Finite Element Modelling Of The Reservoir

As it was discussed in previous Chapter, the hydrodynamic pressure distribution in the reservoir is governed by the pressure wave equation. Assuming that water is linearly compressible and neglecting its viscosity, the small amplitude irrotational motion of water is governed by the two-dimensional wave equation:

where p(x,y,t) is the hydrodynamic pressure in excess of hydrostatic pressure, C is the velocity of pressure wave in water andxand y are the coordinate axes.

The hydrodynamic pressure in the impounded water governed by equation 3.6, is due to the horizontal and the vertical accelerations of the upstream face of the dam, the reservoir bottom as well as the far end of the reservoir in the case of finite reservoir length. The motion of these boundaries is related

to the hydrodynamic pressure by the boundary conditions.

For earthquake excitation, the condition at the boundaries of the damreservoir, reservoir-foundation and the reservoir-far-end are governed by the equation:

wherep is density of water and an(x, y, t) is the component of acceleration on the boundary along the direction of the inward normal n. No wave absorption is considered at the boundaries of the reservoir.

Neglecting the free surface wave, the boundary condition at the free surface is written as:

whereh is the height of the reservoir.

Using finite element discretization of the fluid domain and the discretized formulation of equation 3.6, the wave equation can be written in the following matrix form:

where Gjj = £ Gj, Hij = £ Heej and F = £ Fe. The coefficient Gj, H- and Fj for an individual element are determined using the following expressions:

Se where N is the element shape function, Ae is the element area and se is the prescribed length along the boundary of the elements. In the above formulation, matrices [H] and [G] are constant during the analysis while the force vector {F} and the pressure vector {p} and its derivatives are the variable quantities in equation 3.7.

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