8.4 Open Channel Flow

Open channel flow describes how water moves through channels that aren't completely filled, like rivers, drainage ditches, and stormwater channels. Understanding how to classify these flows, calculate velocities, and predict water surface profiles is essential for designing channels that move water safely and efficiently.

Uniform vs Non-uniform Flow

Characteristics of Uniform and Non-uniform Flow

In uniform flow, the depth, velocity, and cross-sectional area stay constant along the entire channel length. The water surface runs parallel to the channel bed. This only happens in prismatic channels (channels with a consistent cross-section) that have a constant slope and roughness.

In non-uniform flow, depth, velocity, or cross-sectional area change along the channel. This happens whenever the channel geometry changes, the slope shifts, or something obstructs the flow. The water surface profile no longer parallels the bed.

Both types can be either steady or unsteady:

• Steady flow: flow parameters don't change over time at a given point
• Unsteady flow: flow parameters vary with time at a given point

In uniform flow, the energy grade line (EGL) and hydraulic grade line (HGL) have constant slopes that are parallel to the channel bed. In non-uniform flow, those slopes vary along the channel length.

Flow Classification and Analysis

The Froude number ($Fr$) classifies the flow regime, telling you how flow velocity compares to the speed at which a small surface wave would travel:

• Subcritical flow ($Fr < 1$): slow, deep, and tranquil. Most rivers flow subcritically. Disturbances can travel upstream.
• Critical flow ($Fr = 1$): the transitional state between subcritical and supercritical. Depth at this point is called critical depth.
• Supercritical flow ($Fr > 1$): fast, shallow, and rapid. Think of steep mountain streams. Disturbances cannot travel upstream.

For uniform flow, the Manning equation is the primary analysis tool. For non-uniform flow, gradually varied flow (GVF) equations are used instead, since they account for how depth and velocity change along the channel.

Flow Velocity and Discharge Calculation

Manning Equation Components

The Manning equation estimates average flow velocity in an open channel under uniform flow conditions:

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

Where:

• $V$ = average flow velocity (m/s)
• $n$ = Manning's roughness coefficient (dimensionless)
• $R$ = hydraulic radius (m)
• $S$ = channel bed slope (m/m)

Each component deserves a closer look:

Manning's roughness coefficient ($n$) captures how much the channel surface resists flow. Smoother surfaces have lower values. Some typical ranges:

• Finished concrete: ~0.012
• Earth channels (clean): ~0.022
• Earth channels with vegetation: 0.025–0.035
• Natural streams with heavy brush: 0.050+

Channel irregularities, vegetation growth, and obstructions all increase $n$.

Hydraulic radius ($R$) is the ratio of the flow's cross-sectional area to its wetted perimeter:

$R = \frac{A}{P}$

• $A$ = cross-sectional area of the flowing water
• $P$ = wetted perimeter (the length of channel boundary in contact with water)

A larger hydraulic radius means less friction per unit of flowing water, so the flow moves faster.

Channel slope ($S$) is approximated as the bed slope for uniform flow, since the water surface and energy grade line are all parallel.

Discharge Calculation and Applications

Once you have velocity from Manning's equation, discharge follows from the continuity equation:

$Q = V \times A$

Where $Q$ is the volumetric flow rate (m³/s), $V$ is velocity, and $A$ is the cross-sectional flow area.

Here's how to solve a typical Manning equation problem:

1. Identify the channel geometry and calculate the flow area ($A$) and wetted perimeter ($P$)
2. Compute the hydraulic radius: $R = A / P$
3. Look up the Manning's $n$ value for the channel material
4. Determine the bed slope ($S$)
5. Plug values into $V = \frac{1}{n} R^{2/3} S^{1/2}$
6. Multiply velocity by area to get discharge: $Q = V \times A$

Keep in mind that Manning's equation assumes steady, uniform, fully turbulent flow. It works best in rough channels where turbulence dominates. Different forms exist for non-circular conduits, partially full pipes, and the Imperial unit system (which includes a conversion factor of 1.486 in the numerator).

Gradually Varied Flow Profiles

Fundamentals of Gradually Varied Flow

Gradually varied flow (GVF) occurs when depth and velocity change slowly along the channel, so vertical accelerations are negligible. The governing equation for GVF comes from energy conservation and relates the rate of depth change ($dy/dx$) to the channel slope, friction slope, and Froude number.

GVF profiles are classified by comparing three depths at any section:

• Normal depth ($y_n$): the depth that would occur under uniform flow for the given slope and roughness
• Critical depth ($y_c$): the depth at which $Fr = 1$
• Actual depth ($y$): the real water depth at that location

The relationship between these three depths, combined with the channel slope type, produces 12 possible profile types:

The number (1, 2, or 3) indicates where the actual depth falls relative to $y_n$ and $y_c$: Zone 1 is above both, Zone 2 is between them, and Zone 3 is below both.

Backwater and Drawdown Curves

Backwater curves form when the water surface rises above normal depth. The M1 profile is the classic example: a dam or weir downstream raises the water level, and that effect extends upstream as a gradually rising surface. S1 and C1 profiles also produce backwater.

Drawdown curves form when the water surface drops below normal depth. The M2 profile is common near a free overfall (where water drops off the end of a channel) or at a sudden transition to a steeper slope. S2 and C2 profiles are also drawdown curves.

To compute GVF profiles numerically, engineers typically use:

• Standard step method: works along the channel in discrete steps, solving for depth at each station. This is the most common approach for practical problems.
• Direct integration: analytical but limited to simple channel geometries.

Calculations must start from a control point where the depth is known. For subcritical flow, control points are downstream (like a known water surface at a dam). For supercritical flow, control points are upstream. Critical depth sections and gauging stations with known water levels serve as common control points.

Channel Design for Specific Flow Conditions

Channel Design Principles

Designing an open channel means selecting three main things: cross-sectional shape, slope, and lining material.

Common cross-sectional shapes include rectangular, trapezoidal, and circular. Hydraulically efficient sections maximize discharge for a given area by minimizing the wetted perimeter:

• Best hydraulic rectangle: width equals twice the depth ($b = 2y$)
• Best hydraulic trapezoid: side slopes at 60° from horizontal, forming a half-hexagon

Lining materials range from concrete (smooth, low $n$, expensive) to riprap and vegetation (rougher, higher $n$, more natural).

Critical flow conditions matter for specific structures. Hydraulic jumps (the abrupt transition from supercritical to subcritical flow) dissipate energy downstream of spillways. Flow measurement devices like weirs and flumes force flow through critical depth to enable accurate discharge measurement.

Freeboard is the vertical distance between the design water surface and the top of the channel wall. It provides a safety margin against wave action, unexpected flow surges, and debris. Typical freeboard ranges from 0.3 m for small channels to over 1 m for large canals.

Stability and Environmental Considerations

Channel stability analysis ensures the channel won't erode or accumulate sediment over time. Two main approaches:

• Permissible velocity method: compares design velocity against maximum safe velocities for the lining material (e.g., ~0.6 m/s for bare earth, ~6 m/s for reinforced concrete)
• Tractive force method: evaluates shear stress on the channel boundary and compares it to what the lining can withstand

Supercritical flow channels need special design attention. Hydraulic jumps can form unexpectedly, air entrainment increases at high velocities, and specialized structures like stilling basins are often required to safely dissipate energy.

Environmental factors increasingly shape channel design decisions. Fish passage requirements may dictate minimum depths and maximum velocities. Designers often incorporate riffles and pools to support aquatic habitat, preserve riparian vegetation along banks, and consider aesthetics for channels running through urban areas.