#

1 1

( ) ( )

*h h <sup>h</sup> <sup>h</sup>* (85)

2 2 #

a

1 1

2 2 #

a

Having obtained this first adjustment, other data of the literature were also compared to the formulation, obtaining the coefficients of Table 4 (R2 are those of the multilinear regression). Figure 13 shows all predictions using equation (57) plotted against the measured value. The good linear correlation between predicted and measured values shows the adequacy of the present formulation. Further, different experimental arrangements and definitions of relevant parameters influence the coefficients *θ*6 and *θ*7, allowing the study of these influences. For the purposes of this study, which intends to expose the adequacy of the formulation, the literature data were used without distinguishing *Lr* and *Lj* , or the different ways used to obtain these lengths. Other influences, as the growing of the boundary layer at the supercritical flow, width of the channel, bottom roughness, bottom slope, distance to the downstream control (gate or step), use of sluice gates or spillways at the supercritical flow, among other characteristics, may also be studied in order to establish the dependence of *θ*6 and *θ*7 on these variables.

Table 4 shows that the coefficients depend on the experimental conditions. As an example, the data of [36] were obtained for several bed slopes. Table 4 presents the coefficients obtained for the slopes of 1o and 5o , suggesting variations in *θ*6 and *θ*7 (the component of the weight along the channel may influence, for example). Further, the data of [37] were obtained for rough beds, which coefficients may be compared to those for smooth beds of [38], suggesting that roughness also affects the coefficients ([38] also presents data for rough beds, allowing further comparisons). The data of [17] were obtained for flumes having as inlet structure a sluice gate or a spillway, which may be the cause of the different coefficients. The number of "points" of the different series is also shown, in order to allow verifying if the results depend on the "length" of the sample. Because coefficients are adjusted statistically, variations in *θ*6 and *θ*<sup>7</sup> are expected when using only parts of a whole sample (like the two series of [18]). But also similar results were obtained for data of different authors, like [2], for 17.5o , and [17], for spillways, who furnished samples for 35 and 10 "points", respectively.


**Table 4.** Values of *θ*6 and *θ*7 adjusted to data of different authors.

It is of course necessary to have a common definition of the lengths of the roller and the hydraulic jump (for the different studies) in order to allow a definitive comparison. Histori‐ cally, different ways to obtain the lengths were used, which may add uncertainties to this first discussion of the coefficients. However, the good joint behavior of the different sets of data shown in Figure 13 emphasizes that such discussion is welcomed.

#### **3.2. Sequent depths for the hypothesis "The force at the bottom is neglected"**

Figure 3 already shows the good adjustment obtained using data of different authors. Different conditions of the channel may affect the predictions of equation (7) or (61), as shown in the literature, but the ratio between sequent depths, if working with the classical jump, is well predicted, as can be seen in Figure 3.

#### **3.3. Sequent depths for the hypothesis "The force at the bottom is relevant"**

The condition of "not-negligible bottom force" may be relevant, for example, for rough beds, which may affect the length of the jump and the subcritical depth. In order to illustrate this

similar results were obtained for data of different authors, like [2], for 17.5o

[18] Flume F 5.87 -0.586 0.991 22 [18] All data 3.62 -0.157 0.996 120 [34] 4.13 -0.157 0.995 30 [35] 5.10 -0.373 0.998 50

C 3,43 -0,191 0,999 17

C 2,75 -0,103 0,995 35

Bed slope 4,01 -0,177 0,9998 20

 Bed slope 4,30 -0,339 0,9996 14 [37] Data for rough beds 2.76 -0.126 0.998 11 [38] Data for smooth beds 3.60 -0.139 0.994 20 [39] 2.81 -0.187 0.989 72 [15] 4.10 -0.312 0.999 6 [40] 4.65 -0.288 0.9994 31 [17] Spillways 2,79 -0.109 0.9999 10 [17] Sluice gates 2.94 -0.156 0.9999 8

It is of course necessary to have a common definition of the lengths of the roller and the hydraulic jump (for the different studies) in order to allow a definitive comparison. Histori‐ cally, different ways to obtain the lengths were used, which may add uncertainties to this first discussion of the coefficients. However, the good joint behavior of the different sets of data

Figure 3 already shows the good adjustment obtained using data of different authors. Different conditions of the channel may affect the predictions of equation (7) or (61), as shown in the literature, but the ratio between sequent depths, if working with the classical jump, is well

The condition of "not-negligible bottom force" may be relevant, for example, for rough beds, which may affect the length of the jump and the subcritical depth. In order to illustrate this

**Author** *θ<sup>6</sup> θ<sup>7</sup>* **R2 Number of points**

spillways, who furnished samples for 35 and 10 "points", respectively.

**Table 4.** Values of *θ*6 and *θ*7 adjusted to data of different authors.

predicted, as can be seen in Figure 3.

shown in Figure 13 emphasizes that such discussion is welcomed.

**3.2. Sequent depths for the hypothesis "The force at the bottom is neglected"**

**3.3. Sequent depths for the hypothesis "The force at the bottom is relevant"**

[2] 16.5o

98 Hydrodynamics - Concepts and Experiments

[2] 17.5o

[36] 1o

[36] 5o

, and [17], for

**Figure 13.** The predictions of *L*/*h*<sup>1</sup> for different experiments described in the literatures using the coefficients of Table 4 and equation (57).

condition, a multilinear regression analysis was made using equation (67) and the data of [37], furnishing *θ* 4=0.001229, *θ* 8=0.4227, and 2(*θ* 3+*θ* 5)=0.5667. The determinant of equation (66) was negative for 10 experimental conditions. Thus, three real roots were obtained for these conditions. On the other hand, the determinant was positive for one experimental condition, leading to only one real root. When plotting the three calculated roots for *h#* in relation to the measured values, Figure 14 was obtained (several points superpose).

**Figure 14.** The three roots of *h#* predicted with the cubic equation (66) for each experimental condition of [37]. The neg‐ ative roots (in gray) are physically impossible. The positive roots (in white and black) show a constant depth ratio (Ghost Depth) and the reproduction of the observed values (inclined line). The gray point with a circle corresponds to the real root obtained for the positive determinant of equation (66). Several points superpose.

The constant *h#* indicated by the horizontal line was obtained for the different sets of data analyzed here, assuming a proper value for each set. However, because it does not represent an observed result, it was named here "Ghost Depth", *h*GHOST. From equation (66) this root is quantified as:

$$h\_{\rm GHOST} = -\frac{2 - \theta\_8 - 2\left(\theta\_3 + \theta\_5\right)}{2\left(\theta\_3 + \theta\_5\right) - \theta\_8} = \frac{\left(1 - 2\theta\_8\right) \pm \sqrt{\left(1 - 2\theta\_8\right)^2 + 4\theta\_4}}{2\theta\_4} \tag{88}$$

Considering the first equality of equation (88), and dividing the cubic equation (66) by the monomial obtained with this solution, following second degree equation is obtained:

$$h^{\#2} - 2\theta\_{\theta}Fr\_{1}^{2}h^{\#} - \theta\_{10}Fr\_{1}^{2} = 0\tag{89}$$

Where:

$$\Theta\_{9} = -\frac{\Theta\_{4}}{2\left[\mathbb{Z}\left(\theta\_{3} + \theta\_{5}\right) - \theta\_{8}\right]}, \qquad \qquad \Theta\_{10} = \left| \frac{\left(1 - 2\,\theta\_{8}\right)}{\mathbb{Z}\left(\theta\_{3} + \theta\_{5}\right) - \theta\_{8}} + \frac{\Theta\_{4}\left[2 - \theta\_{8} - 2\left\{\theta\_{3} + \theta\_{5}\right\}\right]}{\mathbb{Z}\left(\theta\_{3} + \theta\_{5}\right) - \theta\_{8}\mathbb{Z}^{2}}} \right|$$

**Figure 15.** Calculated values of *h#* obtained with equation (90) and the data of [37]. The number of experimental condi‐ tions is 11, but several points superpose.

As can be seen, also in this case the final equation is a second order equation and depends on only two coefficients, *θ*9 and *θ*10. The solution for *h*# is given by:

Details of Hydraulic Jumps for Design Criteria of Hydraulic Structures http://dx.doi.org/10.5772/58963 101

$$h^{\#} = \theta\_{\phi} Fr\_{\text{l}}^{2} \pm \sqrt{\left[\theta\_{\phi} Fr\_{\text{l}}^{2}\right]^{2} + \theta\_{\text{l0}} Fr\_{\text{l}}^{2}}\tag{90}$$

The adjusted coefficients produced the graph of Figure 15 for the positive root of equation (90) and the data of [37].

The constant *h#*

100 Hydrodynamics - Concepts and Experiments

quantified as:

Where:

GHOST

*<sup>θ</sup>*<sup>9</sup> <sup>=</sup> <sup>−</sup> *<sup>θ</sup>*<sup>4</sup>

tions is 11, but several points superpose.

only two coefficients, *θ*9 and *θ*10. The solution for *h*#

2 2 (*θ*<sup>3</sup> + *θ*5) −*θ*<sup>8</sup>

indicated by the horizontal line was obtained for the different sets of data

( ) ( )

2 (*θ*<sup>3</sup> + *θ*5) −*θ*<sup>8</sup>

+

*h* (88)

8 35 8 84

q

2

(89)

*θ*<sup>4</sup> 2−*θ*<sup>8</sup> −2 (*θ*<sup>3</sup> + *θ*5) 2 (*θ*<sup>3</sup> + *θ*5) −*θ*<sup>8</sup>

<sup>2</sup> }

 qq

q

analyzed here, assuming a proper value for each set. However, because it does not represent an observed result, it was named here "Ghost Depth", *h*GHOST. From equation (66) this root is

35 8 4


2 2

monomial obtained with this solution, following second degree equation is obtained:

#2 2 # 2 9 1 10 1 *h Fr h Fr* - -= 2 0

2 2 12 12 4

Considering the first equality of equation (88), and dividing the cubic equation (66) by the

 q

, *<sup>θ</sup>*<sup>10</sup> ={ (1−<sup>2</sup> *<sup>θ</sup>*8)

**Figure 15.** Calculated values of *h#* obtained with equation (90) and the data of [37]. The number of experimental condi‐

As can be seen, also in this case the final equation is a second order equation and depends on

is given by:

( )

 q

q

( )

qq

 qq

q

The good result of Figure 15 induced to analyze more data of the literature using equations (89) and (90). Table 5 shows the coefficients obtained from multilinear regression analyses applied to the original data, and Figure 16 presents all data plotted together. The values of R2 in Table 5 are those obtained in the multilinear regression.


**Table 5.** *θ*9 and *θ* <sup>10</sup> (and√*θ* 10) adjusted to data of different authors.

The coefficients *θ* 9 and *θ* 10 of Table 5 show that, for most of the cases, *θ* 10>>*θ* 9, which implies *h*# =*θ*9*Fr*<sup>1</sup> 2 +√*θ* <sup>10</sup>*Fr*1 for the usual range of *Fr*1. In many cases, the quadratic term of *Fr*<sup>1</sup> is still much smaller than the linear term. For example, the data of [15], obtained for 4.38≤*Fr*1≤9.26 produce *θ*9*Fr*<sup>1</sup> 2 =0.227 and √*θ* <sup>10</sup>*Fr*1=12.1 as their highest values. This fact suggests to use simpler forms of equation (90) like *h*# =√*θ* <sup>10</sup>*Fr*1+*Constant* or *h*# =√*θ* <sup>10</sup>*Fr*1. Both forms appear in the literature. The mean value of √*θ* 10 in table 5 is 1.19, showing that the coefficients of *Fr*<sup>1</sup> stay around the unity. [44], for example, used *h*# =1.047*Fr*1+0.5902, while [37] simply suggested *h*# =*Fr*<sup>1</sup> for their data. Equation (8) of the present chapter is a further example.

A very positive aspect of equation (90) is that it is based on the physical principles of conser‐ vation of mass, momentum and energy. The simpler forms of the literature follow directly when analyzing the magnitude of the different parcels of the final equation. That is, the simpler forms are in agreement with the conservation principles (being not only convenient results of dimensional analyses).

**Figure 16.** The predictions of *h*2/*h*<sup>1</sup> for different experiments described in the literatures using the coefficients of Table 5 and equation (90).

As an additional information, the coefficient of *Fr*1 in the simpler forms tends to be lower than √2, the value derived from equations (7) or (61) for no shear force at the bottom of the jump.

From the data analysis performed here, and the conclusions that derive from the presence of the Ghost Depth in the cubic equation, the equations for the geometrical characteristics of hydraulic jumps of case ii of section 2.1.2 (The force at the bottom is relevant) simplify to

equation (57) *L r h* 1 = *θ*<sup>6</sup> *Fr*<sup>1</sup> *<sup>h</sup>* \* <sup>+</sup> *<sup>θ</sup>*<sup>7</sup> (*<sup>F</sup> <sup>r</sup>*1)<sup>3</sup> *<sup>h</sup>* \* and equation (90) *<sup>h</sup>* #=*θ*9*<sup>F</sup> <sup>r</sup>*<sup>1</sup> <sup>2</sup> <sup>±</sup> *<sup>θ</sup>*9*Fr*<sup>1</sup> <sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>θ</sup>*10*<sup>F</sup> <sup>r</sup>*<sup>1</sup> 2 , equations already presented in the text.

Considering the deduction procedures followed in this study, the values of *θ*6, *θ*7, *θ*9 and *θ*<sup>10</sup> may depend on the factors that determine the resistance forces (shear forces), similarly to the friction factor in uniform flows. Additionally, because *h*\*=(*h*# -1)-1, the nondimensional length of the roller may be written as a function only of *Fr*1 and the different coefficients.

#### **3.4. Water depth profiles**

of equation (90) like *h*#

[44], for example, used *h*#

102 Hydrodynamics - Concepts and Experiments

dimensional analyses).

and equation (90).

equation (57)

*L r h* 1 = *θ*<sup>6</sup>

equations already presented in the text.

*Fr*<sup>1</sup>

friction factor in uniform flows. Additionally, because *h*\*=(*h*#

=√*θ* <sup>10</sup>*Fr*1+*Constant* or *h*#

Equation (8) of the present chapter is a further example.

mean value of √*θ* 10 in table 5 is 1.19, showing that the coefficients of *Fr*<sup>1</sup> stay around the unity.

A very positive aspect of equation (90) is that it is based on the physical principles of conser‐ vation of mass, momentum and energy. The simpler forms of the literature follow directly when analyzing the magnitude of the different parcels of the final equation. That is, the simpler forms are in agreement with the conservation principles (being not only convenient results of

**Figure 16.** The predictions of *h*2/*h*<sup>1</sup> for different experiments described in the literatures using the coefficients of Table 5

As an additional information, the coefficient of *Fr*1 in the simpler forms tends to be lower than √2, the value derived from equations (7) or (61) for no shear force at the bottom of the jump.

From the data analysis performed here, and the conclusions that derive from the presence of the Ghost Depth in the cubic equation, the equations for the geometrical characteristics of hydraulic jumps of case ii of section 2.1.2 (The force at the bottom is relevant) simplify to

*<sup>h</sup>* \* <sup>+</sup> *<sup>θ</sup>*<sup>7</sup> (*<sup>F</sup> <sup>r</sup>*1)<sup>3</sup> *<sup>h</sup>* \* and equation (90) *<sup>h</sup>* #=*θ*9*<sup>F</sup> <sup>r</sup>*<sup>1</sup>

Considering the deduction procedures followed in this study, the values of *θ*6, *θ*7, *θ*9 and *θ*<sup>10</sup> may depend on the factors that determine the resistance forces (shear forces), similarly to the

of the roller may be written as a function only of *Fr*1 and the different coefficients.

=1.047*Fr*1+0.5902, while [37] simply suggested *h*#

=√*θ* <sup>10</sup>*Fr*1. Both forms appear in the literature. The

=*Fr*<sup>1</sup> for their data.

<sup>2</sup> <sup>±</sup> *<sup>θ</sup>*9*Fr*<sup>1</sup>


<sup>2</sup> <sup>2</sup> <sup>+</sup> *<sup>θ</sup>*10*<sup>F</sup> <sup>r</sup>*<sup>1</sup>

2 ,

#### *3.4.1. Profiles without inflexion points*

The data of [17] and [25], obtained for spillways, are shown in Figure 17. It is possible to admit no inflexion points in this set of surface profiles. So, equation (70) was used to quantify the surface evolution, furnishing equation (91). The adjustment produced *θ*11=2.9, implying that *x*/*L*=1 (the roller length) is attained for *Π*=0.95, that is, for 95% of the final depth difference *h*2 *h*1. Such conditional definition of the length is similar to that used for boundary layer thick‐ nesses, and seems to apply for hydraulic jumps.

**Figure 17.** Normalized measured profiles obtained by [17] and [25], adjusted to the exponential prediction (equation 91), for profiles without inflexion points.

In the present analysis, the normalized profile assumes the form

$$\frac{h - h\_1}{h\_2 - h\_1} = 1 - e^{-2.9 \times L} \tag{91}$$

#### *3.4.2. Profiles with inflexion points*

Equation (82) may be used with any option of *h*# , that is, equation (61) or (90), or the simpler form *h*# =√*θ*10+*Constant*. As example, equation (61) is used, generating equation (92) for *β*=0.99. For comparison, the simpler form of *h*# is used (as proposed by [37]) with √*θ*10=1, *Constant*=0, and *β*=0.99, resulting *h*# =*Fr*1 and equation (93). Figure 18a shows the comparison between equation (92) and the envelope of the measured data of [37]. Figure 18b shows the comparison between equations (92) and (93) for *Fr*1=7. Both equations show the same form, with the results of equation (93) being somewhat greater than those of equation (92).

$$\frac{h - h\_1}{h\_2 - h\_1} = \frac{2(1 - e^{-\lambda h})}{(\sqrt{8Fr\_1^2 + 1} - 1) - (\sqrt{8Fr\_1^2 + 1} - 3)(1 - e^{-\lambda h})} \qquad \qquad Li = -\ln\left[\frac{1}{99(\sqrt{8Fr\_1^2 + 1} - 3)}\right] \tag{92}$$

1 21 1 1 '1 (1 ) <sup>1</sup> ln ( 1)(1 ) 99( 1) - - - - é ù <sup>=</sup> = - ê ú - - -- ë û - *IJs IJs h h <sup>e</sup> IJ h h Fr Fr e Fr* (93)

Figure 19 was obtained for the data of [17] using equation (92), but with *β*=0.97, that is, with the exponent *IJ* given by:

$$IJ = -\ln\left\{6/\left[97(\sqrt{8Fr\_{\parallel}^2 + 1} - 3)\right]\right\}\tag{94}$$

**Figure 18.** a) The lines are the predictions of equation (92) for the values of *Fr*1 shown in the graph (*β*=0.99). The gray region is the envelope of the data of [37]; b) Comparison between equations (92) and (93) for events with and without bed shear force, and *Fr*1=7.0.

**Figure 19.** Normalized measured profiles obtained by [17] and [25], and predictions of equation (93) for profiles with inflexion points (*β*=0.97).

#### *3.4.3. Further normalizations*

1

1

104 Hydrodynamics - Concepts and Experiments

the exponent *IJ* given by:

bed shear force, and *Fr*1=7.0.

inflexion points (*β*=0.97).

2 2 <sup>2</sup> 2 1 1 1 <sup>1</sup>

21 1 1 '1



*IJs h h <sup>e</sup> IJ*


*h h <sup>e</sup> IJ*


*IJs*

*IJs*

2(1 ) <sup>1</sup> ln ( 8 1 1) ( 8 1 3)(1 ) 99( 8 1 3)

 é ù - - <sup>=</sup> = - ê ú - +- - +- - + - ë û

*IJs*

*h h Fr Fr e Fr* (92)

(1 ) <sup>1</sup> ln ( 1)(1 ) 99( 1)

Figure 19 was obtained for the data of [17] using equation (92), but with *β*=0.97, that is, with

**Figure 18.** a) The lines are the predictions of equation (92) for the values of *Fr*1 shown in the graph (*β*=0.99). The gray region is the envelope of the data of [37]; b) Comparison between equations (92) and (93) for events with and without

**Figure 19.** Normalized measured profiles obtained by [17] and [25], and predictions of equation (93) for profiles with

2

<sup>1</sup> *IJ* = -ln{6 / [97( 8 1 3)]} *Fr* + - (94)

*h h Fr Fr e Fr* (93)

[21] presented the experimental water profiles shown in Figure 20b. As can be seen, using their normalization the curves intercept each other for the experimental range of *Fr*1. In order to verify if the present formulation leads to a similar behavior, a first approximation was made using equation (93) for *β*=0.95, that is, with the exponent *IJ* given by

$$II = -\ln\left[5 / \left[\mathbf{95} (Fr\_1 - 1)\right]\right]$$

*x*/*L* was transformed into *x*/(*h*2-*h*1) by multiplying [*x*/(*h*2-*h*1)][*h*1/*L*][*h*# -1]. The coefficients *θ*6 and *θ*7 of Table 4 for the data of [37] were used to calculate *L*/*h*1, together with *h*# =*Fr*1. The calculated curves shown in Figure 20a also intercept each other, obeying similar relative positions when compared to the experimental curves. The results of the formulation thus follow the general experimental behavior.

**Figure 20.** Comparison between observed and calculated surface profiles; a) Calculated curves; b) Experimental curves.

Various values of *β* were used in the present section (3.4), in order to obtain good adjustments. In each figure (18, 19, and 20) *β* was maintained constant for the family of curves, but it may not be always the case, considering previous results of the literature. For example, [45] suggested that the ratio between the final depth of the roller and *h*2 is a function or *Fr*1. Thus, each curve may have its own *β*.

#### **3.5. Depth fluctuations and total depths (wall heights)**

#### *3.5.1. Predictions based on empirical formulation*

As mentioned in item 2.3, oscillations of the free surface may be relevant for the design of side walls. [46] recommended, for the height of the wall, *hwall*=1.25*h*2. [22] performed measurements of water depths and fluctuations along a hydraulic jump in a 40 cm wide horizontal rectangular channel, and imposing *Fr*1=3.0. The measurements were made using ultrasonic acoustic sensors. The mean ratio *h*/*h*<sup>2</sup> attained a maximum value in the roller region, *h*max/*h*2=1.115. The maximum normalized standard error of the fluctuations along the hydraulic jump was *h*'max/ *h*1=0.32 for this condition. [47], based on data of several sources, showed that standard deviation of the fluctuations of the depths in hydraulic jumps increase with the Froude number, suggesting the empirical equation (94), for *Fr*1 between 1.98 and 8.5.

$$h^\*\_{\text{max}} / h\_\text{l} = 0.116(Fr\_\text{l} - 1)^{1.23} \tag{95}$$

This equation furnishes, for the condition *Fr*1=3.0 studied by [22], the prediction *h*'max/*h*1=0.27, about 16% lower of the experimental value, suggesting further studies. In the present section the height *hwall* is quantified following different approximations, and using experimental data. A first evaluation of *hwall* may be given by the sum of the maximum depth and a multiple of the standard deviation (multiple given by the factor *N*), that is:

$$h\_{\text{wall}} \mid h\_2 = h\_{\text{max}} \mid h\_2 + N \, h\_{\text{max}}^\ast \,/\, (h\_\text{i} h^\neq) \tag{96}$$

Using equations (95) and (96), no bed shear force, that is *h*# =[(1+8*Fr*<sup>1</sup> 2 -1)1/2-1]/2, and correcting the predicted value of 0.27 to the observed value of 0.32, we have:

$$\frac{h\_{\text{wall}}}{h\_{\text{2}}} = 1.1115 + \frac{0.275 \left(Fr\_{\text{l}} - 1\right)^{1.23}}{\left(\sqrt{1 + 8Fr\_{\text{l}}^{2}} - 1\right)},\tag{97}$$

$$\frac{h\_{\text{wall}}}{h\_{\text{2}}} = 1.115 + \frac{0.550 \left(Fr\_{\text{l}} - 1\right)^{1.23}}{\left(\sqrt{1 + 8Fr\_{\text{l}}^{2}} - 1\right)},\tag{98}$$

$$\frac{h\_{wall}}{h\_2} = 1.115 + \frac{0.825 \left(Fr\_1 - 1\right)^{1.23}}{\left(\sqrt{1 + 8Fr\_1^2} - 1\right)},\tag{99}$$

As an example, Figure 21 presents different images of surface elevations due to waves, which illustrates a depth fluctuation. Figure 21a presents a sequence of images of a wave passing the hydraulic jump, and Figure 21b shows the relative depth attained by this wave. The approxi‐ mate value *h*max/*h*2=1.40 is compared with equations (97-99).

suggested that the ratio between the final depth of the roller and *h*2 is a function or *Fr*1. Thus,

As mentioned in item 2.3, oscillations of the free surface may be relevant for the design of side walls. [46] recommended, for the height of the wall, *hwall*=1.25*h*2. [22] performed measurements of water depths and fluctuations along a hydraulic jump in a 40 cm wide horizontal rectangular channel, and imposing *Fr*1=3.0. The measurements were made using ultrasonic acoustic sensors. The mean ratio *h*/*h*<sup>2</sup> attained a maximum value in the roller region, *h*max/*h*2=1.115. The maximum normalized standard error of the fluctuations along the hydraulic jump was *h*'max/ *h*1=0.32 for this condition. [47], based on data of several sources, showed that standard deviation of the fluctuations of the depths in hydraulic jumps increase with the Froude

This equation furnishes, for the condition *Fr*1=3.0 studied by [22], the prediction *h*'max/*h*1=0.27, about 16% lower of the experimental value, suggesting further studies. In the present section the height *hwall* is quantified following different approximations, and using experimental data. A first evaluation of *hwall* may be given by the sum of the maximum depth and a multiple of

> ( ) ( ) 1,235

> ( ) ( ) 1,235

> ( ) ( ) 1,235

1

1

1

0.825( 1) 1.115 , 3 18 1 - = + <sup>=</sup> + *wall <sup>h</sup> Fr <sup>N</sup>*

0.550( 1) 1.115 , 2 18 1 - = + <sup>=</sup> + *wall <sup>h</sup> Fr <sup>N</sup>*

0.275( 1) 1.115 , 1 18 1 - = + <sup>=</sup> + *wall <sup>h</sup> Fr <sup>N</sup>*

1.235

max 1 <sup>1</sup> *h h Fr* ' / 0.116( 1) = - (95)
