Hydraulic model structure

Primary computational elements

The hydraulic model consists of four primary computational elements:

• Superjunctions: Represent reservoir-like structures such as retention basins, ponds, lakes, or manholes. There are M superjunctions in a given network, each indexed by j.
• Superlinks: Represent pressurized pipes or open channels. Each superlink consists of a chain of nk+1 junctions and nk links connected together in an interleaved fashion. There are NK superlinks in a given network, each indexed by k.
• Junctions: Represent finite volumes within a superlink (e.g. manholes within a sewer line). There are nk + 1 junctions within each superlink k.
• Links: Represent segments of pressurized conduit or channel. Each link is bounded by junctions on the upstream and downstream ends. There are nk links within each superlink k.

The image below shows a stormwater network consisting of 6 superjunctions and 6 superlinks. Each superlink consists of 5 internal junctions and 4 internal links.

Control volumes

Within the hydraulic solver, a staggered-grid discretization scheme is used, in which the continuity equation is applied around each junction and the momentum equation is applied around each link.

Control volume for continuity equation

Note that the left and right edges of the continuity control volume coincide with the midpoints of the upstream and downstream links, respectively.

Control volume for momentum equation

Note that the left and right edges of the momentum control volume coincide with the edges of the upstream and downstream junctions, respectively.

Indexing schemes

There are four primary indexing schemes used by the hydraulic solver, corresponding to the four primary computational elements.

Indexed by superjunction (j)

State variables that are indexed by superjunction include:

• H_j : Superjunction heads (m)
• A_sj : Surface area of superjunctions (m^2)
• V_sj : Volume stored in superjunctions (m^3)
• z_inv_j : Invert elevation of superjunction (m)
• Q_in : Exogenous inflow to superjunctions (m^3/s)

For the network shown above, an array corresponding to the superjunction indexing scheme would be of length 6, and appear as:

[f_1  f_2  f_3  f_4  f_5  f_6]

Indexed by superlink (k)

State variables that are indexed by superlink include:

• Q_uk : Flow into upstream end of each superlink (m^3/s)
• Q_dk : Flow out of downstream end of each superlink (m^3/s)
• z_inv_uk : Invert elevation of upstream end of superlink (m)
• z_inv_dk : Invert elevation of downstream end of superlink (m)

For the network shown above, an array corresponding to the superlink indexing scheme would be of length 6, and appear as:

[f_1  f_2  f_3  f_4  f_5  f_6]

Indexed by junction (Ik)

State variables that are indexed by junction include:

• h_Ik : Depth in internal junction I within superlink k (m)
• x_Ik : Horizontal position of internal junction I within superlink k (m)
• A_SIk : Surface area of internal junction I within superlink k (m^2)

For the network shown above, an array corresponding to the superlink indexing scheme would be of length 30, and appear as:

[ f_1,1  f_2,1  f_3,1  f_4,1  f_5,1
  f_1,2  f_2,2  f_3,2  f_4,2  f_5,2
  f_1,3  f_2,3  f_3,3  f_4,3  f_5,3
  f_1,4  f_2,4  f_3,4  f_4,4  f_5,4
  f_1,5  f_2,5  f_3,5  f_4,5  f_5,5
  f_1,6  f_2,6  f_3,6  f_4,6  f_5,6 ]

Where the first index corresponds to I, the internal junction index, and the second index corresponds to k, the superlink index. Note the above array is one-dimensional (read from left to right, then top to bottom).

Indexed by link (ik)

State variables that are indexed by junction include:

• Q_ik : Flow in internal link i within superlink k (m^3/s)
• A_ik : Area of flow in internal link i within superlink k (m^2)
• dx_ik : Length of internal link i within superlink k (m)
• B_ik : Top width of flow in internal link i within superlink k (m)
• Pe_ik : Perimeter of flow in internal link i within superlink k (m)
• R_ik : Hydraulic radius of flow in internal link i within superlink k (m)
• S_o_ik : Bottom slope in internal link i within superlink k (-)

For the network shown above, an array corresponding to the link indexing scheme would be of length 24, and appear as:

[ f_1,1  f_2,1  f_3,1  f_4,1
  f_1,2  f_2,2  f_3,2  f_4,2
  f_1,3  f_2,3  f_3,3  f_4,3
  f_1,4  f_2,4  f_3,4  f_4,4
  f_1,5  f_2,5  f_3,5  f_4,5
  f_1,6  f_2,6  f_3,6  f_4,6 ]

Where the first index corresponds to i, the internal link index, and the second index corresponds to k, the superlink index. Note the above array is one-dimensional (read from left to right, then top to bottom).