Applied Scientific Research

, Volume 31, Issue 3, pp 161–186 | Cite as

Semi-similar solutions to the three-dimensional laminar boundary layer

  • James C. WilliamsIII
Article

Abstract

The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: “singular” separation characterized by the vanishing of the total shear along the line of separation and “ordinary” separation characterized by limiting streamlines which become parallel to the line of separation.

Nomenclature

A, B, C, D, E, H, I, J

coefficients ofξ in the reduced momentum equation (Eq. 8)

F(ξ, η)

dimensionless stream function

G(ξ, η)

dimensionless stream function

g(x, y)

scaling function for thez-coordinate

k1,k2,k3

constants in the various flows studied

l

characteristic length for the flow

p

pressure

U

characteristic velocity for the flow

v

y — component of velocity

w

z — component of velocity

x

coordinate direction on body surface

y

coordinate direction on body surface

z

coordinate direction normal to body surface

β

streamline angle

βw

angle of the limiting streamline

η

scaledz-coordinate

ν

kinematic viscosity

ξ

scaledx andy coordinate

ρ

density

τw

wall shear

Subscripts

δ

conditions at the “upper” edge of the boundary layer

w

condition at the body surface (wall)

Superscript

*

dimensionless quantities

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References

  1. [1]
    Hansen, Arthur G. andHoward Z. Herzig, On Possible Similar Solutions for Three-Dimensional, Incompressible, Laminar Boundary Layers, I — Similarity with Respect to Stationary Rectangular Coordinates, NACA TN 3768, 1956.Google Scholar
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    Sears, W. R., JAS15 (1948) 49.Google Scholar
  3. [3]
    Mager, Arthur, JAS21 (1954) 835.Google Scholar
  4. [4]
    Wild, J. M., JAS16 (1949) 41.Google Scholar
  5. [5]
    Dwyer, H. A., Calculation of Unsteady and Three-Dimensional Boundary Layer Flows, AIAA Paper 72–109, presented at the 10th Aerospace Sciences Meeting, San Diego, California, January, 1972.Google Scholar
  6. [6]
    Der, J., AIAA J.9 (1971) 1294.Google Scholar
  7. [7]
    Baker, A. J., Finite Element Computational Theory for Three-Dimensional Boundary Layer Flow, AIAA Paper 72–108, presented at the 10th Aerospace Sciences Meeting, San Diego, California, January, 1972.Google Scholar
  8. [8]
    Blottner, F. G., AIAA J.8 (1970) 193.Google Scholar
  9. [9]
    Maskell, E. C., Flow Separation in Three Dimensions, Royal Aircraft Establishment Report AERO 2565, 1955.Google Scholar

Copyright information

© Martinus Nijhoff 1975

Authors and Affiliations

  • James C. WilliamsIII
    • 1
  1. 1.Dept. of Mech. and Aerospace Eng.North Carolina State UniversityRaleighU.S.A.

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