Chezy’s and Manning’s Formula in Open Channel Flow

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The flow of a liquid through a passage under atmospheric pressure is called an open channel flow. Chezy’s and Manning’s equations are two important formulas used to determine the velocity and discharge of an open channel flow. 

This article explains the important concepts regarding the usage of Chezy’s and Manning’s formula in studying the discharge of open channel flow.

Determination of Discharge through Open Channel Flow

Chezy’s formula was derived in the 1760s. Due to its drawbacks in giving practical results, Manning’s formula was derived in1889 as an improvement to Chezy’s formula.

1. Chezy’s Formula

French engineer Antoine Chezy derived Chezy’s formula in the 1760s to study the flow behavior. The formula is given by:

V = C. R ½ S ½ (Eq.1)

Where C= Chezy’s constant; R= hydraulic radius;

S= slope of the channel bottom

The Chezy’s constant is determined using any of the following equations:

1. Bazin’s Formula ( In MKS Units)

K = Bazin’s constant and depends on the roughness of the surface of the channel; m is the hydraulic mean depth or hydraulic radius.

2. Ganguillet-Kutter Formula

N= Roughness coefficient called as Kutter’s constant, whose value varies for different surfaces

I = slope of the bed; m = hydraulic mean depth ( also represented as R)

Chezy’s coefficient is dependent on the flow’s nature and the nature of the channel surface.

The determination of Chezy’s coefficient is less accurate and not commonly used due to the lack of information to find out equivalent roughness without sufficient experimental and field data. Therefore, it is used only for smooth boundaries.

2. Manning’s Formula

Irish engineer Robert Manning derived manning’s formula in 1889. It is derived from Chezy's formula. 

Manning derived an average empirical relation for Chezy’s coefficient as:

C = 1/nR ⅙ ;

Hence, the final formula (Eq.1) becomes,

V = [(1/n)R ⅙ ] R½  S½ 

Which gives the average velocity of flow as,

V = (1/n) R ⅔ S½ (Eq.2)

V= velocity of flow

n= Manning’s Roughness coefficient

R= Hydraulic radius/hydraulic depth

S= bed slope or slope of the channel bottom

Manning’s roughness coefficient is determined by 

1. Strickler equation

n = [(D50) ⅙ ] / 24 or K ⅙ /24

K = Roughness Height

D50 = 50% of the particles are finer than the size 50 ( bed particles)

2. Meyer Formula

N = [(D90) ⅙ ]/26

D50 = 90% of the particles are finer than the size 90 ( bed particles)

Meyer formula is used when the channel bed contains a higher percentage of coarse-grained particles.

Note: D50 and D90 are obtained from the particle size distribution curve and by studying the grade curve characteristics of the particles.

Manning’s roughness coefficient is dependent on the following:

1. Surface roughness
2. Channel alignment
3. Cross-section irregularities
4. Vegetation

Manning’s formula is the most widely used uniform-flow formula around the world due to the following reasons:

1. Manning’s formula is simple and is proven by practical experience
2. The majority of open channel flows along a rough turbulent region
3. Manning equation is accurate, as manning’s n values are often selected from tables or calculated from field measurements.

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