Open Channel Hydraulics: Manning's Equation and Energy Concepts

This article builds on the fundamentals of open channel flow to explore the hydraulic theory that underlies channel design calculations. Understanding these concepts will help you make better engineering judgments when calculator results need interpretation.

Manning’s Equation in Depth

Manning’s equation is the workhorse of open channel hydraulics:

Where k = 1.486 for US customary units (ft, s) and k = 1.0 for SI units (m, s).

Origin and Limitations

Robert Manning published his equation in 1889 as an empirical formula — it was derived from fitting experimental data, not from first principles. Despite this, it has proven remarkably accurate across a wide range of conditions.

Manning’s equation is valid when:

• Flow is steady and uniform (constant depth along the channel)
• The channel is prismatic (constant cross-section)
• The channel walls are rigid (no deformation under flow)
• Flow is fully turbulent (Reynolds number > ~12,500 based on hydraulic radius)

Manning’s equation is less reliable for:

• Very shallow flow (sheet flow less than about 0.1 ft deep)
• Very smooth surfaces at low velocities (where flow may be transitional)
• Floodplain flow with significant lateral velocity variation
• Channels with rapidly changing geometry

Manning’s Roughness Coefficient

The Manning’s n value is the most subjective parameter in the equation. It represents the combined resistance to flow from:

• Surface roughness (grain size, ripples, vegetation)
• Channel irregularity (variations in cross-section)
• Channel alignment (bends, meanders)
• Obstructions (bridge piers, debris, rocks)
• Vegetation (type, density, submergence)

Typical values:

Selecting the appropriate n value requires experience and judgment. Published tables provide ranges, but the actual value depends on the specific conditions of your channel.

Learn more about selecting Manning’s n →

Hydraulic Radius and Channel Efficiency

The hydraulic radius R = A/P governs how efficiently a cross-section conveys flow. A higher R means more capacity for a given area.

Comparing Channel Shapes

For a given flow area, different shapes produce different hydraulic radii:

Semicircle — Maximum R for any shape. R = D/4 where D is diameter. Used in theory but rarely in practice (hard to build, unstable slopes).

Rectangle — For maximum efficiency, set width = 2 × depth. This gives R = y/2 where y is the depth. Common in concrete-lined channels.

Trapezoid — Most common shape in earthen channels. The most efficient trapezoidal section has a 60° side slope (about 1.73H:1V), but practical side slopes are usually flatter (2H:1V to 4H:1V) for stability and safety.

Energy in Open Channel Flow

The Energy Equation

Total energy at any cross-section is measured relative to a datum:

Where:

• z = elevation of the channel bottom above datum
• y = depth of flow
• V²/2g = velocity head

The energy grade line (EGL) represents the total energy along the channel. The hydraulic grade line (HGL) is the water surface — it represents the sum of elevation head and pressure head (z + y).

In uniform flow, the EGL and HGL are parallel to the channel bed, and the slope of the EGL equals the channel bed slope.

Specific Energy

Specific energy measures energy relative to the channel bottom rather than a fixed datum:

For a given discharge Q, plotting E vs. y produces the specific energy curve. This curve reveals that:

1. Every value of E (above the minimum) corresponds to two possible depths — a subcritical depth and a supercritical depth. These are called alternate depths.

2. The minimum specific energy occurs at critical depth, where Fr = 1.

3. Below the critical energy, the given discharge cannot occur — flow must either increase its energy (upstream backwater) or decrease its discharge (reduced flow).

Energy Losses

In real channels, energy is lost to friction and turbulence. For gradually varied flow, friction losses per unit length equal the friction slope Sf, which can be estimated using Manning’s equation rearranged:

At transitions (expansions, contractions, bends), additional minor losses occur. These are typically expressed as:

Where K is a loss coefficient that depends on the geometry of the transition.

Gradually Varied Flow

When conditions are not uniform — when the slope changes, the cross-section varies, or an obstruction affects the flow — the depth changes along the channel. This is gradually varied flow (GVF).

The governing equation for GVF is:

Where S₀ is the bed slope and Sf is the friction slope. This equation tells us:

• If the numerator and denominator have the same sign, depth increases downstream (backwater curve).
• If they have opposite signs, depth decreases downstream (drawdown curve).
• At critical depth (Fr = 1), the equation is undefined — this is where flow transitions between subcritical and supercritical regimes.

GVF calculations are iterative — they use Manning’s equation at each step to estimate friction losses, then march along the channel computing depth changes. This is exactly what HEC-RAS does with the standard step method.

Momentum and Hydraulic Jumps

When supercritical flow must transition to subcritical flow (for example, at the base of a spillway), a hydraulic jump occurs. The jump is a violent, turbulent transition that dissipates energy.

The momentum equation relates the depths on either side of the jump. For a rectangular channel:

Where y₁ is the upstream (supercritical) depth and y₂ is the downstream (subcritical) depth. These are called sequent depths (not alternate depths — those share the same specific energy, while sequent depths share the same momentum).

Hydraulic jumps are deliberately used in stormwater design as energy dissipators at culvert outlets, dam spillways, and steep channel transitions.

Applying These Concepts

These theoretical concepts have direct practical applications:

1. Channel sizing: Manning’s equation determines the required cross-section for a given flow rate and slope.
2. Backwater analysis: GVF equations predict how downstream obstructions (bridges, culverts) affect upstream water levels.
3. Culvert design: Energy and momentum concepts determine whether a culvert operates under inlet or outlet control.
4. Energy dissipation: Hydraulic jump theory sizes stilling basins and riprap aprons at high-velocity outlets.

Calculate channel capacity → Find critical depth → Find normal depth →

References

1. Chow, V. T. (1959). Open-channel hydraulics. McGraw-Hill.

2. Henderson, F. M. (1966). Open channel flow. Macmillan.

3. Sturm, T. W. (2010). Open channel hydraulics (2nd ed.). McGraw-Hill.

4. French, R. H. (1985). Open-channel hydraulics. McGraw-Hill.