Momentum-Flux Determination for Boundary Layers with Sufficient Fetch Based on Integral Equations

Integral equations for the vertical velocity component and momentum flux within boundary layers with sufficient fetch under neutral stratifications are derived by extending von Kármán’s integral equation. To simplify the equations, a self-similar profile for the streamwise velocity component and a zero-pressure gradient are assumed. The integral equations enable the following calculations. The vertical velocity component and momentum flux are calculated by the integral of the streamwise change in the streamwise velocity component. The vertical velocity component at the boundary-layer top is determined by the drag coefficient and shape factor. The advective flux at the boundary-layer top is of the same order as the turbulent momentum flux near the surface. Accordingly, the derived integral equations are applied to various types of boundary layers with sufficient fetch. For laminar boundary layers, profiles determined by the equations perfectly agree with the profiles determined by the analytical Blasius solution. For turbulent boundary layers, the comparison of the momentum flux between the proposed equations and experimental data verifies that the equations predict the turbulent momentum flux very well. Finally, we employ a power-law approximation for determining both the velocities and the momentum flux. As the streamwise velocity component can be approximated by the power law for both smooth and rough surfaces, the resulting turbulent flux agrees well with that obtained in experiments and numerical simulations.