Chapter 3 Hydraulics: Weir and Notches

Chapter 3 Hydraulics: Weir and Notches#

Introduction: Weir and Notches
Simulation
Self-Assessment

1. Introduction#

✅ Definitions#

Weirs are commonly used as flow-measuring devices in open channels based on the relationship between head over the crest and discharge using empirical formulas.

Term	Description
Weir	A barrier across an open channel that causes water to flow over its crest; used to measure or control flow
Notch	A shaped opening (usually in a vertical plate) through which water flows; often used in lab flumes or small-scale flow measurement

🧱 Best Uses#

Structure	Best Applications
Weirs	- Flow measurement in open channels - Flow control in irrigation and drainage - Energy dissipation and level control
Notches	- Precise flow measurement in lab settings - Calibration of hydraulic models - Educational demonstrations

🔢 Flow Equations#

Rectangular Weir: $ Q = \frac{2}{3} C_d b \sqrt{2g} H^{3/2} $
V-Notch (Triangular): $ Q = \frac{8}{15} C_d \tan\left(\frac{\theta}{2}\right) \sqrt{2g} H^{5/2} $
Trapezoidal (Cipolletti): $ Q = C_d \cdot H^{3/2} $

Where:

$( Q $): discharge (m³/s)
$( C_d $): discharge coefficient
$( b $): crest width
$( H $): head over crest
$( \theta $): notch angle
$( g $): gravitational acceleration

🧠 Design Considerations#

Ensure free-flow (non-submerged) conditions for accurate measurement
Maintain sharp-crested geometry for consistent flow behavior
Use stilling basins upstream to reduce turbulence
Calibrate discharge coefficients based on empirical data

⚠️ Challenges#

Challenge	Description
Submergence Effects	Backwater or downstream control reduces accuracy
Debris Accumulation	Floating materials can block flow or alter head
Sediment Deposition	Affects crest elevation and flow profile
Scale Limitations	Notches are not suitable for high-flow or field-scale applications
Coefficient Sensitivity	Discharge depends heavily on geometry and flow conditions

🧠 Conceptual Insight#

Weirs and notches are simple yet powerful tools for flow measurement and control —
their effectiveness depends on precise geometry, clean flow conditions, and proper calibration.

❓ Reference#

[Chanson, 2004],[Chow, 1959], [Gupta, 2017] provide excellent foundational resources for understanding weirs, notches, their types, design principles, and applications in open-channel hydraulics and flow measurement.

2. Simulation#

Interactive Weir Flow Estimator#

This notebook provides an interactive tool to explore how different types of sharp-crested weirs behave under varying conditions. It calculates and visualizes the discharge ( Q ) as a function of upstream water head ( H ), using empirical equations and standard discharge coefficients $( C_d $).

Types of Weirs Supported#

The tool includes the following weir geometries:

Weir Type	Description
Rectangular	A horizontal crest with uniform width, optionally contracted
Triangular	A V-notch weir with an included angle (e.g., 90°), often used at low flows
Cipolletti	A trapezoidal weir with 1:4 side slopes, designed to self-compensate for end contractions

Each weir formula relates discharge ( Q ) to the upstream head ( H ) as shown below:

Rectangular Weir:
$$ Q = \frac{2}{3} C_d \, b \, \sqrt{2g} \, H^{3/2} $$
Triangular (V-Notch) Weir:
$$ Q = \frac{8}{15} C_d \tan\left(\frac{\theta}{2}\right) \sqrt{2g} \, H^{5/2} $$
Cipolletti Weir:
Same form as the rectangular weir, with a standard Cd for the self-compensating geometry.

Where:

$( C_d $) = discharge coefficient (typically between 0.55 and 0.64)
$( b $) = crest width [m]
$( \theta $) = notch angle in degrees (for triangular weir)
$( H $) = head over crest [m]
$( g = 9.81 \text{ m/s}^2 $)

User Controls#

Weir Type: Select from Rectangular, Triangular, or Cipolletti
Crest Width $( b $): For rectangular/Cipolletti weirs
V-Notch Angle $( \theta $): For triangular weirs
Head Range $( H $): Maximum value for head to plot
Discharge Coefficient $( C_d $): Selected from recommended presets based on field or lab calibration

Output#

The chart dynamically plots the relationship between head $( H $) and discharge $( Q $), allowing you to:

Compare performance across weir types
Visualize sensitivity of flow to small changes in head
Explore how design geometry and material calibration affect flow capacity

import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider, Dropdown
g = 9.81  # m/s²

# Weir formulas
def rectangular_weir_Q(H, b, Cd):
    return (2 / 3) * Cd * b * np.sqrt(2 * g) * H**1.5

def triangular_weir_Q(H, theta_deg, Cd):
    theta = np.radians(theta_deg)
    return (8 / 15) * Cd * np.tan(theta / 2) * np.sqrt(2 * g) * H**(5/2)

def cipolletti_weir_Q(H, b, Cd):
    return (2 / 3) * Cd * b * np.sqrt(2 * g) * H**1.5

# Cd presets
cd_presets = {
    "Rectangular": {
        "Standard (0.60)": 0.60,
        "Lab Calibrated (0.62)": 0.62,
        "Field Rough (0.58)": 0.58
    },
    "Triangular": {
        "Standard (0.58)": 0.58,
        "Sharp-Crested (0.60)": 0.60,
        "Field Calibrated (0.55)": 0.55
    },
    "Cipolletti": {
        "Standard (0.62)": 0.62,
        "High-Precision (0.64)": 0.64,
        "Rounded Crest (0.60)": 0.60
    }
}

# Interactive plotting logic
def plot_weir(weir_type, cd_label, b, theta, H_max):
    Cd = cd_presets[weir_type][cd_label]
    H_vals = np.linspace(0.01, H_max, 300)

    if weir_type == "Rectangular":
        Q_vals = rectangular_weir_Q(H_vals, b, Cd)
        label = f"Rectangular (b = {b:.2f} m, Cd = {Cd})"
    elif weir_type == "Triangular":
        Q_vals = triangular_weir_Q(H_vals, theta, Cd)
        label = f"Triangular (θ = {theta:.0f}°, Cd = {Cd})"
    elif weir_type == "Cipolletti":
        Q_vals = cipolletti_weir_Q(H_vals, b, Cd)
        label = f"Cipolletti (b = {b:.2f} m, Cd = {Cd})"

    plt.figure(figsize=(8, 5))
    plt.plot(H_vals, Q_vals, color='teal', linewidth=2, label=label)
    plt.xlabel("Head H (m)")
    plt.ylabel("Discharge Q (m³/s)")
    plt.title(f"Discharge vs Head – {weir_type} Weir")
    plt.grid(True, linestyle="--", alpha=0.5)
    plt.legend()
    plt.tight_layout()
    plt.show()

# Sync dropdown selection with Cd
def interactive_plot(weir_type, b, theta, H_max):
    cd_choices = list(cd_presets[weir_type].keys())
    
    def plot_with_cd(cd_label):
        plot_weir(weir_type, cd_label, b, theta, H_max)
    
    interact(
        plot_with_cd,
        cd_label=Dropdown(options=cd_choices, value=cd_choices[0], description="Cd")
    )

# Outer interact
interact(
    interactive_plot,
    weir_type=Dropdown(options=["Rectangular", "Triangular", "Cipolletti"], description="Weir Type"),
    b=FloatSlider(value=1.0, min=0.1, max=5.0, step=0.1, description="Width b (m)"),
    theta=FloatSlider(value=90, min=30, max=120, step=1, description="V-Notch θ (°)"),
    H_max=FloatSlider(value=1.0, min=0.2, max=2.0, step=0.05, description="Max Head H (m)")
)

<function __main__.interactive_plot(weir_type, b, theta, H_max)>

3. Self-Assessment#

Conceptual Questions#

Why does the discharge of a weir increase with head in a nonlinear manner? How is this reflected in the exponent of the discharge equation?
How does the shape of the weir (rectangular vs triangular vs Cipolletti) influence its sensitivity to flow depth (H)?
What role does the discharge coefficient C_d play in the weir equation, and why isn’t it constant across all field conditions?
Why are V-notch (triangular) weirs often preferred for measuring small discharges, especially in laboratory and environmental monitoring setups?
How does the geometry of the Cipolletti weir compensate for end contractions present in a standard rectangular weir?

Reflective Questions#

When comparing two weirs of equal crest width and head, which one typically yields higher discharge: a standard rectangular or a Cipolletti weir? Why?
If you increase the notch angle of a triangular weir from 60° to 120°, how would that affect the sensitivity of flow to head changes?
What practical constraints might limit the maximum head that can be used in a weir structure in the field?
Imagine you’re installing a weir in a remote stream to measure stormwater runoff. Which geometry would you choose and why?
How might sediment accumulation or debris at the crest of a weir affect the validity of the weir discharge equation?

✅ Quiz Questions Q1. The discharge equation for a triangular (V-notch) weir includes which head exponent?

1.5
2
2.5
3 🟢 Correct Answer: C

Q2. The discharge coefficient C_d accounts for:

Pipe friction losses
Gravity changes
Real-world effects like viscosity, turbulence, and shape
Unit conversion from feet to meters 🟢 Correct Answer: C

Q3. A Cipolletti weir is designed to:

Measure industrial pipe flows
Eliminate the need for head measurement
Increase surface roughness
Offset end contractions in rectangular weirs 🟢 Correct Answer: D

Q4. If all other parameters are constant, increasing C_d will:

Increase computed flow
Decrease computed flow
Have no effect
Reverse flow direction 🟢 Correct Answer: A

Q5. Which of the following is most appropriate for measuring small, variable flows in environmental monitoring?

Broad-crested weir
V-notch (triangular) weir
Rectangular weir with wide crest
Compound ogee spillway 🟢 Correct Answer: B