**3. Experimental study**

### **3.1 General information**

The experimental results presented in this chapter were obtained in the Laboratory of Environmental Hydraulics of the School of Engineering at São Carlos (University of Sao Paulo). The experiments were performed in a channel with the following characteristics: (1) Width: B = 0.20 m, (2) Length = 5.0 m, 3.5 m was used, (3) Angle between the pseudo bottom and the horizontal: = 45o; (4) Dimensions of the steps s = l = 0.05 m (s = step height l = length of the floor), and (5) Pressurized intake, controlled by a sluice gate. The water supply was accomplished using a motor/pump unit (Fig. 7) that allowed a maximum flow rate of 300 L/s. The flow rate measurements were performed using a thin-wall rectangular weir located in the outlet channel, and an electromagnetic flow meter positioned in the inlet tubes (Fig. 7b), used for confirmation of the values of the water discharge.

Fig. 7. a) Motor/pump system.; b) Schematic drawing of the hydraulic circuit: (1) river, (2) engine room, (3) reservoir, (4) electromagnetic flowmeter, (5) stepped chute, (6) energy sink, (7) outlet channel; (8) weir, (9) final outlet channel.

The position of the free surface was measured using acoustic sensors (ultrasonic sensors), as previously done by Lueker et al. (2008). They were used to measure the position of the free surface of the flows tested in a physical model of the auxiliary spillway of the Folsom Dam, performed at the St. Anthony Falls Laboratory, University of Minnesota. A second study that employed acoustic probes was Murzyn & Chanson (2009), however, for measuring the position of the free surface in hydraulic jumps.

In the present study, the acoustic sensor was fixed on a support attached to a vehicle capable of traveling along the channel, as shown in the sketch of Figure 8. For most experiments, along the initial single phase stretch, the measurements were taken at sections distant 5 cm from each other. After the first 60 cm, the measurement sections were spaced 10 cm from

Stepped Spillways: Theoretical, Experimental and Numerical Studies 247

As can be seen in Figure 4, the positioning of the free surface is complex due to its highly irregular structure, especially downstream from the inception point. One of the characteristics of measurements conduced with acoustic sensors is the detection of droplets ejected from the surface. These values are important for the evaluation of the highest position of the droplets and sprays, but have little influence to establish the mean profiles of the free surface. This is shown in Figure 9a, which contains the relative errors calculated considering the mean position obtained without the outliers (droplets). The corrections were made using standard criteria used for box plots. The maximum percentage of rejected

0

(a) (b)

As mentioned, the starting position of the aeration was set based on the minimum point that characterizes the far end of the S2 profile. In some experiments, this minimum showed a certain degree of dispersion, so that the most probable position was chosen. To quantify the position of the inception point of the aeration, the variables involved in a first instance were

Chanson, 2002; Sanagiotto, 2003) Equation 7 shows the best adjustment obtained for the present set of data, with a correlation coefficient of 0.91. Considering the four variables

\*, adjusting a power law between them, as already used by several authors. (e.g.,

\* (see figure 6a for the definitions of the variables), a second

Fig. 9. (a) Maximum relative deviations corresponding to the eighteen experiments, in which: errh = 100||h(1) – h(2)||/h(2), h(i) = mean value obtained with the acoustic sensor, i = 1 (original sample), i = 2 (sample without outliers) and Fr(0) = Froude number at x = 0; (b) Mean experimental profile due to Exp.18. The deviations were used to obtain the maximum position of the droplets, but were ignored when obtaining the mean profile of the surface. Figure 9b presents an example of a measured average profile obtained in this study. As can be seen, an S2 profile is formed in the one-phase region. The inception point of the aeration is given by the position of the first minimum in the measured curve. It establishes the end of the S2 curve and the beginning of the "transition length", as defined by Simões et al. (2011). As shown by the mentioned authors, the surface of the mixture presents a wavy shape, also used to define the end of the transition length, given by the first maximum of the surface

0 5 10 15

Experimental profile

Exp. 18: q = 0.216 m2/s, h(0) = 7.4 cm

**H [-]**

0.2

0.4

 **[-]**

0.6

samples (droplets) was 8.3% for experiment No 5.

<sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> 5.5 2.5

Fr(0)

**3.2.1 Starting position of the aeration (inception point)** 

3 3.5 4 4.5 5 5.5 6

max(errh) [%]

profile.

**3.2 Results** 

LA/k and Fr

LA/k, h(0)/k, Re(0), and Fr

each other. The sensor was adjusted to obtain 6000 samples (or points) using a frequency of 50 Hz at each longitudinal position. These 6000 points were used to perform the statistical calculations necessary to locate the surface and the drops that formed above the surface. A second acoustic sensor was used to measure the position of the free surface upstream of the thin wall weir, in order to calculate the average hydraulic load and the flow rates used in the experiments. The measured flow rates, and other experimental parameters of the different runs, are shown in Table 1.

Fig. 8. Schematic of the arrangement used in the experiments


Table 1. General data related to experiments

As can be seen in Figure 4, the positioning of the free surface is complex due to its highly irregular structure, especially downstream from the inception point. One of the characteristics of measurements conduced with acoustic sensors is the detection of droplets ejected from the surface. These values are important for the evaluation of the highest position of the droplets and sprays, but have little influence to establish the mean profiles of the free surface. This is shown in Figure 9a, which contains the relative errors calculated considering the mean position obtained without the outliers (droplets). The corrections were made using standard criteria used for box plots. The maximum percentage of rejected samples (droplets) was 8.3% for experiment No 5.

Fig. 9. (a) Maximum relative deviations corresponding to the eighteen experiments, in which: errh = 100||h(1) – h(2)||/h(2), h(i) = mean value obtained with the acoustic sensor, i = 1 (original sample), i = 2 (sample without outliers) and Fr(0) = Froude number at x = 0; (b) Mean experimental profile due to Exp.18. The deviations were used to obtain the maximum position of the droplets, but were ignored when obtaining the mean profile of the surface.

Figure 9b presents an example of a measured average profile obtained in this study. As can be seen, an S2 profile is formed in the one-phase region. The inception point of the aeration is given by the position of the first minimum in the measured curve. It establishes the end of the S2 curve and the beginning of the "transition length", as defined by Simões et al. (2011). As shown by the mentioned authors, the surface of the mixture presents a wavy shape, also used to define the end of the transition length, given by the first maximum of the surface profile.

#### **3.2 Results**

246 Hydrodynamics – Natural Water Bodies

each other. The sensor was adjusted to obtain 6000 samples (or points) using a frequency of 50 Hz at each longitudinal position. These 6000 points were used to perform the statistical calculations necessary to locate the surface and the drops that formed above the surface. A second acoustic sensor was used to measure the position of the free surface upstream of the thin wall weir, in order to calculate the average hydraulic load and the flow rates used in the experiments. The measured flow rates, and other experimental parameters of the different

No Experiment name Q Profile q hc s/hc h(0)

1 Exp. 2 0.0505 S2 0.252 0.187 0.268 0.103 2 Exp. 3 0.0458 S2 0.229 0.175 0.286 0.101 3 Exp. 4 0.0725 S2 0.362 0.238 0.211 0.106 4 Exp. 5 0.0477 S2 0.239 0.180 0.278 0.087 5 Exp. 6 0.0833 **S3** 0.416 0.261 0.192 0.092 6 Exp. 7 0.0504 S2 0.252 0.187 0.268 0.089 7 Exp. 8 0.0073 S2 0.0366 0.051 0.971 0.027 8 Exp. 9 0.0074 S2 0.0368 0.052 0.967 0.024 9 Exp. 10 0.0319 S2 0.159 0.137 0.364 0.058 10 Exp. 11 0.0501 **S3** 0.250 0.186 0.269 0.06 11 Exp. 14 0.0608 S2 0.304 0.211 0.237 0.089 12 Exp. 15 0.0561 S2 0.280 0.200 0.250 0.087 13 Exp. 16 0.0265 S2 0.133 0.122 0.411 0.046 14 Exp. 17 0.0487 S2 0.244 0.182 0.274 0.072 15 Exp. 18 0.0431 S2 0.216 0.168 0.298 0.074 16 Exp. 19 0.0274 S2 0.137 0.124 0.402 0.041 17 Exp. 20 0.0360 S2 0.180 0.149 0.336 0.068 18 Exp. 21 0.0397 S2 0.198 0.159 0.315 0.071

[m3/s] [m2/s] [m] [-] [m]

runs, are shown in Table 1.

Fig. 8. Schematic of the arrangement used in the experiments

Table 1. General data related to experiments

#### **3.2.1 Starting position of the aeration (inception point)**

As mentioned, the starting position of the aeration was set based on the minimum point that characterizes the far end of the S2 profile. In some experiments, this minimum showed a certain degree of dispersion, so that the most probable position was chosen. To quantify the position of the inception point of the aeration, the variables involved in a first instance were LA/k and Fr \*, adjusting a power law between them, as already used by several authors. (e.g., Chanson, 2002; Sanagiotto, 2003) Equation 7 shows the best adjustment obtained for the present set of data, with a correlation coefficient of 0.91. Considering the four variables LA/k, h(0)/k, Re(0), and Fr \* (see figure 6a for the definitions of the variables), a second

Stepped Spillways: Theoretical, Experimental and Numerical Studies 249

was modified by replacing zi by zi' (see Figure 10a). The energy equation was also used, for the region between the critical section (Section 1, represented by the subscript "c") and the initial section of the experiments (Section 2, at x = 0, represented by (0)). The Darcy-Weisbach equation was applied with average values for the hydraulic radius and the velocity. The resulting equation is similar to that proposed by Boes (2000, p.126), who also used the Coriolis coefficient, assumed unity in the present study. Equation (10) is the

c c <sup>c</sup> <sup>3</sup>

h(0)cos h /[2h(0) ] (3 / 2)h z z(0) h (h(0) h ) (B h h(0)) 1 f

2

The calculation of zi' requires the resistance factor, obtained here applying the methodology described by Simões et al. (2010). The obtained equation presented a correlation coefficient

' 0.837

where: zi' = zi + zc – z(0). This equation is valid for the same ranges of the variables shown for equations 7 and 8. Equation 11 furnishes lower values of zi/s when compared to the equations of Boes (2000, p.126) and Boes & Hager (2003b). Two reasons for the difference are mentioned here: (1) The method used to calculate the resistance factor, and (2) the definition of the position of the inception point. The mentioned authors defined the starting position of the aeration as the point where the void fraction at the pseudo-bottom is 1%, while the present definition corresponds to the final section of the S2 profile. Using the equation of Boes & Hager (2003b), zi'/s = 5.9F0.8, it is possible to relate these two positions, as shown by equation 12. The position of the void fraction of 1% at the pseudo-bottom occurs approximately at 1.85 times the position of the final section of the S2 profile, with both

2

z ) 1.85

In equation 12, zi')1% corresponds to the length defined by the equation of Boes & Hager (2003b), and zi')S2 corresponds to the length defined by eq. 11. Transforming equation 11

> \* \* <sup>r</sup> <sup>r</sup> 3 3 q q kk F F F F s s gs sin gk sin

> > \* A \*0.837 r

The behavior of equation 13 in comparison with experimental data is illustrated in Figure 11, which contains experimental data found in the literature, as well as two additional

4.13F

i 1% ' 0.037 i S

'

c

of 0.98 when compared to the measured data, having the form:

lengths having their origin at the crest of the spillway.

and remembering that LA\*/k = (zi'/s)/(sincos), leads then to:

L

considering the variables LA\*/k and Fr

between the Froude numbers is obtained

3 2

(10)

cc c

<sup>i</sup> z /s 3.19F (11)

z) F (12)

\*, in which LA\* correspond to zi', the following relation

3/2 3/2

<sup>k</sup> (13)

(h h(0)) 16sin B

equation adopted in the present study:

equation is presented, as a sum of the powers of the variables. Equation 8 presents a correlation coefficient of 0.98, leading to a good superposition between data and adjusted curve, as can be seen in Fig.10b.

$$\frac{\mathbf{L\_A}}{\mathbf{k}} = 1.61 \mathbf{F\_r^{\*1.06}} \tag{7}$$

$$\frac{\mathbf{L}\_{\text{A}}}{\text{k}} = 699.97 \mathbf{F}\_{\text{r}}^{\*-6.33} + 34.22 \left[ \frac{\mathbf{h}(0)}{\text{k}} \right]^{0.592} - 49.45 \text{Re}(0)^{-0.0379} \tag{8}$$

Fig. 10. Definition of variables related to the start of aeration (a) and comparison between measured data and calculated values using the adjusted equation 8.

Equations 7 and 8 show very distinct behaviors for the involved parameters. For example, the dependence of LA/k on Fr \* shows increasing lengths for increasing Fr \* when using equation 7, and decreasing lengths for increasing Fr \* when using equation 8. Additionally, the influence of h(0)/k appears as relevant, when considering the exponent 0.592. This parameter was used to verify the relevance of Fr \* to quantify the inception point. Although the result points to a possible relevance of the geometry of the flow (h(0)), the adequate definition of this parameter for general flows is an open question. It is the depth of the flow at a fixed small distance from the sluice gate in this study, thus directly related to the geometry, but which correspondent to general flows, as already emphasized, must still be defined. In the present analysis, following restrictions apply: 2.09 Fr \* 20.70, 0.69 h(0)/k 2.99 and 1.15x105 Re(0) 7.04x105.

Equation 7 can be rewritten using zi/s and F, in which zi = LAsin, and F is the Froude number defined by Boes & Hager (2003b) as <sup>3</sup> F q/ gs sin . In this case Fr\* = F/(cos3)1/2. The correlation coefficient is the same obtained for equation 7, and the resulting equation, valid for the same conditions of the previous adjustments, is:

$$\frac{\mathbf{z\_i}}{\mathbf{s}} = \mathbf{1.397F^{1.06}} \tag{9}$$

The power laws proposed by Boes (2000) and Boes & Hager (2003b) were similar to equation 9, but having different coefficients. In order to compare the different proposals, equation 9

equation is presented, as a sum of the powers of the variables. Equation 8 presents a correlation coefficient of 0.98, leading to a good superposition between data and adjusted

> A \*1.06 r

A \* 6,33 0.0379

0.592

0

\* shows increasing lengths for increasing Fr

10

20

**LA/k -** 

(a) (b) Fig. 10. Definition of variables related to the start of aeration (a) and comparison between

Equations 7 and 8 show very distinct behaviors for the involved parameters. For example,

the influence of h(0)/k appears as relevant, when considering the exponent 0.592. This

the result points to a possible relevance of the geometry of the flow (h(0)), the adequate definition of this parameter for general flows is an open question. It is the depth of the flow at a fixed small distance from the sluice gate in this study, thus directly related to the geometry, but which correspondent to general flows, as already emphasized, must still be

Equation 7 can be rewritten using zi/s and F, in which zi = LAsin, and F is the Froude number defined by Boes & Hager (2003b) as <sup>3</sup> F q/ gs sin . In this case Fr\* = F/(cos3)1/2. The correlation coefficient is the same obtained for equation 7, and the

zi 1.06 1.397F

The power laws proposed by Boes (2000) and Boes & Hager (2003b) were similar to equation 9, but having different coefficients. In order to compare the different proposals, equation 9

resulting equation, valid for the same conditions of the previous adjustments, is:

**Experimental**

30

40

<sup>k</sup> (7)

(8)

0 10 20 30 40

\* when using equation 8. Additionally,

\* to quantify the inception point. Although

<sup>s</sup> (9)

**LA/k - Equation 8**

\* 20.70, 0.69 h(0)/k

\* when using

<sup>L</sup> 1.61F

L h(0) 699.97F 34.22 49.45Re(0) k k 

r

measured data and calculated values using the adjusted equation 8.

defined. In the present analysis, following restrictions apply: 2.09 Fr

equation 7, and decreasing lengths for increasing Fr

parameter was used to verify the relevance of Fr

curve, as can be seen in Fig.10b.

the dependence of LA/k on Fr

2.99 and 1.15x105 Re(0) 7.04x105.

was modified by replacing zi by zi' (see Figure 10a). The energy equation was also used, for the region between the critical section (Section 1, represented by the subscript "c") and the initial section of the experiments (Section 2, at x = 0, represented by (0)). The Darcy-Weisbach equation was applied with average values for the hydraulic radius and the velocity. The resulting equation is similar to that proposed by Boes (2000, p.126), who also used the Coriolis coefficient, assumed unity in the present study. Equation (10) is the equation adopted in the present study:

$$\mathbf{z}\_c - \mathbf{z}(0) = \frac{\mathbf{h}(0)\cos\alpha + \mathbf{h}\_c^3 \left/ \left[2\mathbf{h}(0)^2\right] - \left(\mathbf{3} \,' \,\mathbf{2}\right)\mathbf{h}\_c}{\mathbf{1} - \mathbf{f} \frac{\mathbf{h}\_c^3 \left(\mathbf{h}(0) + \mathbf{h}\_c\right)}{\left(\mathbf{h}\_c \mathbf{h}(0)\right)^2 \mathbf{1} \mathbf{6} \sin\alpha} \frac{\left(\mathbf{B} + \mathbf{h}\_c + \mathbf{h}(0)\right)}{\mathbf{B}}} \tag{10}$$

The calculation of zi' requires the resistance factor, obtained here applying the methodology described by Simões et al. (2010). The obtained equation presented a correlation coefficient of 0.98 when compared to the measured data, having the form:

$$\mathbf{z}\_{i}^{\cdot}/\text{s} = \mathbf{3}.19 \mathbf{F}^{0.837} \tag{11}$$

where: zi' = zi + zc – z(0). This equation is valid for the same ranges of the variables shown for equations 7 and 8. Equation 11 furnishes lower values of zi/s when compared to the equations of Boes (2000, p.126) and Boes & Hager (2003b). Two reasons for the difference are mentioned here: (1) The method used to calculate the resistance factor, and (2) the definition of the position of the inception point. The mentioned authors defined the starting position of the aeration as the point where the void fraction at the pseudo-bottom is 1%, while the present definition corresponds to the final section of the S2 profile. Using the equation of Boes & Hager (2003b), zi'/s = 5.9F0.8, it is possible to relate these two positions, as shown by equation 12. The position of the void fraction of 1% at the pseudo-bottom occurs approximately at 1.85 times the position of the final section of the S2 profile, with both lengths having their origin at the crest of the spillway.

$$\frac{\mathbf{z\_i}\mathbf{i\_i}}{\mathbf{z\_i}\mathbf{i\_i}}\mathbf{s\_i} = \frac{\mathbf{1.85}}{\mathbf{F}^{0.037}}\tag{12}$$

In equation 12, zi')1% corresponds to the length defined by the equation of Boes & Hager (2003b), and zi')S2 corresponds to the length defined by eq. 11. Transforming equation 11 considering the variables LA\*/k and Fr \*, in which LA\* correspond to zi', the following relation between the Froude numbers is obtained

$$\mathbf{F} = \boldsymbol{\eta} \mathbf{F}\_r^\* \Rightarrow \frac{\mathbf{q}}{\sqrt{\mathbf{g} \mathbf{s}^3 \sin \alpha}} = \boldsymbol{\eta} \frac{\mathbf{q}}{\sqrt{\mathbf{g} \mathbf{k}^3 \sin \alpha}} \Rightarrow \boldsymbol{\eta} = \left(\frac{\mathbf{k}}{\mathbf{s}}\right)^{3/2} \Rightarrow \mathbf{F} = \left(\frac{\mathbf{k}}{\mathbf{s}}\right)^{3/2} \mathbf{F}\_r^\*$$

and remembering that LA\*/k = (zi'/s)/(sincos), leads then to:

$$\frac{\mathbf{L}\_{\rm A^\*}}{\mathbf{k}} = \mathbf{4.13F\_r^{\*0.837}}\tag{13}$$

The behavior of equation 13 in comparison with experimental data is illustrated in Figure 11, which contains experimental data found in the literature, as well as two additional

Stepped Spillways: Theoretical, Experimental and Numerical Studies 251

The experiments showed that the averaged values of the depths form a free surface profile composed by a decreasing region (S2) followed by a growing region that extends up to a maximum depth, from which a wavy shape is formed downstream, as illustrated by Figure 13a. The maximum value which limits the growing region is denoted by h2. The length of the transition between the minimum (hA) and the maximum (h2) is named here "transition length", and is represented by L, a distance parallel to the pseudo bottom, as shown in Figure 13a. hA/k was related to h2/k using a power law (equation 16), showing a good superposition between experimental data and the adjusted equation, as shown in figure 13b,

0.5

(a) (b)

Fig. 13. a) Definition of the depth h2 and the transition length L, and b) correlation between the depth at the start of aeration (hA/k) and the depth corresponding to the first wave crest

> 0.879 h h 2 A 1.408 k k

for which equation 17 was obtained, with a correlation coefficient of 0.99. The ranges of

2 \*0.553 4 2.1x10

The transition length between the last "full water" section (the last S2 section) and the first "full mixture" section, or, in other words, the section at which the air reaches the pseudobottom, could be well characterized using the ultrasound sensor. From a practical point of view, this length is relevant because it involves a region of the spillway still unprotected, due to the absence of air near the bottom. Experimental verification of void fractions is still necessary to establish the void percentage attained at the pseudo-bottom in the mentioned section. An analysis is presented here considering the hypothesis that the "full mixture" section defined by the maximum of the measured depths corresponds to the 1% void

h h(0) 0.319F 0.529 1.6x10 Re(0) k k

validity of equations 16 and 17 are the same as for equations 7 and 8.

r

fraction defined by Boes (2000) and Boes & Hager (2003b).

0.1 1 10

(16)

<sup>5</sup> 0.744

(17)

\*, h(0)/k and Re(0) were used to quantify h2/k,

hA/k

5

h2/k

**3.2.3 Transition to two-phase flow** 

with a correlation coefficient of 0.99.

(h2/k), expressed by equation 16.

As in previous cases, also the parameters Fr

predictive curves. Observe that, except for the first two points (obtained in the present study), the results are located close to the curve defined by Matos (1999). It is also interesting to note that the equations proposed by Matos (1999) and Sanagiotto (2003) are approximately parallel.

Fig. 11. Starting position of the aeration: a comparison between the experimental data of this research, the equation obtained in this work and data (experimental and numerical) of different authors.

#### **3.2.2 Depths at the end of S2**

As in the previous case, power laws and sum of power laws were used to quantify the flow depth at the inception point, that is, in the final section of the S2 profile. Equations 14 and 15 were then obtained, with correlation coefficients 0.97 and 0.98, respectively. Figure 12 contains a comparison with data from different sources.

$$\frac{\mathbf{h\_A}}{\mathbf{k}} = 0.363 \mathbf{F\_r}^{\*0.609} \tag{14}$$

$$\frac{\mathbf{h\_A}}{\mathbf{k}} = 0.791 \mathbf{F\_r}^{\ast -6.98} + 1.285 \left[ \frac{\mathbf{h(0)}}{\mathbf{k}} \right]^{0.567} - 19.56 \,\mathrm{Re(0)}^{-0.322} \tag{15}$$

Equations 14 and 15 are restricted to: 2.09 Fr \* 20.70, 0.69 h(0)/k 2.99 and 1.15x105 Re(0) 7.04x105.

Fig. 12. Depth in the starting position of the aeration based on the final section of the S2 profile: comparison with experimental and numerical data of different sources.

#### **3.2.3 Transition to two-phase flow**

250 Hydrodynamics – Natural Water Bodies

predictive curves. Observe that, except for the first two points (obtained in the present study), the results are located close to the curve defined by Matos (1999). It is also interesting to note that the equations proposed by Matos (1999) and Sanagiotto (2003) are

> Fr \*

Fig. 11. Starting position of the aeration: a comparison between the experimental data of this research, the equation obtained in this work and data (experimental and numerical) of

As in the previous case, power laws and sum of power laws were used to quantify the flow depth at the inception point, that is, in the final section of the S2 profile. Equations 14 and 15 were then obtained, with correlation coefficients 0.97 and 0.98, respectively. Figure 12

> A \*0.609 r

A \* 6.98 0.322

1 10 100

Fig. 12. Depth in the starting position of the aeration based on the final section of the S2 profile: comparison with experimental and numerical data of different sources.

0.567

Matos (1999) Chanson (2002) Sanagiotto (2003) Arantes (2007); k = 2 cm Arantes (2007); k = 3 cm Arantes (2007); k = 6 cm Experimental data

<sup>h</sup> 0.363F

h h(0) 0.791F 1.285 19.56Re(0) k k 

Matos (1999) Chanson (2002) Sanagiotto (2003) Arantes (2007); k = 2 cm Arantes (2007); k = 3 cm Arantes (2007); k = 6 cm Povh (2000, f.97) L1/k Povh (2000, f.97) L2/k Povh (2000, f.97) L3/k Povh (2000, f.97) L4/k Experimental data Equation 13

<sup>k</sup> (14)

Fr \*

(15)

\* 20.70, 0.69 h(0)/k 2.99 and 1.15x105

1 100

contains a comparison with data from different sources.

Equations 14 and 15 are restricted to: 2.09 Fr

0.1

1

10

hA/k

r

approximately parallel.

different authors.

Re(0) 7.04x105.

1

**3.2.2 Depths at the end of S2**

100

LA\*/k

The experiments showed that the averaged values of the depths form a free surface profile composed by a decreasing region (S2) followed by a growing region that extends up to a maximum depth, from which a wavy shape is formed downstream, as illustrated by Figure 13a. The maximum value which limits the growing region is denoted by h2. The length of the transition between the minimum (hA) and the maximum (h2) is named here "transition length", and is represented by L, a distance parallel to the pseudo bottom, as shown in Figure 13a. hA/k was related to h2/k using a power law (equation 16), showing a good superposition between experimental data and the adjusted equation, as shown in figure 13b, with a correlation coefficient of 0.99.

Fig. 13. a) Definition of the depth h2 and the transition length L, and b) correlation between the depth at the start of aeration (hA/k) and the depth corresponding to the first wave crest (h2/k), expressed by equation 16.

$$\frac{\mathbf{h}\_2}{\mathbf{k}} = \mathbf{1}.408 \left(\frac{\mathbf{h}\_A}{\mathbf{k}}\right)^{0.879} \tag{16}$$

As in previous cases, also the parameters Fr \*, h(0)/k and Re(0) were used to quantify h2/k, for which equation 17 was obtained, with a correlation coefficient of 0.99. The ranges of validity of equations 16 and 17 are the same as for equations 7 and 8.

$$\frac{\text{h}\_2}{\text{k}} = 0.319 \text{F}\_{\text{r}}^{\circ 0.553} + 0.529 \left[ \frac{\text{h}(0)}{\text{k}} \right]^{0.744} - 1.6 \times 10^4 \text{ Re}(0)^{-2.1 \times 10^5} \tag{17}$$

The transition length between the last "full water" section (the last S2 section) and the first "full mixture" section, or, in other words, the section at which the air reaches the pseudobottom, could be well characterized using the ultrasound sensor. From a practical point of view, this length is relevant because it involves a region of the spillway still unprotected, due to the absence of air near the bottom. Experimental verification of void fractions is still necessary to establish the void percentage attained at the pseudo-bottom in the mentioned section. An analysis is presented here considering the hypothesis that the "full mixture" section defined by the maximum of the measured depths corresponds to the 1% void fraction defined by Boes (2000) and Boes & Hager (2003b).

Stepped Spillways: Theoretical, Experimental and Numerical Studies 253

the concentrations at the pseudo-bottom, so that we suggest the present methodology to evaluate the position of the beginning of the "full mixed" region. Of course, the void fraction measurements at the bottom were important to allow the present comparison. Of course, having a first confirmation, it is possible to obtain the same information involving the different axes used in spillway studies. For example, it is possible to LA\*/k with Fr

being the sum of LA\* with L. The result is equation 21, with a correlation coefficient of 0.95, and which behavior is illustrated in Figure 11, where it is compared with data from other sources. In general, there is a good agreement of equation 21 with most of the results of the cited studies. Special mention made be made for the data L4/k obtained by Povh (2000) (the L4 position corresponds to the fully-aerated section of the flow), the data of Chanson (2002),

1 100

\* 1.29

r

Fig. 15. Inception point considering different equations of the literature and equation 21: a

\* LA 0.81 8.4F

Following the previous procedures followed in this section, equation 22 was also obtained, involving the geometrical information of the flow and the Reynolds number, presenting a

A \* 6.36 0.452

The time derivatives of the position of the free surface were used to evaluate the turbulent intensity (w') and, assuming isotropy (as a first approximation), the turbulent kinetic energy

(22)

L h(0) 2397.09F 32.49 0.212Re(0) k k

Fr \* Matos (1999) Chanson (2002) Sanagiotto (2003) Arantes (2007); k = 2 cm Arantes (2007); k = 3 cm Arantes (2007); k = 6 cm Povh (2000, f.97) L1/k Povh (2000, f.97) L2/k Povh (2000, f.97) L3/k Povh (2000, f.97) L4/k Experimental data Equation 21

<sup>k</sup> (21)

\* 20.70, 0.69 h(0)/k 2.99 and 1.15x105 Re(0)

<sup>2</sup> w' w (23)

and Sanagiotto (2003).

100

LA\*/k

1

correlation coefficient of 0.98.

7.04x105.

The restrictions of this study are 2.09 Fr

(ke), defined in equations (23) and (24):

**3.2.4 Turbulence intensity and kinetic energy** 

comparison with data from different authors

\*, LA\*

Combining the transition lengths with the values of LA\* (or zi ' ), the positions of the inception point considering this new origin are then obtained. This length was correlated with the dimensionless parameters (zi ' +Lsin)/s = zL/s and F = q/(gs3sin)0,5. Equation 18 was then obtained, with a correlation coefficient of 0.95. Figure 14 illustrates the behavior of this adjustment in relation to the experimental data. The same figure also shows the curve obtained with the equation of Boes & Hager (2003b), showing that the two forms of analyses generate very similar results.

Fig. 14. Starting position of the aeration considering the present analysis (equation 18), corresponding to the position of the maximum of the measured depths, and the equation of Boes & Hager (2003b), zi'/s = 5.9F0.8 (in this case, zi' corresponds to a mean void fraction of 1% at the pseudo bottom).

$$\frac{\text{Z}\_{\text{L}}}{\text{s}} = 6.4 \text{F}^{0.81} \tag{18}$$

The equations proposed for Boes (2000) and Boes & Hager (2003b) allow to relate zL/s with the position of the 1% void fraction on the pseudo bottom, leading to:

$$\frac{\mathbf{z\_i}}{\mathbf{z\_L}} = 0.73 \mathbf{F}^{0.03} \quad \mathbf{z\_i}' \text{/} \text{from Boes (2000)}\tag{19}$$

$$\frac{\text{(z\textsuperscript{'})}\_{1\%}}{\text{z\textsuperscript{'}}\_{\text{L}}} = \frac{0.92}{\text{F}^{0.01}} \quad \text{z\textsuperscript{'}} \text{)}\_{1\%} \text{ from Boes } \& \text{ Hager (2003b)}\tag{20}$$

As can be seen, the results show different trends in relation to the Froude number. Such differences may be related to the values of the adjusted exponents, which are close, but not the same. Equation 18 is very similar to equation 11, with the Froude number in both equations having similar exponents, and the coefficient of equation 18 being 2 times bigger that the coefficient of equation 11. This result is close to the factor of 1.85 obtained with equation 12. Figure 15 shows that the results obtained with the present analysis are close to those obtained with the equation Boes & Hager (2003b), suggesting to use the maximum depth to locate the beginning of the bottom aeration, or, in other words, the position where there is a void fraction of 1% at the bottom. It is important to emphasize that the measurement of the position of the free surface is much simpler than the measurement of

point considering this new origin are then obtained. This length was correlated with the

obtained, with a correlation coefficient of 0.95. Figure 14 illustrates the behavior of this adjustment in relation to the experimental data. The same figure also shows the curve obtained with the equation of Boes & Hager (2003b), showing that the two forms of analyses

1 10

Fig. 14. Starting position of the aeration considering the present analysis (equation 18), corresponding to the position of the maximum of the measured depths, and the equation of Boes & Hager (2003b), zi'/s = 5.9F0.8 (in this case, zi' corresponds to a mean void fraction of

zL 0.81 6.4F

The equations proposed for Boes (2000) and Boes & Hager (2003b) allow to relate zL/s with

As can be seen, the results show different trends in relation to the Froude number. Such differences may be related to the values of the adjusted exponents, which are close, but not the same. Equation 18 is very similar to equation 11, with the Froude number in both equations having similar exponents, and the coefficient of equation 18 being 2 times bigger that the coefficient of equation 11. This result is close to the factor of 1.85 obtained with equation 12. Figure 15 shows that the results obtained with the present analysis are close to those obtained with the equation Boes & Hager (2003b), suggesting to use the maximum depth to locate the beginning of the bottom aeration, or, in other words, the position where there is a void fraction of 1% at the bottom. It is important to emphasize that the measurement of the position of the free surface is much simpler than the measurement of

the position of the 1% void fraction on the pseudo bottom, leading to:

i 1% 0.03

0.01

'

'

i 1%

L

L

z ) 0.92

Equation 18 Experimental data Boes and Hager (2003b)

'

+Lsin)/s = zL/s and F = q/(gs3sin)0,5. Equation 18 was then

F

z ) 0.73F <sup>z</sup> zi')1% from Boes (2000) (19)

<sup>z</sup> <sup>F</sup> zi')1% from Boes & Hager (2003b) (20)

<sup>s</sup> (18)

), the positions of the inception

Combining the transition lengths with the values of LA\* (or zi

'

1

10

100

zL/s

dimensionless parameters (zi

generate very similar results.

1% at the pseudo bottom).

the concentrations at the pseudo-bottom, so that we suggest the present methodology to evaluate the position of the beginning of the "full mixed" region. Of course, the void fraction measurements at the bottom were important to allow the present comparison.

Of course, having a first confirmation, it is possible to obtain the same information involving the different axes used in spillway studies. For example, it is possible to LA\*/k with Fr \*, LA\* being the sum of LA\* with L. The result is equation 21, with a correlation coefficient of 0.95, and which behavior is illustrated in Figure 11, where it is compared with data from other sources. In general, there is a good agreement of equation 21 with most of the results of the cited studies. Special mention made be made for the data L4/k obtained by Povh (2000) (the L4 position corresponds to the fully-aerated section of the flow), the data of Chanson (2002), and Sanagiotto (2003).

Fig. 15. Inception point considering different equations of the literature and equation 21: a comparison with data from different authors

$$\frac{\text{L}^\*\_{\text{A}}}{\text{k}} = 8.4 \text{F}^{0.81} \tag{21}$$

Following the previous procedures followed in this section, equation 22 was also obtained, involving the geometrical information of the flow and the Reynolds number, presenting a correlation coefficient of 0.98.

$$\frac{\text{L}\_{\text{A}}^{\*}}{\text{k}} = 2397.09 \text{F}\_{\text{r}}^{\*-6.36} - 32.49 \left[ \frac{\text{h}(0)}{\text{k}} \right]^{-1.29} + 0.212 \,\text{Re}(0)^{0.452} \tag{22}$$

The restrictions of this study are 2.09 Fr \* 20.70, 0.69 h(0)/k 2.99 and 1.15x105 Re(0) 7.04x105.

#### **3.2.4 Turbulence intensity and kinetic energy**

The time derivatives of the position of the free surface were used to evaluate the turbulent intensity (w') and, assuming isotropy (as a first approximation), the turbulent kinetic energy (ke), defined in equations (23) and (24):

$$\mathbf{w}' = \sqrt{\mathbf{w}^2} \tag{23}$$

Stepped Spillways: Theoretical, Experimental and Numerical Studies 255

decay region, which is limited downwards around the point z/zi=0.9, (3) Two-phase

Considering the decay region limited by 2.5<z/zi<14, a power law of the type ke=a(z/zi)-n was adjusted, obtaining n = 0.46 with a correlation coefficient of 0.72. Using the terminology of the ke- model, for which the constant C2=(n+1)/n is defined, it implies in a C2=3.7, which is about 1.7 times greater than the value of the standard model, C2=1.92 (Rodi, 1993). This analysis was conducted to verify the possibility of obtaining statistical parameters

Turbulence is a three dimensional and time-dependent phenomenon. If Direct Numerical Simulation (DNS) is planned to calculate turbulence, the Navier-Stokes and continuity equations must be used without any simplifications. Since there is no general analytical solution for these equations, a numerical solution which considers all the scales existing in turbulence must use a sufficiently refined mesh. According to the theory of Kolmogorov, it can be shown that the number of degrees of freedom, or points in a discretized space, is of

where: Lk = characteristic dimension of the large-scales of the movement of the fluid, = Kolmogorov micro-scale of turbulence and Re = Reynolds number of the larger scales. Considering a usual Reynolds number, like Re = 105, the mesh must have about 1011 elements. This number indicates that it is impossible to perform the wished DNS with the current computers. So, we must lower our level of expectations in relation to our results. A next "lower" level would be to simulate only the large scales (modeling the small scales) or the so called large-eddy simulation (LES). This alternative is still not commonly used in problems composed by a high Reynolds number and large dimensions. So, lowering still more our expectations, the next level would be the full modeling of turbulence, which corresponds to the procedures followed in this study. This chapter presents, thus, results obtained with the aid of turbulence models (all scales are modeled), which is the usual way followed to study flows around large structures and subjected to large Reynolds numbers.

In recent years an increasing number of papers related to the use of CFD to simulate flows in hydraulic structures and in stepped spillways has been published. Some examples are Chen et al. (2002), Cheng et al. (2004), Inoue (2005), Arantes (2007), Carvalho & Martins (2009), Bombardelli et al. (2010), Lobosco & Schulz (2010) and Lobosco et al. (2011). Different aspects of turbulent flows were studied in these simulations, such as the development of boundary layers, the energy dissipation, flow aeration, scale effects, among others. The turbulence models k- and RNG k- were used in most of the mentioned studies, and Arantes (2007) also used the SSG Reynolds stress model (Speziale, Sarkar and Gatski, 1991). Some researchers have still adopted commercial softwares to perform their simulations, such as ANSYS CFX® and Fluent®. On the other hand, Lobosco & Schulz (2010) and Lobosco et al. (2011), for example, used a set of free softwares, among which the OpenFOAM®

<sup>3</sup> 9/4 L / Re <sup>k</sup> (27)

growing region, limited by ~0.9<z/zi<~2.11, and (4) Two-phase decay region.

linked to the kinetic energy, similar to those found in the literature of turbulence.

**4. Numerical simulations** 

**4.2 Some previous studies** 

software. In this study we used the ANSYS CFX® software.

the order of (Landau & Lifshitz, 1987, p.134):

**4.1 Introduction** 

$$\mathbf{k}\_{\rm e} = \frac{3}{2} \mathbf{w}^{\prime 2} \tag{24}$$

Also a relative intensity and a dimensionless turbulent kinetic energy were defined, written in terms of the critical kinetic energy (all parameters per unit mass of fluid), which are represented by equations (25) and (26):

$$\text{ir} = \frac{\mathbf{w}^\prime}{\mathbf{V}\_\mathbf{c}} \tag{25}$$

$$\mathbf{k}\_{\mathbf{e}}\,^{\*} = \dot{\mathbf{r}}\mathbf{r}^{2} \tag{26}$$

Vc is given by Vc = (ghc)1/2; and hc = (q2/g)1/3 (critical depth).

Figure 16 contains the results obtained in the present study for the relative intensities and dimensionless kinetic energy, both plotted as a function of the dimensionless position z/zi, where z = vertical axis with origin at x=0 and positive downwards. Four different regions may be defined for the obtained graphs: (1) Single-phase growing region, (2) Single-phase

Fig. 16. Relative turbulent intensity and turbulent kinetic energy plotted against the dimensionless vertical position. The starting position of the aeration is defined as the final section of the S2 profile. (a, b) turbulent intensity; (b, c) dimensionless kinetic energy.

decay region, which is limited downwards around the point z/zi=0.9, (3) Two-phase growing region, limited by ~0.9<z/zi<~2.11, and (4) Two-phase decay region. Considering the decay region limited by 2.5<z/zi<14, a power law of the type ke=a(z/zi)-n was adjusted, obtaining n = 0.46 with a correlation coefficient of 0.72. Using the terminology of the ke- model, for which the constant C2=(n+1)/n is defined, it implies in a C2=3.7, which is about 1.7 times greater than the value of the standard model, C2=1.92 (Rodi, 1993). This analysis was conducted to verify the possibility of obtaining statistical parameters

linked to the kinetic energy, similar to those found in the literature of turbulence.
