Theory

This section presents the theoretical foundations on which the program is based. It explains the concepts, assumptions, and formulations used in the different analysis methods implemented.

The content is aimed at providing a clear understanding of how the calculations are performed within the software, facilitating both the interpretation of results and the technical validation of the models.

Manning’s Equation

Manning’s equation is an empirical formula used in hydraulics to analyze open channel flow. It was developed from experimental observations and allows the calculation of hydraulic parameters such as velocity, flow rate, and flow depth.

V = \frac{1}{n} R_h^{2/3} S^{1/2}

Where:

$Q$ : Flow rate ( $\text{m}^3\text{/s}$ ).
$V$ : Average flow velocity ( $\text{m/s}$ ).
$n$ : Manning’s roughness coefficient, which depends on the channel material characteristics (dimensionless).
$R_h$ : Hydraulic radius ( $\text{m}$ ), defined as $R_h = \frac{A}{P}$ , where $A$ is the flow area and $P$ is the wetted perimeter.
$S$ : Channel slope ( $\text{m/m}$ ).

To calculate the flow rate ( $Q$ ), the following relationship is used:

Q = V A

Where:

$V$ : Average flow velocity ( $\text{m/s}$ ).
$A$ : Flow area ( $\text{m}^2$ ).

Froude Number

The Froude number ( $Fr$ ) determines the flow regime (subcritical, critical, or supercritical):

Fr = \frac{V}{\sqrt{g D}}

Where:

$V$ : Average flow velocity ( $\text{m/s}$ ).
$g$ : Gravitational acceleration ( $\text{m/s}^2$ ).
$D$ : Flow depth ( $\text{m}$ ).

According to its value, the classification is:

Subcritical flow ( $Fr < 1$ )
Critical flow ( $Fr = 1$ )
Supercritical flow ( $Fr > 1$ )

Specific Energy

Specific energy ( $E$ ) in an open channel is calculated as follows:

E = \frac{V^2}{2 g} + h

Where:

$E$ : Specific energy ( $\text{m}$ ).
$V$ : Average flow velocity ( $\text{m/s}$ ).
$g$ : Gravitational acceleration ( $\text{m/s}^2$ ).
$h$ : Water surface elevation relative to a reference ( $\text{m}$ ).

Specific energy has two components:

Kinetic energy: Represented by the term $\frac{V^2}{2g}$ , reflects the energy associated with water movement.
Potential energy: Represented by the term $h$ , reflects the energy due to the flow elevation.

Appendix: Manning’s Roughness Coefficients

Below is a reference table of Manning’s roughness coefficients ( $n$ ), compiled from Ven Te Chow’s work:

Type of Channel and Description	Minimum	Normal	Maximum
Natural streams - minor streams (top width at floodstage < 30 m (100 ft))
1. Main Channels
a. Clean, straight, full stage, no rifts or deep pools	0.025	0.030	0.033
b. Same as above, but more stones and weeds	0.030	0.035	0.040
c. Clean, winding, some pools and shoals	0.033	0.040	0.045
d. Same as above, but some weeds and stones	0.035	0.045	0.050
e. Same as above, lower stages, more ineffective slopes and sections	0.040	0.048	0.055
f. Same as “d” with more stones	0.045	0.050	0.060
g. Sluggish reaches, weedy, deep pools	0.050	0.070	0.080
h. Very weedy reaches, deep pools, or floodways with heavy stand of timber and underbrush	0.075	0.100	0.150
2. Mountain streams, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages
a. Bottom: gravels, cobbles, and few boulders	0.030	0.040	0.050
b. Bottom: cobbles with large boulders	0.040	0.050	0.070
3. Floodplains
a. Pasture, no brush
1. Short grass	0.025	0.030	0.035
2. High grass	0.030	0.035	0.050
b. Cultivated areas
1. No crop	0.020	0.030	0.040
2. Mature row crops	0.025	0.035	0.045
3. Mature field crops	0.030	0.040	0.050
c. Brush
1. Scattered brush, heavy weeds	0.035	0.050	0.070
2. Light brush and trees, in winter	0.035	0.050	0.060
3. Light brush and trees, in summer	0.040	0.060	0.080
4. Medium to dense brush, in winter	0.045	0.070	0.110
5. Medium to dense brush, in summer	0.070	0.100	0.160
d. Trees
1. Dense willows, summer, straight	0.110	0.150	0.200
2. Cleared land with tree stumps, no sprouts	0.030	0.040	0.050
3. Same as above, but with heavy growth of sprouts	0.050	0.060	0.080
4. Heavy stand of timber, a few down trees, little undergrowth, flood stage below branches	0.080	0.100	0.120
5. Same as 4. with flood stage reaching branches	0.100	0.120	0.160
4. Excavated or Dredged Channels
a. Earth, straight, and uniform
1. Clean, recently completed	0.016	0.018	0.020
2. Clean, after weathering	0.018	0.022	0.025
3. Gravel, uniform section, clean	0.022	0.025	0.030
4. With short grass, few weeds	0.022	0.027	0.033
b. Earth winding and sluggish
1. No vegetation	0.023	0.025	0.030
2. Grass, some weeds	0.025	0.030	0.033
3. Dense weeds or aquatic plants in deep channels	0.030	0.035	0.040
4. Earth bottom and rubble sides	0.028	0.030	0.035
5. Stony bottom and weedy banks	0.025	0.035	0.040
6. Cobble bottom and clean sides	0.030	0.040	0.050
c. Dragline-excavated or dredged
1. No vegetation	0.025	0.028	0.033
2. Light brush on banks	0.035	0.050	0.060
d. Rock cuts
1. Smooth and uniform	0.025	0.035	0.040
2. Jagged and irregular	0.035	0.040	0.050
e. Channels not maintained, weeds and brush uncut
1. Dense weeds, high as flow depth	0.050	0.080	0.120
2. Clean bottom, brush on sides	0.040	0.050	0.080
3. Same as above, highest stage of flow	0.045	0.070	0.110
4. Dense brush, high stage	0.080	0.100	0.140
5. Lined or Constructed Channels
a. Cement
1. Neat surface	0.010	0.011	0.013
2. Mortar	0.011	0.013	0.015
b. Wood
1. Planed, untreated	0.010	0.012	0.014
2. Planed, creosoted	0.011	0.012	0.015
3. Unplaned	0.011	0.013	0.015
4. Plank with battens	0.012	0.015	0.018
5. Lined with roofing paper	0.010	0.014	0.017
c. Concrete
1. Trowel finish	0.011	0.013	0.015
2. Float finish	0.013	0.015	0.016
3. Finished, with gravel on bottom	0.015	0.017	0.020
4. Unfinished	0.014	0.017	0.020
5. Gunite, good section	0.016	0.019	0.023
6. Gunite, wavy section	0.018	0.022	0.025
7. On good excavated rock	0.017	0.020
8. On irregular excavated rock	0.022	0.027
d. Concrete bottom float finish with sides of:
1. Dressed stone in mortar	0.015	0.017	0.020
2. Random stone in mortar	0.017	0.020	0.024
3. Cement rubble masonry, plastered	0.016	0.020	0.024
4. Cement rubble masonry	0.020	0.025	0.030
5. Dry rubble or riprap	0.020	0.030	0.035
e. Gravel bottom with sides of:
1. Formed concrete	0.017	0.020	0.025
2. Random stone mortar	0.020	0.023	0.026
3. Dry rubble or riprap	0.023	0.033	0.036
f. Brick
1. Glazed	0.011	0.013	0.015
2. In cement mortar	0.012	0.015	0.018
g. Masonry
1. Cemented rubble	0.017	0.025	0.030
2. Dry rubble	0.023	0.032	0.035
h. Dressed ashlar/stone paving	0.013	0.015	0.017
i. Asphalt
1. Smooth	0.013	0.013
2. Rough	0.016	0.016
j. Vegetal lining	0.030		0.500

References

Chow, V. T. (1959). Open-Channel Hydraulics. McGraw-Hill.