**4. Numerical simulations**

#### **4.1 Introduction**

254 Hydrodynamics – Natural Water Bodies

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

> c w' ir

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

0

0.15

ke\*

water flow

air-water flow

0.00

(c) (d)

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.

0.05

0.10

(a) (b)

0.2

0.4

ir [-]

water flow

air-water flow

e <sup>3</sup> k w'

represented by equations (25) and (26):

0

0.15

ke\*

0.00

0.05

0.10

0.2

0.4

ir [-]

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

0 9 18 27

0 9 18 27

z/zi [-]

z/zi [-]

2

<sup>2</sup> (24)

<sup>V</sup> (25)

<sup>2</sup> k \* ir <sup>e</sup> (26)

0246

0246

z/zi [-]

z/zi [-]

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

$$\left(\mathbf{L}\_{\mathbf{k}} \;/\; \eta\right)^{3} = \mathrm{Re}^{9/4} \tag{27}$$

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.

#### **4.2 Some previous studies**

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

Stepped Spillways: Theoretical, Experimental and Numerical Studies 257

Persistent air cavities below the pseudo-bottom were observed in all simulated results, which is an "uncomfortable" characteristic of the simulations, because the experimental observations did not present such cavities (assuming, as usual, that the numerical solutions converged to the analytical solutions, which, on its turn, is viewed as a good model of the real flows). It must be emphasized that the predictions reproduce the one-phase flow, but that the two-phase flow presents undulating characteristics still not reproduced by numerical simulations. The undulating aspect is observed for different experimental conditions, as shown by Simões et al. (2011), and a complete quantification is still not

The inhomogeneous model and the k- turbulence model were selected to obtain the freesurface profiles using "numerical" scales compatible with prototype scales. The simulations were performed for steady state turbulence, with s = 0.60 m (27 simulations) and s = 2.4 m (four simulations with 1V:0.75H), considering two-dimensional domains, using high resolution numerical schemes and applying boundary conditions similar to those already described in the first example. The inlet condition is the same for all simulations (Figure 18a). The angles between the pseudobottom and the horizontal, and the dimensionless parameter s/hc, chosen for the simulations were: 53.13º (0.133s/hc0.845), 45º (0.11s/hc0.44), 30.96º (0.11s/hc0.44), and 11.31º (0.133s/hc0.44). For each experiment a numerical value for the resistance factor was calculated, as described in item 4.3.1. Figure 18b, shows the analytical solution and the points obtained with the Reynolds Averaged

available.

VE hE

a2 a3

**4.3.2 Simulations using prototype sizes** 

<sup>P</sup> Hdam

a4

showed similar or superior quality to that presented in Figure 18.

0.0

(a) (b)

Navier-Stokes Equations (RANS) for multiphase flows. The axes shown in Figure 18b represent the following nondimensional parameters: =h/hc and H=z/hc. In this case, z has origin at the critical section, where h=hc (close to the crest of the spillway). All the results

Figure 19a shows the distribution of the friction factor values obtained numerically, for all simulations performed for the geometrical conditions described in Figure 18a. Figure 19b

Fig. 18. (a) Domain employed to perform the simulations for =53.13 (colors = void fraction). The values of ai (i=1-5), P, Hdam, hE were chosen for each test (b) S2 profile:

0 8 16

**H [-]**

RANS, Sim. 1.3 Analytical solution

0.5

1.0

 **[-]**

a1

a5

numerical and analytical solution.

#### **4.3 Results**

#### **4.3.1 Free surface comparisons**

The experiments summarized in Table 1 were also simulated, in order to verify the possibilities of reproducing such flows using CFD. The Exp. 15 is the only one shown here, which main characteristics may be found in Table 1, and which was simulated considering the hypothesis of two-dimensional flow for the geometry sketched in Figure 17a. The inlet velocity was set as the mean measured velocity, with the value of 2.91 m/s, the outlet boundary condition was set to extrapolate the volume fractions of air and water. The analytical solution presented by Simões et al. (2010) was used to calculate the theoretical profile of the free surface for the single phase flow, for which f = 0.041 resulted as the adjusted resistance factor. Figure 17b contains experimental data and numerical solutions calculated with different meshes and the following turbulence models: zero equation, k-, RNG k- and SSG. These results were obtained combining the non-homogeneous model and the free-surface model for the interfacial transfers. There is excellent agreement between the experimental points, the numerical results and theoretical curve for the one-phase region. It was found that the use of the mesh denoted by M2 (data indicated in the legend) led to results similar to those obtained with the mesh denoted by M1, two times more refined,

Fig. 17. Exp. 18 simulations: (a) Geometry and dimensions (in cm), (b) Experimental results, numerical solutions and analytical profile (ke=k-; M1 and M2 are unstructured meshes with ~0.5x106 and ~0.25x106 elements, respectively; M3 is a structured mesh with ~0.2x106 elements).

when using the k- model. Further, the results obtained with the models RNG k-, SSG and without transport equations for turbulence (the three calculated using a third mesh denoted by M3) also superposed well the theoretical solution. In addition to the mentioned turbulence models, also the models k-, BSL and k- EARMS were tested. Only the k- EARMS model produced results with quality similar to those shown in Figure 17b. The k and BSL models overestimated the depth of the flow, with maximum relative deviations from the experimental values near 8%. The same deviation was observed when using the mixture model in place of the free-surface model (for simulations using the k-model). Persistent air cavities below the pseudo-bottom were observed in all simulated results, which is an "uncomfortable" characteristic of the simulations, because the experimental observations did not present such cavities (assuming, as usual, that the numerical solutions converged to the analytical solutions, which, on its turn, is viewed as a good model of the real flows). It must be emphasized that the predictions reproduce the one-phase flow, but that the two-phase flow presents undulating characteristics still not reproduced by numerical simulations. The undulating aspect is observed for different experimental conditions, as shown by Simões et al. (2011), and a complete quantification is still not available.

#### **4.3.2 Simulations using prototype sizes**

256 Hydrodynamics – Natural Water Bodies

The experiments summarized in Table 1 were also simulated, in order to verify the possibilities of reproducing such flows using CFD. The Exp. 15 is the only one shown here, which main characteristics may be found in Table 1, and which was simulated considering the hypothesis of two-dimensional flow for the geometry sketched in Figure 17a. The inlet velocity was set as the mean measured velocity, with the value of 2.91 m/s, the outlet boundary condition was set to extrapolate the volume fractions of air and water. The analytical solution presented by Simões et al. (2010) was used to calculate the theoretical profile of the free surface for the single phase flow, for which f = 0.041 resulted as the adjusted resistance factor. Figure 17b contains experimental data and numerical solutions calculated with different meshes and the following turbulence models: zero equation, k-, RNG k- and SSG. These results were obtained combining the non-homogeneous model and the free-surface model for the interfacial transfers. There is excellent agreement between the experimental points, the numerical results and theoretical curve for the one-phase region. It was found that the use of the mesh denoted by M2 (data indicated in the legend) led to results similar to those obtained with the mesh denoted by M1, two times more refined,

11.04

0.2

(a) (b)

Fig. 17. Exp. 18 simulations: (a) Geometry and dimensions (in cm), (b) Experimental results, numerical solutions and analytical profile (ke=k-; M1 and M2 are unstructured meshes with ~0.5x106 and ~0.25x106 elements, respectively; M3 is a structured mesh with ~0.2x106

when using the k- model. Further, the results obtained with the models RNG k-, SSG and without transport equations for turbulence (the three calculated using a third mesh denoted by M3) also superposed well the theoretical solution. In addition to the mentioned turbulence models, also the models k-, BSL and k- EARMS were tested. Only the k- EARMS model produced results with quality similar to those shown in Figure 17b. The k and BSL models overestimated the depth of the flow, with maximum relative deviations from the experimental values near 8%. The same deviation was observed when using the mixture model in place of the free-surface model (for simulations using the k-model).

0 5 10 15

**H [-]**

Experimental Analytical solution M1, ke M2, ke M3, RNG ke M3, SSG M3, Zero eq.

0.3

0.4

 

0.5

**4.3 Results** 

7.42

5

3.61

2.8

elements).

**4.3.1 Free surface comparisons** 

The inhomogeneous model and the k- turbulence model were selected to obtain the freesurface profiles using "numerical" scales compatible with prototype scales. The simulations were performed for steady state turbulence, with s = 0.60 m (27 simulations) and s = 2.4 m (four simulations with 1V:0.75H), considering two-dimensional domains, using high resolution numerical schemes and applying boundary conditions similar to those already described in the first example. The inlet condition is the same for all simulations (Figure 18a). The angles between the pseudobottom and the horizontal, and the dimensionless parameter s/hc, chosen for the simulations were: 53.13º (0.133s/hc0.845), 45º (0.11s/hc0.44), 30.96º (0.11s/hc0.44), and 11.31º (0.133s/hc0.44). For each experiment a numerical value for the resistance factor was calculated, as described in item 4.3.1. Figure 18b, shows the analytical solution and the points obtained with the Reynolds Averaged

Fig. 18. (a) Domain employed to perform the simulations for =53.13 (colors = void fraction). The values of ai (i=1-5), P, Hdam, hE were chosen for each test (b) S2 profile: numerical and analytical solution.

Navier-Stokes Equations (RANS) for multiphase flows. The axes shown in Figure 18b represent the following nondimensional parameters: =h/hc and H=z/hc. In this case, z has origin at the critical section, where h=hc (close to the crest of the spillway). All the results showed similar or superior quality to that presented in Figure 18.

Figure 19a shows the distribution of the friction factor values obtained numerically, for all simulations performed for the geometrical conditions described in Figure 18a. Figure 19b

Stepped Spillways: Theoretical, Experimental and Numerical Studies 259

The authors thank CNPq(141078/2009-0), CAPES and FAPESP, Brazilian research support

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Boes, R.M. & Hager, W.H. (2003a). Hydraulic Design of Stepped Spillways. ASCE, Journal of

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Chamani, M.R. & Rajaratnam, N. (1999a). Characteristic of skimming flow over stepped spillways. ASCE, *Journal of Hydraulic Engineering*. v.125, n.4, p.361-368, April. Chamani, M. R.; Rajaratnam, N. (1999b). Onset of skimming flow on stepped spillways.

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**6. Acknowledgements** 

Portuguese].

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contains the distribution of f considering the experimental and the numerical data together. It was possible to adjust power laws between f and k/hc (k=scos), in the form f = a(k/hc)b. The values of the adjusted "a" and "b", and the limits of validity of the adjusted equations, together with the geometrical information, are given in Table 2.

Fig. 19. Friction factor: (a) Probability distribution function for 11.31º51.13o and numerical results; (b) Probability distribution function considering the numerical (11.31º51.13o) and the experimental results together (=45o). The total area covered by the bars is equal to 1.0 in both figures (R = correlation coefficient; Re = 4q/).


Table 2. Coefficients for f = a(k/hc)b and other details.
