Evaluation of von Karman's Constant from Integral Flow Parameters

Theory and data for turbulent flow in smooth pipes are employed to evaluate von Karman's constant from integral flow parameters. A set of equations for mean velocity and centerline velocity are derived from the equations of motion for steady, uniform turbulent flow in smooth pipes. An eddy‐viscosity closure is used that has a parabolic asymptotic form near the pipe wall and decays near the pipe centerline. The velocity equations depend on von Karman's constant $k$, a coefficient relating the bottom roughness parameter to the scale of the viscous sublayer $\beta =(\nu /u^{*})/z_0$, and a coefficient in the eddy‐viscosity decay term α. These constants are evaluated by fitting the model equations to velocity measurements from 10 high‐quality data sets. A narrowly constrained value, $k=0.408\pm 0.004$ at 95% confidence, is obtained from mean velocity data as a function of Reynolds number. Mean velocity and centerline velocity data give $\alpha =3.34$ and $\beta =8.72$ for the other two parameters. Examining the variation of $k$ over ranges of Reynolds number reveals that $k$ does not vary with Reynolds number to the accuracy of the measurements.