Unified law of turbulent flow velocity and drag coefficient based on turbulent eddy model

The vertical distribution of time-averaged velocity in turbulent flow fields is a critical issue in viscous fluid dynamics and river dynamics. Historically, the community has relied on the log-law to predict velocity distributions. However, despite multiple refinements, the mathematical properties of this method still limit its performance near the wall and the symmetry axis. Zhang Hongwu proposed a new deterministic formula based on the turbulent eddy model, which enables highly accurate prediction of velocity distributions. This approach addresses deficiencies in the Rouse sediment concentration distribution formula, which incorrectly predicts zero concentration at the water surface. However, the formula has not yet achieved unification across all flow regions. Based on this formula, this study derives a functional relationship between the integration constant and both the relative roughness and the roughness Reynolds number. This leads to a unified version of Zhang’s formula valid for smooth, transitional, and fully rough regions. In addition, by conducting cross-sectional integration, a relationship between the integration constant and the drag coefficient is developed, thereby constructing a unified law of turbulent velocity and flow resistance. Using Nikuradse’s experimental data, this unified law is compared with the formulas of Einstein and Dou Guoren, validating the superiority of the proposed theory in terms of velocity distribution prediction and drag coefficient calculation. Additionally, the analysis reveals that the vertical time-averaged velocity distribution across the entire water depth does not necessarily follow a logarithmic distribution law in different flow regions.