modelPenstockKP

Information

Elastic Penstock with KP Method

Note: The older Penstock model using the Staggered Grid scheme is now marked as obsolete. This PenstockKP model, which implements the more accurate and numerically stable KP (Kurganov-Petrova) method, is the recommended choice for all elastic penstock modeling.

This is a detailed model for the pipe that can be used for proper modelling of the penstock or other conduits. Unlike simple conduits, the penstock has considerable pressure variations due to substantial height drop. To make the model more realistic, compressible water and elastic walls are taken into account.

Compressibility

The isothermal compressibility β_T relates density and pressure:

$$ \rho \approx \rho^\mathrm{atm}(1 + \beta_T(p - p^\mathrm{atm})) $$

Similarly for pipe cross-section area due to elastic walls:

$$ A \approx A^\mathrm{atm}(1 + \beta^\mathrm{eq}(p - p^\mathrm{atm})) $$

The total compressibility β^tot = β_T + β^eq is related to the speed of sound in water inside the pipe. The parameter PipeElasticity controls whether elastic walls are included.

PDE Formulation and KP Discretization

The model includes elastic walls and compressible water using the Kurganov-Petrova central upwind scheme (KP). Using the compressibility relationships, the ODEs for mass and momentum balances develop into PDEs:

$$ \frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} = S $$

where \(U = [p_\mathrm{p}, \dot{m}_\mathrm{p}]^T\) is the state vector containing pressure and mass flow rate.

Figure: Geometry of the penstock model.

Conservation laws are solved by finite-volume methods. The pipe is divided into N segments, with inlet and outlet pressures as boundary conditions. The given cell is denoted by j (the j^th cell). Cell averages are calculated at the center of the cell, and U denotes the average values of the conserved variables. The left and right interfaces are denoted by j-1/2 and j+1/2 respectively. At each cell interface, the right(+)/left(-) point values are reconstructed.

Figure: Discretization scheme showing cell j with interfaces and reconstruction points.

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

Implementation

The PenstockKP model uses the KP07.KPmethod function to discretize PDEs into ODEs. The eigenvalues λ_1,2 determine the wave propagation speeds and are related to the speed of sound c = √(A_p/(A_p^atm ρ^atm β^tot)).

Parameters

User specifies:

• Geometry: length L, height difference H, inlet/outlet diameters D_i/D_o, roughness p_eps
• Discretization: number of cells N
• Model fidelity: PipeElasticity checkbox to enable/disable elastic walls
• Initialization: flow rate Vdot_0 and pressure p_p0 vectors, or SteadyState flag

More Information

More info about the KP penstock model can be found in [Vytvytskyi2017] and [Vytvytskyi2019].

Parameters

Connectors

Components