There are also a limited number of methods that integrate the Kármán equation for turbulent boundary layers. On the Kármán momentum-integral approach and the Pohlhausen paradox,” Phys.
The latter was extended recently by Majdalani and Xuan.
Zur naherungsweisen integration der differentialgleichungen der laminaren reilbungsschicht,” ZAMMġ, 252– 260 (1921). For laminar boundary layers, the most well-known integral methods are due to Thwaites 2 2.ī. Using this construct, a number of simplified approaches were developed to produce approximate solutions suitable for estimating the drag of planar aerodynamic shapes. 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. 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. 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. 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 UVP is uniformly valid from the wall to the boundary layer edge and for all Reynolds numbers from zero to infinity. Comparisons with both direct numerical simulation and experimental data demonstrate the ability of the profile to approximate a wide variety of wall-bounded flows. 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.
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