Open Channel Flow: Non-Uniform Flow

Open Channel Flow: Non-Uniform Flow

Learning Objectives

Distinguish rapidly varied, gradually varied, and spatially varied open-channel flow.
Calculate specific energy, critical depth, alternate depths, hydraulic depth, and Froude number.
Apply the general critical-flow condition to nonrectangular channels.
Use the momentum function and conjugate-depth equation to analyze hydraulic jumps.
Classify hydraulic jumps and estimate energy dissipation.
Derive and apply the gradually varied flow equation.
Classify mild, steep, critical, horizontal, and adverse channel profiles and identify their controlling boundaries.
Use direct-step and standard-step methods while respecting their assumptions and numerical limits.

Non-uniform open-channel flow occurs whenever depth changes along a channel because of controls, slope transitions, obstructions, gates, weirs, contractions, expansions, or downstream backwater. The rate of depth change determines whether hydrostatic pressure and one-dimensional gradually varied flow assumptions remain appropriate.

Types of Non-Uniform Flow

Rapidly varied flow, RVF: Depth changes over a short distance comparable with the flow depth. Vertical acceleration and nonhydrostatic pressure may be important. Examples include hydraulic jumps, flow under gates, and spillway transitions.
Gradually varied flow, GVF: Depth changes slowly over many flow depths. Pressure remains approximately hydrostatic and local uniform-flow resistance formulas can be used.
Spatially varied flow, SVF: Discharge changes along the channel because of lateral inflow or outflow, as in side weirs, gutters, and irrigation laterals.

Uniform and Steady Are Different Classifications

A flow may be steady but non-uniform, such as a time-invariant backwater profile. It may also be unsteady and locally uniform only as an approximation. Always state both the time and spatial classifications.

Specific Energy

Specific energy is mechanical energy per unit weight measured relative to the local channel bottom. For a section with hydrostatic pressure and mean velocity,

Specific Energy

Relates flow depth and corrected velocity head at a channel section.

E=y+\alpha\frac{V^2}{2g} =y+\alpha\frac{Q^2}{2gA^2}

Variables

Symbol	Description	Unit
$E$	Specific energy relative to channel bottom	m
$y$	Flow depth	m
$\alpha$	Kinetic-energy correction coefficient	-
$A$	Flow area	$m^2$

Specific-Energy Curve and Alternate Depths

At fixed discharge and geometry, $E(y)$ has a minimum at critical flow. For $E>E_{min}$ , two depths are mathematically possible:

A shallow, high-velocity supercritical depth.
A deep, low-velocity subcritical depth.

These are alternate depths. They share discharge and specific energy, not momentum. A hydraulic jump connects conjugate depths instead and loses energy.

Interactive Specific-Energy Solver

Select a rectangular-channel depth to compute its Froude number and solve the alternate depth at the same discharge and energy.

Specific Energy and Alternate Depths

Rectangular channel with a numerically bracketed alternate-depth solution.

Discharge (

Q

)10.0 m³/s

Channel width (

b

)2.0 m

Selected depth (

y

)1.50 m

Unit discharge: 5.000 m²/s
Critical depth: 1.3659 m
Minimum energy: 2.0489 m
Selected energy: 2.0663 m
Selected regime: Subcritical (Fr=0.869)
Alternate depth: 1.24734 m
Alternate regime: Supercritical (Fr=1.146)
Energy residual: -9.69e-11 m

For any energy above the minimum, the shallow and deep roots lie on opposite sides of critical depth. The logarithmic axes keep both roots visible when their magnitudes differ greatly.

Hydraulic Depth, $D_h$

The flow area divided by top width:

D_h=\frac{A}{T}

It is the characteristic depth used in the general open-channel Froude number.

Froude Number

Classifies open-channel flow from inertia and gravity effects.

Fr=\frac{V}{\sqrt{gD_h}} =\frac{V}{\sqrt{gA/T}}

Variables

Symbol	Description	Unit
$Fr$	Froude number	-
$T$	Water-surface top width	m
$D_h$	Hydraulic depth	m

Flow Regimes

Subcritical: $Fr<1$ . Gravity waves can propagate upstream and downstream; downstream controls influence the profile upstream.
Critical: $Fr=1$ . Specific energy is minimum for the specified discharge.
Supercritical: $Fr>1$ . Surface disturbances cannot propagate upstream against the flow; upstream controls determine the profile.

General Critical-Flow Condition

Applies to any prismatic cross-sectional geometry when alpha is approximately one.

\frac{Q^2T}{gA^3}=1

Equivalently,

Fr=1

Rectangular Critical Depth

Critical depth for a rectangular channel using unit discharge q=Q/b.

y_c=\left(\frac{q^2}{g}\right)^{1/3}

E_{min}=\frac{3}{2}y_c

Variables

Symbol	Description	Unit
$q$	Discharge per unit channel width	$m^2/s$

Critical Sections as Controls

A control section establishes a unique depth–discharge relation. Critical flow commonly occurs near broad-crested weirs, free overfalls, properly designed flumes, and transitions where the available specific energy reaches a minimum. Upstream and downstream profiles must be connected to the physically admissible control.

Choking

If a contraction or bed rise demands a minimum specific energy greater than the energy available upstream, the original discharge-depth state cannot pass unchanged. The upstream depth rises until sufficient energy is available, or the discharge changes. This is open-channel choking.

Momentum Function or Specific Force

Rapidly varied flow is analyzed more reliably with momentum than with energy because energy loss through turbulence is large and difficult to predict. For a hydrostatic section,

General Momentum Function

Combines momentum flux and hydrostatic pressure moment per unit fluid weight.

M=\beta\frac{Q^2}{gA}+A\bar y

Variables

Symbol	Description	Unit
$M$	Momentum function or specific force	$m^3$
$\beta$	Momentum correction coefficient	-
$\bar y$	Depth of the area centroid below the free surface	m

Rectangular Momentum Function per Unit Width

For a rectangular channel with $\beta=1$ ,

M'=\frac{q^2}{gy}+\frac{y^2}{2}

Conjugate depths have equal momentum function when external streamwise forces over the short jump are negligible.

Hydraulic Jump

A hydraulic jump is the rapid transition from supercritical to subcritical flow. It converts organized kinetic energy into turbulence, heat, air entrainment, surface waves, and sound. Stilling basins use this process to protect downstream beds and structures from erosion.

Rectangular Sequent-Depth Relation

Relates upstream and downstream conjugate depths for a horizontal rectangular channel.

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

Variables

Symbol	Description	Unit
$y_1$	Supercritical approach depth	m
$y_2$	Subcritical sequent depth	m
$Fr_1$	Approach Froude number	-

Hydraulic-Jump Energy Loss

Specific-energy loss through a rectangular hydraulic jump.

\Delta E =E_1-E_2 =\frac{(y_2-y_1)^3}{4y_1y_2}

Interactive Hydraulic Jump

The simulator checks whether the approach is supercritical, calculates conjugate depth, classifies the jump, and verifies momentum-function equality.

Hydraulic Jump Simulator

Discharge per unit width (

q

)5.00 m²/s

Upstream depth (

y_1

)0.50 m

A hydraulic jump requires supercritical approach flow ( $Fr_1>1$ ). The rectangular-channel conjugate-depth equation assumes a horizontal prismatic channel and negligible friction through the short jump region.

Steady jump

Upstream

V₁ = 10.00 m/s

Fr₁ = 4.52

Downstream

y₂ = 2.953 m

Fr₂ = 0.31

Specific-energy loss2.498 m

Energy retained55.4%

Momentum-function check8.9e-16

Approximate Jump Classification by $Fr_1$

Approach Froude number	Common description	Typical behavior
$1.0$ – $1.7$	Undular	Surface waves; limited dissipation
$1.7$ – $2.5$	Weak	Small rollers; low energy loss
$2.5$ – $4.5$	Oscillating	Unstable jet and large surface oscillations
$4.5$ – $9$	Steady or stable	Well-developed roller; effective dissipation
$>9$	Strong	Very energetic, rough jump with high loads and aeration

The boundaries are empirical and depend on channel geometry, tailwater, approach turbulence, and basin details.

Tailwater Controls Jump Location

The calculated $y_2$ is the depth required immediately downstream for a free jump. If actual tailwater is lower, the jump may sweep downstream; if higher, it may become submerged or move upstream. Stilling-basin design must compare the sequent-depth curve with the downstream rating curve over the operating range.

Gradually Varied Flow Assumptions

The classical GVF equation assumes steady one-dimensional flow in a prismatic channel, small bed slope, hydrostatic pressure distribution, slowly changing depth, known resistance relation, and no significant lateral inflow or outflow. Curvature and vertical acceleration are neglected.

Gradually Varied Flow Equation

Relates the water-surface slope to bed slope, friction slope, and Froude number.

\frac{dy}{dx} = \frac{S_0-S_f}{1-Fr^2}

Variables

Symbol	Description	Unit
$S_0$	Channel bed slope, positive downward in the flow direction	-
$S_f$	Friction or energy slope	-
$x$	Longitudinal coordinate positive downstream	m

Interpreting the GVF Equation

The numerator compares gravity driving slope with friction demand.
The denominator changes sign at critical flow.
At normal depth, $S_f=S_0$ , so $dy/dx=0$ .
Near critical depth, $1-Fr^2\rightarrow0$ and the gradually varied approximation predicts a very large slope; the flow is approaching a rapidly varied control region.

Normal Depth, $y_n$

The depth of uniform flow for a specified discharge, geometry, roughness, and bed slope. It is calculated from Manning, Chezy, or another resistance equation.

Channel-Slope Classification

For a given discharge and section:

Mild, M: $y_n>y_c$ .
Steep, S: $y_n<y_c$ .
Critical, C: $y_n=y_c$ .
Horizontal, H: $S_0=0$ , so no finite normal depth exists under ordinary uniform-flow resistance.
Adverse, A: Bed rises in the flow direction; no ordinary positive-slope normal depth exists.

Profile Zones

Zone numbers describe depth relative to $y_n$ and $y_c$ :

Zone 1: Depth above both reference depths.
Zone 2: Depth between $y_n$ and $y_c$ .
Zone 3: Depth below both reference depths.

Examples include M1 backwater curves upstream of dams, M2 drawdown curves approaching free overfalls or controls, S1 profiles upstream of high tailwater on steep slopes, and S2 profiles downstream of gates before a hydraulic jump.

Not Every Letter–Number Combination Exists

The relative order of $y_n$ and $y_c$ , and the absence of normal depth on horizontal or adverse slopes, make some profile labels physically impossible. Classify the slope first, then locate the actual depth zone.

Control Direction

Subcritical GVF is controlled from downstream; numerical stepping normally proceeds upstream from a known downstream depth.
Supercritical GVF is controlled from upstream; stepping normally proceeds downstream from an upstream control.
A hydraulic jump or other rapidly varied region may connect profiles but cannot be traversed with the GVF equation.

Direct-Step Method

Computes reach length between two known depths in a prismatic channel.

\Delta x = \frac{E_2-E_1}{S_0-\bar S_f}

Variables

Symbol	Description	Unit
$\bar S_f$	Representative average friction slope between the two sections	-
$\Delta x$	Longitudinal distance between sections	m

Direct-Step Method

STEP-BY-STEP

Choose a sequence of depths between the known control depth and the target reach.
At each depth compute area, hydraulic radius, velocity, Froude number, specific energy, and friction slope.
Average $S_f$ between adjacent sections.
Calculate $\Delta x=(E_2-E_1)/(S_0-\bar S_f)$ with a consistent downstream coordinate and section order.
Accumulate reach lengths and reduce the depth increment near critical depth or rapid changes.

Standard-Step Method

The standard-step method applies the full energy equation between stations of known spacing and solves iteratively for the unknown depth:

z_1+y_1+\alpha_1\frac{V_1^2}{2g} = z_2+y_2+\alpha_2\frac{V_2^2}{2g} +h_L

It is preferred for natural channels, surveyed cross sections, changing roughness, bridges, culverts, and nonprismatic geometry. Modern water-surface-profile software is based on this principle with additional contraction, expansion, and structure models.

Numerical Step Size and Convergence

Large steps can skip controls, conceal multiple roots, or violate the gradually varied assumption. Near critical flow, bridges, contractions, and abrupt geometry changes, use shorter steps and a solver that brackets physically admissible depths rather than relying only on unconstrained Newton iteration.

Spatially Varied Flow

When lateral inflow or outflow changes discharge with distance,

\frac{dQ}{dx}=q_l

where $q_l$ is lateral discharge per unit channel length. The momentum and energy equations require additional terms because entering or leaving water carries momentum. Ordinary constant- $Q$ GVF equations are not sufficient.

Non-Uniform Flow Analysis Workflow

STEP-BY-STEP

Identify whether the depth change is rapid, gradual, or caused by changing discharge.
Calculate $y_c$ , $Fr$ , and, when appropriate, $y_n$ .
Locate physical controls and determine whether information propagates upstream or downstream.
Use energy for alternate-depth and gradual-profile problems.
Use momentum for hydraulic jumps and other short turbulent transitions.
Check tailwater and rating curves before fixing a jump location.
Use direct-step or standard-step calculations only outside rapidly varied regions.
Verify freeboard, velocity, shear, cavitation/aeration concerns, and erosion protection along the complete profile.

Key Takeaways

Specific energy and momentum function answer different questions: alternate depths have equal energy, while conjugate jump depths have equal momentum function under ideal jump assumptions.
The general critical condition is $Q^2T/(gA^3)=1$ .
Hydraulic jumps require supercritical approach flow and dissipate energy while approximately conserving momentum.
The GVF equation is $dy/dx=(S_0-S_f)/(1-Fr^2)$ and becomes singular near critical flow, where the gradual-flow model loses validity.
Profile classification requires both normal and critical depths and an understanding of upstream versus downstream control.
Direct-step and standard-step methods must use physically admissible roots, adequate station spacing, and separate models for rapidly varied structures.

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Learning Objectives

Types of Non-Uniform Flow

Uniform and Steady Are Different Classifications

Specific Energy

Specific Energy

Specific-Energy Curve and Alternate Depths

Interactive Specific-Energy Solver

Specific Energy and Alternate Depths

Hydraulic Depth, DhD_hDh​

Froude Number

Flow Regimes

General Critical-Flow Condition

Rectangular Critical Depth

Critical Sections as Controls

Choking

Momentum Function or Specific Force

General Momentum Function

Rectangular Momentum Function per Unit Width

Hydraulic Jump

Rectangular Sequent-Depth Relation

Hydraulic-Jump Energy Loss

Interactive Hydraulic Jump

Hydraulic Jump Simulator

Approximate Jump Classification by Fr1Fr_1Fr1​

Tailwater Controls Jump Location

Gradually Varied Flow Assumptions

Gradually Varied Flow Equation

Interpreting the GVF Equation

Normal Depth, yny_nyn​

Channel-Slope Classification

Profile Zones

Not Every Letter–Number Combination Exists

Control Direction

Direct-Step Method

Direct-Step Method

Standard-Step Method

Numerical Step Size and Convergence

Spatially Varied Flow

Non-Uniform Flow Analysis Workflow

Hydraulic Depth, $D_h$

Approximate Jump Classification by $Fr_1$

Normal Depth, $y_n$