2. Entropy velocity profiles in open channels

at the same site. On the other hand, the drainage and irrigation channel present a regular cross section which might provide facilities in water discharge assessment and control, inducing also reduction in time and operative costs. That is, the implementation of operative procedures enabling operative charges simplifying the commitment of field activities, indeed, plays a fundamental role in channel monitoring for natural flow and manmade hydraulic structures. The main idea is related to the definition of expeditive procedures for flow field assessment and water discharge evaluation capable to optimize the surveying resources in time and efforts. Thus, the opportunity to manage with a simple and straightforward velocity law, different from the classical logarithm formulation but capable to provide suitable results is all the more technically fruitful. That is, an operative tool for expeditive velocity distribution

Recent theoretical and experimental studies endorse the informational content hold into the distributed velocity measurements following an entropy-probabilistic approach. That is, Chiu [1, 2] drew the correlation between the mean flow velocity and maximum flow velocity defining the entropy parameter, M introducing the velocity ratio Φ(M). Considering the important implication that this finding could have for monitoring of high flows in rivers, many authors investigated the reliability of this relationship using field data [3–7]. Overall, they found M as a river site depending and not influenced by the flood intensity both in terms of amount and duration. Thus, M should be considered a specific factor of the gauged cross section as outlined by Moramarco and Singh [7] exploring the dependence of M on the

The study was able to explain the constancy of M value on the ground that M is not depending on the dynamic of flood, such as expressed by the energy or water surface slope, Sf and to identify a formula expressing M as a function of the hydraulic radius, Manning's roughness and the location, y0, where the horizontal velocity is hypothetically equal to zero. For the latter, it was preliminarily found that if y0 was assessed by distinguishing low flows from high flows, then a better estimation of M would have been obtained across a gauged river site. However, considering that the y0 location is not of simple assessment and then might have high uncertainty, the assessment of M should be addressed, mainly for ungauged river sites, using hydraulic and geometric variables easy to acquire. Such a thought might be discussed introducing the relative submergence D/d (in which, D = average water depth and d = roughness dimension). That is, the velocity distribution in natural rivers depends on several variables like channel geometry, bed and bank roughness, and the vertical velocity distribution generally increases monotonically from 0 at the channel bed, to the maximum at the water surface and can be assumed 1-D flow dominant. Moreover, whenever the channel cannot be considered "wide", that is the aspect ratio (B/D with B channel width and D water depth) is less than 6, besides the presence of the boundary, the velocity varies even transversely and a twodimension distribution occurs, leading G as the 2D entropy parameter. The maximum velocity places below the water surface inducing dip-phenomenon and the position of maximum velocity is also influenced by the aspect ratio [8], which is of simple assessment once channel cross-section geometry is known. Thus, investigating the influence of bed roughness and cross section geometry on medium and maximum velocity ratio at the global scale assumes a

assessment basing on simple and immediate parameters.

6 Irrigation in Agroecosystems

hydraulics and geometries of the river cross sections.

relevant interest in the field of open channel flow.

The concept of informational entropy as a measure of uncertainty associated to a probability distribution was formulated for the first time in the field of hydraulics by Shannon [11]. The principle of maximum entropy introduces the least-biased probability distribution of a random variable constrained by defined information system as well as the theorem of the concentration for hypothesis testing, introducing the informational entropy theory [12]. A direct evaluation of uncertainty related to the probability distribution of a continuous random variable expressed in terms of entropy, H, is defined as follows

$$\mathbf{H} = -\int\_{-\infty}^{+\infty} \mathbf{p}(\mathbf{x}) \log \mathbf{p}(\mathbf{x}) d\mathbf{x} \tag{1}$$

where, p(x) is the continuous probability density function of random variable x.

Using POME, entropy can be maximized through the method of Lagrange multiplier as follows:

$$\mathcal{L} = -\frac{1}{\text{m} - 1} \int\_{-\text{ov}}^{+\text{ov}} \mathbf{p}(\mathbf{x}) \left\{ 1 - \left[ \mathbf{p}(\mathbf{x}) \right]^{\text{m} - 1} \right\} d\mathbf{x} + \sum\_{i=1}^{N} \lambda\_{i} \mathbf{g}\_{i}(\mathbf{x}) \tag{2}$$

in which, m > 0, gi(x) is the ith constraint function and λ<sup>i</sup> is the constrain Lagrange multiplier as a weight in the maximization of entropy.

Chiu [1, 2] applied the concept of entropy to open-channel analysis to model velocity and shear stress distribution as well as sediment concentration. In such a way, the velocity distribution in the probability domain allows to obtain the cross-sectional mean velocity and the momentum and energy coefficients disregarding the geometrical shape of cross sections, which is generally complex in natural channels [2, 13].

Further, an assumption on the probability distribution in the space domain is needed to relate the entropy-based probability distribution to the spatial distribution. Therefore, defining u by the time-averaged velocity placed on an isovelocity curve with the assigned value ξ, the value of u is almost 0 at ξ0, which corresponds to the channel boundary, while u reaches Umax at ξmax, which generally occurs at or below the water surface, depending on the dip-phenomenon. Thus, the velocity u monotonically increases from ξ<sup>0</sup> to ξmax and for each value of the spatial coordinate

greater than ξ, the velocity is greater than u, and the cumulative distribution function can be written as

$$\mathbf{F}(\mathbf{u}) = \frac{\boldsymbol{\xi} - \boldsymbol{\xi}\_0}{\boldsymbol{\xi}\_{\text{max}} - \boldsymbol{\xi}\_0} \tag{3}$$

Eq. (10), in fact, represents the fundamental relationship, from an applied point of view, of the entropy velocity distribution and the assessment of the entropy parameter passing through the

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

In order to identify the dependence of M from the hydraulic and geometric characteristics of channels, that is, the relative submergence and aspect ratio, respectively, the formulation prop-

where u\* is the shear velocity, d is the bed roughness height (i.e., d50), k is the Von Karman

Even the maximum velocity plays an important role in the flow dynamics, and more than it magnitude, a relevant aspect is related to the position of the maximum velocity inside the flow domain. That is, the location of maximum velocity from the channel bottom, ymax, does not always occur at water surface, but a "velocity-dip" may occur as an indicator of secondary currents [18], which represents the circulation in a transverse channel cross section, while the

y0 can be assumed proportional to the characteristic bottom roughness height, d, as suggested by Rouse [19] through the experimental parameter C<sup>ξ</sup> = y0/d. Therefore, Eq. (12) turns into:

Unlike Moramarco and Singh [7], here the ratio between Eq. (11) and Eq. (13), based on logarithm properties, explicitly proposes Φ(M) as a function of the relative submergence D/d:

> αα <sup>C</sup>ξð Þ <sup>1</sup>þ<sup>α</sup> <sup>1</sup>þ<sup>α</sup>

αα <sup>C</sup>ξð Þ <sup>1</sup>þ<sup>α</sup> <sup>1</sup>þ<sup>α</sup> � �; ln <sup>D</sup>

� � h i [13].

where A<sup>Φ</sup> and B<sup>Φ</sup> are the numerical coefficients. Eq. (14) follows under the hypothesis of linear

Eq. (14) highlights, indeed, a possible effect of bed roughness on the entropy velocity distribution in open channel flows, which depends on the roughness scale according to [1]. The dependence between the ratio Φ(M) and the relative submergence, D/d, has been widely

d

<sup>¼</sup> ln <sup>C</sup>0<sup>D</sup> d � �

ln <sup>D</sup> d

þ α k

ln <sup>α</sup><sup>α</sup> Cξð Þ 1 þ α

h i ffi <sup>A</sup>Φln <sup>D</sup>

!

ln <sup>α</sup> 1 þ α

1þα

d

In this context, Moramarco and Singh [7] identified the ratio between Umax and u\* as:

ln <sup>D</sup> y0ð Þ 1 þ α � � ln C<sup>0</sup> (11)

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

9

� � (12)

<sup>d</sup> <sup>þ</sup> <sup>B</sup><sup>Φ</sup> (14)

(13)

knowledge of the ratio between mean and maximum velocities, Φ(M).

Um u∗ ¼ 1 k ln <sup>D</sup> d þ 1 k

osed by Greco [9] for Um is considered:

with α = (D/ymax-1).

constant, and C0 is the dimensionless coefficient.

longitudinal flow component is called the primary flow.

Umax u∗

Umax u∗

<sup>Φ</sup>ð Þ¼ <sup>M</sup> Um

interpolation between the pairs ln C0D

¼ 1 k

¼ 1 k ln <sup>D</sup> d � � þ 1 k

Umax

d � �=ln <sup>D</sup>

Thus, the Shannon entropy of velocity distribution can be written as:

$$H = -\int\_0^{lL\_{\text{max}}} p(u) \log p(u) du\tag{4}$$

Through a similar procedure, the probability density function of the velocity distribution is obtained by maximizing the Shannon entropy equation

$$L = \int\_0^{lL\_{\text{max}}} \frac{f(u)}{m-1} \left\{ 1 - [f(u)]^{m-1} \right\} du + \lambda\_0 \left[ \int\_0^{lL\_{\text{max}}} f(u) du - 1 \right] + \lambda\_1 \left[ \int\_0^{lL\_{\text{max}}} u f(u) du - \overline{u} \right] \tag{5}$$

in which, λ<sup>0</sup> and λ<sup>1</sup> are the Lagrange multipliers and the following constraint equations

$$\mathbf{C}\_{1} = \int\_{0}^{\mathbf{U}\_{\text{max}}} f(u) du = 1 \tag{6}$$

$$\mathbf{C}\_2 = \int\_0^{\mathbf{L}\_{\text{max}}} \mathfrak{u}f(\mathfrak{u})d\mathfrak{u} = \overline{\mathfrak{u}} \tag{7}$$

$$f(\mu) = \exp\left(\lambda\_0 - 1 + \lambda\_1 \mu\right) \tag{8}$$

Thus, Chiu's 1D velocity distribution results as:

$$u = \frac{\mathcal{U}\_{\text{max}}}{M} \ln\left[1 + \left(e^{\mathcal{M}} - 1\right)F(u)\right] = \frac{\mathcal{U}\_{\text{max}}}{M} \ln\left[1 + \left(e^{\mathcal{M}} - 1\right)\frac{\xi - \xi\_0}{\xi\_{\text{max}} - \xi\_0}\right] \tag{9}$$

where M is the dimensionless entropy parameter introduced in the entropy-based derivation [14, 15]. Hence, M can be used as a measure of uniformity of probability and velocity distributions. The value of M can be determined by the mean, Um, and the maximum velocity values are derived from the following equation:

$$\Phi(M) = \frac{\mathcal{U}\_m}{\mathcal{U}\_{\max}} = \left(\frac{e^M}{e^M - 1} - \frac{1}{M}\right) \tag{10}$$

Φ(M) is a relevant parameter which contains relevant information about the flow field asset: the mean velocity value, the location of the mean velocity [14–16], and the energy coefficient [14, 16] can be obtained from M. That is, once known the mean velocity, the flow discharge, sediment transport, and pollutant transport can be derived. Furthermore, mean vs. maximum velocity assumes linear relationship as discovered by Xia collecting velocity data in several cross-sections of the Mississippi River [17].

Eq. (10), in fact, represents the fundamental relationship, from an applied point of view, of the entropy velocity distribution and the assessment of the entropy parameter passing through the knowledge of the ratio between mean and maximum velocities, Φ(M).

In order to identify the dependence of M from the hydraulic and geometric characteristics of channels, that is, the relative submergence and aspect ratio, respectively, the formulation proposed by Greco [9] for Um is considered:

$$\frac{\mathcal{U}\_m}{\mu\_\*} = \frac{1}{k} \ln \frac{D}{d} + \frac{1}{k} \ln \mathbb{C}\_0 \tag{11}$$

where u\* is the shear velocity, d is the bed roughness height (i.e., d50), k is the Von Karman constant, and C0 is the dimensionless coefficient.

Even the maximum velocity plays an important role in the flow dynamics, and more than it magnitude, a relevant aspect is related to the position of the maximum velocity inside the flow domain. That is, the location of maximum velocity from the channel bottom, ymax, does not always occur at water surface, but a "velocity-dip" may occur as an indicator of secondary currents [18], which represents the circulation in a transverse channel cross section, while the longitudinal flow component is called the primary flow.

In this context, Moramarco and Singh [7] identified the ratio between Umax and u\* as:

$$\frac{\mathcal{U}\_{\text{max}}}{\mu\_{\ast}} = \frac{1}{k} \ln \left( \frac{D}{y\_0(1+a)} \right) + \frac{a}{k} \ln \left( \frac{a}{1+a} \right) \tag{12}$$

with α = (D/ymax-1).

greater than ξ, the velocity is greater than u, and the cumulative distribution function can be

F uð Þ¼ <sup>ξ</sup> � <sup>ξ</sup><sup>0</sup>

Through a similar procedure, the probability density function of the velocity distribution is

ðUmax 0

f uð Þdu � 1 � �

<sup>M</sup> ln <sup>1</sup> <sup>þ</sup> <sup>e</sup>

eM � <sup>1</sup> � <sup>1</sup>

� �

M

ðUmax 0

du þ λ<sup>0</sup>

C<sup>1</sup> ¼

C<sup>2</sup> ¼

<sup>M</sup> � <sup>1</sup> � �F uð Þ � � <sup>¼</sup> Umax

<sup>Φ</sup>ð Þ¼ <sup>M</sup> Um

Umax

where M is the dimensionless entropy parameter introduced in the entropy-based derivation [14, 15]. Hence, M can be used as a measure of uniformity of probability and velocity distributions. The value of M can be determined by the mean, Um, and the maximum velocity values

<sup>¼</sup> <sup>e</sup><sup>M</sup>

Φ(M) is a relevant parameter which contains relevant information about the flow field asset: the mean velocity value, the location of the mean velocity [14–16], and the energy coefficient [14, 16] can be obtained from M. That is, once known the mean velocity, the flow discharge, sediment transport, and pollutant transport can be derived. Furthermore, mean vs. maximum velocity assumes linear relationship as discovered by Xia collecting velocity data in several

in which, λ<sup>0</sup> and λ<sup>1</sup> are the Lagrange multipliers and the following constraint equations

ðUmax 0

ðUmax 0

Thus, the Shannon entropy of velocity distribution can be written as:

obtained by maximizing the Shannon entropy equation

<sup>m</sup> � <sup>1</sup> <sup>1</sup> � ½ � f uð Þ <sup>m</sup>�<sup>1</sup> n o

Thus, Chiu's 1D velocity distribution results as:

<sup>M</sup> ln <sup>1</sup> <sup>þ</sup> <sup>e</sup>

<sup>u</sup> <sup>¼</sup> Umax

are derived from the following equation:

cross-sections of the Mississippi River [17].

H ¼ �

ξmax � ξ<sup>0</sup>

p uð Þlog p uð Þdu (4)

ðUmax 0

f uð Þdu ¼ 1 (6)

uf uð Þdu ¼ u (7)

<sup>M</sup> � <sup>1</sup> � � <sup>ξ</sup> � <sup>ξ</sup><sup>0</sup>

� �

ξmax � ξ<sup>0</sup>

f uð Þ¼ exp ð Þ λ<sup>0</sup> � 1 þ λ1u (8)

uf uð Þdu � u � �

þ λ<sup>1</sup>

(3)

(5)

(9)

(10)

written as

8 Irrigation in Agroecosystems

L ¼

ðUmax 0

f uð Þ

y0 can be assumed proportional to the characteristic bottom roughness height, d, as suggested by Rouse [19] through the experimental parameter C<sup>ξ</sup> = y0/d. Therefore, Eq. (12) turns into:

$$\frac{\mathcal{U}\_{\text{max}}}{\mu\_{\ast}} = \frac{1}{k} \ln \left( \frac{D}{d} \right) + \frac{1}{k} \ln \left( \frac{\alpha^{\alpha}}{\mathcal{C}\_{\xi} (1 + \alpha)^{1 + \alpha}} \right) \tag{13}$$

Unlike Moramarco and Singh [7], here the ratio between Eq. (11) and Eq. (13), based on logarithm properties, explicitly proposes Φ(M) as a function of the relative submergence D/d:

$$\mathfrak{O}(M) = \frac{\mathcal{U}\_m}{\mathcal{U}\_{\max}} = \frac{\ln\left(\frac{\mathbb{C}\_\ell D}{d}\right)}{\ln\left[\frac{D}{d} \frac{\alpha^n}{\mathbb{C}\_\ell (1+\alpha)^{1+\alpha}}\right]} \cong A\_\mathcal{O} \ln\frac{D}{d} + B\_\mathcal{O} \tag{14}$$

where A<sup>Φ</sup> and B<sup>Φ</sup> are the numerical coefficients. Eq. (14) follows under the hypothesis of linear interpolation between the pairs ln C0D d � �=ln <sup>D</sup> d αα <sup>C</sup>ξð Þ <sup>1</sup>þ<sup>α</sup> <sup>1</sup>þ<sup>α</sup> � �; ln <sup>D</sup> d � � h i [13].

Eq. (14) highlights, indeed, a possible effect of bed roughness on the entropy velocity distribution in open channel flows, which depends on the roughness scale according to [1]. The dependence between the ratio Φ(M) and the relative submergence, D/d, has been widely studied by Greco [9] using a wide volume of data collected in the field on several cross sections along different rivers and in the laboratory [20–22], showing values of Φ(M) ranging in the [0.5–0.9] interval.

of the measurement reach, and the water depth, D, was assumed as the average value. The velocity was acquired through a micro current-meter with a measuring head diameter of 0.01 m, while the water discharge was measured with a concentric orifice plate installed in the feed pipe and on a laboratory weir placed at the end of the flumes, and compared to the value calculated according to the velocity-area method [23], with a maximum error of around 1–2%. In particular, the adopted velocity-area method must be applied dividing the cross section into a fixed number of verticals and thus, on each vertical, a fixed measurement points are selected. In each point along the vertical, the velocity is acquired in order to compute the mean velocity of the flow along each vertical. Furthermore, the number of measures on each vertical was chosen with respect to the criterion that the difference in velocity between two consecutive points was less than 20%, of the higher measured velocity value, and the points close to the channel bottom and the water surface was fixed according

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

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

11

• RRF: rough rectangular flume, with relative submergence ranging in between 1.89 and

Table 1 synthetically reports the ranges of variation of the main parameters observed during the experiments for the RRF and SRF configuration, while Q is the water discharge, D is the water depth, D/d is the relative submergence, B/D is the aspect ratio, and Φ(M) is the ratio

For each configuration and for all the stages explored, a relevant bulk of velocity measurements was collected in order to provide a detailed reconstruction of the flow field allowing to

Figure 2 shows the linear relationship existing between the pairs (Umax; Um) for the two

From Figure 2, some useful issues arise. Even if the correlation among homogeneous data is very strong in both cases with R<sup>2</sup> greater than 0.95, it is immediately realized a slight different behavior between rough and smooth channels. That is, for the smooth rectangular flow, Φ(M) assumes the value 0.9, while for the rough condition, the value decreases to 0.67. That is, in other terms, it seems to be evident and sufficiently confirmed, the dependence of the velocity ratio on the roughness here represented by the relative submergence D/d as discussed in the

Type Q (mc/sec) D (m) D/d B/D Φ(M) RRF 0.007–0.076 0.07–0.23 1.89–6.43 2.22–7.58 0.52–0.73 SRF 0.025–0.100 0.06–0.40 50–298 2.50–10 0.7–0.93

• SRF: smooth rectangular flume, with relative submergence greater than 50.

to the size of the micro-current meter.

between the mean and maximum velocities.

configurations investigated, RRF and SRF.

previous section for Eq. (14).

6.43; and

In such a way, two roughness configurations were enabled:

obtain mean, Um, and maximum, Umax, cross section velocities.

Table 1. Range of variation for the main parameters of the laboratory experiments.
