# Unified Friction Formulation from Laminar to Fully Rough Turbulent Flow
## 1 Introduction  
In the field of hydraulics, resistance in pipe flow is usually represented by the Darcy flow friction factor \(\lambda\), which depends on the Reynolds number \(Re\) and the relative roughness of the inner pipe surface \(\varepsilon\)[1]. All three quantities are dimensionless. For pipe flow, the Reynolds number \(Re\) usually takes values between 0 and \(10^{8}\), while the relative roughness of the inner pipe surface \(\varepsilon\) ranges from 0 to 0.05. A pipe's material, as well as its condition, is characterized by the relative roughness of the inner pipe surface, but also depends on the hydraulic flow regime ruled by the thickness of a thin laminar boundary sublayer of fluid near the inner pipe surface [2,3] (Figure 1(top)). In general, several different hydraulic regimes can occur during pipe flow (Figure 1): the laminar regime \((a)\), the sharp transition from the laminar regime to the smooth turbulent regime \((b)\), the smooth turbulent regime \((c_{1})\), the non-fully developed partially turbulent regime \((c_{2})\), and the fully developed rough turbulent regime \((c_{3})\). In \((a)\), the laminar regime is characterized by the absence of vortices, while the roughness of the inner pipe surface does not have flow effects. In \((b)\), the transition from the laminar regime to the turbulent regime is sharp and almost immediate, so it can be described by a sudden increase in the friction factor \(\lambda\). The turbulent regimes in \((c)\) are characterized by higher values of the Reynolds number \(Re\), with vortices in flow. In \((c_{1})\), the smooth turbulent regime is characterized by vortices in the middle of the pipe, while in \((c_{2})\) the roughness of the inner pipe surface is covered by the laminarsublayer. In \((c_{3})\), with an increase of the Reynolds number \(Re\), the thickness of the laminar sublayer decreases, and the roughness of the inner pipe surface becomes important. Thus, in the fully developed rough turbulent flow regime, the roughness of the inner pipe surface plays a dominant role. Figure 1(top) includes a physical interpretation of the different hydraulic regimes, while Figure 1(bottom) contains related diagrams. Figure 1: Hydraulic regimes for pipe flow: the laminar regime \((a)\), the sharp transition from the laminar regime to the smooth turbulent regime (critical) \((b)\), the smooth turbulent regime \((c_{1})\), the non-fully developed partially turbulent regime \((c_{2})\), and the fully developed rough turbulent regime \((c_{3})\). (**top**) contains physical interpretations, while (**bottom**) contains related diagrams. Similar to Figure 1, the widely-known Moody diagram [4; 5; 6] provides a representation of different hydraulic regimes. The turbulent part of the Moody diagram is based on the Colebrook equation [7]. While the formula for the laminar regime is theoretical, the formulas for all types of turbulent regimes are empirical [8; 9; 10]. Although coherent unified models for all hydraulic regimes exist [11; 12; 13], they are not as flexible as the model presented here because they are fixed (i.e., they do not allow changing separate formulas valid for different hydraulic regimes). In contrast, the unified formula presented here allows any of the formulas for a particular hydraulic regime to be included: one for the laminar regime \((a)\), one for the hydraulically smooth turbulent regime \((c_{1})\), and one for the fully developed rough turbulent regime \((c_{3})\). Thus, the unified formula presented here uses three switching functions--\(y_{1}\), \(y_{2}\), and \(y_{3}\)--to provide a smooth transition between hydraulic regimes, as seen in Equations (7)-(9) of this paper.