Open Channel Flow: Uniform Flow - Theory & Concepts

Open Channel Flow: Uniform Flow

Learning Objectives

Distinguish open-channel flow from pressure flow and classify uniform flow conditions.
Calculate area, wetted perimeter, top width, hydraulic radius, and hydraulic depth for common channel sections.
Apply the Chezy and Manning equations with consistent units and appropriate roughness values.
Determine normal depth, velocity, and discharge for prismatic channels.
Explain hydraulically efficient rectangular and trapezoidal sections.
Analyze partially full circular conduits without confusing maximum discharge with a semicircular depth.

Open-channel geometry, uniform-flow resistance, normal depth, and hydraulically efficient sections.

Open-Channel Flow

Flow with a free surface exposed to atmospheric pressure. The driving force is primarily the component of gravity acting along the channel slope. Rivers, canals, flumes, roadside drains, and partially full storm or sanitary sewers are common examples.

Open Channel versus Pressure Conduit

A conduit is not classified by shape alone. A circular sewer flowing partially full is an open channel because it has a free surface. The same pipe flowing completely full under pressure is analyzed as closed-conduit flow.

Prismatic and Non-Prismatic Channels

A prismatic channel has constant cross-sectional shape, dimensions, bed slope, and roughness along the reach. Uniform-flow equations are normally applied to prismatic reaches.

A non-prismatic channel changes geometry, slope, or roughness along its length. Its depth generally changes with distance and requires varied-flow analysis.

Uniform Flow Conditions

Uniform flow occurs when depth, area, mean velocity, and discharge remain constant along a channel reach. For steady uniform flow,

S_0=S_w=S_f

where $S_0$ is the bed slope, $S_w$ is the water-surface slope, and $S_f$ is the friction or energy-grade slope. The downslope component of water weight is balanced by boundary resistance.

Normal Depth

The constant depth $y_n$ that satisfies a selected uniform-flow resistance equation for a specified discharge, channel geometry, slope, and roughness. Normal depth is generally found by iteration because area and hydraulic radius both depend on depth.

Velocity Distribution

Velocity varies throughout an open-channel section because of boundary shear and secondary currents.

Velocity is zero at a stationary solid boundary under the no-slip idealization.
Maximum velocity commonly occurs slightly below the free surface rather than exactly at it.
The section-average velocity is $V=Q/A$ .
Field measurements often estimate depth-mean velocity from the velocity at $0.6y$ below the surface, or from the average at $0.2y$ and $0.8y$ .

Geometric Elements

For a flow depth $y$ :

Flow area, $A$ : cross-sectional area occupied by water.
Wetted perimeter, $P$ : channel boundary in contact with water; the free surface is excluded.
Top width, $T$ : width of the free surface.
Hydraulic radius, $R=A/P$ : area available for flow per unit wetted boundary.
Hydraulic depth, $D_h=A/T$ : characteristic depth used in Froude-number and critical-flow analysis.

Hydraulic Radius

Ratio of flow area to wetted perimeter.

R=\frac{A}{P}

Variables

Symbol	Description	Unit
$R$	Hydraulic radius	m
$A$	Flow area	$m^2$
$P$	Wetted perimeter	m

Hydraulic Depth

Ratio of flow area to free-surface top width.

D_h=\frac{A}{T}

Variables

Symbol	Description	Unit
$D_h$	Hydraulic depth	m
$A$	Flow area	$m^2$
$T$	Top width	m

Rectangular Channel Geometry

For bottom width $b$ and flow depth $y$ :

A=by,\qquad P=b+2y,\qquad T=b

R=\frac{by}{b+2y},\qquad D_h=y

Trapezoidal Channel Geometry

Let $m$ be the horizontal projection per unit vertical rise of each side, written $m\text{H}:1\text{V}$ .

A=y(b+my)

P=b+2y\sqrt{1+m^2}

T=b+2my

R=\frac{y(b+my)}{b+2y\sqrt{1+m^2}}

Triangular Channel Geometry

A symmetric triangular section is the special case $b=0$ of a trapezoid:

A=my^2,\qquad P=2y\sqrt{1+m^2},\qquad T=2my

Partially Full Circular Conduit Geometry

For a circular conduit of radius $r$ and wetted central angle $\theta$ in radians:

A=\frac{r^2}{2}(\theta-\sin\theta)

P=r\theta

T=2r\sin\left(\frac{\theta}{2}\right)

y=r\left[1-\cos\left(\frac{\theta}{2}\right)\right]

These relations are used with Manning or Chezy resistance to determine depth and discharge.

Chezy Equation

An empirical uniform-flow relation in which mean velocity depends on hydraulic radius, friction slope, and the Chezy coefficient $C$ .

Chezy Velocity Equation

Mean velocity for steady uniform open-channel flow.

V=C\sqrt{RS_f}

Variables

Symbol	Description	Unit
$V$	Mean velocity	m/s
$C$	Chezy coefficient	$m^{1/2}/s$
$R$	Hydraulic radius	m
$S_f$	Friction slope	m/m

Chezy Discharge Equation

Uniform discharge obtained from Q=AV.

Q=AC\sqrt{RS_f}

Variables

Symbol	Description	Unit
$Q$	Discharge	$m^3/s$
$A$	Flow area	$m^2$
$C$	Chezy coefficient	$m^{1/2}/s$
$R$	Hydraulic radius	m
$S_f$	Friction slope	m/m

Chezy Coefficient from Manning n

Relation obtained by equating the SI Manning and Chezy equations.

C=\frac{1}{n}R^{1/6}

Variables

Symbol	Description	Unit
$C$	Chezy coefficient	$m^{1/2}/s$
$n$	Manning roughness coefficient	$s/m^{1/3}$
$R$	Hydraulic radius	m

Manning Equation

The most widely used empirical resistance relation for steady uniform open-channel flow. The coefficient $n$ represents the combined effects of boundary material, vegetation, irregularity, alignment, obstructions, and maintenance condition.

Manning Velocity Equation — SI

Mean velocity using metric units.

V=\frac{1}{n}R^{2/3}S_f^{1/2}

Variables

Symbol	Description	Unit
$V$	Mean velocity	m/s
$n$	Manning roughness coefficient	$s/m^{1/3}$
$R$	Hydraulic radius	m
$S_f$	Friction slope	m/m

Manning Discharge Equation — SI

Uniform discharge in a channel section.

Q=\frac{1}{n}AR^{2/3}S_f^{1/2}

Variables

Symbol	Description	Unit
$Q$	Discharge	$m^3/s$
$n$	Manning roughness coefficient	$s/m^{1/3}$
$A$	Flow area	$m^2$
$R$	Hydraulic radius	m
$S_f$	Friction slope	m/m

Manning Velocity Equation — US Customary

Mean velocity when R is in feet.

V=\frac{1.486}{n}R^{2/3}S_f^{1/2}

Variables

Symbol	Description	Unit
$V$	Mean velocity	ft/s
$n$	Manning roughness coefficient	$s/ft^{1/3}$
$R$	Hydraulic radius	ft
$S_f$	Friction slope	ft/ft

Unit Consistency

Do not insert metric dimensions into the US customary form or vice versa. Manning $n$ is commonly treated as numerically similar across unit systems only because the conversion factor is carried by the equation coefficient.

Typical Manning Roughness Ranges

Representative values for preliminary work include:

Smooth finished concrete: approximately $0.011$ to $0.014$ .
Ordinary concrete or masonry: approximately $0.013$ to $0.017$ .
Clean straight earth channel: approximately $0.018$ to $0.025$ .
Natural stream with stones, weeds, or irregular banks: often $0.030$ to $0.070$ or higher.

Final design values should come from an accepted reference, field calibration, or agency criteria appropriate to the actual channel condition.

Normal-Depth Calculation

STEP-BY-STEP

Select a trial depth $y$ .
Compute $A(y)$ and $P(y)$ from the channel geometry.
Compute $R(y)=A/P$ .
Evaluate the Manning or Chezy discharge.
Compare calculated discharge with the required discharge.
Adjust depth and repeat until the discharge residual is acceptably small.

Numerical Solution

Bisection is slower than Newton iteration but is robust when a valid depth interval is known. Newton iteration can fail if a trial produces a negative depth, an invalid circular-section angle, or a near-zero derivative. The simulator below uses a bracketed solution to prevent those failures.

Uniform Open-Channel Flow: Manning Equation

Discharge, Q10.0 m³/s

Energy slope, S0.0010

Manning roughness, n0.015

Bottom width, b2.0 m

Normal depth, yₙ = 2.891 m

Mean velocity

1.729 m/s

Hydraulic radius

0.743 m

Hydraulic depth

2.891 m

Froude number

0.325

Subcritical

Q=\frac{1}{n}AR^{2/3}S^{1/2}

Uniform flow assumes a prismatic reach, steady discharge, hydrostatic pressure distribution, and energy slope approximately equal to bed slope. Manning n is empirical and must match lining condition, vegetation, irregularity, and scale.

Conveyance

Manning discharge is often written using conveyance $K$ :

Q=K\sqrt{S_f}

K=\frac{1}{n}AR^{2/3}

Conveyance isolates the effects of geometry and roughness from the energy slope. It is particularly useful for compound and divided sections.

Composite Roughness and Compound Sections

Floodplains, benches, and lined low-flow channels may have different roughnesses. Dividing the section into subsections and summing conveyances is usually more defensible than using a simple arithmetic average of $n$ .

For subsection $i$ :

K_i=\frac{1}{n_i}A_iR_i^{2/3}

Then, when the same energy slope applies,

Q=\left(\sum K_i\right)\sqrt{S_f}

Hydraulically Efficient Section

For a prescribed flow area, slope, and roughness, the hydraulically efficient section minimizes wetted perimeter and therefore maximizes hydraulic radius, velocity, and discharge. Hydraulic efficiency is not automatically the least-cost design because excavation, lining, right-of-way, stability, freeboard, and maintenance also matter.

Most Efficient Rectangular Section

For a rectangular channel with fixed area, the wetted perimeter is minimized when

b=2y

At this condition,

R=\frac{y}{2}

Most Efficient Trapezoidal Section

For a specified side slope, the optimum trapezoid satisfies

\frac{T}{2}=y\sqrt{1+m^2}

In words, half the top width equals one sloping side length. This also gives

R=\frac{y}{2}

If the side slope itself is unconstrained, the theoretical optimum has sides at $60^\circ$ to the horizontal, corresponding to $m=1/\sqrt{3}$ .

Circular Conduit Misconception

A semicircular channel shape chosen freely is hydraulically efficient because it minimizes boundary length for a given area. That result must not be confused with the depth that maximizes flow in a fixed circular conduit.

For a fixed circular pipe flowing partially full under the same $n$ , diameter, and slope:

Maximum mean velocity occurs near $y/D\approx0.81$ .
Maximum discharge occurs near $y/D\approx0.94$ .
The maximum discharge is slightly greater than the full-flow gravity discharge.

Therefore, $y=D/2$ is not the maximum-discharge depth of a circular conduit.

Design Checks beyond Manning Capacity

Provide freeboard for waves, uncertainty, debris, and operational variation.
Check minimum velocity or tractive stress where sediment deposition is possible.
Check maximum permissible velocity and lining stability against erosion.
Account for bends, transitions, structures, backwater, and local losses.
Evaluate rapidly and gradually varied flow where depth is not uniform.
Confirm that downstream controls do not prevent the assumed normal depth from developing.

Key Takeaways

Open-channel flow has a free surface; a conduit may behave as either an open channel or a pressure pipe depending on whether it is full.
Uniform flow requires constant depth and equality of bed, water-surface, and friction slopes.
Hydraulic radius is $A/P$ ; hydraulic depth is $A/T$ .
Manning and Chezy equations are resistance relations, not conservation laws, and depend on defensible roughness values.
Normal depth is generally an implicit numerical solution because geometry depends on depth.
Efficient rectangular and trapezoidal sections minimize wetted perimeter for a given area.
In a fixed circular conduit, maximum velocity and maximum discharge occur at depths greater than one-half diameter; a semicircular depth is not the maximum-discharge condition.

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