**iii. Final equation for CV1 using the results of the global and intrinsic analyses**

From equations (38), (41), and (50 to 52), it follows that:

$$\lg L\_r \left(\frac{H}{\Delta t\_y}\right)\_{up} = \left(\frac{2\theta\_3 + 2\theta\_3 - 1}{\theta\_1}\right) \lg V\_1 H + \frac{\theta\_4}{\theta\_1} V\_1^3 \tag{53}$$

Equations (46) and (53) finally produce:

$$\mathbf{g}^{3/2} H^{1/2} L\_r = \theta\_\mathbf{e} \mathbf{g} V\_\mathbf{i} H + \theta\_\gamma V\_\mathbf{i}^3, \quad \text{with} \quad \theta\_\mathbf{e} = \sqrt{2} \,\theta\_\mathbf{e} \left(\frac{2\theta\_\mathbf{3} + 2\theta\_\mathbf{3} - 1}{\theta\_\mathbf{i}}\right) \text{ and } \quad \theta\_\gamma = \frac{\sqrt{2}}{\theta\_\mathbf{i}} \,\theta\_\mathbf{2} \theta\_\mathbf{i} \tag{54}$$

Nondimensional forms for the length *Lr* obtained from equations (54) are furnished below.

$$\frac{L\_r}{H} = \theta\_\delta \, Fr\_\mathrm{i} \sqrt{h^\star} + \theta\_\gamma \left(Fr\_\mathrm{i} \sqrt{h^\star}\right)^3 \tag{55}$$

Details of Hydraulic Jumps for Design Criteria of Hydraulic Structures http://dx.doi.org/10.5772/58963 89

$$\frac{L\_r}{H} = \Theta\_b \, Fr\_H + \Theta\_\gamma \, Fr\_H^3 \tag{56}$$

$$\frac{L\_r}{h\_1} = \theta\_6 \frac{Fr\_1}{\sqrt{h^\*}} + \theta\_7 \left(Fr\_1\right)^3 \sqrt{h^\*} \tag{57}$$

$$\frac{L\_r}{h\_2} = \Theta\_\delta \frac{Fr\_1}{h^\# \sqrt{h^\star}} + \Theta\_\gamma \frac{Fr\_1^\beta \sqrt{h^\star}}{h^\#} \tag{58}$$

In equations (55-58), *FrH*=*V*1/(*gH*) 1/2, and *h*\*=*h*1/*H*. As can be seen, the set of coefficients generated during the global and intrinsic analyses were reduced to only two, *θ*6 and *θ*7, which must be adjusted using experimental data. The normalizations (57) and (58) showed to be the most adequate for the experimental data analysed here.

#### *2.1.2. Equation for sequent depths and Froude number using control volume 2 (CV2)*

It remains to obtain an equation for the sequent depths. CV2 is now used in the stationary flow under study, as shown in Figure 9.

**Figure 9.** Control volume CV2 used to obtain an equation for the sequent depths.

The forces on CV2 in the contact surfaces between CV1 and CV2 vanish mutually (equilibrium of CV1), thus they are not indicated in Figure 9. In this case, two situations may be considered:


1 1 1 1 1

1 1

*R*

*V V Q Q V BH i i* (49)

, 1 1

14 4

*g i* (51)

& å *<sup>i</sup>*

2

& å *<sup>i</sup>*

2 15 5 2 2 2 1

*g t <sup>i</sup>* (52)

3 5 4 3

1 1

<sup>221</sup> <sup>2</sup> , with 2 and q q

( )

q

3

*<sup>H</sup>* (55)

æ ö + -

1 1

 q

q

*<sup>t</sup>* (53)

rq

*<sup>K</sup> W V gV BH K i* (50)

q

 ¥ ¥ = = <sup>=</sup> = = + + å å *i i*

*i i*

1 3 3

= = + + & å *<sup>i</sup>*

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

, , 22 1

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

<sup>1</sup> , , 2 4 4 1

ç ÷ D + è ø

*<sup>y</sup> <sup>i</sup> up*

= == <sup>+</sup>

( )

**iii. Final equation for CV1 using the results of the global and intrinsic analyses**

221

æ ö æ ö + - ç ÷ = + ç ÷ ç ÷

*<sup>H</sup> g L gV H V*

q

3/2 1/2 3 3 5 2 4 6 1 71 6 2 7

qq

=+ = ç ÷ = è ø *<sup>r</sup> g H L gV H V* (54)

Nondimensional forms for the length *Lr* obtained from equations (54) are furnished below.

 q\* \* *Lr Fr h Fr h*

61 7 1 = +

q

q q

è ø è ø *<sup>r</sup>*

æ ö == = = ç ÷

*H K <sup>K</sup> h K H W V gV BH*

*V V h K W V BH <sup>K</sup>*

rq

ab

1

=

*i*

 q

 q

1 1

q

 q

¥

ab

1

=

*i*

¥

ab

> ab

¥

=

q q

 q

,

 q

( ) 2 3 1 1

b

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

( ) <sup>2</sup>

*down*

88 Hydrodynamics - Concepts and Experiments

*loss*

given by:

r

2

Equations (46) and (53) finally produce:

 q

q

q

From equations (38), (41), and (50 to 52), it follows that:

Δ

*<sup>y</sup> up*

*loss up*

The solutions in the sequence consider both cases.

#### **i. The force at the bottom is neglected**

From the mass conservation equation it follows that:

$$V\_2 = V\_1 \nparallel \text{``}\tag{59}$$

From the momentum equation it follows that:

$$F\_{B\_r} = \rho B \left( h\_2 - h\_1 \right) [V\_1^2 / h^\# - \mathbf{g} \left( h\_1 + h\_2 \right) / \mathcal{Z} ] \tag{60}$$

Assuming *FBτ*=0, equation (7) is reproduced, that is:

$$h^{\#} = \left(\sqrt{1 + 8Fr\_{\parallel}^2} - 1\right) / \,\, 2\tag{61}$$

Using *h*1 as reference depth (for normalization procedures) equations (57) and (61) form the set of equations that relate the "sizes" *h*1, *h*2, and *Lr* in this study. Because *FBτ*=0, dissipation concentrates in the roller region (CV1). It is of course known that the forces at the interface between both volumes dissipate energy in CV2, but because the momentum equation already furnishes a prediction for *h*2/*h*1, it is not necessary to use the energy equation for this purpose. On the other hand, when considering the resistive force at the bottom, power losses must be considered in order to obtain predictions for *h#* .

#### **ii. The force at the bottom is relevant**

In this case, from the energy equation it follows that:

$$
\dot{\rho} - \dot{W} = \rho B V\_1 h\_1 \left( h\_2 - h\_1 \right) \left( \text{g} - \frac{V\_1^2}{h\_2^2} \frac{\left( h\_1 + h\_2 \right)}{2} \right) \tag{62}
$$

The left side of equation (62) must consider the power transferred to CV1, denoted by *W*˙ *VC*1, and the power lost due to the shear with bottom of the flow, denoted by *W*˙ *FBτ*. In mathematical form:

$$
\dot{W} = \dot{W}\_{V1} + \dot{W}\_{F8\tau} \tag{63}
$$

From equations (41), (50), (51) and (52) it follows that:

$$\dot{W}\_{V1} = \rho B V\_1 \left( h\_2 - h\_1 \right) \left[ \mathbf{g} \left( h\_2 - h\_1 \right) \left( \theta\_3 + \theta\_5 \right) + V\_1^2 \theta\_4 / \mathcal{D} \right] \tag{64}$$

The velocity close to the bottom is ideally *V*1. However, because of the expansion of the flow, it does not hold for the whole distance along the bottom. To better consider this variation, the power loss at the bottom was then quantified using a coefficient *θ*8 for *V*1, in the form:

$$\dot{W}\_{FBr} = \Theta\_8 V\_1 F\_{Br} = \Theta\_8 V\_1 \left( \rho B \left( h\_2 - h\_1 \right) [V\_1^2 \;/\; h^\# - \mathbf{g} \left( h\_1 + h\_2 \right) / \;/\; \mathbf{2} \right] \right) \tag{65}$$

Equations (61) to (65) produce the cubic equation:

#

( ) ( ) 2 # 2 11 1 2 [ / / 2]

Using *h*1 as reference depth (for normalization procedures) equations (57) and (61) form the set of equations that relate the "sizes" *h*1, *h*2, and *Lr* in this study. Because *FBτ*=0, dissipation concentrates in the roller region (CV1). It is of course known that the forces at the interface between both volumes dissipate energy in CV2, but because the momentum equation already furnishes a prediction for *h*2/*h*1, it is not necessary to use the energy equation for this purpose. On the other hand, when considering the resistive force at the bottom, power losses must be

.

( ) ( ) <sup>2</sup> 1 2 <sup>1</sup>

t

qq

/ 2] &*W BV h h g h h V VC* (64)

The left side of equation (62) must consider the power transferred to CV1, denoted by *W*˙ *VC*1, and the power lost due to the shear with bottom of the flow, denoted by *W*˙ *FBτ*. In mathematical

> ( ) ( )( ) <sup>2</sup> 1 1 2 1 2 1 3 5 14 = - - ++

The velocity close to the bottom is ideally *V*1. However, because of the expansion of the flow, it does not hold for the whole distance along the bottom. To better consider this variation, the

power loss at the bottom was then quantified using a coefficient *θ*8 for *V*1, in the form:

<sup>2</sup> 2

*W BV h h h g <sup>h</sup>* (62)

&& & *WW W VC FB* (63)

 q

è ø

11 2 1 2

= +<sup>1</sup>

æ ö <sup>+</sup> -= - - ç ÷

& *<sup>V</sup> h h*

*F Bh h V h gh h <sup>B</sup>* = - -+

( ) # 2

From the momentum equation it follows that:

90 Hydrodynamics - Concepts and Experiments

t

Assuming *FBτ*=0, equation (7) is reproduced, that is:

considered in order to obtain predictions for *h#*

In this case, from the energy equation it follows that:

From equations (41), (50), (51) and (52) it follows that:

r

[

r

**ii. The force at the bottom is relevant**

form:

r

2 1 *V Vh* = / (59)

(60)

<sup>1</sup> *h Fr* =+ - 1 8 1 /2 (61)

$$\left[\left[\mathcal{Z}\left(\theta\_{\mathbb{S}}+\theta\_{\mathbb{S}}\right)-\theta\_{\mathbb{S}}\right]h^{\mathbb{S}}\right] + \left[\mathcal{Z}-\theta\_{\mathbb{S}}-\mathcal{Z}\left(\theta\_{\mathbb{S}}+\theta\_{\mathbb{S}}\right)+\theta\_{4}Fr\_{\mathbb{I}}^{2}\right]h^{\mathbb{S}2} - Fr\_{\mathbb{I}}^{2}\left(1-2\,\theta\_{\mathbb{S}}\right)h^{\mathbb{A}} - Fr\_{\mathbb{I}}^{2} = 0\tag{66}$$

The solution of the cubic equation (66) furnishes *h#* as a function of *Fr*1. Equations (57) and (66) form now the set of equations that relate *h*1, *h*2, *Fr*1, and *Lr* for power dissipation occurring also in CV2. In section 3 of this chapter, equation (66) **is reduced to a second order equation**, based on the constancy of one of its solutions. Experimental results are then used to test the proposed equations.

The use of the power dissipation introduced coefficients in the obtained equations that affect all parcels having powers of *h*\* and *h*# . In order to adjust numerical values to *θ*4, *θ*8 and 2(*θ*3+*θ*5), the following form of equation (66) can be used in a multilinear analysis:

$$\left[Fr\_{\rm l}^{2}\left(1+h^{\boldsymbol{\pi}}\right)-2h^{\boldsymbol{\pi}2}=\theta\_{\rm s}\right]\left[2Fr\_{\rm l}^{2}h^{\boldsymbol{\pi}}-h^{\boldsymbol{\pi}2}\left(1+h^{\boldsymbol{\pi}}\right)\right]+\mathcal{2}\left(\theta\_{\rm s}+\theta\_{\rm s}\right)h^{\boldsymbol{\pi}2}\left(h^{\boldsymbol{\pi}}-1\right)+Fr\_{\rm l}^{2}h^{\boldsymbol{\pi}2}\theta\_{\rm 4}\tag{67}$$

Because multilinear analyses depend on the distribution of the measured values (deviations or errors), different arrangements of the multiplying factors (for example, equations (67) and (68)) may generate different numerical values for *θ*4, *θ*8 and 2(*θ*3+*θ*5).

$$\frac{\left(Fr\_{\text{l}}^{2}\left(1+h^{\sharp}\right)-2h^{\sharp 2}\right)}{\left(Fr\_{\text{l}}^{2}h^{\sharp 2}\right)}=\theta\_{8}\left[\frac{2Fr\_{\text{l}}^{2}h^{\sharp}-h^{\sharp 2}\left(1+h^{\sharp}\right)}{Fr\_{\text{l}}^{2}h^{\sharp 2}}\right]+2\left(\theta\_{3}+\theta\_{5}\right)\frac{h^{\sharp 2}\left(h^{\sharp}-1\right)}{Fr\_{\text{l}}^{2}h^{\sharp 2}}+\theta\_{4}\tag{68}$$

#### **2.2. Criteria for the function** *h***(***x***)**

#### *2.2.1. The "depth deficit" criterion*

The focus of the present section is the form of the surface (its profile). Some general charac‐ teristics of classical hydraulic jumps were considered, in order to develop a simple criterion, as follows:


Using *x* as the longitudinal direction pointing downstream and with origin at the toe of the jump, the second characteristic described above (ii) is expressed mathematically as:

$$\frac{d}{d\mathbf{x}}\frac{h}{\mathbf{x}}\alpha \left(h\_2 - h\right) \qquad\qquad\text{or}\qquad\qquad\qquad\frac{d}{d\mathbf{x}}\frac{h}{\mathbf{x}} = \theta\_{\parallel 1} \left(h\_2 - h\right) \tag{69}$$

*θ*11 is a proportionality coefficient with dimension of m-1. The integration of equation (69) using characteristics i and iii as boundary conditions produces:

$$\frac{h - h\_{\text{l}}}{h\_{2} - h\_{\text{l}}} = 1 - e^{-\theta\_{\text{l}}x} \qquad \quad \text{ (we define } \, \, \Pi = \frac{h - h\_{\text{l}}}{h\_{2} - h\_{\text{l}}} \text{ for further calculations)} \tag{70}$$

The surface tends asymptotically to *h*<sup>2</sup> following an exponential function. The proportionality coefficient must be obtained from experimental data, which is done in section 3. Equation (70) is adequate for surface evolutions without inflexion points.

#### *2.2.2. The "air inflow" criterion*

[28] and [29] studied the transition from "black water" to "white water" in spillways and proposed that, if the slope of the surface is produced by the air entrainment, the transfer of air to the water may be expressed in the form:

$$
\dot{\phi} = Kq \frac{d \, h}{d \, \text{x}} \tag{71}
$$

*ċ* [s-1] is the air transfer rate (void generation), *q* [m2 s-1] is the specific water flow rate, *h* [m] is the total depth of the air-water flow, *x* [m] is the longitudinal axis and *K* [m-2] is a proportion‐ ality factor. Equation (71) states that higher depth gradients for the same flow rate imply more air entraining the water, the same occurring for a constant depth gradient and higher flow rates. [30] showed that this equation allows obtaining proper forms of the air-water interfaces in surface aeration (stepped spillways).

Considering hydraulic jumps and a 1D formulation, an "ideal" two-step "expansion" of the flow is followed: 1) the flow, initially at velocity *V*<sup>1</sup> and depth *h*1, aerates accordingly equation (71), expanding to ~*h*<sup>2</sup> ("close" to *h*2); 2) the flow loses the absorbed air bubbles, decelerating to *V*2 and attaining *h*2. The geometrical situation considered in this first analysis is shown in Figure 10.

The evolution of the surface in the first step is obtained using the mass conservation equation:

$$\frac{\partial \mathcal{C}}{\partial t} + u \frac{\partial \mathcal{C}}{\partial x} + \nu \frac{\partial \mathcal{C}}{\partial y} + w \frac{\partial \mathcal{C}}{\partial z} = \left[ \frac{\partial}{\partial x} \left( D\_{\tau} \frac{\partial \mathcal{C}}{\partial x} \right) + \frac{\partial}{\partial y} \left( D\_{\tau} \frac{\partial \mathcal{C}}{\partial y} \right) + \frac{\partial}{\partial z} \left( D\_{\tau} \frac{\partial \mathcal{C}}{\partial z} \right) \right] + \dot{p} \tag{72}$$

**Figure 10.** Geometrical condition of this analysis. 1D formulation. Please see also Figure 2.

Using *x* as the longitudinal direction pointing downstream and with origin at the toe of the

*θ*11 is a proportionality coefficient with dimension of m-1. The integration of equation (69) using

The surface tends asymptotically to *h*<sup>2</sup> following an exponential function. The proportionality coefficient must be obtained from experimental data, which is done in section 3. Equation

[28] and [29] studied the transition from "black water" to "white water" in spillways and proposed that, if the slope of the surface is produced by the air entrainment, the transfer of air

the total depth of the air-water flow, *x* [m] is the longitudinal axis and *K* [m-2] is a proportion‐ ality factor. Equation (71) states that higher depth gradients for the same flow rate imply more air entraining the water, the same occurring for a constant depth gradient and higher flow rates. [30] showed that this equation allows obtaining proper forms of the air-water interfaces

Considering hydraulic jumps and a 1D formulation, an "ideal" two-step "expansion" of the flow is followed: 1) the flow, initially at velocity *V*<sup>1</sup> and depth *h*1, aerates accordingly equation (71), expanding to ~*h*<sup>2</sup> ("close" to *h*2); 2) the flow loses the absorbed air bubbles, decelerating to *V*2 and attaining *h*2. The geometrical situation considered in this first analysis is shown in

The evolution of the surface in the first step is obtained using the mass conservation equation:

*t x y z x xy yz z* (72)

¶¶¶ ¶ ¶¶ ¶¶ ¶¶ é ù æ öæ öæ ö +++ = + + + ê ú ç ÷ç ÷ç ÷ ¶ ¶ ¶ ¶¶ ¶¶ ¶¶ ¶ ë û è øè øè ø

*CCC C C C C uvw D D D p*

*h h h h* (70)

<sup>1</sup> (we define for further calculations) - - -

 q

& <sup>=</sup> *d h c Kq d x* (71)

& *T TT*

s-1] is the specific water flow rate, *h* [m] is

*d x d x* (69)

jump, the second characteristic described above (ii) is expressed mathematically as:

( ) <sup>2</sup> - = or 11 2 ( ) - *d h d h h h h h*

a

92 Hydrodynamics - Concepts and Experiments

q

to the water may be expressed in the form:

in surface aeration (stepped spillways).

Figure 10.

*ċ* [s-1] is the air transfer rate (void generation), *q* [m2

*e*

*2.2.2. The "air inflow" criterion*

characteristics i and iii as boundary conditions produces:

1 1 <sup>11</sup> 2 1 2 1

 = - P = - *h h <sup>x</sup> h h*

(70) is adequate for surface evolutions without inflexion points.

*C* is the air concentration in the mixture. *u*, *v*, and *w*, are the velocities along *x*, *y*, and *z*, respectively. *DT* is the turbulent diffusivity of the mixture (assumed constant), and *p*˙ is the source/sink parcel. The stationary 1D equation simplifies to:

$$
\mu \frac{\partial^{\gamma} C}{\partial \mathbf{x}} = D\_{\mathbf{r}} \frac{\partial^{2} C}{\partial \mathbf{x}^{2}} + \dot{\mathbf{p}} \tag{73}
$$

Equation (73) is normalized using the density of the air *ρair* (the void ratio *ϕ* is defined as *ϕ*=*C*/ *ρair*), and the length of the jump, *Lj* , or roller, *Lr*. Thus we use here simply *L* (defining *s=x/L*). The result is:

$$\frac{\partial \,\phi}{\partial s} = \frac{D\_T}{\mu L} \frac{\partial^2 C}{\partial s^2} + \dot{\phi} \tag{74}$$

By normalizing equation (71) to obtain *ϕ*˙ , the mass discharge of the liquid, *m*˙ , was used, so that:

$$
\phi = \frac{K \dot{m}}{L} \frac{d \ h}{ds} \tag{75}
$$

*K* was adjusted to *K*'. In the 1D formulation the mixture is homogeneous in the cross section. The mean water content in the mixture is expressed as *ϕwater*=*h1/h*. *h1* is the supercritical depth and *h*=*h1+Z*, being *h* the total depth and *Z* the length of the air column in the mixture. *ρair*<<*ρwater*, so that:

$$\rho\_1 \rho\_M = h\_M / h\_1 \,, \qquad \phi = Z / h, \qquad \qquad h = h\_1 / (1 - \phi) \tag{76}$$

The indexes 1 and *M* represent the two sections of the flow shown in Figure 10, and *ρ* is the water density. The integral mass conservation equation furnishes:

$$
\dot{m} = \rho\_l h\_l V\_1 = \rho\_M h\_M V\_M \qquad \qquad \text{or} \qquad \qquad V\_1 = V\_M = \mathbf{u} \tag{77}
$$

From equations (74) to (77) it follows:

$$\frac{\partial \,\phi}{\partial s} = \frac{D\_r}{V\_1} \frac{\partial^2 \phi}{\partial s^2} + \frac{K' \dot{m} h\_l}{L} \frac{\partial \left(1 - \phi\right)^{-1}}{\partial s} \tag{78}$$

*K´* is a factor that may vary along the flow, and indicates the "facility" of the air absorption by the water. Considering arguments about diffusion of turbulence from the bottom to the surface, [30] proposed *K´=K"h-2*. Equation (79) is then obtained:

$$\frac{\partial \phi}{\partial s} = \alpha\_1 \frac{\partial^2 \phi}{\partial s^2} + \alpha\_2 \frac{\partial \phi}{\partial s}, \qquad \text{where} \qquad \alpha\_1 = \frac{D\_T}{V\_1 L}, \qquad \alpha\_2 = \frac{K'' \dot{m} h\_1}{L} \tag{79}$$

The solution of equation (79), using boundary conditions 1) *s*=0, *ϕ* = 0, and 2) *s* →*∞*, *ϕ* →*ϕ∞*, and defining *IJ*=(*ω*2-1)/*ω*1, is given by:

$$
\phi = \phi\_{\alpha} \left( 1 - e^{-Lk} \right) \tag{80}
$$

Equations (76) and (80) produce:

$$h / h\_1 = 1 / \left[1 - \phi\_o \left(1 - e^{-Lk}\right)\right] \tag{81}$$

In order to obtain a nondimensional depth like equation (70), a boundary layer analogy was followed assuming that at *s*=1 (length of the roller) we have *h*/*h*1=*βh*# , where *β* is a constant "close" to the unity (as information, *β*=0.99 is the boundary layer value, but it does not imply its use here). For *s* → ∞ we have *h*/*h*1=*h*# . This implies that *φ*∞=(*h*# -1)/*h*# , so that the solution is given now by:

1 # # # 2 1 (1 ) <sup>1</sup> ln ( 1)(1 ) ( 1) b b - - - - - é ù P= = = - ê ú - --- ë û - *IJs IJs h h <sup>e</sup> IJ hh h h e <sup>h</sup>* (82)

Equation (82) is adequate for surface profiles with inflexion points.

#### **2.3. The Random Square Waves rms criterion**

1 1 1 / / , / , = / (1 )

The indexes 1 and *M* represent the two sections of the flow shown in Figure 10, and *ρ* is the

( ) <sup>1</sup> <sup>2</sup> 1

f-

1 " , where ,

*ss s V L <sup>L</sup>* (79)

' 1

*K´* is a factor that may vary along the flow, and indicates the "facility" of the air absorption by the water. Considering arguments about diffusion of turbulence from the bottom to the surface,

w

The solution of equation (79), using boundary conditions 1) *s*=0, *ϕ* = 0, and 2) *s* →*∞*, *ϕ* →*ϕ∞*,

<sup>1</sup> / 1/ [1 (1 )] f- =-- ¥

In order to obtain a nondimensional depth like equation (70), a boundary layer analogy was

"close" to the unity (as information, *β*=0.99 is the boundary layer value, but it does not imply
