Open Channel Flow: Non-Uniform Flow

Open Channel Flow: Non-Uniform Flow

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 $y$ ) shows two possible depths for any $E > E_{min}$ .

Procedure

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 $Fr > 1$ )

Specific Energy 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 ( $F r$ )

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
$F r$	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

Note

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 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 ( $y_{1}, y_{2}$ )

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
$F r_{1}$	Upstream Froude number ( $F r_{1} > 1$ )	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
	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

Hydraulic Jump Simulation

Interact with the simulation below to visualize a hydraulic jump and see how different flow conditions influence the transition and 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	$m / s^{2}$
$h_{f}$		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. The water surface profile is classified based on the slope ( $S_0$ ) and the depth relative to critical ( $y_c$ ) and normal ( $y_n$ ) depths.

M Profiles (Mild Slope): The normal depth is greater than critical depth ( $y_n > y_c$ $y_{n} > y_{c}$ ).
- 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 in the direction of flow.
S Profiles (Steep Slope): The critical depth is greater than normal depth ( $y_c > y_n$ ). S1, S2, and S3 profiles describe flow adjusting on steep inclines.

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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Non-Uniform Flow Overview

Specific Energy (EEE)

Specific Energy

Specific Energy Curve

Procedure

Specific Energy Simulation

Specific Energy Curve (EEE vs yyy)

Critical Flow

Froude Number (FrFrFr)

Froude Number

Note

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

Hydraulic Jump 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 ( $F r$ )

Sequent Depths ( $y_{1}, y_{2}$ )