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Pressure fluctuations in isotropic turbulence

Published online by Cambridge University Press:  24 October 2008

G. K. Batchelor
G. K. Batchelor
Affiliation:
Trinity CollegeCambridge
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Abstract

This paper considers the correlation P(r) between the fluctuating pressures at two different points distance r apart in a field of homogeneous isotropic turbulence. P(r) can be expressed in terms of the fourth moment of the velocity fluctuation, which is evaluated with the aid of the hypothesis that fourth moments are related to second moments in the same way as for a normal joint distribution of the velocities at any two points. The experimental evidence relevant to this hypothesis (which cannot be exactly true since it gives zero odd-order moments) is examined. The alternative hypothesis made by Heisenberg, that the Fourier coefficients of the velocity distribution are statistically independent, has identical consequences for the fourth moments of the velocity, although it does not lead to such convenient results.

The pressure correlation is worked out in detail for the important special case of very large Reynolds numbers of turbulence; the mean-square pressure fluctuation is found to be . The mean-square pressure gradient is evaluated, from the available data concerning the doublevelocity correlation, for the cases of very small and very large Reynolds numbers, and a simple interpolation between these results is suggested for the general case. Finally, the relation between the mean-square pressure gradient and rate of diffusion of marked fluid particles from a fixed source is established without the neglect of the viscosity effect, and the available observations of diffusion are used to obtain estimates of which are compared with the theoretical values.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 2 , April 1951 , pp. 359 - 374
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Heisenberg, W.Z. Phys. 124 (1948), 624.Google Scholar
(2)Obukhov, A. M.C.R. Acad. Sci. U.R.S.S. 66 (1949), 17 (Translation issued by Ministry of Supply as TPA3/TIB Trans. No. T3757).Google Scholar
(3)Townsend, A. A.Proc. Cambridge Phil. Soc. 43 (1947), 560.CrossRefGoogle Scholar
(4)Stewart, R. W.Proc. Cambridge Phil. Soc. 47 (1951), 146.CrossRefGoogle Scholar
(5)Millionshtchikov, M.C.R. Acad. Sci. U.R.S.S. 5 (1944), 134.Google Scholar
(6)Batchelor, G. K.Proc. Roy. Soc. A, 195 (1949), 513.Google Scholar
(7)Hopf, E.Commun. Appl. Math. 1 (1948), 303.CrossRefGoogle Scholar
(8)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford: Clarendon Press, 1937).Google Scholar
(9)Proudman, I.Proc. Cambridge Phil. Soc. 47 (1951), 158.CrossRefGoogle Scholar
(10)Taylor, G. I.Proc. Roy. Soc. A, 151 (1935), 465.Google Scholar
(11)Batchelor, G. K. and Townsend, A. A.Proc. Roy. Soc. A, 193 (1948), 539.Google Scholar
(12)Batchelor, G. K.Proc. Cambridge Phil. Soc. 43 (1947), 533.CrossRefGoogle Scholar
(13)Chandrasekhar, S.Proc. Roy. Soc. A, 200 (1949), 20.Google Scholar
(14)Robertson, H. P.Proc. Cambridge Phil. Soc. 36 (1940), 209.CrossRefGoogle Scholar
(15)Batchelor, G. K. and Townsend, A. A.Proc. Roy. Soc. A, 190 (1947), 534.Google Scholar
(16)Collis, D. C.Div. of Aeronautics, Coun. Sci. Industr. Res. Australia, Rep. A 55 (1948).Google Scholar