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Computational Mesh and Subgrid Bathymetry (Pipe Flow)
Pipe Network Computational Mesh
In order to take advantage of finite volume numerical methods, the network of drainage conduits and their connecting nodes is discretized into a mesh of computational cells. Unlike the surface 2D area Grid, cells in the pipe mesh may overlap if the pipes occupy different elevations beneath the terrain surface.
Cell Hydraulic Property Curves
Similar to the cells used in 2D surface flow, pipe network cells utilize the subgrid bathymetry approach (Casulli, 2008). Details of the pipe cross-sectional shape and slope are used to pre-compute cell volume at various water levels. This allows far fewer cells to be used to represent the pipe bathymetry, and is a computationally efficient approach.
Cell volume is described as a function of water depth. The derivative of the cell volume is the cell wetted area:
\Omega(H) = \int_{z_b}^{H} A^W(H)dz, \hspace{10pt} A^W(H) = \dfrac{d \Omega(H)}{d H} |
where
z_b is the cell bottom elevation [L],
\Omega(H) is the cell volume [L3],
A^W(H) is the cell wetted area [L2], and
H is the cell hydraulic head (water surface elevation + pressure head) [L].
The cell volumes are calculated at a set of points between the highest and lowest points within a computational cell. Volumes at these points define a continuous, piecewise linear function. The derivative of the volume function, the cell wetted area function, is a piecewise constant curve. For each entry k in the cell property table, the cell wetted surface area is calculated as the discrete derivative of the cell volume entries.
A^W_k = \frac{\Omega_{k+1} - \Omega_k}{H_{k+1} - H_k} |
The cell volume property table is pre-computed at 15 Gaussian quadrature points between the lowest point and highest point in a pipe cell. The Gauss points were chosen to efficiently represent the changing hydraulic properties of a cell with water depth. The calculation of cell volume assumes a flat water level and a sloping pipe section.
Decomposition of the Property Curves
Pressurized flow cannot be handled directly in the original implementation of the semi-implicit solver, because the Newton's Method-like algorithm requires a non-decreasing area-elevation function for convergence. In property tables for cells within closed conduits, the area-elevation curve decreases, something that does not occur with open channel cells. Casulli and Stelling (2013) showed that this can be handled by decomposing the volume- and area-elevation curves into two monotonically increasing functions: one for the positive contribution to volume or area, and one for the negative contribution. Each volume or area contribution can then be linearized independently and solved using a set of nested Newton-like iterations.
The figure below shows an example of a computational cell within a circular cross-section pipe, with a diameter of 1 unit. The total cell surface area- and volume-elevation curves (black) are decomposed into positive (blue) and negative (red) non-decreasing functions.
Formally, the decomposition of a function into positive and negative contributions is known as a Jordan decomposition and can be expressed:
\Omega(H) = \Omega^{+}(H) - \Omega^{-}(H), \hspace{10pt} A(H) = A^{+}(H) - A^{-}(H) |
It can be performed by moving through the property table from the lowest to highest elevations and calculating
A_k^{+} = A_k^{+} + \max(A_k - A_{k-1}, 0), \hspace{10pt} A_k^{-} = A_k^{-} - \min(A_k - A_{k-1}, 0) \newline \Omega_k^{+} = \Omega_{k-1}^{+} + (H_k - H_{k-1}) A_{k-1}^{+}, \hspace{10pt} \Omega_k^{-} = \Omega_{k-1}^{-} + (H_k - H_{k-1}) A_{k-1}^{-} |
with A_0^{+} = A_0 and A_0^{-} = 0.
Faces Hydraulic Property Curves
Faces represent the interface between two computational cells. Property tables for cell faces (cross-sectional area, hydraulic radius, etc.) are not subject to the same convergence requirements in the semi-implicit solver, and do not require the same decomposition. They are computed similarly to the 2D Subgrid Bathymetry at cell faces. Cell and face porosity functions are not considered within the pipe network.