## Open Channel Flow Velocity Empirical Formulae

### flow velocity based on observation or experience

Since the earliest time, people have wanted to know how much water could flow through an open channel or closed pipe. Knowing the velocity of a water flow allowed them to determine how large the pipes or channel needed to be to pass the required flow. This might be to allow water to irrigate crops or to supply drinking water to a city. The Romans knew that to allow water to flow along their aqueducts the structure needed to have a slope in the direction of the flow and probably carried out tests to make sure sufficient water would flow through the aqueducts channel.

Antonia Castelli (1578 – 1643) was an Italian mathematician and a student of Galileo. He took the name Benedetto when he joined the Benedictine order in 1595. He is best known for his work, “Della misura delle acque correnti” (On the measurement of running water) which was published in Rome in 1629. The book was responsible for the initiation of modern hydrodynamics.

Some of the first mathematicians to develop hydrodynamics were Leonhard Euler (1707 – 1783) and Daniel Bernoulli (1700 – 1782), both Swiss. Euler and Bernoulli derived equations of continuity and conservation of momentum of fluid flow. The Bernoulli equation is considered to be a statement of conservation of energy. His equation showed that when fluid flows through a restriction there is a pressure drop in the fluid and the velocity of the fluid increases. This is known as the “Bernoulli effect”.

In 1749 the Dutch hydraulics engineer, Cornelis Velsen published his great work on river management “Rivierkundige Verhandeling”, (River protection discourse). He came to the conclusion that the velocity of flow should be proportional to the square root of the slope.  Water does not flow on a level surface. Once the surface is raised even slightly the water starts to flow. The higher the surface is raised the faster the water flows, or to put it another way the water velocity increases. Since the slope is fundamental to allow water to flow and the angle of the slope will determine the velocity of the water.

Since Cornelis did not have access to modern digital surveyors’ equipment or any satellites to obtain a GPS data from how would he have carried out his experiments? Since I haven’t been able to find any documents that demonstrate how Velsen carried out his observations that made him come to the relationship between velocity and slope, the following is an assumption to an experiment that he may have carried out.

The experiment that Cornelis Velsen may have carried out

The German engineer Albert Brahms (1692 – 1758) was a dike judge elected in 1718 after a Christmas flood in 1717. In 1754 – 1757 he published a two-volume book on dike maintenance, “Anfangsgründe der Deich und Wasser-Baukunst”, (Principles of Dike and Aquatic Engineering). He is responsible for keeping the first records of tide levels on the North Sea coast of Germany. Brahms was also responsible for an equation that defined the relationship between the cross-sectional area of a channel “A”, the wetted perimeter “P” and the hydraulic radius “R”. See Empirical Formulae calculation sheet (EFCS) equation (10).

The French engineer and Hydraulician Antoine de Chézy (1718 – 1798) was a professor and eventually director of École des Ponts et Chassées (French School of Bridges and Highways). He was involved producing calculations to optimize the resistance on the structures for work at Trilport in 1757 and Mantes-la Jolie in 1764. In 1769 he was commissioned by King Louis XV to bring water of the Yvette River into Paris to provide a water supply to the city. He is credited with the first and most lasting equation of resistance in uniform open channel flow.  He devised the Chézy equation V2/R S, but found that this value changed from one stream to another. In order to correct this anomaly, he introduced a constant C known as the Chézy Coefficient into his equation. He worked out the value of C as 31.  See EFCS equation (12).

It was later found that the Chézy coefficient C was not a pure number but has a dimension of  where L and T are units of length and of any measuring system.  See EFCS equation (13).

The French military engineer Pierre Louis Georges Du Buat (1734 - 1809) derived formulae for the discharge of fluids from pipes and open channels. In 1776 he begun studying hydraulics and in 1779 published the first edition of “Principes d’hydraulique”. In 1786 a second volume covering experimental practice was published. In 1787 he was appointed lieutenant du roi (Lieutenant King) having risen to the rank of colonel. During the French revolution he lost his properties and fled with his family to Belgium in 1793 and later to Germany. Du Buat proposed the formula for the average velocity.  See EFCS equation (14).

Johann Albert Eyetelwein (1764 - 1848) was a Prussian hydraulics engineer. After a short career in the Prussian artillery he studied civil engineering and qualified in 1790 when he left the army and entered the Prussian civil service. In 1793 he published a collection of problems in applied mathematics for surveyors and engineers. In 1801 he derived a formula for open channel velocity, which was similar to the Chézy equation, but with the Coefficient of 50.9. See EFCS equation (15).

Julius Lugwig Weisbach (1806 – 1871) was a German mathematician who wrote 59 papers on mechanics, hydraulics, surveying and mathematics. Weisbach collaborated with Henry Darcy on the Darcy – Weisbach friction coefficient. They also developed a formula for the resistance of flow through closed pipes. see EFCS equation (16)

The French engineer Henry Darcy (1803 – 1858), well known for Darcy’s Law, his formula for calculating the flow of water through an aquifer. In 1856 he published a report “The Public Fountains of the City of Dijon”. This report included experiments with pipes that resulted in the Darcy – Weisbach friction coefficient (fD).  This friction coefficient was not a constant value, but depended on the pipe diameter, roughness of the pipe wall, kinetic viscosity and velocity of the fluid flow.  see EFCS equation (17)

During this period the flow velocity equation for open channels was generally accepted to be given by the triple factor formula. see EFCS equation (18)

The coefficient “C” and exponents “x” and “y” were chosen to make the formula conform to the experimental data obtained by each investigator.

Henri Emile Bazin (1829 - 1917) worked as an assistant to Henry Darcy. He conducted experiments to develop his hydraulic work focusing particularly on the flow of water in open channels His formula proposed in 1897 relates to the Chezy coefficient “C”, the hydraulic radius and channel roughness “k”. He observed that the value of “C” increased with an increase in slope, but concluded that this increase was too small to be provided for in the equation. see EFCS equation (19).

Emile Oscar Ganguillet (1818 - 1894) was a swiss engineer. He studied at the Progymnasium in Biel, Obergymnasium in Bern. In 1841 he worked on the construction of bridges, roads and railways in France. In 1847 he was appointed as district engineer in Delémont in Swizerland. In 1858 as Chief engineer he oversaw bridge and hydraulic structures including the Juragas water connection. In 1869 with Wilhelm Kutter he published formulae for the uniform movement of water in canals and rivers.

After an apprenticeship as a surveyor, Willhelm Rudolph Kutter (1818 – 1888), entered the service of the Kt. Bern around 1835, initially in road construction and forestry; In 1851-88 he was secretary of the Dep. for public Buildings. Willhelm recognized the importance of a proper recording of friction losses in rivers and developed together with Cantonal Engineer Emile Ganguillet a formula for the movement of water in rivers and canals (1869 published for the first time). This friction approach for free-drain drains has been world-famous for a long time. see EFCS equation (20).

Phillipe Gaspard Gauckler (1826 - 1905), a German civil engineer trained in Strasbourg before entering the Corps of Engineers of Roads and Bridges in 1848. In 1881 he became the Chief engineer on the French Railways, until he was promoted to Inspector General of Roads and Bridges in 1886. He proposed two formulae for use in different slope ranges based on experiments by Darcy and Bazin. see EFCS equation (21). He also contributed to the Gauckler Manning Strickler Formula developed in 1868. see EFCS equation (22).

In 1889 Robert Manning (1816 – 1897) wrote his scientific paper “On the flow of water in open channels and pipes”. The original Manning’s formula was in the paper. This formula is mostly referred to as the Manning’s formula, but should be more correctly called the Gauckler Manning formula. see EFCS equation (23). Manning later rejected this formula as it required extraction of the cube root and the equation lacked dimensional homogeneity. He proposed a different equation in his 1889 paper. see EFCS equation (24).

Mannings first formula however was more popular and William King (1851 - 1929) a Scottish engineer that brought about widespread acceptance of this and that Manning's coefficient "C" was the inverse of Wilhelm Rudolf Kutter's (1818 - 1888) coefficient.

Albert Strickler (1887 - 1963) was a Swiss mechanical and hydraulic engineer who studied mechanical and electrical engineering at the Swiss Federal Institute of Technology in Zurich

He was one of the authors of the 1868 Gauckler Manning Strickler formula. In 1918 he was elected Section Chief of the Federal Office for Water Management. It is during this period that he developed his formula for velocity. He is also responsible for correlating his coefficient with the roughness of gravel bed rivers. The Strickler equation could be used to determine the Manning's roughness value "n" based on the Strickler value for the median size of the bed material in millimetres. see EFCS equation (25).

The final equation on the Empirical Formulae calculation sheet shows the Manning's equation as it is used currently to calculate the mean flow velocity a channel. see EFCS equation (26).

### Sources used and further reading:

Fluid mechanics, Euler and Bernoulli equations

History of Uniform Flow Velocity and Resistance Factor – Prof. B.S. Thandaveswara

Ben Chie Yen,  Journal of Hydraulic Engineering 2002

Rajesh Srivastava, Indian Institute of Civil Engineering

The History of the Historical Dictionary of Switzerland