**3. Modeling technology**

#### **3.1. Initial conditions**

The adopted mathematical model is presented in Section 2. The general technology of the numerical solution of the adopted mathematical model, the initial and boundary conditions, the model discretization methods are presented in Section 3. Section 4 contains the results of verifying the package OpenFOAM by various dam break test cases. Finally, the examples of using the OpenFOAM package for modeling the breaks of the dams of the Papan and Andijan

The mass and momentum conservation laws for viscous incompressible liquid in the absence

are the mean velocity components, *ρ* is the density, *p*¯ is the mean pressure, ¯

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 writ-

> ∂¯ *u*\_\_\_*<sup>i</sup>* ∂ *xj* + ∂¯ *u*\_\_\_*j* ∂ *xi* ) <sup>−</sup> \_\_<sup>2</sup>

The turbulence kinetic energy *k* and its dissipation rate *ε* are determined from the following

= \_\_\_<sup>∂</sup> ∂ *xj* (*μ* + *μ*\_\_*t <sup>σ</sup>k*) \_\_\_ <sup>∂</sup>*<sup>k</sup>* ∂ *xj*

> \_\_*ε <sup>k</sup>* <sup>−</sup> *<sup>ρ</sup> <sup>C</sup><sup>ε</sup>*<sup>2</sup>

*ε* 2 \_\_ *<sup>k</sup>* <sup>+</sup> \_\_\_<sup>∂</sup> ∂ *xj* ( *μ*\_\_*t σε*) \_\_\_ <sup>∂</sup>*<sup>ε</sup>* ∂ *xj*

is the rate of the generation of the turbulence kinetic energy by mean

= *C<sup>ε</sup>*<sup>1</sup> *Pk*

) is the mean tensor of viscous stresses, *μ* is the dynamic viscosity. The averaging

)/∂ *xi* = 0, (1)

, (2)

<sup>3</sup> *ij k*. (3)

+ *Pk* − , (4)

, (5)

*τij* = *μ*

*ui*

reservoirs are considered in Section 5.

<sup>∂</sup>(*<sup>ρ</sup>*¯

\_\_<sup>∂</sup>

ten in the following form [1]:

transport equations [1]:

<sup>∂</sup>(*k*) \_\_\_\_\_

<sup>∂</sup>() \_\_\_\_\_

( ∂¯ *u*\_\_\_*i* ∂ *xj* + ∂¯ *uj* \_\_\_ ∂ *xi* ) ∂¯ *u*\_\_\_*i* ∂ *xj*

*k*2 \_\_

where *Pk* <sup>=</sup> *μt*

flow, and *μt* <sup>=</sup> *<sup>ρ</sup> Cμ*

<sup>−</sup>*<sup>ρ</sup>*¯

where ¯ *ui*

60 Dam Engineering

( ∂¯ *u*\_\_\_*i* ∂ *xj* + ∂¯ *uj* \_\_\_ ∂ *xi*

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

<sup>∂</sup>*t*(*<sup>ρ</sup>*¯ *ui* ) <sup>+</sup> \_\_\_<sup>∂</sup> ∂ *xj* (*ρ*¯ *ui* ¯ *uj* + *ρ*¯ *ui* ′ *uj* ′ ) <sup>=</sup> <sup>−</sup>\_\_\_ ∂*p*¯ ∂ *xi* + ∂¯ *τ*\_\_\_*ij* ∂ *xj*

> *ui* ′ *uj* ′ <sup>=</sup> *μt*(

∂(*ρ*¯ *uj <sup>k</sup>* \_\_\_\_\_\_) ∂ *xj*

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

∂(*ρ*¯ *uj <sup>ε</sup>*) \_\_\_\_\_\_ ∂ *xj*

*<sup>ε</sup>* is the turbulent viscosity.

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

of mass forces lead to the following unsteady Navier-Stokes equations [1]:

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 integration steps. The initial distribution of the volume fraction *α* is nonuniform because not all the computational cells are filled with water.

**4. The OpenFOAM package verification**

is the gravitational acceleration.

reliability and accuracy of the present numerical modeling.

of hexahedral shape whose length is 3.22 m, height 2 m and width 1 m [5].

*g* = 9.81 m/s2

location.

**Figure 2.** Test problem configuration.

The problem on the liquid column collapse in a horizontal duct of rectangular cross section [4] is the first test problem. At the initial moment of time, the rectangular column of a viscous

Large-Scale Modeling of Dam Break Induced Flows http://dx.doi.org/10.5772/intechopen.78648 63

The numerical results (solid line) are compared with experimental data (markers) of the work [4] in **Figure 1**, where *a* = 0.05715 m is the water column width in the *x*-axis direction,

A good agreement between experiment data and results of numerical computation shows the

**Figure 2** shows the configuration of the next model problem. The model represents a reservoir

**Figure 1.** Comparison of numerical and experimental data. a—water column height change, b—the flow leading front

incompressible fluid is at rest. The column starts collapsing under the gravity force.

## **3.2. Boundary conditions**

The no-slip condition is specified at solid walls of the computational region, which gives the zero components of the velocity vector. The Neumann conditions are specified for the water volume fraction: and ∂*<sup>α</sup>*⁄∂*<sup>n</sup>* = 0; at all wall boundaries, the fixedFluxPressure boundary condition is applied to the pressure (hydrodynamic pressure) field, which adjusts the pressure gradient so that the boundary flux matches the velocity boundary condition for solvers that include body forces such as gravity and surface tension [2].

The boundary conditions for the turbulence kinetic energy *k* and its dissipation rate *ε* were specified with the aid of the technique of wall functions [1]. Systematic calculations performed in this work show that the minimum value of dimensionless distance y<sup>+</sup> for all solid wall greater than 25, so we can use wall functions technique.

The influence of surface tension forces between the solid wall and the gas-liquid mixture were not taken into account.

The top boundary is free to the atmosphere so needs to permit both outflow and inflow according to the internal flow. That is why it is necessary to use a combination of boundary conditions for pressure and velocity that does this while maintaining stability.

#### **3.3. Mesh generation and discretization of governing equations**

BlockMesh utility was used for generation of mesh. The discretization of the computational domain is obtained by the control volume method [3]. The use of an upwind difference scheme for the convective and Gauss linear scheme for diffusion terms yields an acceptable accuracy of numerical computations.

The explicit Euler first-order method was used for the discretization of the unsteady term. Numerical solution of the unsteady equations coupled with the pressure was based on the PIMPLE method [2] with the number of correctors equal to 3.

The iterative solvers PCG and PBiCG—the methods of conjugate gradients and biconjugate gradients with preconditioning were used for solving the obtained system of linear algebraic equations. The procedures based on a simplified Cholesky's incomplete factorization scheme DIC and on the simplified incomplete LU factorization DILU were used as preconditioners.

Mesh sensitivity was analyzed for four mesh of 60 × 25 × 25, 90 × 40 × 40, 135 × 60 × 60 and 150 × 80 × 80, indicating that the mesh size was important only in the vicinity of the leading front location. Since of unimportant differences between the wave fronts using mesh 135 × 60 × 60 and mesh 150 × 80 × 80, a first mesh was adopted to reduce computational effort. In this case, the minimum value of dimensionless distance y<sup>+</sup> was more than 15 for all coordinate axes.

More detailed information about the boundary and initial conditions, discretization techniques, and the solution of systems of algebraic equations may be found in the work [2].
