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On the Calculation of the Boundary Layer*

Published online by Cambridge University Press:  28 July 2016

L. Prandtl Gottingen
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Extract

The field of viscosity forces in fluid currents of low viscosity—i.e., high Reynolds number—is frequently confined to a narrow zone in the vicinity of the surface; consequently, calculation can be simplified by the following assumptions :—

1. In the friction zone, customarily called the “ boundary layer,” only the most effective components of the viscosity force μΔω, are considered—e.g., for the x component of this force, μΔu (calling the direction of main flow parallel to the surface, the x direction, and that perpendicular thereto the y direction) the expressions μ (∂2u/∂x2) and μ (∂2u/∂z2) can be neglected for μ (∂2u/∂y2).

Type
Research Article
Information
The Aeronautical Journal , Volume 44 , Issue 359 , November 1940 , pp. 795 - 801
Copyright
Copyright © Royal Aeronautical Society 1940

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Footnotes

*

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

References

1 Or, more exactly, by a frictionless flow along a surface increased in thickness by the ” displacement “

2 Summary accounts can be found in Tollmien's report in “ Handbuch der Experimentalphysik,“ Vol. IV, 1, p. 241, also, L. Howarth, “ Technical Report of the Aeron. Research Committee “ for 1934, Vol. I, p. 320 (1936). An instructive introduction to the author's paper in Durand, “ Aerodynamic Theory,” Vol. III, pp. 80-119.

3 In the text of the paper (Trans. Int. Math. Congress, Leipzig, 1905—cf. also, reprint cf Prandtl-Betz: Four contributions to Hydrodynamics and Aerodynamics, Gottingen, 1927, pp. 487 and 4) after quoting the above Eq. (1) and (2), it says: “ Let dp/dx be given with certainty, as well as the development of u for the initial section, then any such problem can be numerically evaluated, the appropriate ∂u/∂x being obtained by successively squaring all values of u; then, with the help of any known approximation formula, successive steps can be effected in the X — direction. It is true that a difficulty is presented by sundry singularities occurring at the fixed margin.”

4 Together with u, ∂u/∂x is also = 0, for y = 0; but

5 In the spring of 1914 I occupied myself with the flow through narrow channels of slender section, and when applying the usual boundary layer equations, encountered difficulties in the formulation of the boundary conditions, u=v=0, at the second margin. I attempted then to introduce the function of flow as an independant variable, as both margins could then be expressed simply as ψ = 0, and ψ = Q = const. Eq. (5) was rediscovered by Herr v. Mises, and reported at the Kissinger meeting of the Gesellsch. fur angew. Math, und Mech., in the autumn of 1927—see this journal, Vol. 7 (1927), p. 428. As I had not yet disclosed my method, the “ priority ” in the customary sense, belongs to Herr v. Mises. See further my comment on v. Mises' paper in this journal, Vol. 8 (1928), p. 249, etc., particularly Sec. 2, p. 250.

6 “ What should have been the distribution in. the preceding minute (or hour) to produce the existing irregular temperature distribution.” Evidently, the problem is frequently insoluble!

7 Details can be found in Goldstein's work: “Concerning some solutions of the boundary layer equations in hydrodynamics,” Proc. Cambr. Philos. Sec., Vol. 261, 1930, p. 1.

8 I am obliged to my collaborator, Dr. H. Gortler, for this solution. He has also supplied proof that a solution of f = e+αx, satisfying the boundary conditions, does not exist.

9 The asymptotic formula for J ± 1/3 furnishes a wave length λ proportionate to and a fading constant α=v/ {ay . (λ2π)2}.