The Chézy Formula is a semi-empirical resistance equation[1] [2] which estimates mean flow velocity in open channel conduits.[3] The relationship was conceptualized and developed in 1768 by French physicist and engineer Antoine de Chézy (1718–1798) while designing Paris's water canal system.[4] Chézy discovered a similarity parameter that could be used for estimating flow characteristics in one channel based on the measurements of another. The Chézy formula is a pioneering formula in the field of fluid mechanics that relates the flow of water through an open channel with the channel's dimensions and slope. It was expanded and modified by Irish engineer Robert Manning in 1889.[5] [6] [7] Manning's modifications to the Chézy formula allowed the entire similarity parameter to be calculated by channel characteristics rather than by experimental measurements. Today, the Chézy and Manning equations continue to accurately estimate open channel fluid flow and are standard formulas in various fields related to fluid mechanics and hydraulics, including physics, mechanical engineering, and civil engineering.
The Chézy formula describes mean flow velocity in turbulent open channel flow and is used broadly in fields related to fluid mechanics and fluid dynamics. Open channels refer to any open conduit, such as rivers, ditches, canals, or partially full pipes. The Chézy formula is defined for uniform equilibrium and non-uniform, gradually varied flows.
The formula is written as:
V=C\sqrt{RhS0}
V
Rh
S0
C
For many years following Antoine de Chézy's development of this formula, researchers assumed that
C
The relationship between linear momentum and deformable fluid bodies is well explored, as are the Navier–Stokes equations for incompressible flow. However, exploring the relationships foundational to the Chézy formula can be helpful towards understanding the formula in full.
To understand the Chézy similarity parameter, a simple linear momentum equation can help summarize the conservation of momentum of a control volume uniformly flowing through an open channel:
\sumFcv={\partial\over\partialt}\int\limitsCVV\rho{dV}+\int\limitsCSV\rhoV ⋅ \hat{n}{dA}
Where the sum of forces on the contents of a control volume in the open channel is equal to the sum of the time rate of change of the linear momentum of the contents of the control volume, plus the net rate of flow of linear momentum through the control surface. The momentum principle may always be used for hydrodynamic force calculations.
As long as uniform flow can be assumed, applying the linear momentum equation to a river channel flowing in one dimension means that momentum remains conserved and the forces are balanced in the direction of flow:
\sumFx=0=F1-F2-\tauwPl+W\sin\theta
Here, the hydrostatic pressure forces are F1 and F2, the component (τwPl) represents the shear force of friction acting on the control volume, and the component (ω sin θ) represents the gravitational force of the fluid's weight acting on the sloped channel bottom are held in balance in the flow direction. The free-body diagram below illustrates this equilibrium of forces in open channel flow with uniform flow conditions.
Most open-channel flows are turbulent and characterised by very large Reynolds numbers. Due to the large Reynolds numbers characteristic in open channel flow, the channel shear stress proves to be proportional to the density and velocity of the flow.
This can be illustrated in a series of advanced formulas which identify a shear stress similarity parameter characteristic of all turbulent open channels. Combining this parameter with the Chézy formula, channel components and the conservation of momentum in an open channel flow results in the relationship
V=C\sqrt{RhS0}
Chézy's similarity parameter and formula explain how the velocity of water flowing through a channel has a relationship with the slope and sheer stress of the channel bottom, the hydraulic radius of flow, and the Chézy coefficient, which empirically incorporates several other parameters of the flowing water. This relationship is driven by the conservation of momentum present during uniform flow conditions.
Once this relationship was established by Chézy, many engineers and physicists (see the below section Authors of flow formulas)[9] continued to search for ways to improve Chézy's equation. A slight oversight of Chézy's formula was determined by the research of these colleagues. They determined that the velocity's slope dependence in Chézy's formula (V:S0) was reasonable, but that the velocity's dependence on the hydraulic radius (V:Rh1/2) was not reasonable and that the relationship was closer to (V:Rh2/3). Many formulas based on Chézy's formula have been developed since its discovery by these contemporaries and others, and differing formulas are more suitable in differing conditions.
The Chézy formula provided a substantial foundation for a new flow formula proposed in 1889 by Irish engineer Robert Manning. Manning's formula is a modified Chézy formula that combines many of his aforementioned contemporaries' work. Manning's modifications to the Chézy formula allowed the entire similarity parameter to be calculated by channel characteristics rather than by experimental measurements. The Manning equation improved Chézy's equation by better representing the relationship between Rh and velocity, while also replacing the empirical Chézy coefficient (
C
n
C
n
The Manning formula is described elsewhere but it is included below for comparison purposes. Below, the minor modifications used by the Manning formula to improve upon the Chézy formula are clear.
V=C\sqrt{RhS0}
V=
| {Rh | |
| 2/3 |
| 1/2 | |
| S | |
| 0 |
Chézy formula Manning formula
This similarity between the Chézy and Manning formulas shown above also means that the standardized Manning coefficients may be used to estimate open channel flow velocity with the Chézy formula, by using them to calculate the Chézy's coefficient as shown below. Manning derived the following relationship between Manning coefficient (
n
C
C=k\left[
| 1 | |
| n |
| 1/6 | |
| R | |
| h |
\right]
where
C
R
n
k
Since the Chézy formula and the Manning formula both reference a single control volume location along the channel, neither address friction factor nor head loss directly. However, the change in pressure head may be calculated by combining them with other formulas such as the Darcy–Weisbach equation.
The empirical aspect to the
C
Since partially full pipes aren't pressurized, they are considered open channels by definition. Therefore, the Manning and Chézy formulas can be applied to calculate partially full pipe flow.[10] [11] However, the intended use of these formulas are primarily for considering uniform and turbulent flow. Many other formulas that have been developed since may produce more accurate results, such as the Darcy–Weisbach equation or the Hazen–Williams equation, but lack the simplicity of the Manning or Chézy formulas.
Both formulas continue to be broadly taught and are used in open channel and fluid dynamics research. Today, the Manning formula is likely the most globally used formula for open channel uniform flow analysis, due to its simplicity, proven efficacy, and the fact that most open channel studies are concerned with turbulent flow.[12] Chézy's formula is one of the oldest in the field of fluid mechanics, it applies to a wider range of flows than the Manning equation,[13] and its influence continues to this day.