4. Mathematical model

In order to fully present the hydrodynamics in the dam reservoir, volume of fluid (VOF) model should be developed. Owing to large size and complexity of the computational grid, as well as limited computer capacity, such an approach would be very difficult to apply for a big water bodies. In the literature, VOF technique was adopted only for the CFD modeling of reservoir in a downscale model (1:50) [22].

Sulejow reservoir due to the location, shape of the bowl, and uncovered, flat shores is particularly exposed to the effect of wind. Direction and energy of wind determine the waving movement and mixing of the water, so the impact of the factor, on flow distribution in the CFD simulations, should be taken into consideration.

In the model under wind conditions, the speed, at which the plate was moving on the water surface (which reflects the speed and direction of the wind in the area of the Sulejow reservoir), was determined. For this purpose, independent, 2-D, two-phase problem was resolved. Data concerning wind parameters were provided by the Institute of Meteorology and Water Management in Warsaw and come from the meteorological station (Sulejow-Kopalnia) located near the reservoir. Average speed and direction of wind were selected as 3 m/s and southeast, respectively, based on the daily data from year 2007.

#### 4.1. Simplified VOF CFD model

The 2-D model of a straight channel with a length of 80 m and a width of 6 m (3 m layer of water, 3 m layer of air) was elaborated. Figure 10 shows the hexahedral, structured mesh which has been generated for the simplified geometry. The grid contains 159,242 elements Three-Dimensional CFD Simulations of Hydrodynamics for the Lowland Dam Reservoir http://dx.doi.org/10.5772/intechopen.80377 49

Figure 10. Fragment of the structured, hexahedral mesh generated with Gambit program.

(159,116 hexahedra, 126 wedges) and 321,872 nodes, respectively. A boundary layer was generated consisting of 10 rows.

Wind accelerates surface fluid particles by imparting momentum to the fluid, through surface stresses. In the analysis, the water flows through the air momentum. In the computational domain, the water phase has two outlets that allow for fluid reversing. Analysis of flow in 2-D model was intended to determine which boundary condition best describes the situation that prevails over the water surface by the effect of wind.

#### 4.2. Boundary conditions

The quality of the numerical grid was determined by the shape and the size of the computing field and the total number of elements used in the generated numerical grid and through the position of the first node relative to the plane of the wall. To assess the quality of the numerical

, skewness, and aspect ratio for the analyzed numerical grid.

Parameter Values for numerical grids in this work The limit values in the numerical simulations

—determining the quality of the mesh in the boundary layer

In order to fully present the hydrodynamics in the dam reservoir, volume of fluid (VOF) model should be developed. Owing to large size and complexity of the computational grid, as well as limited computer capacity, such an approach would be very difficult to apply for a big water bodies. In the literature, VOF technique was adopted only for the CFD modeling of reservoir in

Sulejow reservoir due to the location, shape of the bowl, and uncovered, flat shores is particularly exposed to the effect of wind. Direction and energy of wind determine the waving movement and mixing of the water, so the impact of the factor, on flow distribution in the

In the model under wind conditions, the speed, at which the plate was moving on the water surface (which reflects the speed and direction of the wind in the area of the Sulejow reservoir), was determined. For this purpose, independent, 2-D, two-phase problem was resolved. Data concerning wind parameters were provided by the Institute of Meteorology and Water Management in Warsaw and come from the meteorological station (Sulejow-Kopalnia) located near the reservoir. Average speed and direction of wind were selected as 3 m/s and southeast,

The 2-D model of a straight channel with a length of 80 m and a width of 6 m (3 m layer of water, 3 m layer of air) was elaborated. Figure 10 shows the hexahedral, structured mesh which has been generated for the simplified geometry. The grid contains 159,242 elements

, skewness, and aspect ratio for the numerical grid generated in this

grid elements, three parameters were used (Gambit User Guide):

<sup>y</sup><sup>+</sup> 1.8 <sup>≤</sup><sup>2</sup> Skewness 0.8 0–1 Aspect ratio 1.7 ≥1

• "Skewness"—determining the quality of the individual grid elements

• "Aspect ratio"—defining the degree of deformation of the mesh elements

• Parameter y+

48 Dam Engineering

Ranges of parameters: y<sup>+</sup>

Table 1. Values of parameter: y+

work are given in Table 1.

4. Mathematical model

a downscale model (1:50) [22].

4.1. Simplified VOF CFD model

CFD simulations, should be taken into consideration.

respectively, based on the daily data from year 2007.

For two-dimensional CFD model, the following boundary conditions were imposed:.

Inlet boundary conditions applied for the analyzed domain were two tributaries (Pilica and Luciaza rivers) and outlet (dam) as given in Figure 11.

Simulated inflow boundaries were specified with mass flow rates, normal to the boundary. Velocities at each inlet were calculated from the inlet area measurements of stream flow for the Pilica and Luciaza rivers, made by the Regional Board and Water Management in Warsaw in 2007. The monthly values of mass flow rates in the Pilica and Luciaza rivers are presented in Table 2.

The definition of the inlet requires the values of the velocity vectors and turbulence properties. For the air inlet, the simulations were first conducted at a speed of 3 m/s. Velocity profile obtained at the outlet of the computational domain was loaded as an input file to receive the velocity profile at the inlet. This approach allows to obtain a fully developed velocity profile for a small domain.

Implementation of volume of fluid model requires an additional boundary condition to be specified, namely, the turbulence intensity at the inlet and turbulent viscosity ratio. Introduction of disturbance into the flow reflects the real features of the flow pattern.

Another parameter, the turbulent viscosity ratio (the ratio of turbulent to laminar (molecular) viscosity), was used as given in Eq. (2). The default value of 10% was applied in the simulations:

Three-Dimensional CFD Simulations of Hydrodynamics for the Lowland Dam Reservoir

<sup>β</sup> <sup>¼</sup> vt

The air outlet of pressure type was defined in the model. Pressure outlet boundary conditions require specification of a static pressure at the outflow. Convergence difficulties were minimized by specified values for the backflow quantities (backflow turbulence intensity and

Table 3 summarizes the solution conditions and methods used in the modeling process.

Based on the analysis of two-dimensional two-phase model, an appropriate input data for the wall boundary condition, corresponding to real wind conditions at the surface, could be determined. Figure 12 depicts the velocity profile of water at the outlet of the computational domain. As a result, the replacement velocity of wind, which corresponds to the real, average value in the area of the Sulejow reservoir, was obtained. The mean value of 0.147 m/s was used

Due to the fact that the Sulejow reservoir is a shallow, polymictic lake, the wind will be important for the distribution and mixing of the water masses of two tributaries. The results

Space Two-dimensional

Gradient Least squares cell based Pressure Body force weighted

Momentum Second-order upwind

Turbulence energy kinetic Second-order upwind Specific dissipation rate Second-order upwind

Time Steady

Pressure–velocity coupling scheme Coupled

Volume fraction Compressive

viscosity ratio). A no-slip boundary condition was applied on the wall.

in the simulations of flow hydrodynamics in the modeled reservoir.

4.5. Results of CFD calculations at wind conditions

Table 3. Solution conditions and methods for volume of fluid model.

shown in Figure 13 confirm this assumption.

4.3. Solution methods

4.4. Results of VOF model

Model volume of fluid

Discretization method

<sup>v</sup> (2)

http://dx.doi.org/10.5772/intechopen.80377

51

Figure 11. Inlet region in the analyzed domain.


Table 2. The monthly values of mass flow rate in the Pilica and Luciaza rivers in 2007.

The turbulence intensity, I [%], is defined as the ratio of the root-mean-square of the velocity fluctuations u` to the mean flow velocity Uavg. The turbulence intensity at the core of a fully developed duct flow can be estimated from Eq. (1):

$$I = \frac{\mu}{\mathcal{U}\_{avg}} = 0,16\Re^{\ddagger} \tag{1}$$

Kennedy [15] carried out sensitivity analysis on the turbulence intensity in the simple, channel geometry and found that value 4% was able to match the conditions well. The author concluded that this parameter had a noticeable effect on the solution. Following the above findings, turbulence intensity level of 4% was specified at the inlet.

Another parameter, the turbulent viscosity ratio (the ratio of turbulent to laminar (molecular) viscosity), was used as given in Eq. (2). The default value of 10% was applied in the simulations:

$$
\beta = \frac{v\_t}{v} \tag{2}
$$

The air outlet of pressure type was defined in the model. Pressure outlet boundary conditions require specification of a static pressure at the outflow. Convergence difficulties were minimized by specified values for the backflow quantities (backflow turbulence intensity and viscosity ratio). A no-slip boundary condition was applied on the wall.

#### 4.3. Solution methods

Table 3 summarizes the solution conditions and methods used in the modeling process.

#### 4.4. Results of VOF model

The turbulence intensity, I [%], is defined as the ratio of the root-mean-square of the velocity fluctuations u` to the mean flow velocity Uavg. The turbulence intensity at the core of a fully

Kennedy [15] carried out sensitivity analysis on the turbulence intensity in the simple, channel geometry and found that value 4% was able to match the conditions well. The author concluded that this parameter had a noticeable effect on the solution. Following the above find-

<sup>¼</sup> <sup>0</sup>, <sup>16</sup>R�<sup>1</sup>

<sup>8</sup> (1)

/s) Luciaza river (mass flow rate m<sup>3</sup>

/s)

<sup>I</sup> <sup>¼</sup> <sup>u</sup> Uavg

developed duct flow can be estimated from Eq. (1):

Figure 11. Inlet region in the analyzed domain.

50 Dam Engineering

Month Pilica river (mass flow rate m<sup>3</sup>

January 19.86 2.80 February 46.31 5.50 March 22.33 4.92 April 23.04 2.19 May 17.36 1.39 June 15.65 1.64 July 16.45 1.91 August 12.83 1.39 September 13.21 1.59 October 16.37 1.69 November 25.50 2.21 December 25.65 2.55

Table 2. The monthly values of mass flow rate in the Pilica and Luciaza rivers in 2007.

ings, turbulence intensity level of 4% was specified at the inlet.

Based on the analysis of two-dimensional two-phase model, an appropriate input data for the wall boundary condition, corresponding to real wind conditions at the surface, could be determined. Figure 12 depicts the velocity profile of water at the outlet of the computational domain. As a result, the replacement velocity of wind, which corresponds to the real, average value in the area of the Sulejow reservoir, was obtained. The mean value of 0.147 m/s was used in the simulations of flow hydrodynamics in the modeled reservoir.

#### 4.5. Results of CFD calculations at wind conditions

Due to the fact that the Sulejow reservoir is a shallow, polymictic lake, the wind will be important for the distribution and mixing of the water masses of two tributaries. The results shown in Figure 13 confirm this assumption.


Table 3. Solution conditions and methods for volume of fluid model.

Figure 12. The velocity profile at the interface of water-air.

Figure 13. Comparison of velocity field (m/s) in the Sulejow reservoir in October at wind and no-wind conditions.

The findings suggest that when steady flow pattern develops in the basin, large regions of recirculation are formed below the outlet of the reservoir. Figure 14 shows contours of velocity field in the Sulejow reservoir in (a) March (b) July, and (c) December.

Figure 14. Contours of velocity field (m/s) in the Sulejow reservoir in (a) march (b) July, and (c) December.

Three-Dimensional CFD Simulations of Hydrodynamics for the Lowland Dam Reservoir

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53

Three-Dimensional CFD Simulations of Hydrodynamics for the Lowland Dam Reservoir http://dx.doi.org/10.5772/intechopen.80377 53

Figure 14. Contours of velocity field (m/s) in the Sulejow reservoir in (a) march (b) July, and (c) December.

The findings suggest that when steady flow pattern develops in the basin, large regions of recirculation are formed below the outlet of the reservoir. Figure 14 shows contours of velocity

Figure 13. Comparison of velocity field (m/s) in the Sulejow reservoir in October at wind and no-wind conditions.

field in the Sulejow reservoir in (a) March (b) July, and (c) December.

Figure 12. The velocity profile at the interface of water-air.

52 Dam Engineering
