A, Applications of the new series for Case A.
§12. Flow along a plane wall in a divergent channel................... 44
§13. Plow along a flat plate of finite depth, in a convergent channel...... 47
§14. New series solution for the HOWARTH flat plate flow with outer
velocity U(x) = u0 (1 — x/L).................................. 50
B. Application of the new series for Case В
§15. Preliminary remarks concerning the accuracy of the tables of universal functions of the BLASIUS series hitherto available.......... 56
§16. Boundary layer on the circular cylinder with outer velocity distribution according to HIEMENZ' experimental results................ 60
References........................................................ 65
SUMMARY
A new and general method for solving problems of plane and steady laminar boundary layer flows in incompressible fluids with arbitrary outer pressure distribution is developed. This method is based on the introduction of the dimensionless quantities
U(X) dX, 7? = -7—f^-
° \2v I U(x) dx
( Jo
as new independent spatial variables, (x, у are the usual boundary layer coordinates, U(x) the given outer velocit}' distribution, v the kinematic viscosity.) The solution of the boundary layer problem is then given as a power series in ? with coefficient functions depending on tj. This series is a formally exact solution of the boundary layer problem. The new series solution has the following qualities:
1. Whereas the original variables x, у have the significance only of Cartesian coordinates, the influence of wall curvature being neglected in boundary layer theory, the new coordinates are adjusted to the data of the special problem in any case of application. The new variables represent a logical development of former efforts in the field of boundary-layer flow calculation. (Cf. §6.)
2. An important feature, as compared with other series solutions known for some special cases is that the leading term of the new series satisfies exactly the outer boundary condition at all cross-sections along the wall. Therefore, the succeeding terms give corrections only in the inner part of the boundary layer. Accordingly, taking also No. 1 into account, the zero order term by itself gives a good approximation for the boundary layer flow
A New Series for the Calculation of Steady Laminar Boundary Layer Flows
HENRY GORTLER
Dedicated to ALBERT BETZ on his 70th birthday
CONTENTS
Summary........................................................ 2
§ 1. Introduction: Statement of the problem and method of attack. ... 4
PART I: STUDY OF A BOUNDARY LAYER TRANSFORMATION
§ 2. Transformation of the boundary layer equations.................. 7
§ 3. Some useful relations.......................................... 11
§ 4. Series expansions of the outer velocity U(x) and of the principal
function /3(|), and their relations................................ 12
§ 5. Special case of the constant principal function /3(Ј) — /?0 and similar
solutions of the boundary layer equations........................ 22
§ 6. Boundary layers around cylinders in yaw........................ 24
§ 7. Related transformations in the literature......................... 25
PART II: THE NEW SERIES
§ 8. Boundary layer series corresponding to principal functions of Class I 29 § 9. Boundary layer series corresponding to principal functions of the
more general Class II.......................................... 35
§10. Remarks concerning the available tables of universal functions.. . 38
PART III: SOME APPLICATIONS OF THE NEW SERIES
§11. Short description of the numerical work necessary for any application
of the new series.............................................. 40
Journal of Mathematics and Mechanics, Vol. 6, No. 1 (1957).
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