Abstract
The partial differential equation of the boundary layer on a flat plate are simplified by using the universal variables for turbulent flow. For laminar flow this gives boundary layer having a finite thickness and a friction coefficient differing by a few percent from the Blasius value. For a turbulent flow a differential equation for the velocity distribution is obtained with a parameter which varies slowly with the streamwise coordinate. The numerical value of this parameter is determined as an eigenvalue of the differential equations giving a velocity profile which evolves as the boundary layer thickens. Numerical calculations using a simple eddy viscosity model gave results in very good agreement with experiment.
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Abbreviations
- b :
-
a parameter which occurs in the expression for dimensionless eddy viscosity
- c 1, c 2 :
-
constants occurring in the friction law
- C f :
-
friction coefficient = \({{\tau _W } \mathord{\left/ {\vphantom {{\tau _W } {\tfrac{1}{2}}}} \right. \kern-\nulldelimiterspace} {\tfrac{1}{2}}}\rho u^2 _\infty\)
- d :
-
parameter which occurs in the expression for dimensionless eddy viscosity
- H :
-
shape factor = δ*/θ
- K :
-
von Karman's constant
- γ 0 :
-
pipe radius
- Re + :
-
Reynolds' number = u τδ/υ
- R θ :
-
Reynolds number based on momentum thickness = u ∞τ/υ
- R x :
-
Reynolds number = ux/ν
- T :
-
dimensionless shear stress distribution = τ/τ w
- u :
-
mean flow velocity in streamwise direction
- u τ :
-
friction velocity = (τ w/ρ)1/2
- u + :
-
dimensionless mean flow velocity = u/u τ
- v :
-
mean flow velocity in y direction
- y + :
-
dimensionless distance from wall = γu τ/υ
- β :
-
parameter which is defined as (υ/u 2 τ)(du τ/dx)
- δ :
-
boundary layer thickness
- δ*:
-
displacement thickness
- ɛ :
-
eddy viscosity
- η :
-
dimensionless distance from wall = y/δ
- θ :
-
momentum thickness
- ν :
-
molecular viscosity
- ψ :
-
stream function
References
Blasius, H., Zeitschrift für Mathematik und Physik 56 (1908) 1, also NACA Technical Memorandum 1256.
Dhawan, S., Direct Measurements of Skin Friction, NACA Technical Report 1121, 1953.
Kamke, E., Differentialgleigungen, Losungsmethoden und Losungen, Chelsea Publishing Company, N. Y., 1942.
Mei, J. and W. Squire, AIAA J. 10 (1972) 350.
Squire, W., Integration for Engineers and Scientists, American Elsevier Publ. Co., Inc., N. Y., 1970.
Coles, D. E. and E. S. Hirst, Proc. Computations of Turbulent Boundary Layers, AFOSR-IFP-Stanford Conference 2, Stanford Univ., 1968.
Smith, D. W. and J. H. Walker, Skin Friction Measurements in Incompressible Flow, NACA Technical Note 4231, 1958.
Dillon, J. L., A Single Formula for the Velocity Profile Through a Turbulent Incompressible Boundary Layer on a Flat Plate, unpubl. M. S. Thesis, West Virginia University, 1967.
Spalding, D. B., J. of Appl. Mech. Transactions ASME, Ser. E (1961) 445.
Fediaevsky, K., Turbulent Boundary Layer of an Airfoil, NACA Technical Memorandum 822, 1937.
Schubauer, G. B. and P. S. Klebanoff, Investigation of Separation of the Turbulent Boundary Layer, NACA Report 1030, 1951.
Klebanoff, P. S., Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient, NACA Report 1247, 1955.
Ross, D. and J. M. Robertson, J. of Appl. Physics 21 (1950) 557.
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Mei, J., Squire, W. A quasi-similarity analysis of the turbulent boundary layer on a flat plate. Appl. Sci. Res. 29, 461–473 (1974). https://doi.org/10.1007/BF00384166
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DOI: https://doi.org/10.1007/BF00384166