Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-09T14:24:33.640Z Has data issue: false hasContentIssue false
  • English
  • Français

The law of the wake in the turbulent boundary layer

Published online by Cambridge University Press:  28 March 2006

Donald Coles
Donald Coles
Affiliation:
Guggenheim Aeronautical Laboratory, California Institute of Technology, Pasadena
Get access Rights & Permissions [Opens in a new window]

Abstract

After an extensive survey of mean-velocity profile measurements in various two-dimensional incompressible turbulent boundary-layer flows, it is proposed to represent the profile by a linear combination of two universal functions. One is the well-known law of the wall. The other, called the law of the wake, is characterized by the profile at a point of separation or reattachment. These functions are considered to be established empirically, by a study of the mean-velocity profile, without reference to any hypothetical mechanism of turbulence. Using the resulting complete analytic representation for the mean-velocity field, the shearing-stress field for several flows is computed from the boundary-layer equations and compared with experimental data.

The development of a turbulent boundary layer is ultimately interpreted in terms of an equivalent wake profile, which supposedly represents the large-eddy structure and is a consequence of the constraint provided by inertia. This equivalent wake profile is modified by the presence of a wall, at which a further constraint is provided by viscosity. The wall constraint, although it penetrates the entire boundary layer, is manifested chiefly in the sublayer flow and in the logarithmic profile near the wall.

Finally, it is suggested that yawed or three-dimensional flows may be usefully represented by the same two universal functions, considered as vector rather than scalar quantities. If the wall component is defined to be in the direction of the surface shearing stress, then the wake component, at least in the few cases studied, is found to be very nearly parallel to the gradient of the pressure.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 1 , Issue 2 , July 1956 , pp. 191 - 226
Copyright
© 1956 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bauer, W. 1951 The development of the turbulent boundary layer on steep slopes, Thesis: State Univ. of Iowa; abridged in Proc. Amer. Soc. Civ. Engrs., Separate no. 281, 1953.
Clauser, F. 1954 Turbulent boundary layers in adverse pressure gradients, J. Aero. Sci., 21, 91–108.Google Scholar
Coles, D. 1954 The problem of the turbulent boundary layer, Z. angew. Math. Phys., 5, 181–203.Google Scholar
Coles, D. 1955 The law of the wall in turbulent shear flow, 50 Jahre Grenzschichtforschung (ed. H. Görtler & W. Tollmien), pp. 153–163. Braunschweig: F. Vieweg und Sohn.
Gruschwitz, E. 1935 Turbulente Reibungsschichten mit Sekundärströmung, Ing.-Arch., 6, 355–365.Google Scholar
Kármán, T. v. 1921 Über laminare und turbulente Reibung, Z. angew. Math. Mech. 1, 233–252; translated as On laminar and turbulent friction, Nat. Adv. Comm. Aero., Wash., Tech. Mem. no. 1092, 1946.Google Scholar
Kármán, T. v. 1932 Theorie des Reibungswiderstandes, Hydromechanische Probleme des Schiffsantriebs, Hamburg.
Kehl, A. 1943 Untersuchungen über konvergente und divergente turbulente Reibungsschichten. Ing.-Arch. 13, 293–329; translated as Investigations on convergent and divergent turbulent boundary layers, British Ministry of Supply R. T. P. no. 2035, 1946.Google Scholar
Klebanoff, P. & Diehl, Z. 1951 Some features of artificially thickened fully developed turbulent boundary layers with zero pressure gradient, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 2475.Google Scholar
Klebanoff, P. 1954 Characteristics of turbulence in a boundary layer with zero pressure gradient, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 1946.Google Scholar
Kuethea, A. Mckee, P. & Curry, W. 1949 Measurements in the boundary layer of a yawed wing, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 1946.Google Scholar
Laufer, J. 1953 The structure of turbulence in fully developed pipe flow, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 2954.Google Scholar
Lees, L. & Crocco, L. 1952 A mixing theory for the interaction between dissipative flows and nearly isentropic streams, J. Aero. Sci., 19, 649–676.Google Scholar
Liepmann, H. & Laufer, J. 1947 Investigations of free turbulent mixing, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 1257.Google Scholar
Ludwieg, H. & Tillmann, W. 1949 Untersuchungen über die Wandschubspannung in turbulenten Reibungsschichten, Ing.-Arch. 17, 288–299; translated as Investigations of the wall shearing stress in turbulent boundary layers, Nat. Adv. Comm. Aero., Wash., Tech. Mem. no. 1285, 1950.Google Scholar
McCullough, G. & Gault, D. 1949 Boundary-layer and stalling characteristics of the N.A.C.A. 64A006 airfoil section, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 1923.Google Scholar
Millikan, C. 1938 A critical discussion of turbulent flows in channels and circular tubes, Proc. 5th Int. Cong. Appl. Mech., Cambridge, 386–392.
Newman, B. 1951 Some contributions to the study of the turbulent boundary layer near separation, Aust. Dept. Supply, Rep. no. ACA-53.Google Scholar
Nikuradse, J. 1930 Widerstandsgesetz und Geschwindigkeitsverteilung von turbulenten Wasserströmung in glatten und rauhen Rohren, Proc. 3rd Int. Cong. Appl. Mech., Stockholm, 239–248.
Prandtl, L. 1926 über die ausgebildete Turbulenz, Proc. 2nd Int. Cong. Appl. Mech., Zürich, 62–74; translated as Turbulent flow, Nat. Adv. Comm. Aero., Wash., Tech. Mem. no. 435, 1927.
Preston, J. 1954 The determination of turbulent skin friction by means of pitot tubes, J. R. Aero. Soc., 58, 109–121.Google Scholar
Reichardt, H. 1940 Die Wärmeübertragung in turbulenten Reibungsschichten, Z. angew. Math. Mech. 20, 297–328; translated as Heat transfer through turbulent friction layers, Nat. Adv. Comm. Aero., Wash., Tech. Mem. no. 1047, 1943.Google Scholar
Ross, D. & Robertson, J. 1951 A superposition analysis of the turbulent boundary layer in an adverse pressure gradient, J. Appl. Mech., 18, 95–100.Google Scholar
Rotta, J. 1950 Über die Theorie der turbulenten Grenzschichten, Mitt. Max-Planck-Inst., Göttingen, No. 1; translated as On the theory of the turbulent boundary layer, Nat. Adv. Comm. Aero., Wash., Tech. Mem. no. 1344, 1953.Google Scholar
Schubauer, G. & Klebanoff, P. 1950 Investigation of separation of the turbulent boundary layer, Nat. Adv. Comm. Aero., Wash., Tech. Note no. 2133.Google Scholar
Sheppard, P. 1947 The aerodynamic drag of the earth's surface and the value of von Kármán's constant in the lower atmosphere, Proc. Roy. Soc. A, 188, 208–222.Google Scholar
Tillmann, W. 1945 Untersuchungen über Besonderheiten bei turbulenten Reibungsschichten an Platten, Z. W. B., K. W. I., Göttingen, U. & M. 6627; translated as Investigations of some particularities of turbulent boundary layers on plates British R. & T. MAP-VG-34, 1946.Google Scholar
Wieghardt, K. 1943 Über die Wandschubspannung in turbulenten Reibungsschichten bei veränderlichem Aussendruck, Z. W. B., K. W. I., Göttingen, U. & M. 6603. See also Wieghardt, K. & Tillmann, W., Zur turbulenten Reibungsschicht bei Druckanstieg, Z. W. B., K. W. I., Göttingen, U. & M. 6617, 1944; translated as On the turbulent friction layer for rising pressure, Nat. Adv. Comm. Aero., Wash., Tech. Mem. no. 1314, 1951.Google Scholar
Wieghardt, K. 1944 Zum Reibungswiderstand rauher Platten, Z. W. B., K. W. I., Göttingen, U. & M. 6612.Google Scholar