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Subject:   How is the St Venant Equation Solved for in the Dynamic Wave Solution of SWMM 5?

 

An explanation of the four St. Venant Terms in SWMM 5 and how they change for Gravity Mains and Force Mains. The HGL is the water surface elevation in the upstream and downstream nodes of the link. The HGL for a full link goes from the pipe crown elevation up to the rim elevation of the node + the surcharge depth of the node.  The four terms are:

 

dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) / Link Length or

dq2 = Time Step * Awtd * (HGL) / Link Length

Qnew = (Qold – dq2 + dq3 + dq4) / ( 1 + dq1)

when the force main is full dq3 and dq4 are zero and

Qnew = (Qold – dq2) / ( 1 + dq1)

The dq4 term in dynamic.c uses the area upstream (a1) and area downstream (a2), the midpoint velocity, the sigma factor (a function of the link Froude number), the link length and the time step or

dq4 = Time Step * Velocity * Velocity * (a2 – a1) / Link Length * Sigma

the dq3 term in dynamic.c uses the current midpoint area (a function of the midpoint depth), the sigma factor and the midpoint velocity

dq3 = 2 * Velocity * ( Amid(current iteration) – Amid (last time step) * Sigma

dq1 = Time Step * RoughFactor / Rwtd^1.333 * |Velocity|

The weighted area (Awtd) is used in the dq2 term of the St. Venant equation:

dq2 = Time Step * Awtd * (Head Downstream – Head Upstream) / Link Length

 

The four terms change at each iteration and time step to determine the new flow (Figure 1) based on the two equations:

 

Denom = 1 + dq1 + dq5

Q = [Qold – dq2 + dq3 + dq4] / Denom

 

If you look at a table of the values you will see that the terms add up to zero when the flow is constant and to delta Q or the change in Q when the flow is NOT constant (Figure 2).

 

Image003

 

Figure 1.  The four terms define the new flow at each iteration in the dynamic wave solution of SWMM5

 

Image002

 

Figure 2.   The magnitude of the four terms determine the flow at the new iteration and ultimately the new Time Step.  If the flow is constant then the value of the term is constant.

 

Image001

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  • It uses a non linear reservoir equation

     

     

    1058531301.png

     

    Figure 1.   Runoff Alpha value for SWMM 4 or 5 Hydrology.

     

     

    How Runoff is Calculated 

     

    1058531306.png?width=721

  • Thanks for the explanation. Very helpful. Can you also tell me how water in sub-catchments moves into the junctions? Is this by kinematic wave equation?

  • Subject:   Link Iterations in the SWMM 5 Dynamic Wave Solution

     

    Each of the links in the SWMM 5 network can use up to 8 iterations to reach convergence during a time step in the dynamic wave solution of SWMM 5.  The rules governing the number of iterations are:

     

    1.       A minimum of 2 iterations per time step with the 1st iteration NOT using the underrelaxtion parameter of 0.5 (Figure 1)

    2.       If both the downstream and upstream nodes are converged then the link drops out of the iteration process during the time step (Figure 2)

    3.       The number of iterations for each link can vary over the simulation from 2 to 8 depending on how fast the flow is changing.

     

    Figure 1.  A minimum of two and up to eight iterations per time step in the SWMM 5 dynamic wave solution.

    Image002

    Figure 2.  The number of iterations for each link vary through out  the simulation with less iterations being used for constant flows.

    Image001

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