# Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow
## 2 Previous Works and the Source of Their Differences  
Only a few formulas valid for all hydraulic regimes exist. Equations (2)-(4) [11; 12; 13] consist of:  
\[\lambda=8\cdot\sqrt[]{\left(\frac{8}{Re}\right)^{12}+\left[\left(-2.457\cdot \ln\left(\left(\frac{7}{Re}\right)^{0.9}+0.27\cdot\varepsilon\right)\right) \right]^{16}+\left(\frac{37530}{Re}\right)^{16}\right]^{-1.5}} \tag{2}\]  
\[\lambda=\sqrt[]{\left(\frac{64}{Re}\right)^{8}+9.5\cdot\left[\ln\left(\frac{ \varepsilon}{3.7}+\frac{5.74}{Re^{0.9}}\right)-\left(\frac{2500}{Re}\right)^{ 6}\right]^{-16}} \tag{3}\]  
\[\lambda=0.11\cdot\sqrt[]{\frac{\lambda_{1}+\varepsilon+(28\cdot\lambda_{1})^{1 4}}{1+115\cdot(28\cdot\lambda_{1})^{10}}} \tag{4}\]  
In Equations (2)-(4), \(\lambda\) is the Darcy friction factor, \(Re\) is the Reynolds number, and \(\varepsilon\) is the relative roughness of the inner pipe surface (all three quantities are dimensionless). The turbulent parts of Equations (2) and (3) are based on the Colebrook formula [7], while Equation (4) is based on formulas from Russian engineering practice [17]. The turbulent part of Equation (2) [11] is based on the [PERSON] approximation of the Colebrook equation [18], while Equation (3) [12] is based on approximations of the Colebrook equation given by [PERSON] and [PERSON] [19]. As shown in Figure 2, formulas from Russian practices and those based on the Colebrook equation yield almost identical results for low values of relative roughness \(\varepsilon\); in the case of higher values of relative roughness \(\varepsilon\), formulas from Russian practice give lower values for the flow friction factor \(\lambda\) compared with the Colebrook equation. Both the Colebrook equation and formulas from Russian practice have a monotonic shape of flow friction curves (Figure 2), which is disputed not only by new experiments such as those from the Princeton and Oregon laboratories [15], but also by an older [PERSON] experiment [8]. The Nikuradse experiment was conducted in 1932 and 1933, while the Colebrook and White experiment was reported in 1937 [9]). The Nikuradse experiment proposed a shape of curves in the turbulent regime as intervals of declining values of the friction factor \(\lambda\) before they reached the final maximal values for the fully developed rough turbulent flow. This property of flow friction factors is known as [PERSON]'s inflectional shape of transition to the fully turbulent regime, as seen in Figure 3[20; 21]. Figure 3: Different shapes of flow friction factor curves obtained using the Colebrook formula versus those that follow [PERSON]’s inflectional law. Figure 2: Different values of the flow friction factor obtained using the Colebrook formula versus the formulas from Russian practice. The currently available formulas valid for all hydraulic regimes, Equations (2)-(4), do not follow [PERSON]'s inflectional law. The [PERSON] equation can be extended to fit the measured data from [PERSON]'s experiment, as seen in Equation (5) [16]:  
\[\left(c_{2}\right) \sim\frac{1}{\sqrt{\lambda}}=-2\cdot\log_{10}\left(\frac{2.51}{Re} \cdot\frac{1}{\sqrt{\lambda}}+\frac{\varepsilon}{3.71}\cdot e^{E}\right)\] \[E=\frac{-31.13}{Re\cdot\varepsilon}\cdot\frac{1}{\sqrt{\lambda}} \tag{5}\]  
In Equation (5), \(e\) is an exponential function, where \(E=0\) gives the Colebrook equation without the Nikuradse extension. The Colebrook equation is given in the implicit form with respect to the flow friction factor \(\lambda\)[22; 23; 24; 25], while the formulas from Russian practice are primarily given in explicit power-law types of expressions [26]. As shown in Figure 4, the Colebrook equation is developed to unify the von Karman-Prandtl equations for both smooth turbulent flow and fully rough flow in a smooth, asymptotic way.