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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Chézy formula | 1/2 | https://en.wikipedia.org/wiki/Chézy_formula | reference | science, encyclopedia | 2026-05-05T13:31:22.562835+00:00 | kb-cron |
The Chézy formula is a semi-empirical resistance equation which estimates mean flow velocity in open channel conduits. 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. 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. 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.
== Description == 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
R
h
S
0
{\displaystyle V=C{\sqrt {R_{h}S_{0}}}}
where,
V
{\displaystyle V}
is average velocity [length/time];
R
h
{\displaystyle R_{h}}
is the hydraulic radius [length], which is the cross-sectional area of flow divided by the wetted perimeter, for a wide channel this is approximately equal to the water depth;
S
0
{\displaystyle S_{0}}
is the hydraulic gradient, which for uniform normal depth of flow is the slope of the channel bottom [unitless; length/length];
C
{\displaystyle C}
is Chézy's coefficient [length1/2/time]. Values of this coefficient must be determined experimentally. Typically, these range from 30 m1/2/s (small rough channel) to 90 m1/2/s (large smooth channel). For many years following Antoine de Chézy's development of this formula, researchers assumed that
C
{\displaystyle C}
was a constant, independent of flow conditions. However, additional research proved the coefficient's dependence on the Reynolds number as well as a channel's roughness. Accordingly, although the Chézy formula does not appear to incorporate either of these terms, the Chézy coefficient empirically and indirectly represents them.
== Exploring Chézy's similarity parameter ==
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:
∑
F
c
v
=
∂
∂
t
∫
C
V
V
ρ
d
V
+
∫
C
S
V
ρ
V
⋅
n
^
d
A
{\displaystyle \sum F_{cv}={\partial \over \partial t}\int \limits _{CV}V\rho \,{dV}+\int \limits _{CS}V\rho V\cdot {\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:
∑
F
x
=
0
=
F
1
−
F
2
−
τ
w
P
l
+
W
sin
θ
{\displaystyle \sum F_{x}=0=F_{1}-F_{2}-\tau _{w}Pl+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
R
h
S
0
{\displaystyle V=C{\sqrt {R_{h}S_{0}}}}
. 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.