Dynamic Wave A and R calculation slider in SWMM 5.0.013

[Hydraulics]

Purpose:  The purpose of this note is to explain a significant dynamic wave routing difference between EPA SWMM 5.0.013 and EPA SWMM 5.0.011 and before.   A few people have detected a difference.  The previous solution(s) would use only the midpoint area (Amid) and hydraulic radius (Rmid) in the dynamic wave solution.  The new solution will use a slider or linear combination of the midpoint area (Amid) and hydraulic radius (Rmid) and the upstream cross sectional area (A1) and hydraulic radius (R1).  The slider is based on the Froude number in the link.  The change involves the A and R link spacing in the two dominant terms of the St. Venant Equation: 


The new method is a linear combination or slider that weights the value of A and R in the St. Venant Equation based on the value of rho (), or 



where, Rho () is a function of the Froude number.   The effect of this addition is that as the Froude number increases from 0.5 to 1.0 and beyond the area and hydraulic radius used as the pivot point in the St. Venant equation moves from  the midpoint of the link to the upstream end of the link.  When the Froude number is above 1.0 the St. Venant and Normal Flow equation both use the same cross sectional area and hydraulic radius which makes for a more stable model.

Just for reference, the equation for Qnorm or the Manning's Equation flow is 



The equations for the calculation of  Rho () as a function of the Froude Number (Fr) are: 



If ALL of the follow conditions are true Rho () is calculated:

• the pipe is not full,
• h1 >= h2, and
• qLast > 0.

where,
h1 is the head at the upstream end of the link,
h2 is the head at the downstream end of the link and
qLast is the last flow value in the link.  

If any of these conditions are true then rho = 1.0 and the value of A and R are the values Amid and Rmid, respectively.
The next graph shows the relationship between Rho and the Froude Number. 



The value of Awtd and Rwtd move from the midpoint of the link to the upstream end of the link as the Froude number increases from 0.5 to 1.0. 

 

 
 
Conclusion: This change should make the solution more stable because there is no longer an oscillation between the St. Venant Equation A and R and the Normal Flow Equation A and R.