Page - 1476 - in Book of Full Papers - Symposium Hydro Engineering

∇𝑝 2 = 1 𝑐 2 𝜕 2 𝑝 𝜕 𝑡 2 = 𝜌 𝑘 𝜕 2 𝑝 𝜕 𝑡 2 ………………………… (1) Where c= (k/ρ) ½ …………………..……..…... (2) In which p is the pressure function, c is the acoustic wave speed, ρ is the fluid density and k is the fluid compressibility. If the fluid would be incompressible, equation (1) would take the following form: ∇𝑝 2 = 0 ………………………………………. (3) Reservoir upstream boundary (ГR) With the vibration of the dam, volumetric hydrodynamic pressure waves are created in the reservoir and propagate toward the upstream, if the length of the reservoir is assumed to be infinity, then these waves would approach to vanish. It should be noted that the length of reservoir is assumed as a finite length, L, in numerical modeling. Hence, an artificial boundary is applied to simulate effect of infinite reservoir. This boundary is modeled based on the Sommerfeld boundary as 𝜕 𝑝 𝜕 𝑛 (𝑥 ,𝑦 ,𝑧 ) = − 1 𝑐 ( 𝜕 𝑝 𝜕 𝑡 )(𝑥 ,𝑦 ,𝑧 )………….………..…. (4) Reservoir bottom (ГB) According to the rigidity of the reservoir bottom, by assuming the horizontal movement of the earth, the pressure gradient is neglected. ∂p ∂n (x,y,z) = 0 ………………….………………..… (5) Reservoir free surface (ГF) By neglecting the effects of surface waves, the governing boundary condition is as follow: 𝑝 (𝑥 ,𝑦 ,𝑧 ) = 0 …………………………………....… (6) Fluid-structure interface (ГI) In the common boundary between the reservoir and the dam body, an interaction between these two occurs which the result of inertia force is caused by the movement of the dam body. Hence, the applied pressure on the reservoir face caused by the inertial force is as follow; 𝝏 𝒑 𝝏 𝒏 (𝒙 ,𝒚 ,𝒛 ) = − 𝝆 .?̈? 𝒏 (𝒙 ,𝒚 ,𝒛 ) ……………..…….... (7) In which ρ is the density of fluid and ün is the structure's acceleration vector in the direction normal to the common boundary of the fluid and structure. Foundation region boundary conditions The nodes on edges A and B at the end of the foundation region are assumed to be constrained in the vertical direction, while it is free in the horizontal direction. The nodes at the horizontal line at the base (edge C) of the foundation region are assumed to be constrained in both vertical and horizontal directions. The nodes at the interface between the dam body and foundation are coupled 1476