Open Channel Flow: Non-Uniform Flow

Open Channel Flow: Non-Uniform Flow

Learning Objectives

Understand non-uniform flow and specific energy concepts.
Analyze critical flow and the Froude number.
Evaluate hydraulic jumps and their energy dissipation.
Understand gradually varied flow profiles.

Specific energy, critical depth, hydraulic jumps, and gradually varied flow profiles.

Non-Uniform Flow Overview

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 ( $E$ )

Specific energy is the energy per unit weight of fluid relative to the channel bottom.

Specific Energy

Calculates the specific energy as the sum of flow depth and velocity head.

E = y + \frac{V^2}{2g} = y + \frac{Q^2}{2g \cdot A^2}

Variables

Symbol	Description	Unit
$E$	Specific energy	m
$y$	Flow depth	m
$V$	Mean velocity	m/s
$g$	Acceleration due to gravity	$m/s^2$
$Q$	Discharge (volumetric flow rate)	$m^3/s$
$A$	Cross-sectional area of flow	$m^2$

Specific Energy Curve

For a given discharge $Q$ , the specific energy curve ( $E$ vs $y$ ) is an asymptotic curve showing two possible depths for any energy greater than the minimum ( $E > E_{min}$ ). The upper leg of the curve represents subcritical flow, while the lower leg represents supercritical flow.

Flow Depth Types

STEP-BY-STEP

Subcritical Depth ( $y_2 > y_c$ ): Slow, deep flow. (Froude Number $Fr < 1$ )
Supercritical Depth ( $y_1 < y_c$ ): Fast, shallow flow. (Froude Number $Fr > 1$ )

Interactive Simulation

Use the simulation below to explore the relationship between specific energy and flow depth. You can find the critical depth ( $y_c$ ) where specific energy is minimized.

Specific Energy Curve ( $E$ vs $y$ )

Discharge (

Q

): 10 m³/s

Channel Width (

b

): 4 m

Unit Discharge (

q=Q/b

):2.50 m²/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

Critical flow is the flow state where specific energy is minimum for a given discharge. At critical flow, the Froude number is $1$ .

Froude Number ( $Fr$ )

The ratio of inertial forces to gravity forces. In open channels, it determines whether the flow is subcritical, critical, or supercritical.

Froude Number

Calculates the Froude number using flow velocity and hydraulic depth.

Fr = \frac{V}{\sqrt{g \cdot D_h}}

Variables

Symbol	Description	Unit
$Fr$	Froude number	dimensionless
$V$	Mean velocity	m/s
$g$	Acceleration due to gravity	$m/s^2$
$D_h$	Hydraulic depth ( $A/T$ where $T$ is top width)	m

Froude Number Simplification for Rectangular Channels

For a rectangular channel, the hydraulic depth $D_h$ is exactly equal to the flow depth $y$ , which simplifies the Froude number calculation to:

Fr = \frac{V}{\sqrt{g \cdot y}}

Critical Depth Overview

The critical depth ( $y_c$ ) corresponds to the depth of minimum specific energy for a given discharge. The formula depends on the channel's cross-sectional geometry.

Rectangular Channel: Defined in terms of unit discharge $q$ .
Triangular Channel: Defined in terms of total discharge $Q$ and side slope $m$ .

Critical Depth in Rectangular Channels

Calculates critical depth for a rectangular channel using unit discharge.

y_c = \sqrt[3]{\frac{q^2}{g}}

Variables

Symbol	Description	Unit
$y_c$	Critical depth	m
$q$	Discharge per unit width ( $Q/b$ )	$m^2/s$
$g$	Acceleration due to gravity	$m/s^2$

Critical Depth in Triangular Channels

Calculates critical depth for a symmetric triangular channel.

y_c = \left( \frac{2 \cdot Q^2}{g \cdot m^2} \right)^{1/5}

Variables

Symbol	Description	Unit
$y_c$	Critical depth	m
$Q$	Total discharge	$m^3/s$
$g$	Acceleration due to gravity	$m/s^2$
$m$	Side slope ratio (Horizontal to Vertical, H:1)	dimensionless

Hydraulic Jump

A hydraulic jump is a phenomenon where flow transitions abruptly from supercritical (shallow, fast) to subcritical (deep, slow), resulting in significant energy dissipation, high turbulence, and a rapid rise in the water surface.

Applications: Hydraulic jumps are widely used as energy dissipators downstream of spillways, sluice gates, and drop structures to prevent downstream erosion and scouring. They also promote mixing in water treatment facilities.

Sequent Depths ( $y_1, y_2$ )

The depths immediately before and after the hydraulic jump, also referred to as conjugate depths.

Belanger Equation for Sequent Depths

Relates the downstream sequent depth to the upstream depth and Froude number in a rectangular channel.

\frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 \cdot Fr_1^2} - 1 \right)

Variables

Symbol	Description	Unit
$y_1$	Upstream flow depth (supercritical)	m
$y_2$	Downstream flow depth (subcritical)	m
$Fr_1$	Upstream Froude number ( $Fr_1 > 1$ )	dimensionless

Energy Dissipation in Hydraulic Jumps

The primary purpose of engineered hydraulic jumps is to safely dissipate destructive kinetic energy.

Head Loss Concept

The specific energy loss across a hydraulic jump in a horizontal rectangular channel is a function only of the upstream ( $y_1$ ) and downstream ( $y_2$ ) sequent depths.

The power dissipated per unit width ( $P/b$ ) is $\gamma \cdot q \cdot \Delta E$ .
A larger jump (higher $y_2/y_1$ ratio) dissipates exponentially more energy, making it an extremely effective stilling basin mechanism.

Energy Loss in a Hydraulic Jump

Calculates the specific energy head loss across a hydraulic jump in a rectangular channel.

\Delta E = E_1 - E_2 = \frac{(y_2 - y_1)^3}{4 \cdot y_1 \cdot y_2}

Variables

Symbol	Description	Unit
$\Delta E$	Specific energy head loss	m
$E_1$	Specific energy before the jump	m
$E_2$	Specific energy after the jump	m
$y_1$	Upstream sequent depth	m
$y_2$	Downstream sequent depth	m

Interactive Simulation

Interact with the simulation below to visualize a hydraulic jump and see how different upstream flow conditions (Froude number, initial depth) influence the downstream sequent depth and the total energy loss.

Hydraulic Jump Simulator

Input Parameters

Unit Discharge (q)
Volumetric flow per meter width5.0 m²/s

Upstream Depth (y₁)0.50 m

Note: A jump only forms if the upstream flow is supercritical ( $Fr_1 > 1$ ).

Flow Characteristics

Upstream Froude (Fr₁)

0.00

Supercritical

Sequent Depth (

y_2

)

0.00 m

Downstream Froude (Fr₂)

0.00

Subcritical

Energy Loss (ΔE)

0.00 m

ℹ️ Understanding the Hydraulic Jump

A hydraulic jump occurs when a high-velocity, supercritical flow ( $Fr > 1$ ) transitions into a low-velocity, subcritical flow ( $Fr < 1$ ). This abrupt transition causes a sudden rise in water surface elevation and creates significant turbulence.

Engineers use this phenomenon to intentionally dissipate kinetic energy at the base of spillways and dams, preventing downstream erosion. The conjugate depth equation is used to predict the downstream depth ( $y_2$ ) given the upstream conditions.

Water Surface Profiles Computation

Water surface profile computation methods calculate the change in depth over a specific channel length.

Standard Step Method Overview

The Standard Step Method is the most widely used numerical technique for computing gradually varied flow (GVF) profiles. It involves solving the energy equation between two adjacent cross-sections (stations). This method is iterative for unknown depths ( $y_2$ ) given a known starting point ( $y_1$ ) and distance ( $\Delta x$ ).

Standard Step Method Energy Equation

The fundamental energy balance equation between two adjacent channel cross-sections.

y_1 + z_1 + \frac{V_1^2}{2g} = y_2 + z_2 + \frac{V_2^2}{2g} + h_f + h_e

Variables

Symbol	Description	Unit
$y_1, y_2$	Flow depth at stations 1 and 2	m
$z_1, z_2$	Elevation of the channel bottom above a datum at stations 1 and 2	m
$V_1, V_2$	Mean velocity at stations 1 and 2	m/s
$g$	Acceleration due to gravity	$m/s^2$
$h_f$	Friction head loss between sections (calculated as $S_{f,avg} \cdot \Delta x$ )	m
$h_e$	Eddy loss (minor losses due to channel expansion or contraction)	m

Gradually Varied Flow (GVF)

Gradually varied flow occurs where the depth changes gradually over a long distance, meaning the streamlines are practically parallel and hydrostatic pressure distribution holds. The water surface profile is classified based on the bed slope ( $S_0$ ) and the actual depth relative to critical ( $y_c$ ) and normal ( $y_n$ ) depths.

M Profiles (Mild Slope, $y_n > y_c$ ):
- M1 (Backwater curve): Depth is greater than normal depth ( $y > y_n > y_c$ ). Commonly occurs upstream of a dam or a weir where water pools. Depth increases in the direction of flow.
- M2 (Drawdown curve): Depth is between normal and critical depth ( $y_n > y > y_c$ ). Flow accelerates as it approaches a free overfall or sudden channel expansion. Depth decreases in the direction of flow.
- M3: Flow is supercritical on a mild slope ( $y_n > y_c > y$ ), typically occurring immediately downstream of a sluice gate before a hydraulic jump forms.
S Profiles (Steep Slope, $y_c > y_n$ ): Profiles (S1, S2, S3) describe flow adjusting on steep inclines. S1 curves follow a hydraulic jump, while S2 and S3 occur near channel slope transitions.

Key Takeaways

Specific Energy ( $E$ ): The total energy head relative to the channel bottom, composed of depth ( $y$ ) and velocity head ( $V^2/2g$ ).
Froude Number ( $Fr$ ): Measures the flow state ( $Fr = 1$ is critical, $Fr < 1$ is subcritical, $Fr > 1$ is supercritical).
Hydraulic Jump: A highly turbulent, rapid transition from supercritical to subcritical flow, engineered to dissipate energy.
Belanger Equation: Quantifies the relationship between upstream and downstream sequent depths of a hydraulic jump.
Gradually Varied Flow (GVF): Describes slow water surface depth variations along a channel, categorized into profiles (e.g., M1, M2, S2) based on bed slope and relative depths.

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Learning Objectives

Non-Uniform Flow Overview

Specific Energy (EEE)

Specific Energy

Specific Energy Curve

Flow Depth Types

Interactive Simulation

Specific Energy Curve (EEE vs yyy)

Critical Flow

Froude Number (FrFrFr)

Froude Number

Froude Number Simplification for Rectangular Channels

Critical Depth Overview

Critical Depth in Rectangular Channels

Critical Depth in Triangular Channels

Hydraulic Jump

Sequent Depths (y1,y2y_1, y_2y1​,y2​)

Belanger Equation for Sequent Depths

Energy Dissipation in Hydraulic Jumps

Head Loss Concept

Energy Loss in a Hydraulic Jump

Interactive Simulation

Hydraulic Jump Simulator

Input Parameters

Flow Characteristics

ℹ️ Understanding the Hydraulic Jump

Water Surface Profiles Computation

Standard Step Method Overview

Standard Step Method Energy Equation

Gradually Varied Flow (GVF)

Specific Energy ( $E$ )

Specific Energy Curve ( $E$ vs $y$ )

Froude Number ( $Fr$ )

Sequent Depths ( $y_1, y_2$ )