# An Eulerian approach for dynamic analysis of reservoir adjacent to concrete gravity dam

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## Abstract

The present paper deals with the finite element analysis of infinite reservoir adjacent to gravity dam. Two-dimensional eight-node isoparametric elements are used to discretize the domain. In order to reduce the degrees of freedom in the domain, the equation of motion for fluid motion is simulated by pressure-based Eulerian formulation. Different artificial boundary conditions are compared to obtain most suitable boundary condition. In this comparison of boundary conditions, it is noted that almost all the boundary conditions are frequency dependent. Some of the conditions are suitable for the exciting frequency less than the fundamental frequency of reservoir. However, the boundary condition proposed by Gogoi and Maity is suitable for all ranges of exciting frequencies. Further, the force vibration analysis is carried out with and without considering compressibility of water. The hydrodynamic pressure on dam is independent of exciting frequency when the compressibility of water in the reservoir is neglected. However, the effect of exciting frequency on the performances of reservoir is distinct for compressible fluid. Similarly, the magnitude and location of maximum hydrodynamic pressure change continuously, if the inclination of upstream face of dam is considered.

## Keywords

Infinite reservoir Absorbing boundary Compressibility Finite element analysis Time history analysis## 1 Introduction

The precise estimation of hydrodynamic pressure on water retaining structures such as concrete gravity dam is considered to be the major issue in design of these structures. Westergaard [1] evaluated hydrodynamic pressure on the vertical upstream face of dam. Thus, a lot of research work was carried out in this area. A group researcher [2, 3, 4, 5, 6] used displacement-based finite element method to obtain hydrodynamic pressure. Another group of researchers [7, 8, 9, 10] used Eulerian approach as the number of unknowns per node reduces to one and the critical irrotational condition of fluid motion is satisfied automatically. Again, in finite element analysis of reservoir adjacent to concrete gravity dam, difficulties arise mainly due to the infinite extend of reservoir. The first radiation boundary condition in fluid domain was proposed by Sommerfeld [11]. Similarly, Sharan [12] proposed another boundary condition considering the effect of absorption of presser wave at reservoir bottom. Maity and Bhattacharya [13] incorporated a nonreflecting condition at the artificial boundary of infinite reservoir. However, this condition cannot be defined for the case when the exciting frequency is greater than the fundamental frequency of the reservoir. Gogoi and Maity [7] proposed similar frequency-dependent absorbing boundary condition for the analysis of infinite reservoir. The hydrodynamic pressure acting on concrete gravity dam depends on several parameters such as characteristic of reservoir bottom, compressibility of water in the reservoir and the inclination of upstream face of dam. In most of the cases, the reservoir bottom is considered to be rigid and the hydrodynamic pressure on adjacent dam obtained from such cases is overestimated. Lotfi [14] studied the effect of sediment layer at reservoir bottom on the performance of reservoir. In this study, the sediment layer is considered to be viscoelastic and almost incompressible. Some researchers [12, 15, 16] implemented a damping boundary condition at the reservoir bottom to simulate the reservoir bottom absorption. In some cases, the reservoir is considering water to be incompressible [17, 18, 19]. On the other hand, Maity and Bhattacharya [13], Gogoi and Maity [7], Mirzabozorg et al. [8]. Mandal and Maity [9, 10, 20], Adhikary and Mandal [21] considered water to be compressible.

It is apparent from the literatures referred above that for precise estimation of hydrodynamic pressure on concrete gravity dam, finite element analysis is considered to be an efficient numerical tool in which water in the reservoir can be modelled as either compressible or incompressible fluid. Several truncation boundary conditions are available in the existing literatures. However, the suitability of these boundary conditions is not explained. In the present study, a computer code in MATLAB environment has been developed to compare existing boundary conditions. Further, the study is extended to observe the effect of compressibility of water and the inclination of dam upstream face.

## 2 Theoretical formulation

*p*is defined as hydrodynamic pressure whose value is independent explicitly on the rate of deformation, and \(\delta_{ij}\) is Kronecker delta. For isotropic linear elastic material, the most general form of \(T_{ij}^{'}\) is

*μ*and

*λ*are two material constants.

*μ*is known as first coefficient of viscosity or viscosity and (

*λ*+ 2

*μ*/3) is second coefficient of viscosity or bulk viscosity. \(D_{ij}^{{}}\) is the rate of deformation tensor and is expressed as

*λ*+ 2

*μ*/3) is zero. Thus, Eq. (4) becomes

*B*

_{i}is the body force and ρ is the mass density of fluid.Substituting Eq. (6) in Eq. (7), the following relations are obtained.

*u*and

*v*are the velocity components along

*x*and

*y*axes, respectively, and

*f*

_{x}and

*f*

_{y}are body forces along

*x*and

*y*direction, respectively, and if the convective terms are neglected, the equation of motion may be written as

*c*is the acoustic wave speed in fluid. Now, differentiating Eqs. (11) and (12) with respect to

*x*and

*y*, respectively, the following relations are obtained.

- (i)
*At surface*I

- (ii)
*At surface*II

*n*is the outwardly directed normal to the element surface along the interface. \(\rho_{f}\) is the mass density of the fluid.

- (iii)
*At surface*III

- (iii)
*At surface*IV

*L*represents the distance between the structure and the truncation boundary. Incorporating the effect of reservoir bottom absorption, Sharan [12] has incorporated the following condition.

### 2.1 Finite element formulation for fluid domain

*N*

_{rj}is the interpolation function for the reservoir and

*Ω*is the region under consideration. Using Green’s theorem, Eq. (26) may be transformed to

*i*varies from 1 to total number of nodes and

*Γ*represents the boundaries of the fluid domain. The last term of the above equation may be written as

*f*,

*fs*,

*fb*and

*t*stand for the free surface, fluid–structure interface, fluid–bed interface and truncation surface, respectively. For surface wave, Eq. (21a) may be written in finite element form as

*{a}*is the vector of nodal accelerations of generalized coordinates,

*{F*

_{fs}

*}*may be expressed as

*[T]*is the transformation matrix at fluid structure interface and

*N*

_{d}is the shape function of dam. At reservoir–bed interface,

For any given acceleration at the fluid–structure interface, Eq. (40) is solved to obtain the hydrodynamic pressure within the fluid.

### 2.2 Time history analysis of dynamic equilibrium equation

*F*

_{r}}. To obtain the transient response at time

*t*

_{N}, the time axis can be discretized into

*N*equal time intervals (\(t_{N} = \sum\nolimits_{j = 1}^{N} {j\Delta t}\)). The choice of method for time history analysis is strongly problem dependent. Various direct time integration methods exist for time history analysis that are expedient for structural dynamics and wave propagation problem. Amongst these, the Newmark family of methods is most popular and is given by

Here, \(\vartheta\) and \(\gamma\) are chosen to control stability and accuracy. It is evident from the literature that the integration scheme is unconditionally stable if \(2\vartheta \ge \gamma \ge 0.5\).

## 3 Results and discussions

### 3.1 Validation of the algorithm

First five natural frequencies of the reservoir

Mode number | Natural frequency (Hz) | |
---|---|---|

Present study | Sami and Lotfi [22] | |

1 | 3.121 | 3.115 |

2 | 4.812 | 4.749 |

3 | 8.101 | 7.796 |

4 | 9.897 | 9.300 |

5 | 10.324 | 9.958 |

### 3.2 Comparison of various truncation boundary conditions for infinite reservoir

*H)*is considered as 70 m. The mass density and acoustic speed of water are considered as 1000 kg/m

^{3}and 1440 m/sec, respectively. The reservoir bottom refection coefficient is taken as 0.95. The study is carried out for different values of

*T*c

*/H, i.e.*1, 4 and 100. The amplitude of the external sinusoidal excitation,

*i.e. a*, is assumed to be equal to the gravitational acceleration 1.0

*g*. The pressure coefficients at the heel of the rigid dam (

*c*

_{p}=

*P/ρaH*) are determined for different boundary conditions and presented in Table 2. It is noted that the pressure coefficient obtained from all the boundary conditions are almost equal to those obtained from closed-form solution when the truncation of infinite reservoir is at a distance of

*3.0H*. For

*T*c

*/H*= 4, the Sharan boundary [12] gives exact pressure coefficient for reservoir length equal or greater than (

*L*)

*0.5H*. However, for

*T*c

*/H*= 1 and 100, the results are almost equal to the exact results for

*L*≥ 1.0

*H*. On the other hand, for all

*T*c

*/H*values and all truncation lengths, the hydrodynamic pressure obtained from Gogoi and Maity boundary condition [7] is equal to those obtained from closed-form solution. Thus, the remaining study associates with infinite reservoir, Gogoi and Maity boundary [7] condition is implemented at the truncation surface and the infinite reservoir is truncated at a distance of 0.5

*H*from the upstream face of the dam.

Comparison of truncation boundary conditions

| | | Exact | Sommerfeld [11] | Sharan [12] | Maity and Bhattacharya [13] | Gogoi and Maity [7] | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

| % Error | | % Error | | % Error | | % Error | ||||

1 | 0.2 | 0.95 | 0.1073 | 0.8080 | 653.03 | 0.3152 | 193.8 | Not defined | 0.10378 | − 3.30 | |

0.5 | 0.6937 | 546.51 | 0.3171 | 195.5 | 0.10872 | 1.38 | |||||

0.2 | 0.5 | 0.1414 | 4.2313 | 2892.4 | 0.2930 | 107.2 | 0.14609 | 3.316 | |||

0.5 | 0.2657 | 8.80 | 0.2939 | 107.8 | 0.14359 | 1.54 | |||||

4 | 0.2 | 0.95 | 4.4364 | 33.01 | 644.07 | 0.6969 | − 84.2 | Not defined | 4.46194 | 0.57 | |

0.5 | 4.864 | 9.63 | 1.1068 | − 75.0 | 4.46040 | 0.53 | |||||

1.0 | 6.488 | 46.24 | 2.0032 | − 54.9 | 4.43624 | 0.0 | |||||

0.2 | 0.5 | 1.2187 | 56.998 | 4577 | 0.6838 | − 43.8 | 1.86418 | 52.9 | |||

0.5 | 12.658 | 93.86 | 1.0729 | − 9.74 | 1.28262 | 5.19 | |||||

1.0 | 11.661 | 852.90 | 1.3630 | 11.59 | 1.24975 | 2.48 | |||||

100 | 0.02 | 0.95 | 0.7431 | 25.618 | 3347.4 | 0.6550 | − 11.8 | 0.7402 | − 0.04 | 0.74247 | − 0.07 |

0.1 | 0.5 | 0.7430 | 5.1551 | 593.7 | 0.7064 | − 4.93 | 0.7423 | − 0.01 | 0.7431 | 0.00 | |

0.02 | 25.697 | 3347.4 | 0.6550 | − 11.8 | 0.7402 | − 0.04 | 0.74236 | − 0.08 | |||

0.1 | 5.1710 | 593.7 | 0.7064 | − 4.92 | 0.7422 | − 0.01 | 0.74301 | 0.00 |

### 3.3 Analysis of infinite reservoir with different inclinations of upstream face for compressible fluid

*θ*= 30

^{°}, 45

^{°}and 60

^{°}. From these plots, it is evident that with the increase in inclination the pressure coefficient (

*c*

_{p}) is also increased. It is also clear that the distribution for

*T*c

*/H*=

*4100*(Figs. 3, 4) is almost parabolic, but in case of

*T*c

*/H*=

*1 (*Fig. 2

*)*the distribution is slightly different. For earthquake excitation

*(Koyna earthquake),*the distribution of hydrodynamic pressure (Fig. 5) along the face of the dam follows almost similar pattern for all angles of inclination. However, the maximum hydrodynamic pressure occurs when the angle of inclination is 60

^{°}.

*θ*= 60

^{°}and this difference is comparatively higher when

*T*c

*/H*= 1.0. Similarly, for earthquake excitation, the hydrodynamic pressure increases with the increase in angle of inclination, the maximum hydrodynamic pressure occurs when the angle of the upstream face of the dam is 60

^{°}and this increase is more when the reflection coefficient is 0.95.

### 3.4 Analysis of infinite reservoir with different inclinations of upstream face for incompressible fluid

^{3}. Similar to Sect. 3.3, the study is carried out for different upstream slopes and hydrodynamic pressure at point A (Fig. 1) is determined against sinusoidal acceleration of different frequencies (Figs. 11, 12, 13). It is clear from these figure that the variation of hydrodynamic pressure with upstream slope of hydrodynamic pressure is almost similar to the trend obtained in case of compressible fluid, i.e. the hydrodynamic pressure increases with the increases in upstream slope and maximum hydrodynamic pressure occurs when the upstream slope is 60

^{°}and the variation of hydrodynamic pressure coefficient with

*T*c

*/H*is not so much significant when the compressibility of water is neglected.

## 4 Conclusions

The characteristics of hydrodynamic pressures on concrete gravity dam are studied for different exciting frequencies. The governing equation for water is expressed in terms of the pressure variable and is discretized by the finite element method. In finite element modelling of reservoir, a simple but effective boundary condition at the truncation surface of the infinite reservoir is used in the present study. The present far-boundary condition has capable of taking care of reservoir bottom absorption effects. It is observed from the results that the present far-boundary condition produces accurate results for all ranges of excitation frequencies. The results also show that the unbounded reservoir domain may be truncated even at a relatively smaller distance away from the dam, resulting in great computational advantages. The variation of hydrodynamic with different excitations of various frequencies is not significant, i.e. hydrodynamic pressure on dam remains independent of exciting frequency when the water is modelled as incompressible one. On the other hand, the hydrodynamic pressure coefficient very much depends on the frequency of excitation for compressible water. Similarly, the effect of reservoir bottom absorption is found to be small for lower values of excitation frequencies. However, this effect may not be neglected if the excitation frequency becomes equal or more than the fundamental frequency of the reservoir. The hydrodynamic pressure increases with the increase in the slope of the upstream face of the rigid dam. The distribution of hydrodynamic pressure also varies with this slope. The variation is almost parabolic for vertical upstream face, and the maximum pressure occurs at the heel of the dam. However, for other upstream slope the distribution is something different and maximum hydrodynamic pressure occurs just above the hell of rigid dam.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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