have not been settled. It has been supposed that the case of a fluid, moving between fixed walls under the action of a pressure gradient (or of gravity, when the flow is directed downwards), gives a better chance for a stable turbulent motion, than the case considered here. The formulae of § 2 can be extended to this case also; the principal change occurs in the dissipation condition.
Another deduction will be given in Part II. (Note added in the proof).
The question whether it is possible to make use of canonical variables, however, still deserves attention, though it seems that the stream function at any rate leads to more simple formulae. In the case of the motion in an ideal fluid of parallel rectilinear vortices, the diameters of which are very small compared to their distances, canonical variables can be introduced according to a method developed by Kirchhoff and by Lagally (comp. M. Lagally, Sitz. Ber. Munch. Akad. p. 377, 1914). For these coordinates Liouville’s theorem can be proved. In applying statistical methods now the kinetic energy of the motion has to be given.
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Burgers, J.M. (1995). Hydrodynamics. — On the application of statistical mechanics to the theory of turbulent fluid motion. I. 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_5
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