**2.2 Skimming flow**

#### **2.2.1 Energy dissipation**

The energy dissipation of flows along stepped spillways is one of the most important characteristics of these structures. For this reason, several researchers have endeavored to provide equations and charts to allow predictions of the energy dissipation and the residual energy at the toe of stepped spillways and channels. Different studies were performed in different institutions around the world, representing the flows and the related phenomena from different points of view, for example, using the Darcy-Weisbach or the Manning equations, furnishing algebraic equations fitted to experimental data, presenting experimental points by means of graphs, or simulating results using different numerical schemes.

Stepped Spillways: Theoretical, Experimental and Numerical Studies 241

This equation expresses the propagation of the uncertainty of f, for which it was assumed that the errors are statistically independent and that the function f = f (q, h) varies smoothly

> <sup>2</sup> <sup>2</sup> fq h 4 9 fq h

Assuming If = 10 (that is, no uncertainty for If), h = 0.05 0.001 m and q = 0.25 0.005 m2/s, the relative uncertainty of the resistance factor is around 7.2%. The real difficulty in defining the position of the free surface imposes higher relative uncertainties. So, for h = 3 mm, we have f/f = 18.4% and for h = 5 mm, the result is f/f = 30.3%. These h values

Figure 4 contains sequential images of a multiphase flow, obtained by Simões (2011). They illustrate a single oscillation of the mean position of the surface with amplitude close to 15 mm. The first three pictures were taken under ambient lighting conditions, generating images similar to the perception of the human eye. The last two photographs were obtained with a high speed camera, showing that the shape of the surface is highly irregular, with portions of fluid forming a typical macroscopic interface under turbulent motion. It is evident that the method used to measure the depth of such flows may lead to incorrect results if these aspects

Figure 4 shows that it is difficult to define the position of the free surface. Simões et al. (2011) used an ultrasonic sensor, a high frequency measurement instrument for data acquisition, during a fairly long measurement time, and presented results of the evolution of the two-phase flow that show a clear oscillating pattern, also allowing to observe a transition length between the "full water" and "full mixture" regions of the flows along stepped spillways. Details on similar aspects for smooth spillways were presented by

are not well defined and the measurement equipment is not adequate.

(3)

with respect to the error propagation.

are possible in laboratory measurements.

Fig. 4. Behavior of the free surface (>1)

#### **Darcy-Weisbach resistance function ("friction factor")**

The Darcy-Weisbach resistance function has been widely adopted in studies of stepped spillways. It can be obtained following arguments based on physical arguments or based on a combination of experimental information and theoretical principles. In the first case, dimensional analysis is used together with empirical knowledge about the energy evolution along the flow. In the second case, the principle of conservation of momentum is used together with experimental information about the averaged shear stress on solid surfaces. Of course, the result is the same following both points of view. The dimensional analysis is interesting, because it shows that the "resistance factor" is a function of several nondimensional parameters. The most widespread resistance factor equation, probably due to its strong predictive characteristic, is that deduced for flows in circular pipes. For this flows, the resistance factor is expressed as a function of only two nondimensional parameters: the relative roughness and the Reynolds number. When applying the same analysis for stepped channels, the resistance factor is expressed as dependent on more nondimensional parameters, as illustrated by eq. 1:

$$\mathbf{f} = \Phi\_1 \Big( \text{Re}, \text{Fr}, \alpha, \frac{\mathbf{k}}{\mathbf{L}\_\text{c}}, \frac{\mathbf{e}\_\text{p}}{\mathbf{L}\_\text{c}}, \frac{\mathbf{e}\_\text{e}}{\mathbf{L}\_\text{c}}, \frac{\mathbf{e}\_\text{m}}{\mathbf{L}\_\text{c}}, \frac{\mathbf{s}}{\mathbf{L}\_\text{c}}, \frac{1}{\mathbf{L}\_\text{c}}, \frac{\mathbf{L}\_\text{c}}{\mathbf{B}}, \mathbf{C} \Big) \tag{1}$$

f is the resistance factor. Because the obtained equation is identical to the Darcy-Weisbach equation, the name is preserved. The other variables are: Re = Reynolds number, Fr = Froude number, = atg(s/l), k = scos, Lc = characteristic length, = sand roughness (the subscripts "p", "e "and "m" correspond to the floor of the step, to the vertical step face and the side walls, respectively), s = step height, l = step length, B = width of the channel, C = void fraction.

Many equations for f have been proposed for stepped channels since 1990. Due to the practical difficulties in measuring the position of the free surface accurately and to the increasing of the two-phase region, the values of the resistance factor presented in the literature vary in the range of about 0.05 to 5! There are different causes for this range, which details are useful to understand it. It is known that, by measuring the depth of the mixture and using this result in the calculation of f, the obtained value is higher than that calculated without the volume of air. This is perhaps one of the main reasons for the highest values. On the other hand, considering the lower values (the range from 0.08 to 0.2, for example), they may be also affected by the difficulty encountered when measuring depths in multiphase flows. Even the depths of the single-phase region are not easy to measure, because high-frequency oscillations prevent the precise definition of the position of the free surface, or its average value. Let us consider the following analysis, for which the Darcy-Weisbach equation was rewritten to represent wide channels

$$\mathbf{f} = \frac{8\mathbf{g}\mathbf{h}^3\mathbf{I}\_\mathbf{f}}{\mathbf{q}^2} \tag{2}$$

in which: g = acceleration of the gravity, h = flow depth, If = slope of the energy line, q = unit discharge. The derivative of equation (2), with respect to f and h, results 3 f 3 f 16gh I q q 

$$\text{and } \frac{\partial \mathbf{f}}{\partial \mathbf{h}} = \frac{24 \text{gh}^2 \mathbf{I}\_{\mathbf{f}}}{\mathbf{q}^2}, \text{ respectively, which are used to obtain equation 3.1.}$$

The Darcy-Weisbach resistance function has been widely adopted in studies of stepped spillways. It can be obtained following arguments based on physical arguments or based on a combination of experimental information and theoretical principles. In the first case, dimensional analysis is used together with empirical knowledge about the energy evolution along the flow. In the second case, the principle of conservation of momentum is used together with experimental information about the averaged shear stress on solid surfaces. Of course, the result is the same following both points of view. The dimensional analysis is interesting, because it shows that the "resistance factor" is a function of several nondimensional parameters. The most widespread resistance factor equation, probably due to its strong predictive characteristic, is that deduced for flows in circular pipes. For this flows, the resistance factor is expressed as a function of only two nondimensional parameters: the relative roughness and the Reynolds number. When applying the same analysis for stepped channels, the resistance factor is expressed as dependent on more

<sup>p</sup> e c <sup>m</sup> <sup>1</sup>

f is the resistance factor. Because the obtained equation is identical to the Darcy-Weisbach equation, the name is preserved. The other variables are: Re = Reynolds number, Fr = Froude number, = atg(s/l), k = scos, Lc = characteristic length, = sand roughness (the subscripts "p", "e "and "m" correspond to the floor of the step, to the vertical step face and the side walls, respectively), s = step height, l = step length, B = width of the channel, C = void fraction. Many equations for f have been proposed for stepped channels since 1990. Due to the practical difficulties in measuring the position of the free surface accurately and to the increasing of the two-phase region, the values of the resistance factor presented in the literature vary in the range of about 0.05 to 5! There are different causes for this range, which details are useful to understand it. It is known that, by measuring the depth of the mixture and using this result in the calculation of f, the obtained value is higher than that calculated without the volume of air. This is perhaps one of the main reasons for the highest values. On the other hand, considering the lower values (the range from 0.08 to 0.2, for example), they may be also affected by the difficulty encountered when measuring depths in multiphase flows. Even the depths of the single-phase region are not easy to measure, because high-frequency oscillations prevent the precise definition of the position of the free surface, or its average value. Let us consider the following analysis, for which the Darcy-

> 3 f 2 8gh I <sup>f</sup>

in which: g = acceleration of the gravity, h = flow depth, If = slope of the energy line, q =

unit discharge. The derivative of equation (2), with respect to f and h, results

, respectively, which are used to obtain equation 3.

<sup>q</sup> (2)

<sup>k</sup> s l <sup>L</sup> f Re,Fr, , , , , , , , ,C LLLLLL B 

cccccc

(1)

3 f 3 f 16gh I q q 

**Darcy-Weisbach resistance function ("friction factor")** 

nondimensional parameters, as illustrated by eq. 1:

Weisbach equation was rewritten to represent wide channels

and

2 f 2 f 24gh I h q

This equation expresses the propagation of the uncertainty of f, for which it was assumed that the errors are statistically independent and that the function f = f (q, h) varies smoothly with respect to the error propagation.

$$\frac{\Delta \mathbf{f}}{\mathbf{f}} = \sqrt{4 \left(\frac{\Delta \mathbf{q}}{\mathbf{q}}\right)^2 + 9 \left(\frac{\Delta \mathbf{h}}{\mathbf{h}}\right)^2} \tag{3}$$

Assuming If = 10 (that is, no uncertainty for If), h = 0.05 0.001 m and q = 0.25 0.005 m2/s, the relative uncertainty of the resistance factor is around 7.2%. The real difficulty in defining the position of the free surface imposes higher relative uncertainties. So, for h = 3 mm, we have f/f = 18.4% and for h = 5 mm, the result is f/f = 30.3%. These h values are possible in laboratory measurements.

Fig. 4. Behavior of the free surface (>1)

Figure 4 contains sequential images of a multiphase flow, obtained by Simões (2011). They illustrate a single oscillation of the mean position of the surface with amplitude close to 15 mm. The first three pictures were taken under ambient lighting conditions, generating images similar to the perception of the human eye. The last two photographs were obtained with a high speed camera, showing that the shape of the surface is highly irregular, with portions of fluid forming a typical macroscopic interface under turbulent motion. It is evident that the method used to measure the depth of such flows may lead to incorrect results if these aspects are not well defined and the measurement equipment is not adequate.

Figure 4 shows that it is difficult to define the position of the free surface. Simões et al. (2011) used an ultrasonic sensor, a high frequency measurement instrument for data acquisition, during a fairly long measurement time, and presented results of the evolution of the two-phase flow that show a clear oscillating pattern, also allowing to observe a transition length between the "full water" and "full mixture" regions of the flows along stepped spillways. Details on similar aspects for smooth spillways were presented by

Stepped Spillways: Theoretical, Experimental and Numerical Studies 243

The flows along smooth spillways have some characteristics that coincide with those presented by flows along stepped channels. The initial region of the flow is composed only by water ("full water region" 1 in Figure 5a), with a free surface apparently smooth. The position where the thickness of the boundary layer coincides with the depth of flow defines the starting point of the superficial aeration, or inception point (see Figure 5). In this position the effects of the bed on the flow can be seen at the surface, distorting it intensively. Downstream, a field of void fraction C(xi,t) is generated, which depth along x1 (longitudinal

The flow in smooth channels indicates that the region (1) is generally monophasic, the same occurring in stepped spillways. However, channels having short side entrances like those used for drainage systems, typically operate with aerated flows along all their extension, from the beginning of the flow until its end. Downstream of the inception point a twodimensional profile of the mean void fraction C is formed, denoted by C \* . From a given position x1 the so called "equilibrium" is established for the void fraction, which implies that C \* C \*(x ) <sup>1</sup> . Different studies, like those of Straub & Anderson (1958), Keller et al. (1974), Cain & Wood (1981) and Wood et al. (1983) showed results consistent with the above descriptions, for flows in smooth spillways. Figure 5b shows the classical sketch for the evolution of two-phase flows, as presented by Keller et al. (1974). Wilhelms & Gulliver (2005) introduced the concepts of entrained air and entrapped air, which correspond respectively to the air flow really incorporated by the water flow and carried away in the form of bubbles, and to the air surrounded by the twisted shape of the free surface, and not

coordinate) increases from the surface to the bottom, as illustrated in Figure 5.

Fig. 5. Skimming flow and possible classifications of the different regions

One of the first studies describing coincident aspects between flows along smooth and stepped channels was presented by Sorensen (1985), containing an illustration indicating the inception point of the aeration and describing the free surface as smooth upstream of this point (Fig. 6a). Peyras et al. (1992) also studied the flow in stepped channels formed by gabions, showing the inception point, as described by Sorensen (1985) (see Figure 6b).

Sources: (a) Simões (2011), (b) Keller *et al.* (1974)

**2.2.2 Two phase flow** 

incorporated by the water.

Wilhelms & Gulliver (2005), while reviews of equations and values for the resistance factor were presented by Chanson (2002), Frizell (2006), Simões (2008), and Simões et al. (2010).

#### **Energy dissipation**

The energy dissipated in flows along stepped spillways can be defined as the difference between the energy available near the crest and the energy at the far end of the channel, denoted by H throughout this chapter. Selecting a control volume that involves the flow of water between the crest (section 0) and a downstream section (section 1), the energy equation can be written as follows:

$$\mathbf{z}\_0 + \frac{\mathbf{p}\_0}{\gamma} + \mathbf{a}\_0 \frac{\mathbf{V}\_0^2}{2\mathbf{g}} = \mathbf{z}\_1 + \frac{\mathbf{p}\_1}{\gamma} + \mathbf{a}\_1 \frac{\mathbf{V}\_1^2}{2\mathbf{g}} + \Delta \mathbf{H} \tag{4}$$

According to the characteristics of flow and the channel geometry, the flows across these sections can consist of air/water mixtures. Assuming hydrostatic pressure distributions, such that p0/ = h0 and p1/ = h1cos (Chow, 1959), the previous equation can be rewritten as:

$$\begin{split} \mathbf{A} \mathbf{H} &= \overbrace{\mathbf{z}\_{0} - \mathbf{z}\_{1}}^{\mathbf{H}\_{\text{dam}}} + \mathbf{h}\_{0} + \alpha\_{0} \frac{\mathbf{q}^{2}}{2 \text{g} \mathbf{h}\_{0}^{2}} - \left( \mathbf{h}\_{1} \cos \alpha + \alpha\_{1} \frac{\mathbf{q}^{2}}{2 \text{g} \mathbf{h}\_{1}^{2}} \right) = \\ &= \left( \mathbf{H}\_{\text{dam}} + \mathbf{h}\_{0} + \alpha\_{0} \frac{\mathbf{h}\_{\text{c}}^{3}}{2 \text{h}\_{0}^{2}} \right) \left[ 1 - \left( \mathbf{h}\_{1} \cos \alpha + \alpha\_{1} \frac{\mathbf{q}^{2}}{2 \text{g} \mathbf{h}\_{1}^{2}} \right) \Big/ \left( \mathbf{H}\_{\text{dam}} + \mathbf{h}\_{0} + \alpha\_{0} \frac{\mathbf{h}\_{\text{c}}^{3}}{2 \text{h}\_{0}^{2}} \right) \right] \end{split}$$

Denoting 3 <sup>c</sup> dam 0 0 <sup>2</sup> 0 <sup>h</sup> H h 2h by Hmax, the previous equation is replaced by:

$$\frac{\Delta \mathbf{H}}{\mathbf{H}\_{\text{max}}} = 1 - \left(\frac{\mathbf{h}\_1}{\mathbf{h}\_c}\right) \left(\frac{\cos \alpha + \alpha\_1 \frac{\mathbf{h}\_c^3}{2\mathbf{h}\_1^3}}{\frac{\mathbf{H}\_{\text{dam}}}{\mathbf{h}\_c} + \frac{\mathbf{h}\_0}{\mathbf{h}\_c} + \alpha\_0 \frac{\mathbf{h}\_c^2}{2\mathbf{h}\_0^2}}\right) \tag{5}$$

Taking into account the width of the channel, and using the Darcy-Weisbach equation for a rectangular channel in conjunction with equation 5, the following result is obtained:

$$\frac{\Delta \mathbf{H}}{\mathbf{H}\_{\text{max}}} = 1 - \left\{ \frac{\left[\frac{8 \mathbf{I}\_{\text{f}}}{(1 + 2 \mathbf{h}\_{1} / \text{B}) \mathbf{f}}\right]^{-1/3} \cos \alpha + \frac{\alpha\_{1}}{2} \left[\frac{8 \mathbf{I}\_{\text{f}}}{(1 + 2 \mathbf{h}\_{1} / \text{B}) \mathbf{f}}\right]^{2/3}}{\frac{\mathbf{H}\_{\text{dam}}}{\mathbf{h}\_{\text{c}}} + \frac{\mathbf{h}\_{0}}{\mathbf{h}\_{\text{c}}} + \alpha\_{0} \frac{\mathbf{h}\_{\text{c}}^{2}}{2 \mathbf{h}\_{0}^{2}}} \right\} \tag{6}$$

Rajaratnam (1990), Stephenson (1991), Hager (1995), Chanson (1993), Povh (2000), Boes & Hager (2003a), Ohtsu et al. (2004), among others, presented conceptual and empirical equations to calculate the dissipated energy. In most of the cases, the conceptual models can be obtained as simplified forms of equation 6, which is considered a basic equation for flows in spillways.

#### **2.2.2 Two phase flow**

242 Hydrodynamics – Natural Water Bodies

Wilhelms & Gulliver (2005), while reviews of equations and values for the resistance factor were presented by Chanson (2002), Frizell (2006), Simões (2008), and Simões et al. (2010).

The energy dissipated in flows along stepped spillways can be defined as the difference between the energy available near the crest and the energy at the far end of the channel, denoted by H throughout this chapter. Selecting a control volume that involves the flow of water between the crest (section 0) and a downstream section (section 1), the energy

According to the characteristics of flow and the channel geometry, the flows across these sections can consist of air/water mixtures. Assuming hydrostatic pressure distributions, such that p0/ = h0 and p1/ = h1cos (Chow, 1959), the previous equation can be rewritten

0 1

 

1 1

f 1 f 1 1

8I 8I cos

<sup>0</sup> <sup>2</sup> c c <sup>0</sup>

h h 2h

cos

2gh 2gh

c c dam 0 0 2 22 <sup>1</sup> <sup>1</sup> dam 0 0

h h <sup>q</sup> H h 1 h cos /H h 2h 2gh 2h

by Hmax, the previous equation is replaced by:

<sup>2</sup> max <sup>c</sup> dam 0 c

Taking into account the width of the channel, and using the Darcy-Weisbach equation for a

<sup>H</sup> (1 2h /B)f 2 (1 2h /B)f <sup>1</sup>

Rajaratnam (1990), Stephenson (1991), Hager (1995), Chanson (1993), Povh (2000), Boes & Hager (2003a), Ohtsu et al. (2004), among others, presented conceptual and empirical equations to calculate the dissipated energy. In most of the cases, the conceptual models can be obtained as simplified forms of equation 6, which is considered a basic equation for flows

rectangular channel in conjunction with equation 5, the following result is obtained:

<sup>2</sup> max dam 0 c

H Hh h

H h 2h <sup>1</sup> H h Hh h

3 3 2

0 10

<sup>0</sup> <sup>2</sup> c c <sup>0</sup>

h h 2h

 

1/3 2/3

h

Hdam 2 2 01 0 0 1 1 2 2

q q Hz z h h cos

3

0

2h

2 2 0 0 1 1 0 01 1 p V p V zz H 2g 2g 

(4)

(5)

(6)

**Energy dissipation** 

as:

Denoting

in spillways.

equation can be written as follows:

<sup>c</sup> dam 0 0 <sup>2</sup>

<sup>h</sup> H h

The flows along smooth spillways have some characteristics that coincide with those presented by flows along stepped channels. The initial region of the flow is composed only by water ("full water region" 1 in Figure 5a), with a free surface apparently smooth. The position where the thickness of the boundary layer coincides with the depth of flow defines the starting point of the superficial aeration, or inception point (see Figure 5). In this position the effects of the bed on the flow can be seen at the surface, distorting it intensively. Downstream, a field of void fraction C(xi,t) is generated, which depth along x1 (longitudinal coordinate) increases from the surface to the bottom, as illustrated in Figure 5.

The flow in smooth channels indicates that the region (1) is generally monophasic, the same occurring in stepped spillways. However, channels having short side entrances like those used for drainage systems, typically operate with aerated flows along all their extension, from the beginning of the flow until its end. Downstream of the inception point a twodimensional profile of the mean void fraction C is formed, denoted by C \* . From a given position x1 the so called "equilibrium" is established for the void fraction, which implies that C \* C \*(x ) <sup>1</sup> . Different studies, like those of Straub & Anderson (1958), Keller et al. (1974), Cain & Wood (1981) and Wood et al. (1983) showed results consistent with the above descriptions, for flows in smooth spillways. Figure 5b shows the classical sketch for the evolution of two-phase flows, as presented by Keller et al. (1974). Wilhelms & Gulliver (2005) introduced the concepts of entrained air and entrapped air, which correspond respectively to the air flow really incorporated by the water flow and carried away in the form of bubbles, and to the air surrounded by the twisted shape of the free surface, and not incorporated by the water.

Fig. 5. Skimming flow and possible classifications of the different regions Sources: (a) Simões (2011), (b) Keller *et al.* (1974)

One of the first studies describing coincident aspects between flows along smooth and stepped channels was presented by Sorensen (1985), containing an illustration indicating the inception point of the aeration and describing the free surface as smooth upstream of this point (Fig. 6a). Peyras et al. (1992) also studied the flow in stepped channels formed by gabions, showing the inception point, as described by Sorensen (1985) (see Figure 6b).

Stepped Spillways: Theoretical, Experimental and Numerical Studies 245

As can be seen, stepped chutes are a matter of intense studies, related to the complex

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

(a) (b)

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

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

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,

phenomena that take place in the flows along such structures.

tubes (Fig. 7b), used for confirmation of the values of the water discharge.

(7) outlet channel; (8) weir, (9) final outlet channel.

position of the free surface in hydraulic jumps.

10. Use of spaced steps; 11. Inclined step and end sills; 12. Side walls converging; 13. Use of precast steps; 14. Length of stilling basins.

**3. Experimental study 3.1 General information** 

Fig. 6. Illustration of the flow Reference: (a) Sorensen (1985, p.1467) and (b) Peyras *et al.* (1992, p.712).

The sketch of Figure 6b emphasizes the existence of rolls downstream from the inception position of the aeration. Further experimental studies, such as Chamani & Rajaratnam (1999a, p.363) and Ohtsu et al. (2001, p.522), showed that the incorporated air flow distributes along the depth of the flow and reaches the cavity below the pseudo-bottom, where large eddies are maintained by the main flow.

The mentioned studies of multiphase flows in spillways (among others) thus generated predictions for: (1) the position of the inception point of aeration, (2) profiles of void fractions (3) averages void fractions over the spillways, (4) characteristics of the bubbles. As mentioned, frequently the conclusions obtained for smooth spillways were used as basis for studies in stepped spillways. See, for example, Bauer (1954), Straub & Anderson (1958), Keller & Rastogi (1977), Cain & Wood (1981), Wood (1984), Tozzi (1992), Chanson (1996), Boes (2000), Chanson (2002), Boes & Hager (2003b) and Wilhelms & Gulliver (2005).
