**4. The OpenFOAM package verification**

integration steps. The initial distribution of the volume fraction *α* is nonuniform because not

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

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

The influence of surface tension forces between the solid wall and the gas-liquid mixture were

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

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

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

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,

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].

was more than 15 for all coordinate axes.

for all solid wall

in this work show that the minimum value of dimensionless distance y<sup>+</sup>

conditions for pressure and velocity that does this while maintaining stability.

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

PIMPLE method [2] with the number of correctors equal to 3.

the minimum value of dimensionless distance y<sup>+</sup>

all the computational cells are filled with water.

body forces such as gravity and surface tension [2].

greater than 25, so we can use wall functions technique.

**3.2. Boundary conditions**

62 Dam Engineering

not taken into account.

accuracy of numerical computations.

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

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, *g* = 9.81 m/s2 is the gravitational acceleration.

A good agreement between experiment data and results of numerical computation shows the reliability and accuracy of the present numerical modeling.

**Figure 2** shows the configuration of the next model problem. The model represents a reservoir of hexahedral shape whose length is 3.22 m, height 2 m and width 1 m [5].

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

**Figure 2.** Test problem configuration.

The water column of height *H* = 0.6 m, length 1.2 m and width 1 m lies in the left lower corner of the reservoir. To measure the pressure force of fluid flow on the right reservoir wall, the corresponding pressure probe was placed at point Р with coordinates *x* = 3.22 m and *y* = 0.16 m. Besides, the levels *h*<sup>1</sup> and *h*<sup>2</sup> of the water free surface were measured in two sections at *x*<sup>1</sup> = 2.725 m and *x*<sup>2</sup> = 2.228 m. The density of water was equal to 998.2 kg/m3 , and that of air was equal to 1.225 kg/m3 .

water with the formation of a typical bend of its surface. The reverse wave formed in such a way reaches the main flow, impinges onto it and forms the secondary wave, and so on (at *t* = 1.5 s and *t* = 1.8 s). After the moment of time *t* = 2.2 s, the water inertia drops significantly,

**Figure 4** shows the numerical (solid line) and corresponding experimental (markers) data [5] on the water column height in sections with coordinates *х*<sup>1</sup> = 2.725 m and *х*<sup>2</sup> = 2.228 m. The coincidence between these data is satisfactory up to the moment of time *t* ≈ 1.5 s for section

After this moment of time, the reverse wave moving oppositely to the main stream impinges onto the free surface, and this gives rise to certain inaccuracies both in numerical and experimental data. The inaccuracies of such a kind were also observed in the work [6], where the wellknown commercial package FLUENT was used for numerical modeling. The results of this work

nondimensionalized here by quantity *Н*, and the time is presented in the dimensionless form

initial water column height. One can conclude from a comparison of **Figures 4** and **5** that the free surface levels in two different sections have been predicted more accurately in the present work. The problem of determining the pressure *P* on the obstacle is very important at the solution of unsteady problems with free surface, in particular, at the interaction of forming waves with various obstacles. **Figure 6** shows the numerical results (solid line) for the pressure of the fluid on the right wall at point P with coordinates (*x* = 3.2 m, *y* = 0.16 m) and the corresponding

**Figure 5.** Water column height from the work [6] at *х* = 2.725 m (a) and 2.228 m (b). 1—experiment [5], 2—computation

, *h*2

is the free-fall acceleration, *Н* = 0.6 m is the

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

) has been

65

are presented in **Figure 5** in a form similar to **Figure 4**. The water column height (*h*<sup>1</sup>

0.5, where *t* is the physical time, *g* = 9.81 m/s2

**Figure 4.** Water column height in different sections. *х*<sup>1</sup> = 2.725 m (a) and *x*<sup>2</sup> = 2.228 m (b).

and the further consideration of the motion is of no practical interest.

*х*<sup>1</sup> = 2.725 m (**Figure 4a**).

*τ* = *t*(*g*/*H*)

of the work [6].

As initial configuration of the modeling with OpenFOAM, the water column in the left bottom part of the domain is at rest. When diaphragm is removed suddenly simulation is started, due to gravity water column drops and starts to move into the empty part of the domain. The total duration of modeling in time amounted to 2.5 s. In **Figure 3**, we have several snapshots of some stages of the simulation.

The diaphragm is suddenly removed at the moment of time *t* = 0, and the water column runs under the gravity force into the right empty part of the reservoir. At the moment of time *t* ≈ 0.65 s, the water reaches the right wall and impinging on it under the inertia force, moves upward. The flow is thinned as it moves upward along the right wall and at the moment of time *t* = 1.3 s, when the gravity force exceeds the inertia force; there occurs a reverse flow of

**Figure 3.** The motion of water column at different time step.

water with the formation of a typical bend of its surface. The reverse wave formed in such a way reaches the main flow, impinges onto it and forms the secondary wave, and so on (at *t* = 1.5 s and *t* = 1.8 s). After the moment of time *t* = 2.2 s, the water inertia drops significantly, and the further consideration of the motion is of no practical interest.

**Figure 4** shows the numerical (solid line) and corresponding experimental (markers) data [5] on the water column height in sections with coordinates *х*<sup>1</sup> = 2.725 m and *х*<sup>2</sup> = 2.228 m. The coincidence between these data is satisfactory up to the moment of time *t* ≈ 1.5 s for section *х*<sup>1</sup> = 2.725 m (**Figure 4a**).

After this moment of time, the reverse wave moving oppositely to the main stream impinges onto the free surface, and this gives rise to certain inaccuracies both in numerical and experimental data. The inaccuracies of such a kind were also observed in the work [6], where the wellknown commercial package FLUENT was used for numerical modeling. The results of this work are presented in **Figure 5** in a form similar to **Figure 4**. The water column height (*h*<sup>1</sup> , *h*2 ) has been nondimensionalized here by quantity *Н*, and the time is presented in the dimensionless form *τ* = *t*(*g*/*H*) 0.5, where *t* is the physical time, *g* = 9.81 m/s2 is the free-fall acceleration, *Н* = 0.6 m is the initial water column height. One can conclude from a comparison of **Figures 4** and **5** that the free surface levels in two different sections have been predicted more accurately in the present work.

The problem of determining the pressure *P* on the obstacle is very important at the solution of unsteady problems with free surface, in particular, at the interaction of forming waves with various obstacles. **Figure 6** shows the numerical results (solid line) for the pressure of the fluid on the right wall at point P with coordinates (*x* = 3.2 m, *y* = 0.16 m) and the corresponding

**Figure 4.** Water column height in different sections. *х*<sup>1</sup> = 2.725 m (a) and *x*<sup>2</sup> = 2.228 m (b).

**Figure 3.** The motion of water column at different time step.

*y* = 0.16 m. Besides, the levels *h*<sup>1</sup>

of some stages of the simulation.

was equal to 1.225 kg/m3

64 Dam Engineering

The water column of height *H* = 0.6 m, length 1.2 m and width 1 m lies in the left lower corner of the reservoir. To measure the pressure force of fluid flow on the right reservoir wall, the corresponding pressure probe was placed at point Р with coordinates *x* = 3.22 m and

As initial configuration of the modeling with OpenFOAM, the water column in the left bottom part of the domain is at rest. When diaphragm is removed suddenly simulation is started, due to gravity water column drops and starts to move into the empty part of the domain. The total duration of modeling in time amounted to 2.5 s. In **Figure 3**, we have several snapshots

The diaphragm is suddenly removed at the moment of time *t* = 0, and the water column runs under the gravity force into the right empty part of the reservoir. At the moment of time *t* ≈ 0.65 s, the water reaches the right wall and impinging on it under the inertia force, moves upward. The flow is thinned as it moves upward along the right wall and at the moment of time *t* = 1.3 s, when the gravity force exceeds the inertia force; there occurs a reverse flow of

of the water free surface were measured in two sections

, and that of air

and *h*<sup>2</sup>

.

at *x*<sup>1</sup> = 2.725 m and *x*<sup>2</sup> = 2.228 m. The density of water was equal to 998.2 kg/m3

**Figure 5.** Water column height from the work [6] at *х* = 2.725 m (a) and 2.228 m (b). 1—experiment [5], 2—computation of the work [6].

experimental data (markers). The exact pressure value at point Р cannot be measured because the pressure probes have a finite size—a circle about 90 mm in diameter.

The numerical pressure at the pressure probe center (see **Figure 6a**) increases slowly with time, after the moment of time *t* = 1.5 s or after the second maximum the coincidence of experimental data with numerical results improves. The character of the numerical pressure at the lower point of the probe (see **Figure 6b**) agrees fairly well with the character of the variation of corresponding experimental data; however, the maximum values are slightly underestimated.

It is assumed at the numerical modeling that the diaphragm is removed suddenly that is its vertical velocity is infinite. On the other hand, there can be also different physical conditions, which are hard to take into account at the numerical modeling. A detailed analysis of the conditions for conducting the experiment shows that this velocity has a finite value. The verification experimental data under the same conditions give different results, which do not coincide with one another [5]. In addition, the above discrepancies between the computation and experiment after the moment of time *t* = 1.5 s can probably be explained by the two-dimensionality of the employed model. It is possible that the flow acquires a threedimensional character at some points of the computational region.

Comparing the data of the present work (**Figure 4**) and the work [6] (**Figure 5**), one can assert that the numerical results of modeling the task under consideration, which were obtained with the aid of the open package OpenFOAM, are closer to the experimental data than the results obtained with the aid of the commercial package FLUENT.

water—Н4, and the remaining ones Н1, Н2, and Н3 were located in the reservoir empty part.

**Figure 8.** Comparison of numerical (on the left) and experimental data [7] (on the right) at the moment of time *t* = 0.4 s.

Large-Scale Modeling of Dam Break Induced Flows http://dx.doi.org/10.5772/intechopen.78648 67

The container was supplied with eight pressure probes: four probes on the exterior surface at points with coordinates *z* = 0.025, 0.063, 0.099 and 0.136 m, and the remaining four probes were located on the container upper side with coordinates *x* = 0.806, 0.769, 0.733 and 0.696 m. The probes on the exterior surface were located at the distance of 0.026 m to the left of the central line (*y* = 0), and the probes on the upper surface were located at the distance of 0.026 m of this line. The force exerted on the container on the water stream side was also measured in experiment. The water was at rest up to the moment of time *t* = 0. At the moment of time *t* = 0, the separating wall was removed suddenly, and the water column ran into the reservoir empty part under the gravity force. The 180 × 60 × 80 computational grid was used, and the CPU time amounted to 6 s. The initial time step was 0.001 s, and it was varied further depending on the

**Figures 8** and **9** show the comparison of numerical and experimental data for the moments of time *t* = 0.4 and 0.6 s, respectively. The pictures of shooting done during the experiment are shown on the right. One can notice a fairly good visual coincidence of the numerical results

The time of reaching the container by water flow both in experiment and at the numerical simulation is the same. Besides, the free surface shapes forming after the flow impact onto the container also coincide. One can note, however, that there are at the numerical modeling some

**Figure 10** shows the water flow height at two different points: in the reservoir and in the immediate proximity of the container. There is a fairly good agreement between them until

imperfections of the free surface between water and the ambient medium—air.

The coordinates of these probes are *x* = 0.5, 1.0, 1.5 and 2.66 m, respectively.

**Figure 7.** Configuration (on the left) and location of measurement pressure probes (on the right).

Courant number, which was equal to 0.85.

with experimental data.

The problem of the fluid column breakdown in a reservoir of rectangular shape with an obstacle [7] is the next, more complex test problem. The chosen coordinate system and the problem diagram without the geometric proportion preservation are shown in **Figure 7**.

An open reservoir 3.22 m in length with cross section of 1 × 1 m2 has been used in experiment. The reservoir was initially partitioned into two unequal parts by a vertical wall located in section *х* = 2 m. The water 0.55 m in height is located behind this wall, another reservoir part is empty. A container 40 cm in length with cross section of 16 × 16 cm2 is located in this empty part of reservoir. The container left-face coordinate equals *х* = 0.67 m.

At the execution of experiment, the water column height and the fluid pressure on the container surface were measured. The location of measuring probes is shown in **Figure 8**. Four probes were used to measure the water column height: the one on the part filled with

**Figure 6.** Pressure variation at point Р. Pressure at the center (a) and at the lower point (b) of the probe.

**Figure 7.** Configuration (on the left) and location of measurement pressure probes (on the right).

experimental data (markers). The exact pressure value at point Р cannot be measured because

The numerical pressure at the pressure probe center (see **Figure 6a**) increases slowly with time, after the moment of time *t* = 1.5 s or after the second maximum the coincidence of experimental data with numerical results improves. The character of the numerical pressure at the lower point of the probe (see **Figure 6b**) agrees fairly well with the character of the variation of corresponding experimental data; however, the maximum values are slightly

It is assumed at the numerical modeling that the diaphragm is removed suddenly that is its vertical velocity is infinite. On the other hand, there can be also different physical conditions, which are hard to take into account at the numerical modeling. A detailed analysis of the conditions for conducting the experiment shows that this velocity has a finite value. The verification experimental data under the same conditions give different results, which do not coincide with one another [5]. In addition, the above discrepancies between the computation and experiment after the moment of time *t* = 1.5 s can probably be explained by the two-dimensionality of the employed model. It is possible that the flow acquires a three-

Comparing the data of the present work (**Figure 4**) and the work [6] (**Figure 5**), one can assert that the numerical results of modeling the task under consideration, which were obtained with the aid of the open package OpenFOAM, are closer to the experimental data than the

The problem of the fluid column breakdown in a reservoir of rectangular shape with an obstacle [7] is the next, more complex test problem. The chosen coordinate system and the problem

The reservoir was initially partitioned into two unequal parts by a vertical wall located in section *х* = 2 m. The water 0.55 m in height is located behind this wall, another reservoir part

At the execution of experiment, the water column height and the fluid pressure on the container surface were measured. The location of measuring probes is shown in **Figure 8**. Four probes were used to measure the water column height: the one on the part filled with

has been used in experiment.

is located in this empty

diagram without the geometric proportion preservation are shown in **Figure 7**.

**Figure 6.** Pressure variation at point Р. Pressure at the center (a) and at the lower point (b) of the probe.

the pressure probes have a finite size—a circle about 90 mm in diameter.

dimensional character at some points of the computational region.

results obtained with the aid of the commercial package FLUENT.

An open reservoir 3.22 m in length with cross section of 1 × 1 m2

is empty. A container 40 cm in length with cross section of 16 × 16 cm2

part of reservoir. The container left-face coordinate equals *х* = 0.67 m.

underestimated.

66 Dam Engineering

**Figure 8.** Comparison of numerical (on the left) and experimental data [7] (on the right) at the moment of time *t* = 0.4 s.

water—Н4, and the remaining ones Н1, Н2, and Н3 were located in the reservoir empty part. The coordinates of these probes are *x* = 0.5, 1.0, 1.5 and 2.66 m, respectively.

The container was supplied with eight pressure probes: four probes on the exterior surface at points with coordinates *z* = 0.025, 0.063, 0.099 and 0.136 m, and the remaining four probes were located on the container upper side with coordinates *x* = 0.806, 0.769, 0.733 and 0.696 m. The probes on the exterior surface were located at the distance of 0.026 m to the left of the central line (*y* = 0), and the probes on the upper surface were located at the distance of 0.026 m of this line. The force exerted on the container on the water stream side was also measured in experiment.

The water was at rest up to the moment of time *t* = 0. At the moment of time *t* = 0, the separating wall was removed suddenly, and the water column ran into the reservoir empty part under the gravity force. The 180 × 60 × 80 computational grid was used, and the CPU time amounted to 6 s. The initial time step was 0.001 s, and it was varied further depending on the Courant number, which was equal to 0.85.

**Figures 8** and **9** show the comparison of numerical and experimental data for the moments of time *t* = 0.4 and 0.6 s, respectively. The pictures of shooting done during the experiment are shown on the right. One can notice a fairly good visual coincidence of the numerical results with experimental data.

The time of reaching the container by water flow both in experiment and at the numerical simulation is the same. Besides, the free surface shapes forming after the flow impact onto the container also coincide. One can note, however, that there are at the numerical modeling some imperfections of the free surface between water and the ambient medium—air.

**Figure 10** shows the water flow height at two different points: in the reservoir and in the immediate proximity of the container. There is a fairly good agreement between them until the water returns from the back wall after the moment of time *t* ≈ 1.8 s. After that, the numerical data (solid lines) prove to be somewhat higher than the experimental ones (markers). At the moment of time of *t* ≈ 5 s, the secondary wave reaches the neighborhood of the probe Н2. This time is, however, equal to about 5.3 s at the numerical modeling. The general character of the variations of the numerical and experimental data nevertheless coincides.

**Figure 11** shows the moment of time *t* = 0.5 s when the wave reaches the container has been predicted with a good accuracy; however, the computed pressure value (solid lines) is slightly

Large-Scale Modeling of Dam Break Induced Flows http://dx.doi.org/10.5772/intechopen.78648 69

The numerical values of the second pressure maximum at point P2 are, however, shifted in comparison with experiment to the right by 0.6 s, and at point P7, they are shifted by 0.5 s. As the experiment has shown, the moment of time when the flow reaches repeatedly the container (≈4.7 s) is seen in these figures fairly well. Besides, when comparing the numerical and experimental pressure values at points Р7 (the right figure) one can notice some differences. After *t* = 1.3 s, a small oscillation lasting for about 0.2 s takes place in numerical computations,

To illustrate the techniques of the application of numerical modeling of large-scale hydrodynamic computations, we consider the problem of computing the flood process in the areas

It is to be emphasized here that the situation of a real breakthrough of the dam and the flood of the areas at the lower level is not modeled here but the fundamental possibility of using the above technology under the availability of necessary topography data is demonstrated. The topography data of Digital Terrain Elevation Data [8] were used in computations, which were converted subsequently into the STL format. The hexahedral background grid generated with the aid of the utilities blockMesh and snappyHexMesh of the OpenFOAM package was transformed into a three-dimensional surface, which is employed for modeling the flood process (**Figure 13**).

For the Andijan reservoir, the computational field had the sizes 6000 × 4000 × 1500 m, the physical modeling time amounted to about 9 h for the 120 × 120 × 80 grid. **Figures 14** and **15** show different stages of the flood in areas with real topology. The red corresponds to a pure

overestimated as compared to the experimental value (markers) (the left figure).

**5. Dam break flooding flow modeling in real region**

**Figure 12.** Maps of the Andijan (left) and Papan (right) reservoirs.

near the dams of the Andijan and the Papan reservoirs (see **Figure 12**).

which is not observed in experiment.

**Figure 9.** Comparison of numerical (left) and experimental [7] (right) data at the moment of time *t* = 0.6 s.

**Figure 10.** Water flow heights at points Н2 (left) and Н4 (right).

**Figure 11.** Pressure at points Р2 (left) and Р7 (right).

**Figure 11** shows the moment of time *t* = 0.5 s when the wave reaches the container has been predicted with a good accuracy; however, the computed pressure value (solid lines) is slightly overestimated as compared to the experimental value (markers) (the left figure).

The numerical values of the second pressure maximum at point P2 are, however, shifted in comparison with experiment to the right by 0.6 s, and at point P7, they are shifted by 0.5 s. As the experiment has shown, the moment of time when the flow reaches repeatedly the container (≈4.7 s) is seen in these figures fairly well. Besides, when comparing the numerical and experimental pressure values at points Р7 (the right figure) one can notice some differences. After *t* = 1.3 s, a small oscillation lasting for about 0.2 s takes place in numerical computations, which is not observed in experiment.
