Normal Flow Criterion and Upstream Weighted A and R

[cbuahin]

@dickinsonre, should the upstream weighting approach for A and R not only be used when normal flow limit criterion is turned on?

[dickinsonre]

Good question - however, most models probably do not use NONE

[dickinsonre]

Short Answer:
In newer versions of SWMM5, when solving the Saint-Venant (dynamic wave) equations, the hydraulic properties of a conduit—such as its flow cross-sectional area (A) and hydraulic radius (R)—are sometimes computed using a weighted approach that places more emphasis on the upstream end of the conduit. Instead of simply averaging the upstream and downstream values, the software may now apply an upstream-weighted interpolation to determine the conditions within the conduit. This upstream weighting can improve numerical stability, better handle rapid flow changes, and produce more accurate hydraulic results under certain conditions.

Detailed Explanation:
When SWMM5 applies the dynamic wave solution, it uses a finite difference scheme to solve the full one-dimensional form of the Saint-Venant equations. These equations relate flow (Q), cross-sectional area (A), hydraulic radius (R), and water depth at discrete points along the drainage network. Because the flow and depth vary along a conduit (such as a pipe or channel), the model must define how to represent hydraulic conditions (like area and hydraulic radius) at the computational points where the equations are solved.

Originally, SWMM often computed the hydraulic properties (A, R) at an interior point of a conduit using a simple average (or midpoint approach) of the upstream and downstream node conditions. However, this averaging approach can sometimes lead to numerical instabilities or less accurate results when conditions change rapidly—mainly if there is a strong flow direction and significant variation in depth or velocity from upstream to downstream.

To address these issues, an upstream-weighted scheme may be employed. With this approach:

1. Identify Upstream and Downstream Conditions:
   At each time step, the model knows the water depth, flow, and other hydraulic conditions at both the upstream and downstream ends of the conduit.

2. Apply a Weighting Factor:
   Instead of simply averaging A and R, the algorithm uses a weighting factor that tilts the balance towards the upstream conditions. This might mean, for example, that the hydraulic radius at the computational point within the conduit is calculated as:
   [
   R = w \times R_\text{upstream} + (1 - w) \times R_\text{downstream},
   ]
   where ( w > 0.5 ) (greater than 0.5) places more emphasis on upstream conditions.

3. Benefit of Upstream Weighting:

   • Stability: By looking more closely at upstream conditions, the model can anticipate changes in flow depth and velocity that are just entering the conduit. This can prevent “overshoot” or unrealistic fluctuations in the computed results.
   • Accuracy for Rapid Transitions: If there’s a surge, backwater effect, or quickly changing hydraulic grade line, relying more on upstream information can help capture the dynamic response more faithfully.
   • Reduced Numerical Noise: Weighted calculations help minimize oscillations or numerical noise that can arise in fully implicit solutions.

4. Overall Improvement in the Solution Quality:
   With upstream weighting of A and R, SWMM5’s dynamic wave routing becomes more robust and can handle a wider range of hydraulic conditions. While the details (such as the exact weighting factor) can depend on the specific version of the software and internal solver settings, the concept remains: by leaning toward upstream conditions, the model can produce smoother, more stable, and often more accurate results.

In summary, upstream-weighted A and R calculations in SWMM5’s dynamic wave routing scheme represent a numerical technique designed to improve stability, accuracy, and robustness in simulating rapidly changing flow conditions within stormwater and wastewater collection systems.

[dickinsonre]

Now, is this still true? Lew added the NONE criteria for Ben Hodeges' UT team. I sometimes use NONE as an option for models, and sometimes it is fine, but other times it causes a continuity error.