Empirical

Chezy Formula (1775); $V=C\sqrt{RS}$

where, v = mean velocity (m/sec), C=Chezy's constant, it is only a factor of flow resistance, R=mean hydraulic radius or hydraulic mean depth (m), i.e., cross-sectional area/wetted perimeter, S=sine of water surface slope.

This is a practical and fundamental formula for uniform flow in open channels and is the basis of all modern velocity formulae. It is based on the assumptions that (i) force resisting flow per unit of area of channel bed is proportional to the square of mean velocity, and (ii) the effective component of gravity force causing the flow must be balanced by total force of resistance to flow.

Advantages :(i) Recognized as a classical equation of flow, and (ii) Applicable in all cases where uniform flow is assumed or is in existence.

Limitation (i) It is not dimensionally correct, since for a given channel, C and S having numerical values, V varies as $\sqrt{R}$, and the dimensional equation in m/sec = $\sqrt{m}$, (ii) It does not take into account the physical character of the channel, (iii) It is derived from a limited number of observations, but aim at application to all types of rugosities, (iv) Its assumption that hydraulic gradient runs parallel to energy gradient is a condition which is realized in fairly straight regular channels in steady or nearly steady discharge rate, and (v) It is valid only for channel cross section resembling wide rectangles with roughly level bottom and parallel flow at right angle thereto.

Kutter's Formula (1869)

C= $[\frac{\frac{1}{N}+(23+\frac{0.00155}{S})}{1+(23+\frac{0.00155}{S})\frac{N}{\sqrt{R}}}]$

or $V= [\frac{\frac{1}{N}+(23+\frac{0.00155}{S})}{1+(23+\frac{0.00155}{S})\frac{N}{\sqrt{R}}}]\sqrt{RS}$

where, N= Kutter's roughness factor(coefficient of rugosity) having different values for various grades of roughness of channels. Value of table depend on the roughness of channel, depth is considered not to effect N to any appreciable extent in alluvial soils, S= water surface slope, and R = mean hydraulic radius or depth (m).

Central Water and Power Commission recommended values of Kutter's N applicable for the canals taking off from reservoirs carry silt free water flowing through rocky cuts range between 0.035 - 0.05 and for the soils other than rock varies from 0.020-0.030 corresponding to discharge between 0.14 and over 14 cumecs.

Limitations : Kutter's formula is widely used but it suffers from limitations (i) It is unnecessary complicated, (ii) Coefficient C increases with slpe for values of hydraulic radius (R) less tha 1 m but drops when the slope rises for R greater than 1 m, (iii) It is not more accurate than Manning's formula, and (iv) Coefficient of N is assumed to depend only on the nature of the material.

Manning's Formula (1889)

Manning, an Irish engineer, evolved the formula: $V=\frac{1}{N}R^\frac{2}{3}S^\frac{1}{2}$

It gives better results than Chezy formula. According to this formula, the value of Chezy coefficient is given by the realtion: $C=R^\frac{1}{6}/N$

where, C=Chezy coefficient. It is considered dependentonly on the roughness and hydraulic radius, thereby simplifying their computations, and N= coefficient of rugosity. Its value in normal ranges of slopes and hydraulic mean radius generally seem to be numerically very close to Kutter's N for practical purposes.

Manning's formula, in terms of discharge, for trapezoidal cross section can be written as

Q= $ \frac{PR^\frac{5}{3}S^\frac{1}{2}}{N}$

A simplified form of Manning's formula is $Q=KD^\frac{5}{3}$, where D is the depth of water in the channel. It is commonly used for the formulation of discharge tables.

Limitations: (i) Exponent of R (in relation $C=R^\frac{1}{6}/N$) is not a constant but varies depending mainly upon the channel shape and roughness, and (ii) Proper selection of value of N to be used is a complex problem. Great difficulty lies in judicious selection of value of N.

Bazin Formula (1897)

Bazin considered Chezy C to be a function of R but not of S. He evolved the following formula for the determination of Chezy coefficient C. The formula gives fairly comparable results.

C=$\frac{87}{1+M/\sqrt{R}}$

where, M=coefficient of roughness which depends on the roughness of channel surface; values for different materials in which the channel is constructed are given in the table.

The values of M are more nearly constant than N in Kutter's formula for smooth channels under variations of R and V under identical conditions but reverse is true in the case of erodible channels.

The formula is derived from data collected from small experimental channels and is less satisfactory than Kutter's formula.

Pavlovsky's Formula (1925)

Pavlovsky proposed the following uniform flow equation as an improvement over Manning's formula. The formula is widely used in U.S.S.R.

C=$\frac{1}{n}R^y$

where, $y=2.5\sqrt{n}-0.13 -0.75\sqrt{R}(\sqrt{n}-0.010)$. It depends on the roughness coefficient and hydraulic radius. The formula is valid for values of R between 0.1 m and 3.0 m and n between 0.011 and 0.040.

These formulas have been extensively used in the design of artificial channels but they are deficient as they have no reference to silt which the water carries in the earthen irrigation channels. The result was that channels designed with these formulas regularly silted badly requiring frequent silt clearance at huge annual expenditure.

Regime Approach

Kennedy Formula (1895)*

Kennedy, a punjab engineer, have a clasic emperical equation correlating mean velocity V in a regime canal with the vertical depth D, as measured on the approximately horizontal silted bed.

According this relation: $V_0 = C.m. D^n$

or $V)0 = 0.55mD^0.64$

where, $V_0$= critical(non silting, non-scouring) velocity (m/sec). Fine silt has lower critical velocity than coarse silt, D=fully supply depth (m) of water over the bed portion of the canal, Kennedy's modified equation: $V_0 =0.55mD^0{}^.{}^6{}^4$.

Kennedy gave hydraulic diagrams which are extensively used for designing irrigation channels in India. From these diagrams it is possoble to design any number of channel sections for a given slope with different bed width and depth. It was considered doubtful if all these would be equally efficient regarding stability.

Lacey's Formula (1929)

Lacey's a U.P. engineer, developed relationships for determing regime slope and channel dimensions. Lacey's set of fromulae uniquely fix the size, shape and slope of stable alluvial channels, as under:

Silt factor (f) : Lacey introduced silt factor f which is very important in the use of his formulae, to denote silt grade by linear ratio rather than by velocity ratio and dependent on the following factors:

(a) Average size of boundary material in the channel, $f=1.76\sqrt{m_r}$

where, $m_r$ is the average particle size of boundary material (mm). The value of 'f' is estimated from this relation after determining $m_r$ size of the boundry material. Suggested values of 'f' for various values of boundary material are given in table below.

when related to Kennedy's critical velocity ratio $f=(V/V_0)^2$.

(b) f is proportional to $V^2/R$ or $f=0.5V^2/R$.

where, V= average velocity (m/sec), and R = hydraulic mean depth (m).

Coefficient of Rugosity N -f Relatio: Lacey defined $N_a$ as the coefficient of absolute rugosity which is dependent only on grade and density of boundary material. The standard grade of silt is that which connotes a rugosity coefficient of 0.0225 corresponding to hydraulic mean depth of 1 m. It is ana overall rugosity coefficient, which in addition to boundry friction allows for shock losses in a channel due to irregularities and bends.

$N_a = 0.0225f^1{}^/{}^4$

$N/N_a =1.104R^1{}^/{}^1{}^2$

When related to Manning's N, value of $N_a$ for silt factor f1.0 is 0.0225 for f0.9, $N_a=0.0223$ and corresponding to f0.08, $N_a=0.020$.

Suggested values of $N_a$ for different earthen channels are as under: