Chapter 3 Hydraulics:Broad-Crested Weir

Chapter 3 Hydraulics:Broad-Crested Weir#

  1. Introduction: Broad Crested Weir

  2. Simulation: Broad Crested Weir

  3. Self-Assessment

1. Introduction#

✅ What Is a Broad-Crested Weir?#

A broad-crested weir is a hydraulic structure where water flows over a wide, flat crest that is long enough for the flow to become nearly parallel to the crest surface. Unlike sharp-crested weirs, the flow adjusts gradually, making it suitable for stable and high-flow conditions. —

📐 Characteristics#

Feature

Description

Crest Width

Typically ≥ 2× flow depth; allows uniform velocity profile

Flow Profile

Subcritical upstream, critical over crest, supercritical downstream

Measurement Accuracy

Lower than sharp-crested; requires calibration

Construction

Often made of concrete or earth; integrated into channel or spillway

⚠️ Problems and Limitations#

Issue

Description

Lower Accuracy

Flow measurement less precise due to gradual velocity changes

Submergence Sensitivity

Backwater effects can distort flow profile and reduce discharge

Large Footprint

Requires more space and excavation compared to sharp-crested weirs

Costly Construction

Wide crest demands more material and structural support

Limited Low-Flow Use

Not ideal for small discharges or laboratory settings

💰 Why They’re Not Widely Used for Measurement#

  • High construction cost due to large crest width and structural requirements

  • Lower measurement precision makes them unsuitable for small-scale or regulatory flow monitoring

  • Site constraints often favor compact weir types like sharp-crested or V-notch

🏞️ Best Use Cases for Broad-Crested Weirs#

Application

Reason for Suitability

Spillways in Dams

Stable flow control under high discharge; integrated into dam crest

Irrigation Channels

Durable and low-maintenance flow regulation

Flood Control Structures

Handles large volumes with minimal turbulence

Energy Dissipation

Smooth transition to downstream supercritical flow

🌊 Broad-Crested Weir: Equation, Design Procedure, and Discharge Coefficient Estimation#

✅ Governing Equation#

For free-flow (non-submerged) conditions, the discharge over a broad-crested weir is given by:

\( Q = C_d \cdot b \cdot H^{3/2} \)

Where:

  • \(( Q \)): discharge (m³/s)

  • \(( C_d \)): discharge coefficient (dimensionless)

  • \(( b \)): crest width (m)

  • \(( H \)): head over the crest (m)

📐 Design Steps#

Step 1: Define Design Flow#

  • Determine design discharge \(( Q \)) based on hydrologic analysis (e.g., design storm, return period)

Step 2: Select Crest Geometry#

  • Choose crest width \(( b \)) and height based on site constraints and flow characteristics

  • Ensure crest is long enough for parallel flow (typically \(( b \geq 2H \)))

Step 3: Estimate Head Over Crest#

  • Rearrange the weir equation to solve for \(( H \)): \( H = \left( \frac{Q}{C_d \cdot b} \right)^{2/3} \)

Step 4: Check Flow Conditions#

  • Ensure free-flow (critical depth over crest)

  • Avoid submergence (tailwater elevation < crest + head)

Step 5: Refine Crest Shape#

  • Use rounded upstream edge and flat crest to minimize flow separation

  • Consider aeration slots or vents for high-head flows

🔢 Estimating Discharge Coefficient \(( C_d \))#

Method / Source

Empirical Relation / Notes

Empirical (USBR)

\(( C_d \approx 1.7 \)) for well-shaped broad-crested weirs

Based on Froude Number

\(( C_d = 1.705 - 0.011 \cdot Fr \)), where \(( Fr = \frac{V}{\sqrt{gH}} \))

Based on Crest Length

\(( C_d = 1.66 + 0.1 \cdot \frac{L}{H} \)) for \(( L/H \leq 5 \))

Laboratory Calibration

Use flume tests or CFD models to refine \(( C_d \))

Note: \(( C_d \)) typically ranges from 1.6 to 1.8 depending on crest shape, approach flow, and aeration.

⚠️ Design Considerations#

  • Submergence: Reduces effective head and alters flow profile

  • Approach Velocity: High approach flow may require velocity head correction

  • Sediment and Debris: May accumulate on crest and affect flow

  • Structural Integration: Often built into spillways, requiring geotechnical and structural design

🧠 Conceptual Insight#

Broad-crested weirs are ideal for high-flow, stable discharge control,
but require careful design of crest geometry and flow conditions to ensure accuracy.

Broad-crested weirs are robust and hydraulically stable,
but their cost, size, and lower precision limit their use to high-flow, structural applications.

References#

[Gupta, 2017] and [Chanson, 2004] provide valuable but distinct treatments of broad-crested weir design, each suited to different educational and professional contexts. While [Gupta, 2017] is ideal for undergraduate learners [Chanson, 2004] offers a rigorous, research-based analysis of broad-crested weirs, including pressure distribution, velocity profiles, and boundary shear stress, highlights critical flow conditions, non-hydrostatic pressure zones, and rounded crest effects using physical modeling and advanced instrumentation. The description is suitable for graduate-level study and professional hydraulic design, especially in dam and spillway applications.

2. Simulation#

Interactive Broad-Crested Weir Flow Visualizer#

This tool demonstrates how discharge \(( Q \)) varies with upstream head \(( H \)), crest width \(( b \)), and the discharge coefficient \(( C_d \)) for a broad-crested weir.

The flow rate is calculated using the empirical equation:

\[ Q = C_d \cdot b \cdot H \cdot \sqrt{2gH} \]

Where:

  • \(( Q \)): discharge [m³/s]

  • \(( C_d \)): discharge coefficient (typically between 0.4 and 0.7)

  • \(( b \)): crest width [m]

  • \(( H \)): head over the crest [m]

  • \(( g = 9.81 \), \(text{m/s}^2 \)): gravitational acceleration

Parameters Controlled with Sliders#

  • Discharge Coefficient \(( C_d \)): Adjust between 0.4 and 0.7

  • Crest Width \(( b \)): Choose values from 0.1 m to 5.0 m

  • Maximum Head \(( H \)): Define the head range up to 2.0 m

Output#

The resulting plot shows:

  • A continuous curve of discharge vs head

  • How changes in geometry or \(( C_d \)) influence capacity

  • Useful visual insight for preliminary design or learning

import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider

g = 9.81  # gravitational acceleration (m/s²)

# Broad-crested weir discharge equation
def broad_crested_weir_Q(b, H, Cd):
    return Cd * b * H * np.sqrt(2 * g * H)

# Interactive plot
def plot_broad_weir(Cd, b, H_max):
    H_vals = np.linspace(0.01, H_max, 300)
    Q_vals = broad_crested_weir_Q(b, H_vals, Cd)
    
    plt.figure(figsize=(8, 5))
    plt.plot(H_vals, Q_vals, color='royalblue', linewidth=2)
    plt.xlabel("Head over Crest H (m)")
    plt.ylabel("Discharge Q (m³/s)")
    plt.title(f"Broad-Crested Weir: Discharge vs Head\nWidth b = {b:.2f} m, Cd = {Cd}")
    plt.grid(True, linestyle="--", alpha=0.5)
    plt.tight_layout()
    plt.show()

# Interactive controls
interact(
    plot_broad_weir,
    Cd=FloatSlider(value=0.5, min=0.4, max=0.7, step=0.01, description="Discharge Coeff. Cd"),
    b=FloatSlider(value=1.0, min=0.1, max=5.0, step=0.1, description="Crest Width b (m)"),
    H_max=FloatSlider(value=1.0, min=0.1, max=2.0, step=0.05, description="Max Head H (m)")
)

<function __main__.plot_broad_weir(Cd, b, H_max)>

3. Self-Assessment#

Interactive Broad-Crested Weir Flow Estimator#

This notebook provides an interactive tool for exploring how the discharge over a broad-crested weir varies with upstream head, crest width, and discharge coefficient.

Conceptual Questions#

  1. Why is the discharge through a broad-crested weir proportional to ( H^{1.5} ), and what physical principles explain this relationship?

  2. How does the discharge coefficient ( C_d ) reflect the impact of surface roughness, approach flow, and weir geometry?

  3. Why is the weir crest made “broad” compared to sharp-crested weirs? What advantage does this provide in real applications?

  4. Explain why weir width ( b ) has a linear relationship with discharge, while head ( H ) has a nonlinear one.

  5. What assumptions underlie the standard broad-crested weir equation used in this tool?

Reflective Questions#

  1. If your goal is to minimize flow over a flood control structure, would increasing \(( b \)) or decreasing \(( C_d \)) be more effective? Why?

  2. In what situations might a broad-crested weir be preferred over a V-notch or Cipolletti weir? What trade-offs are involved?

  3. How would sediment buildup on the crest affect the accuracy of the discharge prediction?

  4. What factors might cause the actual \(( C_d \)) in the field to differ from the nominal value used in design?

  5. Imagine you must design a weir for a fish passage. How might your weir geometry or flow regime need to change?

General Problem & Solution#

Problem:
A broad-crested weir has a crest width \(( b = 2.0 \)) m. If the upstream head \(( H \)) is 0.6 m and the discharge coefficient is estimated at \(( C_d = 0.5 \)), calculate the flow rate \(( Q \)) over the weir.

Solution:

\[ Q = C_d \cdot b \cdot H \cdot \sqrt{2gH} \]

Substitute values:

\[ Q = 0.5 \cdot 2.0 \cdot 0.6 \cdot \sqrt{2 \cdot 9.81 \cdot 0.6} \approx 0.6 \cdot \sqrt{11.772} \approx 0.6 \cdot 3.43 \approx \boxed{2.06\ \text{m³/s}} \]

✅ Quiz Questions#

Q1. Which term in the broad-crested weir equation has the most significant impact on discharge for small changes?
A. Crest width \(( b \))
B. Discharge coefficient \(( C_d \))
C. Head over crest \(( H \))
🟢 Correct Answer: C

Q2. The exponent on head \(( H \)) in the weir equation is:
A. 1
B. 1.5
C. 2
D. 0.5
🟢 Correct Answer: B

Q3. The function of a broad-crested weir is primarily to:
A. Increase flow turbulence
B. Reduce head losses
C. Measure or regulate flow
D. Control sediment transport
🟢 Correct Answer: C

Q4. Increasing the discharge coefficient \(( C_d \)) causes discharge to:
A. Decrease
B. Stay the same
C. Increase
D. Become zero
🟢 Correct Answer: C

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