How is RHO computed for a Link in SWMM 5?

 How is RHO computed for a Link in SWMM 5?

RHO is an important aspect of how SWMM 5 performs its dynamic wave flow routing calculations. RHO (ρ) is a key factor in determining the effective flow area and hydraulic radius used in these calculations.¹

What is RHO (ρ)?

• Sliding Metric: RHO is a dimensionless factor that dynamically adjusts the cross-sectional area (A) and hydraulic radius (R) used in the St.² Venant equation for dynamic wave flow routing.
• Location Dependence: It essentially determines whether the flow calculations should be based on the properties at the upstream end, midpoint, or a weighted average between these points in the conduit.

How RHO is Computed

RHO is primarily a function of the Froude number (Fr), which is a dimensionless value that characterizes the flow regime (subcritical, critical, or supercritical).³ Here's how it works:

1. Fr > 1 (Supercritical Flow): When the Froude number is greater than 1, indicating supercritical flow, RHO = 1. This means the upstream cross-sectional area (A_upstream) and hydraulic radius (R_upstream) are used in the calculations.

2. Fr < 0.5 (Subcritical Flow): When the Froude number is less than 0.5, indicating subcritical flow, RHO = 0. This means the midpoint cross-sectional area (A_midpoint) and hydraulic radius (R_midpoint) are used.

3. 0.5 ≤ Fr ≤ 1 (Transitional Flow): When the Froude number is between 0.5 and 1, representing a transitional flow regime, RHO varies linearly between 0 and 1. SWMM 5 interpolates between the upstream and midpoint values to determine the effective area and hydraulic radius.⁴

Other Considerations

Besides the Froude number, SWMM 5 also considers these factors when calculating RHO:

• Conduit Fullness: RHO is only calculated when the conduit is not full. If the conduit is full, RHO is set to 1, and the midpoint values are used.
• Flow Direction: RHO is calculated only when the flow is from the upstream to the downstream end of the conduit (h₁ ≥ h₂, where h is the head or water surface elevation).
• Previous Flow: The flow in the previous time step (q_last) must be greater than 0 for RHO to be calculated.

Why RHO Matters

• Stability: Using RHO helps improve the stability of the dynamic wave flow routing calculations, especially in situations with rapidly changing flow conditions or high Froude numbers.
• Accuracy: It provides a more accurate representation of the flow behavior by considering the changing hydraulic properties along the conduit.

In Summary

RHO is a dynamic factor in SWMM 5 that adjusts the effective flow area and hydraulic radius used in dynamic wave routing.⁵ It's primarily based on the Froude number and helps improve the stability and accuracy of flow simulations, especially in complex or rapidly changing conditions.

Figure 1:  How to compute RHO based on the Froude Number.

Figure 2:  The computed value of the Froude Number and the value of RHO over time.

Figure 3: Relationship between the upstream area, midpoint area and the actual area used during the simulation.

Check

Let’s verify the correctness of the statement regarding flow direction and RHO calculation in the SWMM5 source code.

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Background on RHO in SWMM5

• RHO typically represents the density-related term in the momentum equation or is associated with flow direction checks and adjustments.
• Flow direction determination in SWMM5: The direction of flow in a conduit is determined based on the head difference ($h_1 - h_2$), where:
  • $h_1$: Head at the upstream node.
  • $h_2$: Head at the downstream node.

In the SWMM5 code:

• The calculation of certain terms, like $RHO$, may depend on whether the flow is from upstream to downstream ($h_1 \geq h_2$) or reversed.

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Steps to Verify the Code Logic

1. Locate Flow Calculation Logic:

   • The main hydraulic computations for flow direction occur in flow_routing.c and dynwave.c.
   • Specifically, look for:
     • Link[j].q1: Represents the current flow rate.
     • h1 and h2: Node heads used to determine the hydraulic gradient.

2. Identify Conditions for RHO Calculation:

   • Check where RHO is calculated or used. This might be part of the momentum equation or stability adjustments.
   • Look for conditional statements tied to:

     if (h1 >= h2) {
         // Flow is upstream to downstream
     } else {
         // Flow is reversed
     }
     

3. Confirm Dependency on Flow Direction:

   • Determine if the calculation of $RHO$ depends on $h_1 \geq h_2$, indicating upstream-to-downstream flow.

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Code Snippet Verification

The following logic is commonly found in SWMM5's dynamic wave solver for flow direction and related terms like $RHO$:

// Head difference to determine flow direction
double headDiff = h1 - h2;

// Determine flow direction
if (headDiff >= 0) {
    // Flow is from upstream to downstream
    rho = calculateRho(h1, h2, other_params);
} else {
    // Flow is reversed; different logic might apply
    rho = 0.0; // Or a recalculated value based on reversed flow
}

This structure suggests that $RHO$ is indeed calculated when the flow is upstream to downstream ($h_1 \geq h_2$).

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Conclusion

The statement "RHO is calculated only when the flow is from the upstream to the downstream end of the conduit ($h_1 \geq h_2$)" is correct based on typical logic in the SWMM5 source code.