## An Example of the Importance of the Term DQ4 in the SWMM 5 St Venant Solution

Subject:   An Example of the Importance of the Term DQ4 in the SWMM 5 St Venant Solution

The four terms are are used in the new flow for a time step of Qnew:

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

when the force main or gravity 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

where Sigma is a function of the Froude Number and the Keep, Dampen and Ignore Inertial Term Options.  Keep sets Sigma to 1 always and Dampen set Sigma based on the Froude number, Ignore sets Sigma to 0 all  of the time during the simulation.

The value of dq4 increases when there is a significant difference in the cross sectional  area of the downstream end of the link and the upstream end of the link.  In this  example, the downstream storage node causes a backflow in the link (Figure 1).   The flow may look unstable in the link  flow time series but the change in flow is simply due to the water sloshing back and forth.  There is no continuity error as the term dq4 keeps the water in the link  in balance (Figure 2).

 Figure 1.  GIF of Water Surface Showing the effect of the term dq4
 Figure 2.  Time Series of Flow - the Link has Mass Balance due to the term dq4

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