**2. The mathematical model of the dam break flows**

The mass and momentum conservation laws for viscous incompressible liquid in the absence of mass forces lead to the following unsteady Navier-Stokes equations [1]:

$$
\partial (\rho \mathbf{\overline{r}\_i}) / \partial \mathbf{x\_i} = \mathbf{0},\tag{1}
$$

The coefficients of the model have the following standard values [1]: *Cμ* <sup>=</sup> 0.09, *<sup>C</sup><sup>ε</sup>*<sup>1</sup> <sup>=</sup> 1.44, *<sup>C</sup><sup>ε</sup>*<sup>2</sup> <sup>=</sup> 1.92,

The method of determining the interface between two phases—water and air occupies special place at the modeling of the flow class under consideration. According to the main idea of the volume of fluid (VOF) method [1], one determines for each computational cell a scalar quantity, which represents the degree of filling the cell with one phase, for example, water. If this quantity is equal to 0, it is empty; if it is equal to 1, then it is filled completely. If its value lies between 0 and 1, then one can say, respectively, that this cell contains the free (interphase) boundary. In other words, the volume fraction of water *α* is determined as the ratio of the water volume in the cell to the total volume of the given cell. The quantity 1 − *α* represents, respectively, the volume fraction of the second phase—air in the given cell. At the initial moment of time, one specifies the distribution of the field of this quantity, and its further temporal and spatial evolutions are computed from the following transport

<sup>∂</sup>*<sup>t</sup>* <sup>+</sup>

∂(*α*¯*ui* ) \_\_\_\_\_ ∂ *xi*

The free boundary location is determined by the equation *α*(*x*, *y*, *z*, *t*) = 0. Therefore, the physical properties of the gas-liquid mixture are determined by averaging with the corresponding

The essence of the VOF method implemented in the solver interFoam of the OpenFOAM package [2] lies in the fact that the interface between two phases is not computed explicitly, but is determined, to some extent, as a property of the field of the water volume fraction. Since the volume fraction values are between 0 and 1, the interphase boundary is not determined accurately; however, it occupies some region, where a sharp interphase boundary must exist

In the case of unsteady problem, it is necessary to specify for the initial values for all dependent variables. The values of all velocity components are equal to zero because according to the condition of the problem under study, there is no motion until the moment of time *t* = 0. The hydrodynamic pressure is also equal to zero since the used solver—interFoam calculates hydrodynamic pressure [2]. The turbulence kinetic energy and its dissipation rate have some small value, which ensures a good convergence of the numerical solution at the first

, *μ* = <sup>1</sup> + (1 − *α*) *μ*<sup>2</sup>

= 0. (6)

Large-Scale Modeling of Dam Break Induced Flows http://dx.doi.org/10.5772/intechopen.78648 61

. (7)

*<sup>σ</sup><sup>k</sup>* <sup>=</sup> 1.0, *<sup>σ</sup><sup>k</sup>* <sup>=</sup> 1.3.

equation [1]:

weight coefficient:

in the proximity.

**3. Modeling technology**

**3.1. Initial conditions**

\_\_\_ <sup>∂</sup>*<sup>α</sup>*

*ρ* = <sup>1</sup> + (1 − *α*) *ρ*<sup>2</sup>

Here, the subscripts 1 and 2 refer to the liquid and gaseous phases.

$$
\partial(\rho \mathbf{\overline{x}\_{i}}) \partial \mathbf{x}\_{i} = 0,\tag{1}
$$

$$
\frac{\partial}{\partial t}(\rho \mathbf{\overline{x}\_{i}}) + \frac{\partial}{\partial \mathbf{x}\_{j}}(\rho \mathbf{\overline{u}\_{i}} \mathbf{\overline{x}\_{j}} + \rho \mathbf{\overline{u\_{i}} \mathbf{\overline{u}\_{j}}}) = -\frac{\partial \overline{p}}{\partial \mathbf{x}\_{i}} + \frac{\partial \overline{\mathbf{r}\_{i}}}{\partial \mathbf{x}\_{j}^{\prime}}.\tag{2}
$$

where ¯ *ui* are the mean velocity components, *ρ* is the density, *p*¯ is the mean pressure, ¯ *τij* = *μ* ( ∂¯ *u*\_\_\_*i* ∂ *xj* + ∂¯ *uj* \_\_\_ ∂ *xi* ) is the mean tensor of viscous stresses, *μ* is the dynamic viscosity. The averaging is done in time, and the prime denotes the fluctuation part of the velocity. In the presence of external forces, it is necessary to augment these equations by the corresponding terms.

To close the systems of Eqs. (1) and (2), it is necessary to introduce some turbulence model. Most turbulence models employed in practice are based on the concepts of turbulent viscosity and turbulent diffusion. For the flows of general form, the turbulent viscosity introduced by Boussinesq, which couples the Reynolds stresses with the mean flow gradients, may be written in the following form [1]: <sup>−</sup>*<sup>ρ</sup>*¯

$$-\rho \overrightarrow{u\_i u\_j} = \mu\_i \left(\frac{\partial \overline{u}\_i}{\partial x\_j} + \frac{\partial \overline{u}\_j}{\partial x\_i}\right) - \frac{2}{3} \,\rho \delta\_{ij}\,k. \tag{3}$$

The turbulence kinetic energy *k* and its dissipation rate *ε* are determined from the following transport equations [1]:

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho \mathbf{\overline{u}} \%)}{\partial \mathbf{x}\_{j}} = \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \mu + \frac{\mu\_{t}}{\sigma\_{k}} \right) \frac{\partial k}{\partial \mathbf{x}\_{j}} + P\_{k} - \rho \mathbf{e}\_{\prime} \tag{4}$$

$$\frac{\partial(\rho\varepsilon)}{\partial t} + \frac{\partial(\rho\overline{u}/\varepsilon)}{\partial x\_{\dot{\gamma}}} = \mathbf{C}\_{\varepsilon1} P\_k \frac{\varepsilon}{k} - \rho \mathbf{C}\_{\varepsilon2} \frac{\varepsilon^2}{k} + \frac{\partial}{\partial x\_{\dot{\gamma}}} \left(\frac{\mu\_t}{\sigma\_\ast}\right) \frac{\partial \varepsilon}{\partial x\_{\dot{\gamma}}} \tag{5}$$

where *Pk* <sup>=</sup> *μt* ( ∂¯ *u*\_\_\_*i* ∂ *xj* + ∂¯ *uj* \_\_\_ ∂ *xi* ) ∂¯ *u*\_\_\_*i* ∂ *xj* is the rate of the generation of the turbulence kinetic energy by mean flow, and *μt* <sup>=</sup> *<sup>ρ</sup> Cμ k*2 \_\_ *<sup>ε</sup>* is the turbulent viscosity.

The coefficients of the model have the following standard values [1]: *Cμ* <sup>=</sup> 0.09, *<sup>C</sup><sup>ε</sup>*<sup>1</sup> <sup>=</sup> 1.44, *<sup>C</sup><sup>ε</sup>*<sup>2</sup> <sup>=</sup> 1.92, *<sup>σ</sup><sup>k</sup>* <sup>=</sup> 1.0, *<sup>σ</sup><sup>k</sup>* <sup>=</sup> 1.3.

The method of determining the interface between two phases—water and air occupies special place at the modeling of the flow class under consideration. According to the main idea of the volume of fluid (VOF) method [1], one determines for each computational cell a scalar quantity, which represents the degree of filling the cell with one phase, for example, water. If this quantity is equal to 0, it is empty; if it is equal to 1, then it is filled completely. If its value lies between 0 and 1, then one can say, respectively, that this cell contains the free (interphase) boundary. In other words, the volume fraction of water *α* is determined as the ratio of the water volume in the cell to the total volume of the given cell. The quantity 1 − *α* represents, respectively, the volume fraction of the second phase—air in the given cell. At the initial moment of time, one specifies the distribution of the field of this quantity, and its further temporal and spatial evolutions are computed from the following transport equation [1]:

$$\frac{\partial a}{\partial t} + \frac{\partial (a \text{tr}\_{\cdot})}{\partial x\_{i}} = 0. \tag{6}$$

The free boundary location is determined by the equation *α*(*x*, *y*, *z*, *t*) = 0. Therefore, the physical properties of the gas-liquid mixture are determined by averaging with the corresponding weight coefficient:

$$
\rho = a\rho\_1 + (1 - a)\,\rho\_2\,\mu = a\mu\_1 + (1 - a)\,\mu\_2. \tag{7}
$$

Here, the subscripts 1 and 2 refer to the liquid and gaseous phases.

The essence of the VOF method implemented in the solver interFoam of the OpenFOAM package [2] lies in the fact that the interface between two phases is not computed explicitly, but is determined, to some extent, as a property of the field of the water volume fraction. Since the volume fraction values are between 0 and 1, the interphase boundary is not determined accurately; however, it occupies some region, where a sharp interphase boundary must exist in the proximity.
