4. Entropy velocity profile approach for rating curve assessment

The wide bulk of measurements obtained through the laboratory experiments allows us to perform a robust analysis in order to obtain suitable information for the use in the operative chain of water discharge assessment as well as in numerical flow dynamics modeling in regular open channel flow.

In Eq. (10), the mean velocity can be evaluated using Manning's formula:

$$\mathcal{U}\_m = \frac{1}{n} \mathbb{R}^{2/3} \sqrt{\mathcal{S}\_f} \tag{15}$$

Umax <sup>¼</sup> <sup>u</sup><sup>∗</sup>

Φð Þ¼ M

Um <sup>¼</sup> <sup>Φ</sup>ð Þ <sup>M</sup>

about hydraulic and geometric characteristics of the flow.

se ¼

and geometric characteristics of a river:

derived:

smooth channel.

relationship:

<sup>k</sup> ln <sup>3</sup> 4 D y0 � �

Therefore, inserting Eqs. (15) and (18) in Eq. (10), Φ(M) can be expressed in terms of hydraulic

Informational Entropy Approach for Rating Curve Assessment in Rough and Smooth Irrigation Ditch

1 nR<sup>2</sup>=<sup>3</sup> ffiffiffiffiffi Sf p

From this latter equation, a new formulation of Manning's roughness, ne, based on Φ(M) is

Therefore, if Φ(M) is available, then Eq. (20) allows us to estimate the n value in the cross-section.

which takes into account the variation of a flow hydraulic and geometric characteristics following the change of the water discharge. Eq. (20) computes Manning's roughness once the values of Φ(M) are known and the values of y0 are calibrated. Once the Manning's coefficient, ne, was evaluated, the mean velocity was recalculated according to Eq. (21).

Figure 4 shows the correspondence between Qcalc, computed through the Eq. (21), and those observed Qobs, for both cases RRF and SRF. The result shows the perfect correlation between the observed and computed values and enforces the use of the proposed Manning's Eq. (20), derived by the entropy velocity theory and the assumption of a constant value of the dip velocity. The approach leads to get water discharge assessment by integrating the information

Finally, the following Figures 5 and 6 report the theoretical rating curves obtained by the modified Manning's equation and the experimental data collected for both cases rough and

Defining the standard error, Se, as suggested by the ISO 1100-2 [28], through the following

<sup>P</sup> ln Qobs � ln Qcalc ð Þ<sup>2</sup> N � 2

where N is the number of available measures, the computed Se is permanently less than 5% for the rectangular rough flow (RRF), while increases up to 15%, with a generalized

" #<sup>0</sup>:<sup>5</sup>

= ffiffiffi g p

� <sup>0</sup>:<sup>4621</sup> � � ffiffiffiffiffiffiffiffiffiffi

<sup>k</sup> ln <sup>3</sup> 4 D y0

ffiffiffiffiffiffiffi gRSf p

ne <sup>¼</sup> <sup>R</sup><sup>1</sup>=<sup>6</sup>

Replacing Eq. (20) in Eq. (15), the modified form of the Manning's equation is obtained:

<sup>k</sup> ln <sup>3</sup> 4 D y0 � �

Φð Þ M <sup>k</sup> ln <sup>3</sup> 4 D y0

� <sup>0</sup>:<sup>4621</sup> � � (18)

http://dx.doi.org/10.5772/intechopen.78975

� � � <sup>0</sup>:<sup>4621</sup> h i (19)

� � � <sup>0</sup>:<sup>4621</sup> h i (20)

(21)

15

(22)

gRSf q

where n is the Manning's roughness, R is the hydraulic radius, and Sf is the energy slope.

To determine the maximum velocity of the cross-section, Umax, along the y-axis assumed perpendicular to the bottom, the dip-modified logarithmic law for the velocity distribution in a smooth uniform open channel flow, proposed by Yang et al. [8], is considered:

$$u(y) = u\_\* \left[ \frac{1}{k} \ln \frac{y}{y\_0} + \frac{\alpha}{k} \ln \left( 1 - \frac{y}{D} \right) \right] \tag{16}$$

where <sup>u</sup><sup>∗</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi gRSf p is the shear velocity (g = gravity acceleration); k is the von Karman constant equal to 0.41; y0 is the distance at which the velocity is hypothetically equal to zero; α is the dip-correction factor, depending only on the ratio between the relative distance of the maximum velocity location from the river bed, ymax, and the water depth, D, along the y-axis, where Umax is sampled.

The location of the maximum velocity, supporting the dip-phenomenon hypothesis, can be obtained by differentiating Eq. (16) and equating du/dy = 0, which gives:

$$\frac{y\_{\text{max}}}{D} = \frac{1}{1+\alpha} \tag{17}$$

Experimental studies [2–9] have shown that, for channels at different shapes of the crosssection, the velocity maximum is below the free surface around the 20–25% of the maximum depth. Thus, considering ymax equal to ¾ of the maximum depth, D, according to Eq. (17), α becomes equal to 1/3. Replacing the value of α in Eq. (16), and after a few algebraic manipulation, the maximum flow velocity can be expressed as:

Informational Entropy Approach for Rating Curve Assessment in Rough and Smooth Irrigation Ditch http://dx.doi.org/10.5772/intechopen.78975 15

$$\mathcal{U}\_{\max} = \frac{\mu\_\*}{k} \left[ \ln \left( \frac{3}{4} \frac{D}{y\_0} \right) - 0.4621 \right] \tag{18}$$

Therefore, inserting Eqs. (15) and (18) in Eq. (10), Φ(M) can be expressed in terms of hydraulic and geometric characteristics of a river:

velocity. Thus, assuming the value of Φ(M) equal to 0.9, the mean velocity can be computed and the water discharge as well. The benefit even deals with the reduction of measurement time and costs. On the other side, once performed velocity measurements in a cross section following the above mentioned procedure, the observed value of Φ(M) can suggest whether or

Finally, the use of the entropy velocity profile gives a robust feedback in terms of operative assessment of water discharge, due to the easy and immediate evaluation of the M parameter.

The wide bulk of measurements obtained through the laboratory experiments allows us to perform a robust analysis in order to obtain suitable information for the use in the operative chain of water discharge assessment as well as in numerical flow dynamics modeling in

> ffiffiffiffiffi Sf q

<sup>k</sup> ln <sup>1</sup> � <sup>y</sup>

� � � �

p is the shear velocity (g = gravity acceleration); k is the von Karman

D

<sup>1</sup> <sup>þ</sup> <sup>α</sup> (17)

(15)

(16)

4. Entropy velocity profile approach for rating curve assessment

In Eq. (10), the mean velocity can be evaluated using Manning's formula:

Um <sup>¼</sup> <sup>1</sup> n R<sup>2</sup>=<sup>3</sup>

a smooth uniform open channel flow, proposed by Yang et al. [8], is considered:

1 k ln <sup>y</sup> y0 þ α

u yð Þ¼ u<sup>∗</sup>

obtained by differentiating Eq. (16) and equating du/dy = 0, which gives:

tion, the maximum flow velocity can be expressed as:

where n is the Manning's roughness, R is the hydraulic radius, and Sf is the energy slope.

To determine the maximum velocity of the cross-section, Umax, along the y-axis assumed perpendicular to the bottom, the dip-modified logarithmic law for the velocity distribution in

constant equal to 0.41; y0 is the distance at which the velocity is hypothetically equal to zero; α is the dip-correction factor, depending only on the ratio between the relative distance of the maximum velocity location from the river bed, ymax, and the water depth, D, along the y-axis,

The location of the maximum velocity, supporting the dip-phenomenon hypothesis, can be

Experimental studies [2–9] have shown that, for channels at different shapes of the crosssection, the velocity maximum is below the free surface around the 20–25% of the maximum depth. Thus, considering ymax equal to ¾ of the maximum depth, D, according to Eq. (17), α becomes equal to 1/3. Replacing the value of α in Eq. (16), and after a few algebraic manipula-

ymax <sup>D</sup> <sup>¼</sup> <sup>1</sup>

not some changes in bed roughness occurred.

regular open channel flow.

14 Irrigation in Agroecosystems

where <sup>u</sup><sup>∗</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

where Umax is sampled.

gRSf

$$\Phi(M) = \frac{\frac{1}{n} R^{2/3} \sqrt{S\_f}}{\frac{\sqrt{gRS\_f}}{k} \left[ \ln \left( \frac{3}{4} \frac{D}{y\_0} \right) - 0.4621 \right]} \tag{19}$$

From this latter equation, a new formulation of Manning's roughness, ne, based on Φ(M) is derived:

$$m\_{t} = \frac{R^{1/6} / \sqrt{g}}{\frac{\mathcal{O}(M)}{k} \left[ \ln \left( \frac{3}{4} \frac{D}{y\_0} \right) - 0.4621 \right]} \tag{20}$$

Therefore, if Φ(M) is available, then Eq. (20) allows us to estimate the n value in the cross-section. Replacing Eq. (20) in Eq. (15), the modified form of the Manning's equation is obtained:

$$\Delta U\_m = \frac{\Phi(M)}{k} \left[ \ln \left( \frac{3}{4} \frac{D}{y\_0} \right) - 0.4621 \right] \sqrt{gRS\_f} \tag{21}$$

which takes into account the variation of a flow hydraulic and geometric characteristics following the change of the water discharge. Eq. (20) computes Manning's roughness once the values of Φ(M) are known and the values of y0 are calibrated. Once the Manning's coefficient, ne, was evaluated, the mean velocity was recalculated according to Eq. (21).

Figure 4 shows the correspondence between Qcalc, computed through the Eq. (21), and those observed Qobs, for both cases RRF and SRF. The result shows the perfect correlation between the observed and computed values and enforces the use of the proposed Manning's Eq. (20), derived by the entropy velocity theory and the assumption of a constant value of the dip velocity. The approach leads to get water discharge assessment by integrating the information about hydraulic and geometric characteristics of the flow.

Finally, the following Figures 5 and 6 report the theoretical rating curves obtained by the modified Manning's equation and the experimental data collected for both cases rough and smooth channel.

Defining the standard error, Se, as suggested by the ISO 1100-2 [28], through the following relationship:

$$\mathbf{s}\_{\mathbf{e}} = \left[ \frac{\sum \left( \ln Q\_{obs} - \ln \mathbf{Q}\_{\text{calc}} \right)^{2}}{\text{N} - 2} \right]^{0.5} \tag{22}$$

where N is the number of available measures, the computed Se is permanently less than 5% for the rectangular rough flow (RRF), while increases up to 15%, with a generalized

describing the geometric and hydraulic characteristics of a rectangular ditch, should allow us

Informational Entropy Approach for Rating Curve Assessment in Rough and Smooth Irrigation Ditch

http://dx.doi.org/10.5772/intechopen.78975

17

This approach was tested, in a first phase, on a suitable data set of water discharge measures collected in the laboratory on both rough and smooth rectangular cross section proposing

The rating curve evaluation, derived for the rough rectangular flow, underlines a standard error less than 5%, generally, favoring an expeditive assessment of the flow stage with a sufficient level

of reliability, while such an error increase up to 15% in case of smooth cross section.

2 Regional Environmental Observatory Research Foundation of Basilicata, Italy

[1] Chiu C-L. Entropy and probability concepts in hydraulics. Journal of Hydraulic Engineer-

[2] Chiu C-L. Velocity distribution in open channel flow. Journal of Hydraulic Engineering,

[3] Greco M, Mirauda D, Volpe Plantamura A. Manning's roughness through the entropy parameter for steady open channel flows in low submergence. Procedia Engineering, ISSN 1877-7058, Published by Elsevier Ltd. 2014;70:773-780. DOI: 10.1016/j.proeng.2014.02.084. 2-

[4] Greco M, Mirauda D. An entropy based velocity profile for steady flows with large-scale roughness. In: Lollino G, Arattano M, Rinaldi M, Giustolisi O, Marechal JC, Grant G, editors. Engineering Geology for Society and Territory. Vol. 3. Cham: Springer International Publishing; 2015. pp. 641-645. Print; ISBN: 9 978-331909054-2; 978-331909053-5, DOI: 10.1007/

[5] Mirauda D, Greco M, Moscarelli P. Practical method for flow velocity measurements in fluvial sections. WIT Transactions on Ecology and the Environment. 2011;146:355-366.

[6] Moramarco T, Saltalippi C, Singh VP. Estimation of mean velocity in natural channels based on Chiu's velocity distribution equation. Journal of Hydrologic Engineering, ASCE.

978-3-319-09054-2\_128, WOS: 000358990300128, EID: 2-s2.0-84944599725

ISBN:978-1-84564-516-8 e ISSN 17433541. DOI: 10.2495/RM110301

the improvement of water discharge assessment.

Address all correspondence to: michele.greco@unibas.it

ing, ASCE. 1987;113(5):583-600

ASCE. 1989;115(5):576-594

s2.0-84899680421

2004;9(1):42-50

1 Engineering School, University of Basilicata, Potenza, Italy

practical and common flow conditions.

Author details

Greco Michele

References

Figure 5. Observed data and calculated rating curves for roughness rectangular flow.

Figure 6. Observed data and calculated rating curves for smooth rectangular flow.

overestimation, in case of smooth rectangular flow. In both cases, the results support the use of this expeditive methodology in the chain of operative procedures leading a good assessment of the rating curve.

#### 5. Conclusion

The use of a rating curve formulation derived from the entropy velocity theory complained to the assumption of a constant value of the dip velocity and taking into account the variables describing the geometric and hydraulic characteristics of a rectangular ditch, should allow us the improvement of water discharge assessment.

This approach was tested, in a first phase, on a suitable data set of water discharge measures collected in the laboratory on both rough and smooth rectangular cross section proposing practical and common flow conditions.

The rating curve evaluation, derived for the rough rectangular flow, underlines a standard error less than 5%, generally, favoring an expeditive assessment of the flow stage with a sufficient level of reliability, while such an error increase up to 15% in case of smooth cross section.
