# Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow
## 3 Switching Functions, Friction Factors, New Formulation, and Comparative Analysis
### New Unified Flow Friction Formulation  
A novel formula for all hydraulic regimes is developed in order to easily encapsulate separate formulas for different hydraulic regimes into one coherent hydraulic regime. The structure of the proposed formula, \(\lambda=(1-y_{1})\cdot(a)+(y_{1}-y_{3})\cdot(c_{1})+y_{2}\cdot(c_{3})\), is based on using three switching functions--\(y_{1},y_{2}\), and \(y_{3}\)--and formulas for laminar flow \((a)\), smooth turbulent flow \((c_{1})\), and fully developed rough turbulent flow \((c_{3})\). Consequently, all hydraulic regimes can be simulated in one simple formula that includes a sharp transition from the laminar to the smooth turbulent \((b)\) and the non-fully developed partially turbulent regimes \((c_{2})\). As already explained, the switching functions are set in such a way so as to always produce Nikuradse's inflectional curve shape [34]. Two possible encapsulations in the unified coherent hydraulic model are given in Equation (15), with related diagrams contained in Figures 9 and 10:  
\[\begin{array}{l}\lambda=\overbrace{\frac{64}{Re}}^{(a)}\cdot(1-y_{1})+ \overbrace{\frac{0.316}{Re^{0.25}}}^{(c_{1})}\cdot(y_{1}-y_{3})+\overbrace{ \frac{0.25}{\log_{10}^{2}\left(\frac{\varepsilon}{3.71}\right)}}^{(c_{3})}\cdot y _{2}\\ \lambda=\overbrace{\frac{64}{Re}}_{(a)}\cdot(1-y_{1})+\underbrace{\left(0.003 2+\frac{0.221}{Re^{0.259}}\right)}_{(c_{1})}\cdot(y_{1}-y_{3})+\underbrace{0. 11\cdot\varepsilon^{0.25}}_{(c_{3})}\cdot y_{2}\end{array} \tag{15}\]  
Figure 8: A comparison of some correlations for the hydraulically smooth turbulent regime.  
In Figure 9, the shape outlined by the blue dots results from using the Colebrook equation with the extension for Nikuradse's inflectional law from Equation (5). Laminar flow \((a)\) is comprised of \(\lambda=\frac{64}{Re}\), whereas the hydraulically smooth turbulent regime \((c_{1})\) with the Blasius expression is represented as \(\lambda=\frac{0.316}{Re^{2}\cdot 0^{-5}}\). Finally, the fully developed turbulent regime \((c_{3})\) with the von Karman-Prandtl equation \(\lambda=\frac{0.25}{\log(\frac{1}{37})}\) can be seen in Equation (15)-I. In Figure 10, a similar approach is outlined using different formulas, as indicated in Equation (15)-II.  
Figure 10: The unified hydraulic model II; Equation (15).  
Figure 9: The unified hydraulic model I; Equation (15).  
Using the same separate formulas for different hydraulic regimes from Equation (15), the following additional encapsulation is also possible, seen in Equation (16):  
\[\lambda=\overbrace{\frac{64}{Re}}^{(a)}\cdot(1-y_{1})+\underbrace{ \overbrace{0.316}^{(c_{1})}}_{(c_{1})}\cdot(y_{1}-y_{3})+\overbrace{0.11\cdot \epsilon^{0.25}}_{(c_{3})}\cdot y_{2} \tag{16}\] \[\lambda=\underbrace{\frac{64}{Re}}_{(a)}\cdot(1-y_{1})+ \underbrace{\left(0.0032+\frac{0.221}{Re^{0.237}}\right)}_{(c_{1})}\cdot(y_{1} -y_{3})+\underbrace{\frac{0.25}{\log_{10}^{2}\left(\frac{5}{371}\right)}}_{(c_ {3})}\cdot y_{2}\]  
Using this method, and after performing numerical tests, the most appropriate equation (comprised of separate available formulas for different hydraulic regimes) can be optimally encapsulated depending upon the particular circumstances of each engineering project.