**Details of Hydraulic Jumps for Design Criteria of Hydraulic Structures**

Harry Edmar Schulz, Juliana Dorn Nóbrega, André Luiz Andrade Simões, Henry Schulz and Rodrigo de Melo Porto

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/58963

#### **1. Introduction**

The sudden transition from supercritical to subcritical flow, known as hydraulic jump, is a phenomenon that, although being studied along decades, still presents aspects that need better quantification. Geometrical characteristics, such as the length of the roller (or the jump itself), still have no definitive formulation for designers of hydraulic structures. Even predictions of the sequent depths, usually made considering no shear forces, may present deviations from the observed values.

In the present chapter the geometrical characteristics of hydraulic jumps are obtained follow‐ ing different deductive schemes. Firstly, two adequate control volumes and the principles of conservation of mass, momentum and energy were used to obtain the length of the roller and the sequent depths. The conditions of presence or absence of bed shear forces are discussed. Secondly, two ways are used to propose the form of surface profiles: i) a "depth deficit" criterion and ii) the mass conservation principle using an "air capture" formulation. The presence or absence of inflexion points is discussed considering both formulations. Finally, the height attained by surface fluctuations (water waves and drops), useful for the design of the lateral walls that confine the jumps, is considered using empirical information and an approx‐ imation based on results of the Random Square Waves method (RSW).

Experimental data from the literature were used for comparisons with the proposed theoretical equations, allowing the adjustment of coefficients defined in these equations.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

#### **1.1. General aspects**

A supercritical flow in a long horizontal channel (or having a small slope) has an inexpressive component of the weight as impulsive force. As a result, the power dissipation induces the depth of the liquid to increase along the flow, tending to its critical value. But a sudden change, a hydraulic jump, forms before the flow reaches the critical condition (which corresponds to the condition of minimum specific energy). In other words, transitions to the subcritical condition do not occur gradually in such channels. Hydraulic jumps have been studied over the years, providing a good theoretical understanding of the phenomenon, as well as experi‐ mental data that can be used to check new conceptual proposals.

The characteristics of hydraulic jumps in free flows are of interest for different applications. For example, they are used in structures designed for flow rate measurements, or for energy dissipation, like the stilling basins downstream from spillways. Although simple to produce and to control in the engineering praxis, hydraulic jumps still present aspects which are not completely understood. Interestingly, the geometrical characteristics of the jumps are among the features whose quantification still not have consensus among researchers and professio‐ nals. Perhaps the length of the jump (or the roller) is the dimension that presents the highest level of "imprecision" when quantifying it. Such difficulties may be a consequence of the simplifying hypotheses made to obtain the usual relationships. In the most known equation for the sequent depths, the shear force is neglected, and a very simple functional dependence is obtained between the depths and the Froude number (*Fr*). However, by neglecting the shear force, difficulties arise to calculate the length of the jump, because it does not appear in the formulation.

Many authors observed this difficulty since the beginning of the studies on hydraulic jumps. As a consequence, also a large number of solutions were presented for the sequent depths and for the length of the jump (or roller). Good reviews were presented, for example, by [1] and [2].

The hydraulic jump may be viewed as a "shock" between the supercritical and the subcritical flows. As result of this "shock", the water of the higher depth "falls down" on the water of the lower depth. Because of the main movement of the flow, and the resulting shear forces, a roller is formed in this "falling region", characterized as a two phase turbulent zone with rotating motion. This description is sketched approximately in Figure 1a. Figure 1b shows an example of this highly turbulent region, responsible for most of the energy dissipation in a hydraulic jump. Despite the fact of being turbulent (thus oscillating in time) and 3D, the hydraulic jump may be quantified as a permanent 1D flow when considering mean quantities.

Table 1 presents some studies related to the theme around the world, while Table 2 lists some academic studies in Portuguese language, developed in universities of Brazil, as a consequence of the increasing interest on hydroelectric energy in the country, and the corresponding projects of spillways with dissipation basins. The present chapter uses the background furnished by the previous studies.

Details of Hydraulic Jumps for Design Criteria of Hydraulic Structures http://dx.doi.org/10.5772/58963 75

**1.1. General aspects**

74 Hydrodynamics - Concepts and Experiments

formulation.

furnished by the previous studies.

A supercritical flow in a long horizontal channel (or having a small slope) has an inexpressive component of the weight as impulsive force. As a result, the power dissipation induces the depth of the liquid to increase along the flow, tending to its critical value. But a sudden change, a hydraulic jump, forms before the flow reaches the critical condition (which corresponds to the condition of minimum specific energy). In other words, transitions to the subcritical condition do not occur gradually in such channels. Hydraulic jumps have been studied over the years, providing a good theoretical understanding of the phenomenon, as well as experi‐

The characteristics of hydraulic jumps in free flows are of interest for different applications. For example, they are used in structures designed for flow rate measurements, or for energy dissipation, like the stilling basins downstream from spillways. Although simple to produce and to control in the engineering praxis, hydraulic jumps still present aspects which are not completely understood. Interestingly, the geometrical characteristics of the jumps are among the features whose quantification still not have consensus among researchers and professio‐ nals. Perhaps the length of the jump (or the roller) is the dimension that presents the highest level of "imprecision" when quantifying it. Such difficulties may be a consequence of the simplifying hypotheses made to obtain the usual relationships. In the most known equation for the sequent depths, the shear force is neglected, and a very simple functional dependence is obtained between the depths and the Froude number (*Fr*). However, by neglecting the shear force, difficulties arise to calculate the length of the jump, because it does not appear in the

Many authors observed this difficulty since the beginning of the studies on hydraulic jumps. As a consequence, also a large number of solutions were presented for the sequent depths and for the length of the jump (or roller). Good reviews were presented, for example, by [1] and [2].

The hydraulic jump may be viewed as a "shock" between the supercritical and the subcritical flows. As result of this "shock", the water of the higher depth "falls down" on the water of the lower depth. Because of the main movement of the flow, and the resulting shear forces, a roller is formed in this "falling region", characterized as a two phase turbulent zone with rotating motion. This description is sketched approximately in Figure 1a. Figure 1b shows an example of this highly turbulent region, responsible for most of the energy dissipation in a hydraulic jump. Despite the fact of being turbulent (thus oscillating in time) and 3D, the hydraulic jump

Table 1 presents some studies related to the theme around the world, while Table 2 lists some academic studies in Portuguese language, developed in universities of Brazil, as a consequence of the increasing interest on hydroelectric energy in the country, and the corresponding projects of spillways with dissipation basins. The present chapter uses the background

may be quantified as a permanent 1D flow when considering mean quantities.

mental data that can be used to check new conceptual proposals.

**Figure 1.** a) The "shock" between supercritical and subcritical flows, and the "falling region" forming the roller; b) An example of the "shock" sketched in Figure 1a.



**Table 1.** Studies related to hydraulic jumps. Table considering information contained in the studies of [1] and [2].

Having so many proposals, it may be asked why is there no consensus around one of the possible ways to calculate the geometrical characteristics. A probable answer could be that different solutions were obtained by detailing differently some of the aspects of the problem. But perhaps the most convincing answer is that the presented mechanistic schemes are still not fully convincing. So, it does not seem to be a question about the correctness of the used concepts, but on how they are used to obtain solutions. In other words, it seems that there are still no definitive criteria, generally accepted.

This chapter shows firstly the traditional quantification of the sequent depths of hydraulic jumps, which does not consider the bed shear force. In the sequence, using two adequate control volumes and the principles of conservation of mass, momentum and energy, an equation for the length of the roller is furnished. Further, the sequent depths are calculated considering the presence and the absence of bed shear forces, showing that both situations lead to different equations. Still further, the surface profiles and the height attained by the surface fluctuations (drops and waves) are discussed, and equations are furnished using i) a "depth deficit" criterion, ii) the mass conservation principle using an "air capture" formula‐


tion, and iii) an approximation that uses results of the Random Square Waves method (RSW). Literature data are used to quantify coefficients of the equations and to compare measured and calculated results.

**Table 2.** Academic studies conducted in Brazil about hydraulic jumps. \*USP: University of São Paulo; \*\*UFRGS: Federal University of Rio Grande do Sul.

#### **1.2. Traditional quantification of sequent depths**

**Author (quoted by [1] and [2]) Contribution** Einwachter (1933) Length of the jump. Ludin & Barnes (1934) Length of the jump. Woycieki (1934) Length of the jump. Smetana (1934) Length of the jump. Douma (1934) Length of the jump. Aravin (1935) Length of the jump.

76 Hydrodynamics - Concepts and Experiments

Kinney (1935) Experiments and Length of the jump.

Willes (1937) Length of the jump and air entrainment. Goodrum & Dubrow (1941) Influence of surface tension and viscosity.

Bakhmeteff & Matzke (1936) Extensive project for the generalization of equations.

**Table 1.** Studies related to hydraulic jumps. Table considering information contained in the studies of [1] and [2].

Having so many proposals, it may be asked why is there no consensus around one of the possible ways to calculate the geometrical characteristics. A probable answer could be that different solutions were obtained by detailing differently some of the aspects of the problem. But perhaps the most convincing answer is that the presented mechanistic schemes are still not fully convincing. So, it does not seem to be a question about the correctness of the used concepts, but on how they are used to obtain solutions. In other words, it seems that there are

This chapter shows firstly the traditional quantification of the sequent depths of hydraulic jumps, which does not consider the bed shear force. In the sequence, using two adequate control volumes and the principles of conservation of mass, momentum and energy, an equation for the length of the roller is furnished. Further, the sequent depths are calculated considering the presence and the absence of bed shear forces, showing that both situations lead to different equations. Still further, the surface profiles and the height attained by the surface fluctuations (drops and waves) are discussed, and equations are furnished using i) a "depth deficit" criterion, ii) the mass conservation principle using an "air capture" formula‐

Page (1935) Length of the jump. Chertoussov (1935) Length of the jump.

Ivanchenko (1936) Length of the jump.

Posey (1941) Length of the jump. Wu (1949) Length of the jump. Bureau of Reclamation (1954) Experimental study. Schröder (1963) Length of the jump. Rajaratnam (1965) Length of the jump. Malik (1972) Length of the jump. Sarma & Newnham (1973) Length of the jump. Hager *et al*. (1990) [2] Length of the roller.

still no definitive criteria, generally accepted.

The Newton's second law for a closed system establishes that:

$$\sum \vec{F} = d\left(m\vec{V}\right) / \left.dt\right|\_{S} \tag{1}$$

where *F* <sup>→</sup> is the force, *m* is the mass of the system, *<sup>V</sup>* <sup>→</sup> is the speed, and *<sup>t</sup>* is the time. The system is indicated by the index *S*. Considering the Euler formulation, for open systems or control volumes, equation (1) is rewritten using the Reynolds transport theorem, furnishing:

$$
\sum \vec{F} = \frac{d}{dt} \iiint\_{\mathcal{V}} \vec{V} \, \rho dVol + \iiint\_{\mathcal{S}} \vec{V} \left( \rho \vec{V} \cdot \vec{n} dA \right) \tag{2}
$$

where *ρ* is the density, *n* <sup>→</sup> is the unit vector normal to the control surface and pointing outside of the control volume, and *A* is the area of the control surface.

Hydraulic jumps in horizontal or nearly horizontal channels are usually quantified using a control volume as shown in Figure 2, with inlet section 1 and outlet section 2, no relevant shear forces at the bottom and the upper surface (*F*shear), no effects of the weight (*W*x), and hydrostatic pressure distributions at sections 1 and 2.

**Figure 2.** Hydraulic jump and adopted control volume. The photograph was taken to show the air bubbles ascending, and the outlet section located where no air bubbles are present.

For steady state conditions, equation (2) is written as:

$$\mathbb{P}\left\{F\_1 - F\_2 + W\_x - F\_{\text{shear}} \equiv F\_1 - F\_2 = \right\} \quad \sum \vec{F} = \iiint \vec{V} \left(\rho \vec{V} \cdot \vec{n} dA\right) \quad \left\{= \rho \left(V\_2^2 A\_2 - V\_1^2 A\_1\right)\right\} \tag{3}$$

In the sequence, from *Q*=*V*1*A*1=*V*2*A*2:

$$F\_1 - F\_2 = \rho Q^2 \left(\frac{1}{A\_2} - \frac{1}{A\_1}\right) \tag{4}$$

*Q* is the water flow rate, *ρ* is the water density, *F*1 and *F*<sup>2</sup> are the forces due to the pressure at the inlet and outlet sections, respectively. The hydrostatic distribution of pressure leads to:

$$F\_1 = \rho \text{g} \mathbf{g} \mathbf{h}\_1 \mathbf{A}\_1 / \, \mathbf{2} \qquad \text{and} \qquad F\_2 = \rho \mathbf{g} \mathbf{g} \mathbf{h}\_2 \mathbf{A}\_2 / \, \mathbf{2} \tag{5}$$

*g* is the acceleration of the gravity, *h*1 and *h*2 are the water depths, and *A*1 and *A*2 are the areas of the cross sections 1 and 2, respectively. Substituting equations (5) into equation (4) produces:

$$\mathbf{Q}^2 \mid \mathbf{g}A\_1 + A\_1\mathbf{h}\_1 / \mathbf{\hat{z}} = \mathbf{Q}^2 \mid \mathbf{g}A\_2 + A\_2\mathbf{h}\_2 / \mathbf{\hat{z}} \tag{6}$$

For a rectangular channel *A*1=*Bh*1, *A*2=*Bh*2, where *B* is the width of the channel. Further, the definition *q*=*Q*/*B* is of current use. Algebraic steps applied to equation (6) lead to:

*h*1 <sup>2</sup> <sup>−</sup>*h*<sup>2</sup> <sup>2</sup> <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *<sup>g</sup>* ( <sup>1</sup> *h*2 − 1 *h*1 ) <sup>⇒</sup>(*h*<sup>2</sup> <sup>−</sup>*h*1)(*h*<sup>2</sup> <sup>+</sup> *<sup>h</sup>*1) <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *<sup>g</sup>* ( <sup>1</sup> *h*2 − 1 *h*1 ) <sup>⇒</sup>(*h*<sup>2</sup> <sup>+</sup> *<sup>h</sup>*1) <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *gh*1*h*<sup>2</sup> ⇒ ⇒*h* <sup>1</sup> 2 *h* <sup>2</sup> + *h* <sup>2</sup> 2 *<sup>h</sup>* <sup>1</sup> <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *g* ⇒ *h* 2 2 *h* 1 <sup>2</sup> + *h* 2 *h* 1 <sup>−</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *gh* <sup>1</sup> <sup>3</sup> =0 or ( *<sup>h</sup>* <sup>2</sup> *h* 1 ) 2 + *h* 2 *h* 1 −2*F r* <sup>1</sup> <sup>2</sup> =0

Solving the final obtained equation for *h*2/*h*1, the result is:

forces at the bottom and the upper surface (*F*shear), no effects of the weight (*W*x), and hydrostatic

**Figure 2.** Hydraulic jump and adopted control volume. The photograph was taken to show the air bubbles ascending,

r

2 1 1 1

> r

*<sup>g</sup>* ( <sup>1</sup> *h*2 − 1 *h*1

<sup>3</sup> =0 or ( *<sup>h</sup>* <sup>2</sup>

*h* 1 ) 2 + *h* 2 *h* 1

1 11 2 22 *Q gA A h Q gA A h* / /2 / /2 += + (6)

è ø

r rr r

*SC F F W F F F F V V ndA V A V A* (3)

 r

*FF Q A A* (4)

/ 2 and *F gh A* = / 2 (5)

22 11

) <sup>⇒</sup>(*h*<sup>2</sup> <sup>+</sup> *<sup>h</sup>*1) <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup>

−2*F r* <sup>1</sup> <sup>2</sup> =0

*gh*1*h*<sup>2</sup>

⇒

{ } ( ) { ( )} 2 2 1 2 -+ - @-= = × = - 1 2 å òò

2

æ ö -= - ç ÷

*Q* is the water flow rate, *ρ* is the water density, *F*1 and *F*<sup>2</sup> are the forces due to the pressure at the inlet and outlet sections, respectively. The hydrostatic distribution of pressure leads to:

*g* is the acceleration of the gravity, *h*1 and *h*2 are the water depths, and *A*1 and *A*2 are the areas of the cross sections 1 and 2, respectively. Substituting equations (5) into equation (4) produces:

For a rectangular channel *A*1=*Bh*1, *A*2=*Bh*2, where *B* is the width of the channel. Further, the

r

1 11 2 22 *F gh A* =

2 2

*h* 1 <sup>2</sup> + *h* 2 *h* 1

definition *q*=*Q*/*B* is of current use. Algebraic steps applied to equation (6) lead to:

) <sup>⇒</sup>(*h*<sup>2</sup> <sup>−</sup>*h*1)(*h*<sup>2</sup> <sup>+</sup> *<sup>h</sup>*1) <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup>

<sup>−</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *gh* <sup>1</sup>

pressure distributions at sections 1 and 2.

78 Hydrodynamics - Concepts and Experiments

and the outlet section located where no air bubbles are present.

In the sequence, from *Q*=*V*1*A*1=*V*2*A*2:

*h*1 <sup>2</sup> <sup>−</sup>*h*<sup>2</sup>

⇒*h* <sup>1</sup> 2 *h* <sup>2</sup> + *h* <sup>2</sup> 2 *<sup>h</sup>* <sup>1</sup> <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *g* ⇒ *h* 2 2

<sup>2</sup> <sup>=</sup> <sup>2</sup>*<sup>q</sup>* <sup>2</sup> *<sup>g</sup>* ( <sup>1</sup> *h*2 − 1 *h*1

For steady state conditions, equation (2) is written as:

r

*x shear* Ò

1 2

$$h^\* = h\_2 \mid h\_1 = \left(\sqrt{8Fr\_1^2 + 1} - 1\right) / 2 \qquad \text{where} \quad Fr\_1 = V\_1 \mid \sqrt{\text{g}h\_1} \tag{7}$$

Figure 3 shows that equation (7) compares well with experimental results.

**Figure 3.** Sequent depths ratio as a function of the Froude number. Data of [2, 18, 19].

The experimental data of [2] and [19] show that predictions of equation (7) may generate relative errors in the range of 0.10% to 12.2%. Considering simpler equations, different proposals may be found in the literature. For example. for *Fr*1>2 [20] suggests the simplified form *h*# =√2*Fr*1-1/2, (where *h*# =*h*2/*h*1). But other linear equations are also suggested. Equation (8), for example, is valid for 2.26<*Fr*1<15.96, has a correlation coefficient of 0.99, maximum relative error of 5.4%, for the data of Figure 3, excluding [18].

$$h^\* = 1.29Fr\_1 - 0.116\tag{8}$$

Although alternative equations exist in the literature, equation (7) is one of the most known and accepted conclusions for hydraulic jumps. Interestingly, hydraulic jumps are used as dissipative singularities, but their most known design equation implies absence of dissipative shear forces. This contradiction is one of the reasons of the continuing discussion on the theme. In this chapter we present (see items 2.1.1, 2.1.2, 2.1.3) a way to conciliate the dissipative characteristic of the jump and the adequacy of the predictions given by equation (7).

#### **1.3. Equations of the literature for the lengths of the roller and the hydraulic jump**

The experimental determination of the length of the hydraulic jump is not a simple task. The intense turbulence and the occurrence of single-phase and two-phase flows adds difficulties to the measurement of flow depths, velocity fields, pressure distributions and the lengths of the roller and the hydraulic jump. A further difficulty is related to the definition of the end of the hydraulic jump. Accordingly to [1], the earliest formulation for the length of the hydraulic jump was proposed by, Riegel and Beebe, in 1917, while [2] suggest that the first systematic study of the length of the roller was made by Safranez, in 1927-1929. [21] defined the end of the hydraulic jump as the position where the free surface attains its maximum height, and the upper point of the expanding main flow (located between the roller and the bottom of the channel) coincides with the surface, beginning to decline towards the subsequent gradually varied flow. This definition led to lengths greater than the rollers.

[2] mention Schröder, who in 1963 used visualizations of the free surface to quantify jump lengths. However, such visual procedures depend on personal decisions about different aspects of the moving and undulating surface. [2] also cite Malik, who in 1972 employed a probe to measure forces and to locate the superficial region with zero mean force. The position of this region corresponds to the roller length, and the experimental error was about 8%.

The results of [18] are among the most used for calculating lengths of hydraulic jump stilling basins, downstream of spillways. The end of the jump was assumed as the position where the high velocity jet starts to peel off the bottom, or the section immediately downstream of the roller.

[19] measured pressure distributions along the bottom of a horizontal channel for hydraulic jumps with *Fr*<sup>1</sup> between 4.9 and 9.3. The pressure records were used to calculate coefficients of skewness and kurtosis of the measured data along the channel. The authors obtained values of coefficients of kurtosis around 3.0, which became practically constant for distances greater than *x*/(*h*2-*h*1)=8.5. The authors then defined the end of the hydraulic jump as *Lj* =8.5(*h*2-*h*1).

[22, 23] measured water depths along a hydraulic jump in a rectangular channel, for *Fr*1=3.0, using an ultrasonic sensor to locate the water surface. The sensor was moved along the longitudinal axis of the channel. Vertical turbulent intensities and related Strouhal numbers were calculated for each measurement position. The authors suggested to estimate the length of the jump using the final decay of the vertical turbulent intensity at the free surface, obtaining *L*j =9.5(*h*2-*h*1). Similarly, [24, 25] and [17] generated and analyzed data of ultrasonic sensors and high speed cameras to evaluate comparatively the results of the sensors and to better locate the surface.

Some equations for the length of the hydraulic jump and the roller are resumed in Table 3. Most of them can be written as *L*<sup>j</sup> /*h*2=f(*Fr*1) when using equation (7). A qualitative comparison between some of these equations is shown in Figure 4, assuming the interval 2≤*Fr*1≤20 as valid for all equations.


**1.3. Equations of the literature for the lengths of the roller and the hydraulic jump**

varied flow. This definition led to lengths greater than the rollers.

80 Hydrodynamics - Concepts and Experiments

roller.

*L*j

the surface.

for all equations.

Most of them can be written as *L*<sup>j</sup>

The experimental determination of the length of the hydraulic jump is not a simple task. The intense turbulence and the occurrence of single-phase and two-phase flows adds difficulties to the measurement of flow depths, velocity fields, pressure distributions and the lengths of the roller and the hydraulic jump. A further difficulty is related to the definition of the end of the hydraulic jump. Accordingly to [1], the earliest formulation for the length of the hydraulic jump was proposed by, Riegel and Beebe, in 1917, while [2] suggest that the first systematic study of the length of the roller was made by Safranez, in 1927-1929. [21] defined the end of the hydraulic jump as the position where the free surface attains its maximum height, and the upper point of the expanding main flow (located between the roller and the bottom of the channel) coincides with the surface, beginning to decline towards the subsequent gradually

[2] mention Schröder, who in 1963 used visualizations of the free surface to quantify jump lengths. However, such visual procedures depend on personal decisions about different aspects of the moving and undulating surface. [2] also cite Malik, who in 1972 employed a probe to measure forces and to locate the superficial region with zero mean force. The position of this region corresponds to the roller length, and the experimental error was about 8%.

The results of [18] are among the most used for calculating lengths of hydraulic jump stilling basins, downstream of spillways. The end of the jump was assumed as the position where the high velocity jet starts to peel off the bottom, or the section immediately downstream of the

[19] measured pressure distributions along the bottom of a horizontal channel for hydraulic jumps with *Fr*<sup>1</sup> between 4.9 and 9.3. The pressure records were used to calculate coefficients of skewness and kurtosis of the measured data along the channel. The authors obtained values of coefficients of kurtosis around 3.0, which became practically constant for distances greater

[22, 23] measured water depths along a hydraulic jump in a rectangular channel, for *Fr*1=3.0, using an ultrasonic sensor to locate the water surface. The sensor was moved along the longitudinal axis of the channel. Vertical turbulent intensities and related Strouhal numbers were calculated for each measurement position. The authors suggested to estimate the length of the jump using the final decay of the vertical turbulent intensity at the free surface, obtaining

=9.5(*h*2-*h*1). Similarly, [24, 25] and [17] generated and analyzed data of ultrasonic sensors and high speed cameras to evaluate comparatively the results of the sensors and to better locate

Some equations for the length of the hydraulic jump and the roller are resumed in Table 3.

between some of these equations is shown in Figure 4, assuming the interval 2≤*Fr*1≤20 as valid

/*h*2=f(*Fr*1) when using equation (7). A qualitative comparison

=8.5(*h*2-*h*1).

than *x*/(*h*2-*h*1)=8.5. The authors then defined the end of the hydraulic jump as *Lj*

**Table 3.** Equations for the length of the roller and the hydraulic jump. The table uses citations of [1, 2, 18]. *Vc*=critical velocity; *E*=specific energy. \$ *α*r=20 if *h*1/*B*<0.1; *α*r=12,5 if 0.1≤*h*1/*B*≤0.7. *B*=channel width. If *Fr*1<6, following approximation may eventually be used: *L*r/*h*1=8*Fr*1-12.

**Figure 4.** a) Comparison of equations of nondimensional jump lengths proposed by different authors cited in Table 3, as a function of *Fr*1; b) The region of the graph covered by the equations.

Experimental data of [18] for *L*<sup>j</sup> /*h*2, conducted for *Fr*1 between about 2 and 20, are shown in Figure 5. The middle curve corresponds to the adjusted equation (29) proposed by [26].

**Figure 5.** Data of *Lj* /*h*<sup>2</sup> obtained by [18], and equation (29) presented by[26] (see also [27]). The upper and lower curves are shown as envelopes.

#### **2. Proposed formulation**

#### **2.1. The criterion of two control volumes**

The focus of this item is to present the equations obtained for the length of the roller and for the sequent depths using only one deduction schema. A detailed explanation of the forces that maintain the equilibrium of the expansion is presented firstly, followed by their use to quantify the power dissipation. The integral analysis is used for the two control volumes shown in Figure 6.

**Figure 6.** The two control volumes used in the present formulation, for the quantification of the length of the roller and the sequent depths of the hydraulic jump.

The two control volumes allow considering details, such as the vortex movement of the roller and the recirculating flow rate. The regime of the flow is permanent (stationary conditions), and both control volumes are at rest. In order to simplify the calculations, control volume 1 (CV1) contains the roller of the jump and ideally does not exchange mass with control volume 2 (CV2). But, as a result of the shear forces at the interface between both volumes, they exchange momentum and energy. The analyses of the flows in both volumes are made here separately.

*2.1.1. Length of the roller using control volume 1 (CV1)*

#### **i. Global Analysis of CV1**

**Figure 4.** a) Comparison of equations of nondimensional jump lengths proposed by different authors cited in Table 3,

Figure 5. The middle curve corresponds to the adjusted equation (29) proposed by [26].

/*h*2, conducted for *Fr*1 between about 2 and 20, are shown in

/*h*<sup>2</sup> obtained by [18], and equation (29) presented by[26] (see also [27]). The upper and lower curves

The focus of this item is to present the equations obtained for the length of the roller and for the sequent depths using only one deduction schema. A detailed explanation of the forces that maintain the equilibrium of the expansion is presented firstly, followed by their use to quantify

as a function of *Fr*1; b) The region of the graph covered by the equations.

Experimental data of [18] for *L*<sup>j</sup>

82 Hydrodynamics - Concepts and Experiments

**Figure 5.** Data of *Lj*

are shown as envelopes.

**2. Proposed formulation**

**2.1. The criterion of two control volumes**

**Figure 7.** a) CV1 and the forces acting on it. *Fxτ* and *Fyτ* are shear forces acting in the *x* and *y* directions, respectively; b) The three regions of influence of the characteristic velocities (*V*1, *Vup*, and *Vdown*) defined for CV1.

Figure 7a shows CV1 already isolated from CV2. Because there is no mass exchange between CV1 and the main flow (CV2), the mass conservation equation (30) reduces to the elementary form (31):

$$\frac{d}{dt} \iiint\limits\_{CV1} \rho dVol + \iint\_{S} \rho \vec{V} \, d\vec{A} = 0 \tag{30}$$

$$\frac{dM}{dt} = 0\tag{31}$$

*CS* is the control surface and *M* is the mass in CV1. For the momentum conservation, the general integral equation also reduces to the simplest form:

$$\vec{F} = \frac{d}{dt} \iiint\_{CV1} V \,\rho dVol + \iiint\_{CS} \vec{V} \,\rho \vec{V} \,d\vec{A} = 0 \tag{32}$$

Because there are no mass fluxes across the surfaces, and the flow is stationary, the resultant of the forces must vanish. It implies that, for the coordinate directions *x* and *y*:

$$F\_x = 0 \qquad\qquad\text{and}\qquad\qquad F\_y = 0 \tag{33}$$

The mass in CV1 is constant and the volume does not change its form (that is, it will not slump and flow away). It implies that the shear forces of the flow must equate the pressure forces in the *x* direction, and the weight in the *y* direction. So, from Figure 7a, and equations (33) it follows that:

$$F\_{\rm rr} = \frac{\rho \text{ g } H^2 B}{2} \tag{34}$$

$$F\_{\rm yr} = \frac{\rho \text{ g } L\_r B \, H}{2} \theta\_{\rm l} \tag{35}$$

*θ*1 is a constant that corrects the effects of using the inclined straight line instead of the real form of the water surface (understood as the water equivalent depth). *H*=*h*2-*h*1 and *Lr* is the length of the roller.

The shear forces presented in Figure 7a induce movement to the water in CV1, so that, in a first step, power is inserted into CV1. No mass exchanges occur across the control surface, and this energy is converted into thermal energy in CV1. But, further, this thermal energy is lost to the environment (main flow and atmosphere), so that the equilibrium or the stationary situation is maintained, and the energy inserted into the volume is released as heat to the environment.

Figure 7a shows CV1 already isolated from CV2. Because there is no mass exchange between CV1 and the main flow (CV2), the mass conservation equation (30) reduces to the elementary

> r+ = . 0 òòò òò <sup>r</sup> <sup>r</sup>

*CS* is the control surface and *M* is the mass in CV1. For the momentum conservation, the general

Because there are no mass fluxes across the surfaces, and the flow is stationary, the resultant

The mass in CV1 is constant and the volume does not change its form (that is, it will not slump and flow away). It implies that the shear forces of the flow must equate the pressure forces in the *x* direction, and the weight in the *y* direction. So, from Figure 7a, and equations (33) it

2

<sup>1</sup> 2

*θ*1 is a constant that corrects the effects of using the inclined straight line instead of the real form of the water surface (understood as the water equivalent depth). *H*=*h*2-*h*1 and *Lr* is the

The shear forces presented in Figure 7a induce movement to the water in CV1, so that, in a first step, power is inserted into CV1. No mass exchanges occur across the control surface, and this energy is converted into thermal energy in CV1. But, further, this thermal energy is lost

q*<sup>r</sup>*

2

r

r=

t

t

*y*

 r

*dt* (30)

*dt* (32)

= = 0 and 0 *F F x y* (33)

*<sup>x</sup>* <sup>=</sup> *gH B <sup>F</sup>* (34)

*gLBH <sup>F</sup>* (35)

*dt* (31)

1 r

1

of the forces must vanish. It implies that, for the coordinate directions *x* and *y*:

integral equation also reduces to the simplest form:

Ò *CV CS <sup>d</sup> dVol V dA*

<sup>=</sup> <sup>0</sup> *dM*

= += r

 . 0 òòò òò <sup>r</sup> r r <sup>r</sup> Ò *CV CS <sup>d</sup> F V dVol V V dA*

form (31):

84 Hydrodynamics - Concepts and Experiments

follows that:

length of the roller.

Considering the energy equation, for stationary conditions and no mass exchange between the control volumes, it follows that:

$$
\vec{Q} - \vec{W} = \frac{d}{dt} \iiint\_{CV} e\rho dVol + \oint\_{CS} (e + \frac{p}{\rho}) \rho \vec{V} \, d\vec{A} = 0 \qquad \text{where} \qquad e = \frac{V^2}{2} + \text{gy} + u \tag{36}
$$

*W*˙ and *Q*˙ are the work and the thermal energy exchanged with the surroundings per unit time, *u* is the specific internal energy and *y* is shown in Figure 7b. The surface integral vanishes because there is no mass flow across the control surface. From the first member of equation (36) it follows that:

$$
\dot{W} = \dot{\underline{Q}} = Power\ loss\tag{37}
$$

Equation (37) means that, in this stationary case, all power furnished to CV1 is lost as heat. Equations (34, 35) and (37) allow writing:

$$\frac{\rho \text{ g } H^2 B V\_1}{2} + \frac{\rho \text{ g } L B \, H}{2} \left(\frac{H}{\Delta t\_y}\right)\_{up} \theta\_1 = Power \text{ loss} \tag{38}$$

(*H*/∆*ty*) is the mean vertical velocity (*Vup* in Figure 7b) that allows calculating the power furnished by the upwards force. ∆*ty* is the mean travel time of the water in the vertical direction (positive *y* axis, see Figure 7b). The mean upwards flow rate equals the mean downwards flow rate (because no mass is lost in CV1), so that, considering mean quantities for the involved areas, it follows that:

$$Area\_{up} \left( \frac{H}{\Delta t\_{\circ}} \right)\_{up} = Area\_{down} \left( \frac{H}{\Delta t\_{\circ}} \right)\_{down} \tag{39}$$

Or, because *H* is the same for both directions (up and down):

$$
\Delta t\_{\text{yup}} = \left(\frac{Area\_{\text{up}}}{Area\_{\text{down}}}\right) \Delta t\_{\text{ydown}} \tag{40}
$$

Equations (31) through (40) were obtained from integral conservation principles. In order to quantify the geometrical characteristics of the hydraulic jump, the variables *Power loss* and ∆ *tyup* still need to be expressed as functions of the basic flow parameters. It is necessary, now, to consider the movement inside of the CV1, that is, to perform an intrinsic analy‐ sis in addition to the global analysis.

#### **ii. Intrinsic Analysis of CV1**

The total power loss in CV1 is the result of losses occurring in regions subjected to different characteristic velocities.Easily recognizedare the velocity*V*<sup>1</sup> andthe upwards velocity*H*/∆*tyup* . Furthermore, from equation (40) it follows that ∆ *tyup* can be obtained from the downwards movement. From these considerations, three regions influenced by different "characteristic velocities" may be defined in CV1, as shown in Figure 7b. The *Powerloss* of equation (38) is thus calculated as a sum of the losses in each region of Figure 7b, that is:

$$Power\,\text{loss} = \dot{W}\left(V\_1\right) + \dot{W}\left(V\_{up}\right) + \dot{W}\left(V\_{down}\right) \tag{41}$$

The downwards region is considered firstly, following a particle of fluid moving at the "falling" boundary of the control surface, as shown in Figure 8a. (See also Figures 1a and 1b).

**Figure 8.** a) The "falling" of a particle in the "downwards region", between points A and B; b) Transversal sections S1, Sup, and Sdown in the roller.

Taking the movement along *A*¯ *B* in Figure 8a, equation (42) applies:

$$\mathbf{y}\_A + \frac{\mathbf{p}\_A}{\rho \, \mathbf{g}} + \frac{V\_A^2}{2\mathbf{g}} = \mathbf{y}\_B + \frac{\mathbf{p}\_B}{\rho \, \mathbf{g}} + \frac{V\_B^2}{2\mathbf{g}} + h\_{\text{loss}}\tag{42}$$

*p* is the pressure, which does not vary at the surface. *VA* equals zero, so that the characteristic velocity is a function of *VB*. Considering *VB*=*Vdown* and *yB*=0, it follows:

Details of Hydraulic Jumps for Design Criteria of Hydraulic Structures http://dx.doi.org/10.5772/58963 87

$$H = \frac{V\_{don}^2}{2\text{g}} + h\_{loss} \tag{43}$$

For local dissipations *hloss* is represented as proportional to the kinetic energy, considering a representative velocity. For the region of *Vdown* (see Figures 7b and 8) it follows that:

$$h\_{\rm loss} = K \frac{V\_{\rm down}^2}{2 \text{g}}, \qquad \qquad V\_{\rm down} = \sqrt{\frac{2 \text{g } H}{1 + K}} = \sqrt{2 \text{g } ^\circ H}, \qquad \qquad \text{g } \text{'} = \frac{\text{g}}{1 + K} \tag{44}$$

The "falling time" along *A*¯ *B* is thus given by:

$$
\Delta t\_{y\_{dON}} = \sqrt{\frac{2H}{\mathbf{g}}} \tag{45}
$$

Equations (40) and (45) lead to:

and ∆ *tyup*

Sup, and Sdown in the roller.

Taking the movement along *A*¯

sis in addition to the global analysis.

Furthermore, from equation (40) it follows that ∆ *tyup*

calculated as a sum of the losses in each region of Figure 7b, that is:

**ii. Intrinsic Analysis of CV1**

86 Hydrodynamics - Concepts and Experiments

still need to be expressed as functions of the basic flow parameters. It is necessary,

.

(42)

can be obtained from the downwards

now, to consider the movement inside of the CV1, that is, to perform an intrinsic analy‐

The total power loss in CV1 is the result of losses occurring in regions subjected to different characteristic velocities.Easily recognizedare the velocity*V*<sup>1</sup> andthe upwards velocity*H*/∆*tyup*

movement. From these considerations, three regions influenced by different "characteristic velocities" may be defined in CV1, as shown in Figure 7b. The *Powerloss* of equation (38) is thus

The downwards region is considered firstly, following a particle of fluid moving at the "falling" boundary of the control surface, as shown in Figure 8a. (See also Figures 1a and 1b).

**Figure 8.** a) The "falling" of a particle in the "downwards region", between points A and B; b) Transversal sections S1,

2 2

 r 2 2 + + =+ + + *AA BB A B loss pV pV y yh gg gg*

*p* is the pressure, which does not vary at the surface. *VA* equals zero, so that the characteristic

r

velocity is a function of *VB*. Considering *VB*=*Vdown* and *yB*=0, it follows:

*B* in Figure 8a, equation (42) applies:

=+ + ( <sup>1</sup> ) ( ) ( ) && & *Power loss W V W V W V up down* (41)

$$
\Delta t\_{\text{up}} = \left(\frac{Area\_{\text{up}}}{Area\_{\text{down}}}\right) \sqrt{\frac{2\,\text{H}}{\text{g}\,\text{"}}} = \theta\_2 \sqrt{\frac{2\,\text{H}}{\text{g}}}\tag{46}
$$

The coefficient *θ* 2 accounts for the proportionality constant *K* of equation (44), the ratio between the mean areas of equation (40), and the use of the characteristic velocity. The power loss of the mean descending flow in the region of *Vdown* is given by:

$$\dot{W}\left(V\_{\rm down}\right) = \rho \text{g } \underline{Q}\_{R} h\_{\rm loss} = K \,\rho \,\underline{Q}\_{R} \frac{V\_{\rm down}^2}{2} = \frac{K}{1+K} \,\rho \,\underline{\text{g}} \,\underline{Q}\_{R} H \tag{47}$$

The rotating flow rate *QR* in the roller is the remaining parameter to be known. Because no variation of mass exists, it may be quantified at any position (or transversal section "S") **of the roller**(S1, Sup, and Sdown) as shown in Figure 8b. That is, *QR*=*Q*1=*Qup*=*Qdown*. Section S1 is used here to quantify *QR*. A general equation for the velocity profile in section S1 is used, following a power series of *y*, as:

$$\frac{V}{V\_1} = \sum\_{i=1}^{\circ} \mathcal{J}\_i \left(\frac{y}{\alpha \, H}\right)' \tag{48}$$

*αΗ* expresses the maximum value of *y* in S1, as shown in Figure 8b. *β<sup>i</sup>* are coefficients of the power series, in which *β*0=1. The mean velocity in section S1 is obtained through integration, and *QR* is given by multiplying the mean velocity by the area *αBH*, furnishing, respectively:

$$
\overline{V} = V\_1 \sum\_{l=1}^{n} \frac{\beta\_l}{l+1}, \quad \overline{Q}\_{\overline{\alpha}} = \underline{Q}\_l = V\_l B H \sum\_{l=1}^{n} \frac{\alpha \beta\_l}{l+1} \tag{49}
$$

Combining equations (47) and (49) results in:

$$
\dot{W} \left( V\_{down} \right) = \rho \mathbf{g} V\_1 B H^2 \,\theta\_3,\tag{5}
$$

$$
\theta\_3 = \frac{K}{1+K} \sum\_{l=1}^n \frac{a \beta\_l}{l+1} \tag{50}
$$

Following similar steps for the flow in the region of *V*1, the power consumption is given by:

$$h\_{\rm loss} = K \frac{V\_1^2}{2g}, \quad \qquad \qquad \dot{W} \left( V\_1 \right) = \rho \frac{V\_1^3}{2} BH \theta\_4, \quad \qquad \qquad \theta\_4 = K \sum\_{l=1}^{\frac{n}{2}} \frac{a \beta\_l}{l+1} \tag{51}$$

Finally, repeating the procedures for the flow in the region of *Vup*, the power consumption is given by:

$$h\_{\rm loss} = K \frac{1}{2g} \left(\frac{H}{\Delta t\_{\rm y}}\right)\_{\rm up}^2 = \frac{K}{4\theta\_2^2} H, \qquad \qquad \dot{W} \left(V\_{\rm up}\right) = \rho g V\_1 B H^2 \theta\_5, \qquad \qquad \theta\_5 = \frac{K}{4\theta\_2^2} \sum\_{l=1}^{\rm v} \frac{a\rho \theta\_l}{l+1} \tag{52}$$
