8.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Chézy formula | 2/2 | https://en.wikipedia.org/wiki/Chézy_formula | reference | science, encyclopedia | 2026-05-05T13:31:22.562835+00:00 | kb-cron |
== Chézy's formula inspires the Manning formula == Once this relationship was established by Chézy, many engineers and physicists (see the below section Authors of flow formulas) 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
{\displaystyle C}
) with the Manning resistance coefficient (
n
{\displaystyle n}
), which is also referenced in places as the Manning roughness coefficient. Unlike the Chézy coefficient (
C
{\displaystyle C}
) which could only be determined by field measurements, the Manning coefficient (
n
{\displaystyle n}
) was determined to remain constant based on the material of the wetted perimeter, allowing for a standardized table of values to be developed that could reasonably estimate flow velocity. While field measurements remain the most precise way to obtain either Chézy or Manning coefficients, the standardized values that were developed with the use of the Manning formula provided a much-desired simplicity to open-channel flow estimates.
=== Chézy formula vs Manning formula === 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
R
h
S
0
{\displaystyle V=C{\sqrt {R_{h}S_{0}}}}
V
=
R
h
2
/
3
S
0
1
/
2
n
{\displaystyle V={\frac {{R_{h}}^{2/3}S_{0}^{1/2}}{n}}\,}
Chézy formula Manning formula
=== Using Chézy formula with Manning coefficient === 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
{\displaystyle n}
) to Chézy coefficient (
C
{\displaystyle C}
) based upon experiments:
C
=
k
[
1
n
R
h
1
/
6
]
{\displaystyle C=k\left[{\frac {1}{n}}R_{h}^{1/6}\right]}
where
C
{\displaystyle C}
is the Chézy coefficient [length1/2/time], a function of relative roughness and Reynolds number;
R
{\displaystyle R}
is the hydraulic radius, 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) [m];
n
{\displaystyle n}
is Manning's coefficient [time/length1/3]; and
k
{\displaystyle k}
is a constant; k = 1 when using SI units and k = 1.49 when using BG units.
== Modern use of Chézy and Manning formulas == 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
{\displaystyle C}
coefficient indirectly addresses friction factor and Reynold's number and is the reason why the Chézy formula remains most accurate in certain conditions, such as river channels with non-uniform channel dimensions. Additionally, both equations are explicitly used with uniform or "steady-state" flow where the hydraulic depth is constant, due to their derivation from the conservation of momentum. In contrast, if the hydraulic conditions fluctuate in open channel flow, they are then described as gradually or rapidly varied flow, and will require further analyses beyond these two formula methods. 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. 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. 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, and its influence continues to this day.
== See also == Hydrology Hydraulic engineering
=== Authors of flow formulas === Albert Brahms (1692–1758) Antoine de Chézy (1718–1798) Claude-Louis Navier (1785–1836) Adhémar Jean Claude Barré de Saint-Venant (1797–1886) Gotthilf Heinrich Ludwig Hagen (1797–1884) Jean Léonard Marie Poiseuille (1797–1869) Henri P. G. Darcy (1803–1858) Julius Ludwig Weisbach (1806–1871) Charles Storrow (1809–1904) Robert Manning (1816–1897) Wilhelm Rudolf Kutter (1818–1888) Emile Oscar Ganguillet (1818–1894) Sir George Stokes (1819–1903) Philippe Gaspard Gauckler (1826–1905) Henri-Émile Bazin (1829–1917) Alphonse Fteley (1837–1903) Frederic Stearns (1851–1919) Ludwig Prandtl (1875–1953) Paul Richard Heinrich Blasius (1883–1970) Albert Strickler (1887–1963) Cyril Frank Colebrook (1910–1997)
== References ==
== External links == History of the Chézy Formula