## ABSTRACT

Several formulae of the coefficient of friction have emerged after that of Nikuradse such as the Colebrook–White formula adopted for the calculation of the coefficient of friction in penstocks. However, this requires enormous computing resources because it suffers from an implicit nature. Other authors have proposed explicit formulae for this in order to reduce calculation times and make applications easy. Among these authors, the most used are the formulae of Swamee–Jain and Haaland. In this work, we set out to study the evolution of friction on the top, bottom and side walls of a penstock using these two formulas. The objective is to find which one best characterizes the friction and gives results close to the implicit Colebrook–White formula. To carry out this work, the study was made by a numerical approach in FLUENT. It appears that the Swamee–Jain correlation gives values closer to that of Colebrook–White along the walls. On the side walls, Haaland's formula better describes the constancy of friction with a much larger range of values than Swamee–Jain. On the upper and lower walls, the friction has a linear character.

## HIGHLIGHTS

The Colebrook–White equation is adopted for calculating the coefficient of friction in pipelines.

The Colebrook–White equation serves as a reference formula for the calculation of the coefficient of friction in the conduits.

We propose to determine from the Haaland and Swamee-Jain formulae.

The Three Gorges Dam in China was studied.

## INTRODUCTION

A dam is an engineering structure built across a watercourse and intended to regulate its flow and/or to store water to allow a desired flow. The water retained by the dam is channeled to the turbine using the penstock, which is the site of several physical phenomena. The flow is still turbulent there (Elie 2014) and the pressure drops in the case of small dams are around 15%, of which the most important are due to friction (Chapallaz *et al.* 1995). As a result, friction becomes a key parameter to be controlled in order to guarantee and maintain the structure's performance.

The coefficient of friction is associated with this key parameter and depends both on the Reynolds number that identifies the flow regime and on the roughness of the internal wall of the penstock. The 1937 Colebrook–White equation is adopted for calculating the coefficient of friction in pipelines, given its applicability over a wide range of flow regimes and roughness values. Note that this equation follows from the development of Prandtl for smooth pipes (Prandtl 1932) and from that of Von Karman for rough pipes (Von Karman 1934). Since its publication, it has become the reference formula for calculating the coefficient of friction in large-diameter pipes. However, its use is tedious because it is an implicit equation.

In order to overcome the implicit form of this equation, a certain number of explicit functions have been developed to approach it. These equations are all constant functions and do not take into account any other parameter apart from Reynolds number, length and relative roughness. In fact, this aspect is the subject of a work we are developing and which will be published in the near future.

In this article, we propose to determine from the Haaland formulae, then from Swamee–Jain's, the coefficient of friction in the penstock of a dam by a numerical approach (via the CFD method). The aim is to determine which of these two explicit expressions best approximates the Colebroock–White formula and also to observe which appears to be sensitive to any variation in the penstock. To this end, the dam used for the application of our approach is the Three Gorges Dam in China, with a diameter of 12.4 m and made up of three sections. Note that the flow is not symmetrical in all directions in this penstock (Wang *et al.* 2012).

## MATERIALS AND METHODS

The work is done on the three sections constituting the penstock with its maximum flow rate of 970 m^{3}/s:

The first section is located between the water intake and the first bend;

The second section is located between the two elbows;

The third section is located between the second bend and the turbine inlet.

Our work is based on the CFD method, with modeling of our structure in Gambit 2.4 software and simulation in Fluent 6.3.26 software. The turbulence model used is *k*– *ε* realizable because it is well suited to boundary layers with a strong adverse pressure gradient, to strong curvature and vortex flows, and finally is considered isotropic. The second order equations are solved with the velocity-pressure coupling method SIMPLEC and a convergence criterion 10^{−6}. The approach flow is stationary, the function near walls function considered in the software is the standard wall function, and the discretization scheme is Body Force Weighted.

### General equation of the problem

*et al.*(2014) and Tchawe

*et al.*(2015, 2018):The dynamic conservation is:where

*U*represents the velocities in

_{i}*x*coordinate directions;

_{i}*p*is the static pressure load,

*ρ*the constant density, and

*τ*the viscous stress tensor. For a Newtonian fluid:

_{ij}*i*= 1, 2, 3). In the same way for the pressure and the other scalar values:where

*ϕ*represents a scalar such as pressure, energy, or other concentration.

represents the degree of influence of volume forces, is the component of the flow velocity parallel to the gravitational vector, and *u* is the component of the flow velocity perpendicular to the gravitational vector.

In these equations, represents the generator term of the kinetic energy of turbulence due to the mean of the calculated velocity gradient, is the generator term of the kinetic energy of turbulence due to the volume forces, *Y _{M}* represents the fluctuation of the expansion in compressible turbulence.

*C*

_{1}and

*C*

_{2}are their constants.

*σ*et

_{k}*σ*

_{ε}are the turbulent Prandtl numbers for

*k*and ε respectively.

*S*and

_{k}*S*

_{ε}are user-defined terms.

### Friction coefficient equation

*ε/D*), from the Colebrook–White relation according to the following equation:Haaland (1983) also proposes a correlation for the calculation of friction by:

## RESULTS

The results thus obtained will be presented on the lower (bottom), upper (top) and side walls as a function of the Reynolds number, for each of the three sections that constitute our penstock.

### First section

We first note that the maximum coefficient of friction is not observed at the same height in the penstock on each wall considered in the section.

#### Bottom wall

We note from these curves that the evolution of the coefficient of friction at this scale is substantially linear for the Swamee–Jain formula and that of Haaland. However, Haaland's formula is more sensitive to the effects on the walls.

#### Upper wall

We also observe that the observation made for the lower wall is practically the same for the upper wall.

#### Side wall

The sensitivity noted above in Haaland's formula is more noticeable at this level. The exact value of the coefficient of friction is difficult to observe for a given velocity. Although the Swamee–Jain formula presents a more attenuated form of the coefficient of friction at the same given velocity, it is no less unpredictable.

It thus emerges that the maximum height for a maximum value of coefficient of friction for this section is approximately ≤20 mm.

### Second section

This section has the particularity of being the most inclined of the sections. Thus, the structure of the inlet flow is conditioned by the geographical arrangement of the first section and the shape of the elbow.

#### Bottom wall

We note from these curves that the evolution of the coefficient of friction is still substantially linear for the two formulas. The results are even more or less identical for the two formulas on this scale.

#### Upper wall

From these figures, the finding is the same as that noted on the lower wall of this portion, with a slight fluctuation observed for the maximum velocity in Haaland's formula.

#### Side wall

The same observation is made when determining the friction coefficient on the side wall of the first section. Haaland's formula still presents its more sensitive character than Swamee–Jain's. The maximum height for a value of the maximum friction coefficient is substantially ≤60 mm in this section.

### Third section

Unlike the other two sections, this one is laid out horizontally and has the particularity of being just upstream of the turbine. The efficiency of the turbine depends on the characteristics resulting directly from it.

#### Bottom wall

The two formulas describe substantially the same behavior on this wall as noted in the case of the preceding sections.

#### Upper wall

Likewise, a sensitivity was observed in the result obtained from Haaland's formula compared to Swamee–Jain.

#### Side wall

As for the first two sections, the observation is the same when we highlight the coefficient of friction on the side wall. Haaland's formula always presents a more sensitive character than that of Swamee–Jain. The average height for a value of the maximum coefficient of friction is this time substantially ≤5 mm in this section.

To better appreciate the different results, a numerical calculation was carried out with the formula proposed by Colebrook–White (1937, 1939). The different results are shown in Tables 1–3, with the thicknesses of the boundary layers found for each correlation.

Walls . | Velocity (m/s) . | Formulas . | Colebrook–White . | Difference . | ||||
---|---|---|---|---|---|---|---|---|

Haaland . | Swamee–Jain . | Haaland . | Swamee–Jain . | |||||

Thickness (mm) . | f_{max}
. | Thickness (mm) . | f_{max}
. | |||||

Bottom | 7 | 20 | 9.32337 × 10^{–3} | 20 | 9.30871 × 10^{–3} | 9.30201 × 10^{–3} | 2.136 × 10^{–5} | 6.696 × 10^{–6} |

8 | 20–25 | 9.32262 × 10^{–3} | 20 | 9.30779 × 10^{–3} | 9.30201 × 10^{–3} | 2.061 × 10^{–5} | 5.785 × 10^{–6} | |

9 | 20–25 | 9.32251 × 10^{–3} | 20 | 9.30713 × 10^{–3} | 9.30200 × 10^{–3} | 2.051 × 10^{–5} | 5.132 × 10^{–6} | |

Upper | 7 | 0.05–2 | 9.32304 × 10^{–3} | 0.05–2 | 9.30810 × 10^{–3} | 9.30201 × 10^{–3} | 2.103 × 10^{–5} | 6.086 × 10^{–6} |

8 | 0.05–3 | 9.32259 × 10^{–3} | 0.05–3 | 9.30727 × 10^{–3} | 9.30201 × 10^{–3} | 2.058 × 10^{–5} | 5.265 × 10^{–6} | |

9 | 0.05–4 | 9.32226 × 10^{–3} | 0.05–1.2 | 9.30665 × 10^{–3} | 9.30200 × 10^{–3} | 2.026 × 10^{–5} | 4.652 × 10^{–6} | |

Side | 7 | 0.05–4 | 9.32285 × 10^{–3} | 0.05–4 | 9.30775 × 10^{–3} | 9.30201 × 10^{–3} | 2.084 × 10^{–5} | 5.736 × 10^{–6} |

8 | 0.05–4 | 9.32243 × 10^{–3} | 0.05–4 | 9.30696 × 10^{–3} | 9.30201 × 10^{–3} | 2.042 × 10^{–5} | 4.955 × 10^{–6} | |

9 | 0.05–4 | 9.32212 × 10^{–3} | 0.05–4 | 9.30637 × 10^{–3} | 9.30200 × 10^{–3} | 2.012 × 10^{–5} | 4.372 × 10^{–6} |

Walls . | Velocity (m/s) . | Formulas . | Colebrook–White . | Difference . | ||||
---|---|---|---|---|---|---|---|---|

Haaland . | Swamee–Jain . | Haaland . | Swamee–Jain . | |||||

Thickness (mm) . | f_{max}
. | Thickness (mm) . | f_{max}
. | |||||

Bottom | 7 | 20 | 9.32337 × 10^{–3} | 20 | 9.30871 × 10^{–3} | 9.30201 × 10^{–3} | 2.136 × 10^{–5} | 6.696 × 10^{–6} |

8 | 20–25 | 9.32262 × 10^{–3} | 20 | 9.30779 × 10^{–3} | 9.30201 × 10^{–3} | 2.061 × 10^{–5} | 5.785 × 10^{–6} | |

9 | 20–25 | 9.32251 × 10^{–3} | 20 | 9.30713 × 10^{–3} | 9.30200 × 10^{–3} | 2.051 × 10^{–5} | 5.132 × 10^{–6} | |

Upper | 7 | 0.05–2 | 9.32304 × 10^{–3} | 0.05–2 | 9.30810 × 10^{–3} | 9.30201 × 10^{–3} | 2.103 × 10^{–5} | 6.086 × 10^{–6} |

8 | 0.05–3 | 9.32259 × 10^{–3} | 0.05–3 | 9.30727 × 10^{–3} | 9.30201 × 10^{–3} | 2.058 × 10^{–5} | 5.265 × 10^{–6} | |

9 | 0.05–4 | 9.32226 × 10^{–3} | 0.05–1.2 | 9.30665 × 10^{–3} | 9.30200 × 10^{–3} | 2.026 × 10^{–5} | 4.652 × 10^{–6} | |

Side | 7 | 0.05–4 | 9.32285 × 10^{–3} | 0.05–4 | 9.30775 × 10^{–3} | 9.30201 × 10^{–3} | 2.084 × 10^{–5} | 5.736 × 10^{–6} |

8 | 0.05–4 | 9.32243 × 10^{–3} | 0.05–4 | 9.30696 × 10^{–3} | 9.30201 × 10^{–3} | 2.042 × 10^{–5} | 4.955 × 10^{–6} | |

9 | 0.05–4 | 9.32212 × 10^{–3} | 0.05–4 | 9.30637 × 10^{–3} | 9.30200 × 10^{–3} | 2.012 × 10^{–5} | 4.372 × 10^{–6} |

Walls . | Velocity (m/s) . | Formulas . | Colebrook–White . | Difference . | ||||
---|---|---|---|---|---|---|---|---|

Haaland . | Swamee–Jain . | Haaland . | Swamee–Jain . | |||||

Thickness (mm) . | f_{max}
. | Thickness (mm) . | f_{max}
. | |||||

Bottom | 7 | 0.05–70 | 9.32349 × 10^{–3} | 0.05–70 | 9.30894 × 10^{–3} | 9.30201 × 10^{–3} | 2.148 × 10^{–5} | 6.926 × 10^{–6} |

8 | 0.05–70 | 9.32265 × 10^{–3} | 0.05–70 | 9.30801 × 10^{–3} | 9.30201 × 10^{–3} | 2.064 × 10^{–5} | 6.005 × 10^{–6} | |

9 | 0.05–70 | 9.32262 × 10^{–3} | 0.05–70 | 9.30732 × 10^{–3} | 9.30200 × 10^{–3} | 2.062 × 10^{–5} | 5.322 × 10^{–6} | |

Upper | 7 | 0.05–1.2 | 9.32296 × 10^{–3} | 0.05–1,2 | 9.30795 × 10^{–3} | 9.30201 × 10^{–3} | 2.095 × 10^{–5} | 5.936 × 10^{–6} |

8 | 0.05–1.2 | 9.32195 × 10^{–3} | 0.05–1,2 | 9.30713 × 10^{–3} | 9.30201 × 10^{–3} | 1.994 × 10^{–5} | 5.125 × 10^{–6} | |

9 | 0.05–1.2 | 9.32220 × 10^{–3} | 0.05–1,2 | 9.30653 × 10^{–3} | 9.30200 × 10^{–3} | 2.020 × 10^{–5} | 4.532 × 10^{–6} | |

Side | 7 | 0.05–4 | 9.32277 × 10^{–3} | 0.05–4 | 9.30761 × 10^{–3} | 9.30201 × 10^{–3} | 2.076 × 10^{–5} | 5.596 × 10^{–6} |

8 | 0.05–70 | 9.32236 × 10^{–3} | 0.05–70 | 9.30683 × 10^{–3} | 9.30201 × 10^{–3} | 2.035 × 10^{–5} | 4.825 × 10^{–6} | |

9 | 5 | 9.32206 × 10^{–3} | 5 | 9.30627 × 10^{–3} | 9.30199 × 10^{–3} | 2.006 × 10^{–5} | 4.272 × 10^{–6} |

Walls . | Velocity (m/s) . | Formulas . | Colebrook–White . | Difference . | ||||
---|---|---|---|---|---|---|---|---|

Haaland . | Swamee–Jain . | Haaland . | Swamee–Jain . | |||||

Thickness (mm) . | f_{max}
. | Thickness (mm) . | f_{max}
. | |||||

Bottom | 7 | 0.05–70 | 9.32349 × 10^{–3} | 0.05–70 | 9.30894 × 10^{–3} | 9.30201 × 10^{–3} | 2.148 × 10^{–5} | 6.926 × 10^{–6} |

8 | 0.05–70 | 9.32265 × 10^{–3} | 0.05–70 | 9.30801 × 10^{–3} | 9.30201 × 10^{–3} | 2.064 × 10^{–5} | 6.005 × 10^{–6} | |

9 | 0.05–70 | 9.32262 × 10^{–3} | 0.05–70 | 9.30732 × 10^{–3} | 9.30200 × 10^{–3} | 2.062 × 10^{–5} | 5.322 × 10^{–6} | |

Upper | 7 | 0.05–1.2 | 9.32296 × 10^{–3} | 0.05–1,2 | 9.30795 × 10^{–3} | 9.30201 × 10^{–3} | 2.095 × 10^{–5} | 5.936 × 10^{–6} |

8 | 0.05–1.2 | 9.32195 × 10^{–3} | 0.05–1,2 | 9.30713 × 10^{–3} | 9.30201 × 10^{–3} | 1.994 × 10^{–5} | 5.125 × 10^{–6} | |

9 | 0.05–1.2 | 9.32220 × 10^{–3} | 0.05–1,2 | 9.30653 × 10^{–3} | 9.30200 × 10^{–3} | 2.020 × 10^{–5} | 4.532 × 10^{–6} | |

Side | 7 | 0.05–4 | 9.32277 × 10^{–3} | 0.05–4 | 9.30761 × 10^{–3} | 9.30201 × 10^{–3} | 2.076 × 10^{–5} | 5.596 × 10^{–6} |

8 | 0.05–70 | 9.32236 × 10^{–3} | 0.05–70 | 9.30683 × 10^{–3} | 9.30201 × 10^{–3} | 2.035 × 10^{–5} | 4.825 × 10^{–6} | |

9 | 5 | 9.32206 × 10^{–3} | 5 | 9.30627 × 10^{–3} | 9.30199 × 10^{–3} | 2.006 × 10^{–5} | 4.272 × 10^{–6} |

Walls . | Velocity (m/s) . | Formulas . | Colebrook–White . | Difference . | ||||
---|---|---|---|---|---|---|---|---|

Haaland . | Swamee–Jain . | Haaland . | Swamee–Jain . | |||||

Thickness (mm) . | f_{max}
. | Thickness (mm) . | f_{max}
. | |||||

Bottom | 7 | 0.05–2 | 9.32329 × 10^{–3} | 0.05–1.2 | 9.30859 × 10^{–3} | 9.30201 × 10^{–3} | 2.128 × 10^{–5} | 6.576 × 10^{–6} |

8 | 0.05–60 | 9.32282 × 10^{–3} | 0.05 | 9.30770 × 10^{–3} | 9.30201 × 10^{–3} | 2.081 × 10^{–5} | 5.695 × 10^{–6} | |

9 | 0.05–2 | 9.32247 × 10^{–3} | 0.05–2 | 9.30701 × 10^{–3} | 9.30200 × 10^{–3} | 2.047 × 10^{–5} | 5.012 × 10^{–6} | |

Upper | 7 | 0.05–2 | 9.32296 × 10^{–3} | 0.05–3 | 9.30797 × 10^{–3} | 9.30201 × 10^{–3} | 2.095 × 10^{–5} | 5.956 × 10^{–6} |

8 | 0.05–3 | 9.32253 × 10^{–3} | 0.05–3 | 9.30715 × 10^{–3} | 9.30201 × 10^{–3} | 2.052 × 10^{–5} | 5.145 × 10^{–6} | |

9 | 0.05–1.2 | 9.32221 × 10^{–3} | 0.05–3 | 9.30655 × 10^{–3} | 9.30200 × 10^{–3} | 2.021 × 10^{–5} | 4.552 × 10^{–6} | |

Side | 7 | 0.05–70 | 9.32279 × 10^{–3} | 0.05–70 | 9.30765 × 10^{–3} | 9.30201 × 10^{–3} | 2.078 × 10^{–5} | 5.636 × 10^{–6} |

8 | 0.05–70 | 9.32238 × 10^{–3} | 0.05–70 | 9.30686 × 10^{–3} | 9.30201 × 10^{–3} | 2.037 × 10^{–5} | 4.855 × 10^{–6} | |

9 | 5 | 9.32208 × 10^{–3} | 5 | 9.30631 × 10^{–3} | 9.30200 × 10^{–3} | 2.008 × 10^{–5} | 4.312 × 10^{–6} |

Walls . | Velocity (m/s) . | Formulas . | Colebrook–White . | Difference . | ||||
---|---|---|---|---|---|---|---|---|

Haaland . | Swamee–Jain . | Haaland . | Swamee–Jain . | |||||

Thickness (mm) . | f_{max}
. | Thickness (mm) . | f_{max}
. | |||||

Bottom | 7 | 0.05–2 | 9.32329 × 10^{–3} | 0.05–1.2 | 9.30859 × 10^{–3} | 9.30201 × 10^{–3} | 2.128 × 10^{–5} | 6.576 × 10^{–6} |

8 | 0.05–60 | 9.32282 × 10^{–3} | 0.05 | 9.30770 × 10^{–3} | 9.30201 × 10^{–3} | 2.081 × 10^{–5} | 5.695 × 10^{–6} | |

9 | 0.05–2 | 9.32247 × 10^{–3} | 0.05–2 | 9.30701 × 10^{–3} | 9.30200 × 10^{–3} | 2.047 × 10^{–5} | 5.012 × 10^{–6} | |

Upper | 7 | 0.05–2 | 9.32296 × 10^{–3} | 0.05–3 | 9.30797 × 10^{–3} | 9.30201 × 10^{–3} | 2.095 × 10^{–5} | 5.956 × 10^{–6} |

8 | 0.05–3 | 9.32253 × 10^{–3} | 0.05–3 | 9.30715 × 10^{–3} | 9.30201 × 10^{–3} | 2.052 × 10^{–5} | 5.145 × 10^{–6} | |

9 | 0.05–1.2 | 9.32221 × 10^{–3} | 0.05–3 | 9.30655 × 10^{–3} | 9.30200 × 10^{–3} | 2.021 × 10^{–5} | 4.552 × 10^{–6} | |

Side | 7 | 0.05–70 | 9.32279 × 10^{–3} | 0.05–70 | 9.30765 × 10^{–3} | 9.30201 × 10^{–3} | 2.078 × 10^{–5} | 5.636 × 10^{–6} |

8 | 0.05–70 | 9.32238 × 10^{–3} | 0.05–70 | 9.30686 × 10^{–3} | 9.30201 × 10^{–3} | 2.037 × 10^{–5} | 4.855 × 10^{–6} | |

9 | 5 | 9.32208 × 10^{–3} | 5 | 9.30631 × 10^{–3} | 9.30200 × 10^{–3} | 2.008 × 10^{–5} | 4.312 × 10^{–6} |

We notice in these tables on one hand the differences between the results of the coefficient of friction obtained by the Colebrook–White formula and those of the two other formulae. It is of the order of 10^{−6} for the Swamee–Jain formula and of the order of 10^{−5} for the Haaland formula. Regarding the thicknesses at which the friction is maximum, the Swamee–Jain correlation gives substantially identical results for different velocities on the bottom and upper walls. On the other hand, these thicknesses vary as a function of the velocity in the Haaland correlation. Regarding the side wall, the two formulas give the same height for the friction coefficient.

## DISCUSSION

Winning & Coole (2015) propose an improved method to determine the coefficient of friction in pipes by mathematical calculations. They calculate the mean square error (MSE), the mean relative error in percentage (ERMP) and the mean time required to perform 300 million calculations using random input values obtained in 28 explicit equations examined. These equations and the results of the calculations are listed in Table 4.

Equation . | MSE . | ERMP . | Mean time . |
---|---|---|---|

Sonnad & Goudar (2006) | 1.05 × 10^{–8} | 0.174 | 0.5999 |

Schorle et al. (1980) | 3.83 × 10^{–9} | 0.125 | 0.6224 |

Zigrang & Sylvester (1982), Eq. (11) | 3.10 × 10^{–9} | 0.141 | 0.6343 |

Seghides (1984), Eq. (3) | 1.89 × 10^{–10} | 0.037 | 0.6463 |

Jain (1976) | 1.33 × 10^{–7} | 0.452 | 0.6016 |

Haaland (1983) | 2.67 × 10^{–8} | 0.373 | 0.6326 |

Fang et al. (2011) | 6.55 × 10^{–9} | 0.156 | 0.6411 |

Swamee & Jain (1976) | 1.61 × 10^{–7} | 0.478 | 0.5931 |

Seghides (1984), Eq. (2) | 7.51 × 10^{–12} | 0.006 | 0.6582 |

Churchill (1973) | 1.68 × 10^{–7} | 0.492 | 0.5829 |

Chen (1979) | 2.03 × 10^{–9} | 0.097 | 0.6531 |

Eck (1973) | 4.64 × 10^{–7} | 1.503 | 0.559 |

Barr (1981) | 1.95 × 10^{–9} | 0.063 | 0.6616 |

Zigrang & Sylvester (1982) Eq. (12) | 7.07 × 10^{–11} | 0.019 | 0.6735 |

Churchill (1977) | 2.19 × 10^{–7} | 0.475 | 0.604 |

Wood (1966) | 5.91 × 10^{–6} | 3.876 | 0.525 |

Manadilli (1997) | 1.28 × 10^{–7} | 0.393 | 0.6514 |

Round (1980) | 1.84 × 10^{–5} | 4.474 | 0.5795 |

Buzzelli (2008) | 5.62 × 10^{–12} | 0.005 | 0.6922 |

Avci & Karagoz (2009) | 3.35 × 10^{–6} | 1.716 | 0.6122 |

Moody (1947) | 7.14 × 10^{–5} | 6.098 | 0.4892 |

Romeo et al. (2002) | 1.33 × 10^{–9} | 0.057 | 0.711 |

Tsal (1989) | 1.75 × 10^{–4} | 8.894 | 0.5675 |

Brkić (2011) Eq. (B) | 1.35 × 10^{–7} | 0.479 | 0.6735 |

Altshul cited in Genić et al. (2011) | 1.75 × 10^{–4} | 11.449 | 0.4773 |

Papaevangelou et al. (2010) | 4.09 × 10^{–8} | 0.23 | 0.7399 |

Brkić (2011), Eq. (A) | 1.51 × 10^{–7} | 0.721 | 0.6939 |

Rao & Kumar (2007) | 5.16 × 10^{–5} | 13.27 | 0.6854 |

Equation . | MSE . | ERMP . | Mean time . |
---|---|---|---|

Sonnad & Goudar (2006) | 1.05 × 10^{–8} | 0.174 | 0.5999 |

Schorle et al. (1980) | 3.83 × 10^{–9} | 0.125 | 0.6224 |

Zigrang & Sylvester (1982), Eq. (11) | 3.10 × 10^{–9} | 0.141 | 0.6343 |

Seghides (1984), Eq. (3) | 1.89 × 10^{–10} | 0.037 | 0.6463 |

Jain (1976) | 1.33 × 10^{–7} | 0.452 | 0.6016 |

Haaland (1983) | 2.67 × 10^{–8} | 0.373 | 0.6326 |

Fang et al. (2011) | 6.55 × 10^{–9} | 0.156 | 0.6411 |

Swamee & Jain (1976) | 1.61 × 10^{–7} | 0.478 | 0.5931 |

Seghides (1984), Eq. (2) | 7.51 × 10^{–12} | 0.006 | 0.6582 |

Churchill (1973) | 1.68 × 10^{–7} | 0.492 | 0.5829 |

Chen (1979) | 2.03 × 10^{–9} | 0.097 | 0.6531 |

Eck (1973) | 4.64 × 10^{–7} | 1.503 | 0.559 |

Barr (1981) | 1.95 × 10^{–9} | 0.063 | 0.6616 |

Zigrang & Sylvester (1982) Eq. (12) | 7.07 × 10^{–11} | 0.019 | 0.6735 |

Churchill (1977) | 2.19 × 10^{–7} | 0.475 | 0.604 |

Wood (1966) | 5.91 × 10^{–6} | 3.876 | 0.525 |

Manadilli (1997) | 1.28 × 10^{–7} | 0.393 | 0.6514 |

Round (1980) | 1.84 × 10^{–5} | 4.474 | 0.5795 |

Buzzelli (2008) | 5.62 × 10^{–12} | 0.005 | 0.6922 |

Avci & Karagoz (2009) | 3.35 × 10^{–6} | 1.716 | 0.6122 |

Moody (1947) | 7.14 × 10^{–5} | 6.098 | 0.4892 |

Romeo et al. (2002) | 1.33 × 10^{–9} | 0.057 | 0.711 |

Tsal (1989) | 1.75 × 10^{–4} | 8.894 | 0.5675 |

Brkić (2011) Eq. (B) | 1.35 × 10^{–7} | 0.479 | 0.6735 |

Altshul cited in Genić et al. (2011) | 1.75 × 10^{–4} | 11.449 | 0.4773 |

Papaevangelou et al. (2010) | 4.09 × 10^{–8} | 0.23 | 0.7399 |

Brkić (2011), Eq. (A) | 1.51 × 10^{–7} | 0.721 | 0.6939 |

Rao & Kumar (2007) | 5.16 × 10^{–5} | 13.27 | 0.6854 |

From this table, it can be seen that the Swamee–Jain equation, although more widely used in industry according to Achour & Bedjaoui (2006), is neither more efficient nor faster. The same is true for the Haaland equation. However, they are the ones that offer the mean of the weakest squares. These results show us that Haaland's formula is precise with a low ERMP, but less efficient and slower compared to Swamee–Jain. On the other hand, Gregory & McEnery (2017) developed and tested against data generated from the accepted standard for predicting the coefficient of friction in Moody's diagram, an equation for accurately predicting the coefficient of friction without iteration. Rather, they rely on the Haaland expression, as it is more precise than the Swamee–Jain equation.

Thus from the results obtained in this work, the Swamee–Jain equation gives numerical results closer to the results calculated from the Colebrook–White expression. On the side walls of all the sections, we find the maximum friction in a range of the same thickness (4–6 mm); which results in the fact that the friction on this wall does not evolve and does not take into account the position of the geometry. Using Haaland's formula, we observe its sensitivity to the thickness of the boundary layer as a function of the velocity and the wall chosen, which is not the case for the Swamee–Jain formula. Haalands formula turns out to be more interesting for the description of the phenomenon of turbulent friction along the walls. However, the Swamee–Jain formula comes closest to the Colebrook–White equation. It should be noted that this work is carried out with clear water, and without sediment transport.

## CONCLUSIONS

In this work, the focus was on determining the coefficient of friction using two approaches and finding which one best characterizes the friction and/or which gives results close to the implicit formula of Colebrook–White. Using a numerical approach based on the CFD method, we first started with the expression developed by Haaland, and second, with the formula developed by Swamee–Jain. It appears that the Haaland equation better describes the evolution of friction because it has a more sensitive character than that of Swamee–Jain on phenomena near the walls. However, the Swamee–Jain equation gives numerical results closer to the results calculated from the Colebrook–White expression. Regarding the thicknesses at which the friction is maximum, the Swamee–Jain correlation gives substantially identical results for different velocities on the lower and upper walls. On the other hand, these thicknesses vary as a function of the velocity in the Haaland correlation. Regarding the side wall, the two formulas give the same height for the coefficient of friction. Likewise, we note that the average height for a value of the maximum coefficient of friction varies according to the geographical disposition of the penstock portion.

## ACKNOWLEDGEMENTS

We sincerely acknowledge IUG-Cameroon that eased the realization of this work, and pay homage to its founder Louis-Marie DJAMBOU (rest in peace) who opened the door to his institution.

## FUNDING

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

## AUTHOR CONTRIBUTIONS

All authors have read and approved the final manuscript.

## ETHICAL STATEMENT

Free and informed consent of the participants or their legal representatives was obtained.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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