This is a more detailed model for the pipe that mostly can be used for proper modelling of the penstock or other conduits.

The model could include the elastic walls and compressible water and use discretization method based on Kurganov-Petrova central upwind scheme (KP). The geometry of the penstock is shown in the figure below:

Conservation laws are usually solved by the finite-volume methods. With the finite-volume method, we divide the grid into small control volumes or control cells and then apply the conservation laws. Here the pipe is divided in N segments, with input and output pressure as a boundary conditions. The given cell is denoted by j, i.e., it is the jth cell. Cell average is calculated at the center of the cell and U denotes the average values of the conserved variables. The left and the right interfaces of the cell are denoted by j-1/2 and j+1/2 respectively. At each cell interface, the right(+)/left(-) point values are reconstructed. The letter adenotes the right and the left sided local speeds of propagation at the left/right interface of the cell.

In order to determine the fluxes at the cell interface H and the source term S, the KP scheme is used, which is a second order scheme that is well balanced.

More info about the KP pipe model can be found in can be found in [Vytvytskyi2017].

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