The universal velocity profile provides a description of the mean velocity within a turbulent boundary layer. If u(y) defines the velocity at distance y above a solid boundary located at y = 0, dimensional analysis suggests that immediately above the viscous sublayer the velocity within the so-called inner region (or wall layer) is given by:
where uτ is the friction velocity or shear velocity defined by uτ = (τ/ρ)1/2 where t is the shear stress, ρ the fluid density and ν the fluid kinematic viscosity. Within this region measurements suggest that the eddy stress or eddy viscosity (ε) is approximately constant. Using this assumption it can be shown, using several different approaches (momentum transport theory, similarity theory, and dimensional analysis), that the appropriate form of the function f is logarithmic and hence:
where A and B are constants which are typically given values of 5.75 and 5.5, respectively.
Although this equation is often referred to as the inner velocity law or the law of the wall, experimental measurements suggest that in the case of a turbulent boundary layer over a flat plate it may be applied with reasonable success over a larger region of the flow field. Indeed, this region of applicability is increased further in the case of pipe flow. However, if a detailed description of the velocity distribution within the so-called outer region is required, it is reasonable to assume that the velocity defect (u1 - u), where u1 is the free stream velocity, is independent of the viscosity. Dimensional analysis thus suggests
Von Karman considered this case and, having noted the apparent success of the logarithmic distribution, suggested that a universal velocity profile describing the flow over the entire boundary layer might be given by
where δ is the overall thickness of the boundary layer, and (y/δ) is a universal function which reduces to B in the inner region. In a review of boundary layer theory Duncan et al. (1970) suggest that for the case of flow over a flat plate, a good fit to the available data is obtained from
where K = 2.8 and (0)=B=5.5. Further details of this and other boundary layer theory is given by Schlichting (1968).
Duncan, W. J., Thom, A. S. and Young, A. D. (1970) Mechanics of Fluids, Arnold, London.
Schlichting, H. (1968) Boundary Layer Theory, McGraw-Hill, New York.