A recently developed mixing length model of the turbulent shearing stress in wall bounded flows has been used to formulate a universal velocity profile (UVP) that provides an effective replacement for the widely used Coles wall-wake formulation. Comparisons with both direct numerical simulation and experimental data demonstrate the ability of the profile to approximate a wide variety of wall-bounded flows. The UVP is uniformly valid from the wall to the boundary layer edge and for all Reynolds numbers from zero to infinity. There is no presumption of logarithmic dependence of the velocity profile outside the viscous wall layer so the profile can accurately approximate low Reynolds number turbulent boundary layers. The effect of a pressure gradient is included in the UVP through the introduction of a modified Clauser parameter that correlates well with the parameters that determine the wake portion of the velocity profile. The inherent dependence of the UVP on Reynolds number, extended to include the effect of pressure gradient, enables it to be used as the basis of a new method for integrating the von Kármán boundary layer integral equation for a wide variety of attached wall bounded flows. To illustrate its application, the UVP is used to determine the zero-lift drag coefficient of the Joukowsky 0012 and NACA (National Advisory Committee for Aeronautics) 0012 airfoils over a wide range of chord Reynolds numbers.

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The universal velocity profile (UVP), defined initially for a smooth pipe, is extended to pipe flow with a rough wall. Data from pipe experiments of fixed roughness fraction and varying Reynolds number are used to formulate a simple correction to the smooth-wall UVP using a function of the roughness Reynolds number. A good fit to the data is achieved using an additive velocity correction combined with a new roughness transition function that accounts for the delayed onset of roughness friction at low roughness Reynolds numbers. It is concluded that the UVP can represent rough-wall pipe flow with a very small error comparable to the smooth-wall case.

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