Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T05:09:34.362Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Calculation of the Boundary Layer*

Published online by Cambridge University Press:  28 July 2016

L. Prandtl Gottingen
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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}.