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On a New Theory of free Turbulence*

Published online by Cambridge University Press:  28 July 2016

H. Reichardt
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Extract

Measurements of momentum in regions of turbulent mixing show that an analogy exists between the processes of turbulent and molecular balance which enables a rational application of the differential equation of thermal conduction to the propagation of turbulent momentum to be made. This fact forms the basis of a new phenomenological theory of free turbulent flows.

Type
Research Article
Information
The Aeronautical Journal , Volume 47 , Issue 390 , June 1943 , pp. 167 - 176
Copyright
Copyright © Royal Aeronautical Society 1943

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Footnotes

*

Reprinted by permission of the Ministry of Aircraft Production (R.T.P.).

References

1 Prandtl, L.. Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z.A.M.M., Vol. 5, 1925, p. 136 Google Scholar.

2 cf. Footnote 1.

3 The error curve naturally can no longer hold on the outermost boundaries, because the potential motion of the outer space sets a limit on the turbulent propagation.

The velocity distributions measured in the two-dimensional free jet by Forthmann were recently represented by Okaya and Hasegawa by the equation

where λ=y/x and k=const. The authors discuss the compatibility of this velocity formula with the mixing path theorem (2). (Proc. Physico. Math. Soc. Japan, No. 3, Vol. 22, 1940.)

4 The velocity as such is not physically effective, but the momentum ρu 2. When ρu 2 is represented by an error curve, the distribution of u is also an exponential function. In wake regions the loss of momentum is approximately –2Uu 1 (U = constant principal velocity, u 1,= velocity loss). Here therefore (with moderately large values of u 1) loss of momentum and loss of velocity are proportional to each other.

5 H. Muttray. Die experimentallen Tatsachen des Widerstandes ohne Auftrieb. Hdbk. d. Exper.Phys. von Wien-Harms. IV. 2, p. 325, 1932.

6 cf. Ergeb. d. Aerodyn. Vers. Göttingen, 2nd Ed., 1923, p. 69; further, Cordes, G., Ing. Arch., Vol. 8, 1937, p, 245 CrossRefGoogle Scholar; v., Bohl, Ing-Arch., Vol. 11, 1940, p. 295 Google Scholar. (Cordes' measurement points deviate only slightly from the error integral curve.)

7 Tollmien, W.. Berechnung turbulenter Ausbreitungsvorgänge. Z.A.M.M., Vol. 6, 1926, p. 468 Google Scholar.

8 It is unreasonable to demand too great a degree of accuracy, because with intense turbulence (particularly in the boundry regions) systematic measurement errors are bound to appear. Besides this, the difference between the measured total pressure and the momentum ρu 2 is of the order of value of the deviations of the measurement points from the error curve or the error integral curve.

It might moreover also be conceivable that p + ρu 2 and not ρu 2 is statistically distributed (equation (10) would then be modified accordingly). Such refinements, which in themselves are of fundamental significance, cannot, however, be assessed at the moment, since a sufficiently exact method for the determination of the static pressure is still lacking.

9 A detailed exposition of the laws governing free turbulence on the basis of the theory developed here will be published shortly as a VDI-Forschungshef t. In this paper also the suitability of the statistical functions for the representation of the variation of momentum (impulse) is. proved in detail on the strength of a large number of ’ measurements.

10 Gran Olsson, R.. Geschwindigkeits-und Temperaturverteilung hinter einem Gitter bei turbulenter Strömung. Z.A.M.M., Vol. 16, 1936, p. 257 Google Scholar.

11 From the statistical standpoint the factors and are obviously equivalent and the characteristics of these terms are solely of a hydrodynamical nature. The equality of and seems plausible in as much as the steady v-motion also is a direct result of the action of the turbulence on the fluid. (Without the turbulence, with the postulated absence of friction, there would be no v-component and only the u-velocity would vary discontinuously at the jet boundaries).

12 Tollmein, W.. Uber die Korrelation der Geschwindigkeitskomponenten von periodisch schwankenden Wirbelverteilungen. Z.A.M.M., Vol. 15, 1935, p. 96 Google Scholar.

13 Schlichting, H.. Uber das ebene Windschattenproblem Ing-Arch, Vol. 1, 1930, p. 533 CrossRefGoogle Scholar.