Abstract
A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B. J. Cantwell, “A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)]. More recently, the same profile was shown to accurately approximate velocity profiles in channel flow, the zero pressure gradient boundary layer, and the boundary layer in an adverse pressure gradient [M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows,” AIAA Paper No. 2021-0061, 2021 and M. A. Subrahmanyam, B. J. Cantwell, and J. J. Alonso, “A universal velocity profile for turbulent wall flows including adverse pressure gradient boundary layers,” J. Fluid Mech. (unpublished) (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity. At a low Reynolds number, the profile approaches the laminar channel/pipe flow limit. The primary measure of the Reynolds number used in this work is the friction Reynolds number Rτ=uτδ/ν. It is a little unusual to use Rτ for the boundary layer since it requires that the velocity profile be cutoff using an arbitrarily defined overall boundary layer thickness, δ. Because of the slow approach of the velocity to the free stream, different conventions used to define the thickness lead to different values of Rτ assigned to a given flow. It will be shown in this paper that, through its connection to channel/pipe flow, the universal velocity profile can be used to define a practically useful, unambiguous, measure of overall boundary layer thickness, called here the equivalent channel half height, δh. For Rτ>≈5000, the universal velocity profile defines a Reynolds number independent shape function that can be used to generate explicit expressions for the infinite Reynolds number behavior of all the usual integral boundary layer measures; displacement thickness, momentum thickness, energy thickness, overall boundary layer thickness, and skin friction. The friction coefficient Cf(Rδ2) generated by the universal velocity profile accurately approximates data over a wide range of momentum thickness Reynolds numbers collected by Nagib et al. [“Can we ever rely on results from wall-bounded turbulent flows without direct measurements of wall shear stress?,” AIAA Paper No. 2004-2392, 2004]. The universal velocity profile is used to integrate the von Kaŕmań boundary layer integral equation [T. von Kármán, “Uber laminaire und turbulente reibung,” Z. Angew. Math. Mech. 1, 233–252 (1921)] in order to generate the various thicknesses and friction velocity as functions of the spatial Reynolds number, Rx=uex/ν.
Highlights
The simplification of the incompressible Navier–Stokes equations in the early 1900s to the boundary layer approximation for high Reynolds number flow over a wall by Prandtl6 and his students was one of the most important advances in the history of mechanics
A recently developed mixing length model of the turbulent shearing stress has been shown to generate a universal velocity profile that provides an accurate approximation to incompressible pipe flow velocity profiles over a wide Reynolds number range [B
Fluid Mech. (2021)] The universal velocity profile is uniformly valid from the wall to the free stream at all Reynolds numbers from zero to infinity
Summary
The simplification of the incompressible Navier–Stokes equations in the early 1900s to the boundary layer approximation for high Reynolds number flow over a wall by Prandtl and his students was one of the most important advances in the history of mechanics. A number of simplified approaches were developed to produce approximate solutions suitable for estimating the drag of complex aerodynamic shapes such as airships and wings These methods all used the von Karman boundary layer integral Eq (1) derived by integrating the boundary layer equations over the height of the boundary layer. One of the simplest to apply is the method due to Head.9 He assumes the existence of two universal functions, F and G, for the turbulent boundary layer. The first function relates a modified boundary layer shape factor to the conventional shape factor, and the second relates the conventional shape factor to the entrainment velocity at the outer edge of the boundary layer When these functions are combined with the friction law of Ludweig and Tillman, all of the basic characteristics of the boundary layer can be determined. As will be seen from the Figures there is considerable scatter of the points, and in Fig. 1, there is a fairly marked and consistent discrepancy between the two sets of results, which makes the drawing of a hypothetical common curve, representing the function FðHdÀdà Þ, a somewhat arbitrary procedure.”
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