On the solution of the laminar boundary layer equations

  • Itiro Tani


In the first part of paper, the momentum equation of the incompressible laminar boundary layer is solved for the velocity distribution outside the boundary layer of the form U = U 0 — bxn. It is found from results of solution that the frequently used parameter λ = (θ2/v) (d U/d x) cannot exactly fix the velocity profile, its value at separation becoming less negative as d2 U/x2 becomes more positive. In the second part, a simple approximate method of solution is developed, in which the velocity profile is expressed as a member of a family of curves, with the parameter α different from λ. It is found that results of sufficient accuracy can be obtained by calculating λ from a quadrature formula and determining the relation between λ and α from the momentum and energy integrals of the boundary layer. The method of solution can easily be extended to the boundary layer of a compressible fluid.


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  1. [1]
    Pohlhausen, K.: Zur näherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht. Z.A.M.M., vol. 1 (1921), pp. 252–268.zbMATHGoogle Scholar
  2. [2]
    Schlichting, H., and Ulrich, A.: Zur Berechnung des Umschlages laminar-turbulent. Jahrbuch der deutschen Luftfahrtforschung, 1—8 (1942), pp. 8—36.Google Scholar
  3. [3]
    Howarth, L.: On the Solution of the Laminar Boundary Layer Equations. Proc. Boy. Soc, London, A, vol. 164 (1938), pp. 547–579.CrossRefzbMATHADSGoogle Scholar
  4. [4]
    Falkner, V. M., and S. Jcan S. W.: Some Approximate Solutions of the Boundary Layer Equations. A. R. C, R. & M. no. 1314 (1930).Google Scholar
  5. [5]
    Hartree, D. R.: On the Equation Occurring in Falkner and Skan’s Approximate Treatment of the Equations of the Boundary Layer. Proc. Cambridge Phil. Soc, vol. 33 (1937), pp. 223–239.CrossRefADSGoogle Scholar
  6. [6]
    Tani, I.: On the Solution of the Laminar Boundary Layer Equations. Jour. Phys. Soc, Japan, vol. 4 (1949), pp. 149–154.CrossRefMathSciNetADSGoogle Scholar
  7. [7]
    Hudimoto, B.: An Approximate Method for Calculating the Laminar Boundary Layer (in Japanese). Jour. Soc. Aero. Sci., Japan, vol. 8 (1941), pp. 279—282.Google Scholar
  8. [8]
    Tani, I.: A Simple Method for Determining the Laminar Separation Point (in Japanese). Jour. Aero. Res. Inst., Tokyo Imp. Univ., no. 199 (1941).Google Scholar
  9. [9]
    Young, A. D., and Winterbottom, N. E.: Note on the Effect of Compressibility on the Profile Drag of Aerofoils in the Absence of Shock Waves. A. R. C. Report no 4697 (1940).Google Scholar
  10. [10]
    Walz, A.: Ein neuer Ansatz für das Geschwindigkeitspröfil der laminaren Grenzschicht. Lilienthal-Bericht no! 141 (1941).Google Scholar
  11. [11]
    Thwaites, B.: Approximate Calculation of the Laminar Boundary Layer. Aero. Quart., vol. 1 (1949), pp. 245–280.MathSciNetGoogle Scholar
  12. [12]
    Tani, I., and, Tsuji, H.: On the Solution of the Laminar Boundary Layer Equations (in Japanese; abstract only). Proc. Phys. Soc, Japan, vol. 2 (1947), p. 86.Google Scholar
  13. [13]
    Wieghardt, K.: Über einen Energiesatz zur Berechnung laminarer Grenzschichten. Ingenieur-Archiv, vol. 16 (1948), pp. 231—242.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    Timman, B.: A One-Parameter Method for the Calculation of Laminar Boundary Layers. Report F-35, Nationaal Luchtvaartlaboratorium (1949).Google Scholar
  15. [15]
    Morduchow, M., and Clarke, J. H.: Method for Calculation of Compressible Laminar Boundary-Layer Characteristics in Axial Pressure Gradient with Zero Heat Transfer. N. A. C. A. Tech. Note no. 2784 (1952).Google Scholar
  16. [16]
    Walz, A.: Anwendung des Energiesatzes von Wieghardt auf einparametrige Geschwindigkeitsprofile in laminaren Grenzschichten. Ingenieur-Archiv, vol. 16 (1948), pp. 243—248.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    Truckeribrodt, E.: Ein Quadraturverfahren zur Berechnung der laminaren und turbulenten Reibungsschicht bei ebener und rotationssymmetrischer Strömung. Ingenieur-Archiv, vol. 20 (1952), pp. 211–228.CrossRefGoogle Scholar
  18. [18]
    Tani, I.: On the Approximate Solution of the Laminar Boundary Layer Equations. Jour. Aero. Sci., vol. 21 (1954), pp. 487–495.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1955

Authors and Affiliations

  • Itiro Tani
    • 1
  1. 1.Institute of Science and TechnologyUniversity of TokyoJapan

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