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On the velocity and temperature distributions in the turbulent wake behind a heated body of revolution

Published online by Cambridge University Press:  24 October 2008

S. Goldstein
S. Goldstein
Affiliation:
St John's College
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Extract

The velocity distribution in the wake behind a body of revolution calculated by Miss Swain on the momentum transfer theory with l constant over a cross-section of the wake is not in accordance with experiment.

On the generalized vorticity transfer theory the equation for the first approximation to the velocity defect u far downstream is

There are two methods of simplifying this equation on the analogy of the successful simplification of the corresponding equation for the two-dimensional case. One method gives the same equation as if the turbulence were symmetrical; the other gives the equation of the modified vorticity transfer theory.

For the former there is no real solution if l is constant over a cross-section; for a real solution l must be infinite on the axis of symmetry. The equation was integrated with l ∝ r−1 over a cross-section; this variation of l was suggested by putting the change in the area enclosed by a circular vortex-line proportional to the area of the section of the wake. The agreement with experiment is fairly good over the middle portion of the wake, but not near the edge, where the theoretical value of ∂u/∂r is not zero. If l2 ∝ rp, ∂u/∂r vanishes at the edge of the wake only if p = − 3. The solution is carried through for this case; it shows only fair agreement with experiment over the middle of the wake, and none at the edge.

The calculation on the modified vorticity transfer theory is carried out with l constant over a section; in this case there is no definite edge to the wake, nor does u fall off sufficiently rapidly to make the momentum integral for the drag converge. (u ∝ r−2 for large r.) Even over the middle of the wake there is a discrepancy with experimental results, rather like that shown by the momentum transfer theory.

The temperature distribution behind a heated body was calculated according to each of the foregoing theories.

The temperature distribution was also calculated from the experimental velocity distribution, without any assumption being made concerning the kinematic coefficient of turbulent diffusion, which was eliminated between the equations for the velocity and temperature. The vorticity transfer theory with symmetrical turbulence gives an impossible result. The modified vorticity transfer theory gives θ/θmax < u/umax when the experimental velocity distribution is used.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 34 , Issue 1 , January 1938 , pp. 48 - 67
Copyright
Copyright © Cambridge Philosophical Society 1938

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References

* Prandtl, , Zeits. f. angew. Math. u. Mech. 5 (1925), 137, 138;Google ScholarVerhandl. d. 2ten internat. Kongresses f. techn. Mechanik (Zürich, 1926), pp. 62–74.Google Scholar

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* Goldstein, , Proc. Cambridge Phil. Soc. 31 (1935), 357,Google Scholar equation (35).

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* Goldstein, , Proc. Cambridge Phil. Soc. 31 (1935), 358,Google Scholar equation (43).

* Proc. Roy. Soc. 151 (1935), 495–7;Google Scholar 159 (1937), 499–502.

* Fage, , Phil. Mag. (7), 21 (1936), 95–105.CrossRefGoogle Scholar

* If in place of (34) we had taken the last terms in (80) and (81) would have been absent. The velocity distribution would have been the same as on the momentum transfer theory, and the temperature distribution would have been the square root of the velocity distribution.

* It is hoped to publish experimental results shortly. The only results so far published appear in a small-scale diagram in a paper by R. Gran Olsson, Zeits. f. angew. Math. u. Mech. 16 (1936), 273, Fig. 18, where θ/θ max is plotted against r/R′, R′ being the value of r at which θ/θmax = ½.