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Concerning some Solutions of the Boundary Layer Equations in Hydrodynamics

Published online by Cambridge University Press:  24 October 2008

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

The boundary layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated). This initial velocity distribution is assumed expressible as a polynomial in the distance from the wall. Three cases are considered: first, when in the initial distribution the velocity vanishes at the wall but its gradient along the normal does not; second, when the velocity in the initial distribution does not vanish at the wall; and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary). The solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations. Some progress is made in the numerical calculation of these coefficients, especially in the first case. The main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious.

The same mathematical method is used to calculate the flow behind a flat plate along a stream. The results are shown in Figures 1 and 2, drawn from Tables III and IV.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 1 , January 1930 , pp. 1 - 30
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

* The equations were first given by Prandtl, , Verhandl. d. III intern. Math.-Kongresses, Heidelberg, 1904; reprinted in Vier Abhandlungen zur Hydrodynamik und Aerodynamik, L. Prandtl and A. Betz, Göttingen, 1927.Google Scholar Prandtl's method is more fully given by Blasius, , Zeitschrift f. Math. u. Phys. 56, 1 (1908).Google Scholar Concerning the derivation of the equations, see also Kármán, v., Zeitschrift f. angewandte Math. und Mech. (Z.A.M.M.) 1, 233 (1921);CrossRefGoogle ScholarPolhausen, , Z.A.M.M. 1, 252 (1921);Google ScholarBairstow, , Journal Roy. Aeronautical Society, 29, 3 (1925);CrossRefGoogle Scholar and Mises, v., Z.A.M.M. 7, 425 (1927).Google Scholar

* Terms of order multiplied by the gradient of the curvature along the surface are also neglected. See Bairstow, loc. cit.

† For recent attempts, see Oseen's, Hydrodynamik (Leipzig, 1927), including as appendix two lectures by Zeilon at the International Congress for Technical Mechanics, Zurich, 1926.Google Scholar Reference may also be made to Burgers, , Proc. Roy. Acad. Sci. Amsterdam, 31, 433 (1928).Google Scholar

‡ In flow along a curved wall there is a region in which the pressure increases in the direction of motion; in this region the forward flow in the boundary layer is forced to leave the wall, and the fluid in it, having acquired vorticity, mixes in the main stream. See Prandtl, , loc. cit. and Journal Roy. Aeronautical Soc. 31, 720 (1927).Google Scholar Also Blasius, loc. cit. Many popular and semi-popular expositions have been published, mainly in connection with the rotor ship (flow past rotating cylinders), and the effects of suction on the boundary layer. References are given in the Vier Abhandlungen.

* For experiments on flow along flat plates, containing measurements in the laminar and turbulent regions, and also in the region of transition, see Burgers and van der Hegge Zynen, Mededeellng No. 5 uit het laboratorium voor aerodynamica en hydrodynamica der tecknische hoogeschool te Delft; van der Hegge Zynen, Mededeeling, No. 6; Burgers, , Proceedings of the First International Congress for Applied Mechanics, Delft (1924), p. 113;Google ScholarHansen, , Z.A.M.M. 8, 185 (1928).Google Scholar The flow past a circular cylinder in the critical Reynolds number interval has been examined by Fage, , Phil. Mag. (7) 7, 253 (1929).CrossRefGoogle Scholar The discovery that the flow in the boundary layer may become turbulent apparently dates back to Blasius, , Mitteilungen über Forschungsarbeiten herausg. vom Verein deutsch. Ing. Heft 131, p. 1 (1913)Google Scholar, and Prandtl, , Göttinger Nachrichien (1914), p. 177.Google Scholar

† Blasius, loc. cit.

‡ Polhausen, loc. cit. The exact solution for converging or diverging flow between non-parallel straight walls was given by Hainel, (Jahresbericht der deutscher Math.-Vereinigung, 25 (1916), p. 34) as a special case of flow in which the stream-lines are logarithmic spirals, which is the only possible form if they are to coincide with the stream-lines of a potential flow.Google Scholar see also Oseen, , Arkiv för Math.-Astron. och Fys. 20, 1927, No. 14;Google ScholarMillikan, , Math. Ann. 101 (1929), p. 446;CrossRefGoogle Scholar and Kármán, v., Vortrádge aus dem Gebiete der Hydro- und Aerodynamik (Innsbruck, 1922), p. 150.Google Scholar

§ Polhausen, loc. cit.

∥ Blasius, loc. cit.; Hiemenz, , Dinglers Polytechn. Journal, Bd. 326 (1911). The latter used an experimentally determined pressure distribution. See Polhausen's remarks on Hiemenz's solution, and v. Mises's remarks on Polhausen's solution.Google Scholar An approximate numerical solution has also been given by A. Thorn, Reports and Memoranda of the Aeronautical Research Committee, No. 1176 (1928). The region in which the solution holds appears to be almost the same as that for Hiemenz's solution. Measurements of the velocity distribution in the boundary layer at the surface of a circular cylinder are also given in the paper cited.

* Trans. Camb. Phil. Soc. 20, 253–269 (1908).Google Scholar

* Barnes, loc. cit.

* ψ is the recognised symbol for both the stream function in hydrodynamics and the logarithmic derivative of the gamma function. When it occurs in this paper with the second meaning, its numerical argument is always specified, so no confusion can be caused.

* The process adopted for the numerical solution of differential equations was that of Adams, described in Chap, xiv of Whittaker and Eobinson's Calculus of Observations, and by Kriloff, in the Proceedings of the First International Congress for Applied Mechanics, Delft (1924), p. 212.Google Scholar The method is so much less laborious than others in use (that of Ruuge and Kutta for example) that the numerical work in this paper was possible only because it was available.

* Zeitschrift f. Math. u. Physik, 60, 397–98 (1912).

* A glance at equation 1.5 (5) shows that when, as here, a 1 is zero, the conditions 1.5 (3) for the absence of singularities are by no means sufficient. The conditions required are complicated. If we suppose u expanded in a power series in x, u=u 0 + u 1x + u 2x 2 + …, with u 1, u 2 as power series in y, u 1 = b 1y + b 2y 2 + …, u 2 = c 1y + c 2y 2 + …, we find that we must have

and so on. Only a 8, a 12, a 16, a 20, … are at our disposal. In addition b 1c 1d 1… are determined, not from the equations for u 1u 2u 3,… respectively, but from the conditions for the absence of singularities in u 2, u 3, u 4,… respectively. Further, there is an ambiguity of sign which can be decided only from physical considerations. A little more light, but not much, is shed on the matter by considering the equation in the form

* It is hot suggested that the above solution of the problem is always valid. When f 0 has the asymptotic expansion given by 4.11 (2), in which A 0 must not vanish, then the equation for f 1 has three asymptotic solutions, giving f 1 of order n 4, of order η2, and exponentially small respectively. The method used fails if the general solution for f 1 with a double zero at the origin does not involve the solution of order η 4 at infinity. This happens, for example, when p 0 + 2a 2 = 0 or ½π0 = 64a 2. The solution for f 0 is then π0η3/3!, and the general solution for f 1, with a double zero at the origin is any arbitrary multiple of n 2. This difficulty may occur, of course, in solving for any function fr, and, though the matter may be tested by numerical computation in any given case for the first few of the fr, a theoretical discussion of the general equation appears too difficult to be possible.

* Blasius, loc. cit.