# Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow
## 3 Switching Functions, Friction Factors, New Formulation, and Comparative Analysis
### Friction Factors
#### 3.2.2 Smooth Turbulent Flow \((c_{1})\)  
In the hydraulically smooth regime, the friction factor \(\lambda\) is only a function of the Reynolds number \(Re\), and the resistance to flow is independent of the inner pipe surface relative roughness \(\varepsilon\). This regime is restricted to relatively small values of the Reynolds number \(Re\), where the roughness of the inner pipe surface is completely hidden in the laminar boundary sublayer. The literature on the hydraulically smooth regime abounds with reliable friction factor equations. In general, with some extensions and modifications, there are Blasius-form relationships (power-law relationships), as seen in Equation (11)-up, and von Karman-Prandtl-form relationships (logarithmic relationships), as seen in Equation (11)-down:  
\[\left. \begin{array}{c}(c_{1})\,\sim\lambda=A\cdot\!Re^{-B}\\ (c_{1})\,\sim\frac{1}{\sqrt{\lambda}}=\ C\cdot\!\log_{10}\left(Re\cdot\!\sqrt{ \lambda}\right)-D\end{array} \right\} \tag{11}\]  
Some possible values for coefficient \(A\) and exponent \(B\) in Blasius-form power-law relationships [17; 26; 31], as seen in Equation (11), are given in Table 1 below:  
Figure 7: Functions \(y_{1}\)–\(y_{3}\), with the main purpose of providing Nikuradse’s inflectional shape of flow friction curves. The right side consists of a magnified detail from the left side; the blue area represents the difference between the Colebrook–White monotonic and the Nikuradse inflectional shapes. With regard to von Karman-Prandtl-form logarithmic relationships, the below displays their basic form, which is used in a smooth part of the Colebrook equation, as seen in Equation (12). This formula is an updated version of that given by [PERSON] et al. (2016), seen in Equation (12):  
\[ \begin{array}{l}(c_{1})\,\sim\,\frac{1}{\sqrt{\lambda}}=\,2\cdot\log_{10}\left( Re\cdot\sqrt{\lambda}\right)-0.8=-2\cdot\log_{10}\left(\frac{2.51}{Re}\cdot\frac{1}{ \sqrt{\lambda}}\right)\\ (c_{1})\,\sim\,\frac{1}{\sqrt{\lambda}}=\,1.884\cdot\log_{10}\left(Re\cdot \sqrt{\lambda}\right)-0.331\end{array}  \tag{12}\]  
In addition, for the hydraulically smooth turbulent regime, among others, it is possible to use Equation (13)--by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] (2007, 2010, 2011)--from top to bottom:  
\[ \begin{array}{l}(c_{1})\,\sim\,\frac{1}{\sqrt{\lambda}}=\,1.6\cdot\log_{10} \left(\frac{Re\cdot\sqrt{\lambda}}{2}\right)-0.6\\ (c_{1})\,\sim\,\lambda=0.0072+\frac{0.612}{Re^{0.35}}\\ (c_{1})\,\sim\,\lambda=0.0032+\frac{0.221}{Re^{0.227}}\\ (c_{1})\,\sim\,\lambda=0.0054+\frac{0.936}{Re^{0.35}}\\ (c_{1})\,\sim\,\lambda=\frac{1.02}{\left(\log_{10}\left(Re\right)\right)^{2.5} }\\ (c_{1})\,\sim\,\lambda=\frac{1}{\left(1.81\,\cdot\,\log_{10}\left(Re\right)-1. 5\right)^{2}}\end{array}  \tag{13}\]  
Though the Blasius-form power-law relationships, as seen in Equation (11)-up, have the merit of simplicity, they also have certain disadvantages; one of these disadvantages is that the relationships can only be applied over a limited range of hydraulically smooth regimes. Extrapolations beyond this range cannot be made with confidence. Figure 8 gives a comparison of some of the friction coefficients \(\lambda\) used in the hydraulically smooth turbulent regime. The friction factor \(\lambda\) corresponding to the Hodanovic equation has higher values than those in Panhandle B, as seen in Table 1. This demonstrates the limits of the application of several of the available equations, as many of them were developed for particular situations. For example, Panhandle B is valid for large-diameter pipelines, while Renoard is suitable for the distribution of PVC pipeline networks in urban areas. In addition, the main constraint in using von Karman-Prandtl-form logarithmic relationships, as seen in Equation (11)-down, is their implicit form with respect to flow friction \(\lambda\).