Open Channel Flow: Non-Uniform Flow
Specific energy, critical depth, hydraulic jumps, and gradually varied flow profiles.
Non-uniform flow occurs when the depth of flow changes along the length of the channel. This happens due to changes in slope, cross-section, or obstructions.
Specific Energy ()
Specific energy is the energy per unit weight of fluid relative to the channel bottom.
Specific Energy ($E$)
Specific energy, critical depth, hydraulic jumps, and gradually varied flow profiles.
For a given discharge , the specific energy curve ( vs ) shows two possible depths for any :
Procedure
- Subcritical Depth (): Slow, deep flow. (Froude Number < 1)
- Supercritical Depth (): Fast, shallow flow. (Froude Number > 1)
Specific Energy Simulation:
Explore the relationship between specific energy and depth. Find the critical depth () where specific energy is minimum.
Specific Energy Curve ($E$ vs $y$)
Unit Discharge ($q=Q/b$):2.50 m$^2$/s
Critical Depth ($y_c$):0.860 m
Min Specific Energy ($E_min$):1.291 m
Hover over graph to inspect
The Specific Energy curve shows two possible depths for a given energy $E > E_min$: a subcritical depth (slow, deep) and a supercritical depth (fast, shallow). $y_c$ represents the transition point.
Critical Flow
The flow state where specific energy is minimum for a given discharge. At critical flow, the Froude number is 1.
Froude Number ()
Ratio of inertial forces to gravity forces.
Froude Number ($Fr$)
- = Hydraulic Depth (, where is top width).
- For Rectangular Channel: .
Critical Depth Formulas
Rectangular Channel:
Critical Depth Formulas
- (discharge per unit width).
Variables
| Symbol | Description | Unit |
|---|---|---|
| Side slope (H:V). | - |
Hydraulic Jump
A hydraulic jump is a phenomenon where flow transitions abruptly from supercritical to subcritical, resulting in significant energy dissipation and a rise in water surface. This is often used to dissipate energy downstream of spillways.
Sequent Depths ()
The depths before and after the jump. For a rectangular channel:
Sequent Depths ($y_1, y_2$)
Energy Dissipation in Hydraulic Jumps
The primary purpose of engineered hydraulic jumps is to safely dissipate destructive kinetic energy.
Head Loss ()
The specific energy loss across a hydraulic jump in a horizontal rectangular channel is a function only of the upstream () and downstream () sequent depths.
Head Loss ($\Delta E$)
The primary purpose of engineered hydraulic jumps is to safely dissipate destructive kinetic energy.
- The power dissipated per unit width () is .
- A larger jump (higher ratio) dissipates exponentially more energy, making it an extremely effective stilling basin mechanism.
Hydraulic Jump Simulator
Input Parameters
5.0 m²/s
0.50 m
Note: A jump only forms if the upstream flow is supercritical (Fr₁ > 1).
Flow Characteristics
Upstream Froude (Fr₁)
0.00
Supercritical
Sequent Depth (y₂)
0.00 m
Downstream Froude (Fr₂)
0.00
Subcritical
Energy Loss (ΔE)
0.00 m
Water Surface Profiles Computation
Methods to calculate the change in depth over a specific channel length.
Standard Step Method
The most widely used numerical technique for computing GVF profiles. It involves solving the energy equation between two adjacent cross-sections (stations).
Standard Step Method
Methods to calculate the change in depth over a specific channel length.
Variables
| Symbol | Description | Unit |
|---|---|---|
| Flow depth. | - | |
| Elevation of the channel bottom above a datum. | - | |
| Friction head loss between sections (calculated using a representative friction slope $S_f$, typically the average of $S_{f1}$ and $S_{f2}$, and multiplying by the reach length $\Delta x$). | - | |
| Eddy loss (minor losses due to expansion or contraction, often taken as zero for straight prismatic channels). | - |
This method is iterative for unknown depths () given a known starting point () and distance ().
Gradually Varied Flow (GVF)
Flow where the depth changes gradually over a long distance. The water surface profile is classified based on the slope () and the depth relative to critical () and normal () depths.
- M Profiles (Mild Slope): The normal depth is greater than critical depth ().
- M1: Backwater curve (e.g., flow approaching a dam). Depth increases in the direction of flow.
- M2: Drawdown curve (e.g., flow approaching a free overfall). Depth decreases.
- S Profiles (Steep Slope): The critical depth is greater than normal depth (). S1, S2, S3 profiles describe flow adjusting on steep inclines.
Key Takeaways
- Specific Energy is the total energy head relative to the channel bottom. It is composed of the flow depth () and the velocity head ().
- Standard Step Method: An iterative numerical approach to predict water surface profiles in GVF by balancing energy between adjacent cross-sections.
- For a constant discharge, plotting specific energy against depth yields a parabolic curve with a distinct minimum point.
- Any specific energy greater than the minimum can occur at two possible alternate depths: one subcritical (deep and slow) and one supercritical (shallow and fast).
- Critical flow is the state of flow at which specific energy is a minimum for a given discharge.
- It is characterized by a Froude Number () equal to 1, where inertial forces and gravity forces are perfectly balanced.
- Flow with is subcritical (tranquil), and is supercritical (rapid).
- A Hydraulic Jump is a sudden, highly turbulent transition from supercritical flow to subcritical flow.
- It is frequently engineered at the base of spillways to act as an energy dissipator, preventing downstream erosion.
- The depths before and after the jump are called sequent depths (or conjugate depths), and they can be related through the momentum equation.
- In Gradually Varied Flow (GVF), the depth and velocity change slowly over a long distance, meaning the energy line, water surface, and channel bottom are not parallel.
- Water surface profiles are classified based on the channel's bed slope (Mild, Steep, Critical, Horizontal, Adverse) and the actual flow depth relative to normal () and critical () depths.