What This Solves
Calculates energy losses at channel transitions (contractions, expansions, drops) using standard loss coefficients for various transition geometries.
Best Used When
- You are designing a channel width change and need to estimate the head loss
- You need to evaluate energy losses at a bridge approach or culvert inlet transition
- You want to compare head losses between abrupt, gradual, and warped transition geometries
Do NOT Use When
- You need to calculate the full water surface profile through a reach of varying geometry — Use Gradually Varied Flow Calculator
- You need to calculate uniform flow capacity in a constant-section channel — Use Manning's Channel Calculator
Key Assumptions
- Head loss coefficients are empirical values from FHWA HDS-4 and USACE EM 1110-2-1601
- Flow is subcritical through the transition (no hydraulic jumps)
- Energy equation applies between upstream and downstream sections
- Velocity distribution is approximately uniform at each section
Input Quality Notes
Loss coefficients vary with transition angle and geometry. Use published values for the closest matching transition type. For unusual geometries, physical or numerical modeling may be needed.
Estimate the head loss through an open-channel contraction or expansion from the upstream and downstream velocities and an empirical loss coefficient. Supports abrupt, gradual, warped, cylindrical-quadrant and wedge geometries using coefficients from FHWA HEC-22 and USACE EM 1110-2-1601.
Contraction Coefficients (Cc)
| Geometry | Cc |
|---|---|
| Abrupt | 0.5 |
| Gradual (typical) | 0.2 |
| Warped | 0.1 |
| Cylindrical Quadrant | 0.15 |
| Wedge (1:4) | 0.3 |
Expansion Coefficients (Ce)
| Geometry | Ce |
|---|---|
| Abrupt | 1 |
| Gradual (typical) | 0.5 |
| Warped | 0.2 |
| Cylindrical Quadrant | 0.25 |
| Wedge (1:4) | 0.5 |
Ready to Calculate
Select transition type and enter velocities to calculate head loss.
For educational purposes only. Not a substitute for professional engineering judgment.
How it works
A change in channel cross-section changes the flow velocity, and therefore the velocity head V²/2g. Some of that energy is lost to turbulence and, in expansions, to flow separation. The standard transition-loss model expresses this loss as the magnitude of the change in velocity head times a dimensionless coefficient.
Velocity head
hv = V² / (2g)
Contraction (flow accelerates, V2 > V1)
hL = Cc × | (V2² − V1²) / (2g) |
Expansion (flow decelerates, V1 > V2)
hL = Ce × | (V1² − V2²) / (2g) |
Transition length for a given flare angle
L = ΔW / (2 × tan(θ / 2))
Variables
- hL — transition head loss (ft or m)
- hv — velocity head, V²/2g (ft or m)
- V1, V2 — upstream and downstream mean velocities (ft/s or m/s)
- Cc, Ce — contraction / expansion loss coefficient (dimensionless)
- g — gravitational acceleration, 32.2 ft/s² (9.81 m/s²)
- L — transition length (ft or m); ΔW — width change; θ — total flare angle
Because the loss scales with the change in velocity head, transitions between sections of similar velocity lose little energy, while large velocity changes — or abrupt geometry — produce significant losses that should be added to the channel's energy grade line in a backwater profile.
Transition loss coefficients
Typical empirical loss coefficients by geometry, after FHWA HEC-22 (Table 9-2) and USACE EM 1110-2-1601. Contraction coefficients (Cc) apply where the channel narrows; expansion coefficients (Ce) apply where it widens. Values are indicative — verify against the governing manual for your project.
| Transition geometry | Contraction Cc | Expansion Ce |
|---|---|---|
| Abrupt (square-edged) | 0.5 | 1.0 |
| Gradual, well designed | 0.1 | 0.2 |
| Gradual, typical | 0.2 | 0.5 |
| Warped | 0.1 | 0.2 |
| Cylindrical quadrant | 0.15 | 0.25 |
| Wedge (1:4 taper) | 0.3 | 0.5 |
Coefficient by flare angle (gradual transitions)
For a gradual transition, the loss coefficient rises with the wall flare angle. The calculator interpolates from this schedule when you supply a transition angle.
| Flare angle θ (up to) | Contraction Cc | Expansion Ce |
|---|---|---|
| 10° | 0.10 | 0.15 |
| 20° | 0.20 | 0.30 |
| 30° | 0.30 | 0.50 |
| 45° | 0.40 | 0.70 |
| 60° | 0.45 | 0.85 |
| 90° | 0.50 | 1.00 |
Sources: FHWA HEC-22, Urban Drainage Design Manual (3rd ed., 2009), Table 9-2; USACE EM 1110-2-1601, Hydraulic Design of Flood Control Channels (1994); Chow, V.T., Open-Channel Hydraulics (McGraw-Hill, 1959).
Contraction vs. expansion design
Contractions
The channel narrows and flow accelerates, so the pressure gradient is favorable and losses are modest. Flare angles up to about 12.5° generally perform well. Common at culvert and bridge inlets.
Expansions
The channel widens and flow decelerates against an adverse pressure gradient, so it tends to separate and form eddies. Keep the divergence angle below about 6°, and prefer warped or cylindrical-quadrant outlets to recover energy.
The standard coefficients assume steady, subcritical, gradually varied flow in prismatic approach and departure channels. Supercritical flow, sediment transport and waves can change the loss substantially and require specialized analysis.
Frequently asked questions
How is head loss in a channel transition calculated?
The loss is the change in velocity head multiplied by an empirical loss coefficient. For a contraction, h_L = C_c × |(V2² − V1²) / 2g|; for an expansion, h_L = C_e × |(V1² − V2²) / 2g|. V1 is the upstream velocity, V2 the downstream velocity, and g is gravitational acceleration (32.2 ft/s² or 9.81 m/s²). The coefficient C_c or C_e depends on how gradual the transition geometry is.
What is the difference between a contraction and an expansion loss?
In a contraction the channel narrows and flow accelerates (V2 > V1), so velocity head increases and losses are relatively small — typical coefficients run from about 0.1 for a warped transition to 0.5 for an abrupt one. In an expansion the channel widens and flow decelerates (V1 > V2); the decelerating flow tends to separate from the boundary, producing eddies and larger losses, so expansion coefficients are higher, from about 0.2 (warped) up to 1.0 (abrupt).
What transition angle should I use to minimize losses?
Gentler transitions lose less energy. For contractions, wall flare angles up to about 12.5 degrees generally perform well. Expansions are more sensitive to flow separation, so the divergence angle should not exceed roughly 6 degrees. The angle-based coefficient table below shows how C_c and C_e grow as the transition gets steeper.
How do I find the transition length for a chosen flare angle?
Use L = ΔW / (2 × tan(θ/2)), where ΔW is the change in channel width between the upstream and downstream sections and θ is the total flare angle. The calculator includes a helper that converts between width change, transition length and angle in either direction.
Related calculators
Manning's Channel Calculator
Solve open-channel flow, depth and velocity for the upstream and downstream sections.
Gradually Varied Flow
Compute water-surface profiles where transition losses enter the energy balance.
Compound Channel
Analyze main channel and floodplain subsections feeding a transition.
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Last verified: February 2026