Flow in Open Channels - Cheat Sheet
Your final 30-minute exam companion for open-channel hydraulics. Every formula, critical concept, and exam pitfall condensed. Use this to verify your understanding of Manning's equation, critical flow, specific energy, and hydraulic jumps.
Uniform Flow & Manning's Equation
Formulas
v = (1/n) × R^(2/3) × S^(1/2)
Meaning
v = velocity (m/s); n = Manning's roughness; R = hydraulic radius (m); S = bed slope (m/m, decimal)
Watch out
S must be entered as a decimal (0.001, not 0.1% or 1:1000). This is SI form — NOT the 1.49 US version. R = A/P, not geometric radius.
When to use
Calculate mean velocity in uniform (constant-depth) open-channel flow.
Q = A × v
Meaning
Q = discharge (m³/s); A = flow cross-sectional area (m²); v = mean velocity (m/s)
Watch out
A is the area of water, not the channel—must account for actual depth and cross-section geometry.
When to use
Convert velocity to discharge; fundamental continuity equation.
R = A / P
Meaning
R = hydraulic radius (m); A = flow area (m²); P = wetted perimeter (m)
Watch out
P includes only surfaces in contact with water—NOT the free surface. For rectangular: P = b + 2y (width + 2×depth).
When to use
Every Manning calculation; link between geometry and flow capacity.
A = b × y (rectangular); A = (b + m×y) × y (trapezoidal)
Meaning
b = bottom width (m); y = depth (m); m = side slope ratio (horizontal:vertical)
Watch out
Trapezoidal: m is the horizontal distance for 1 unit vertical. Side slope 1.5:1 means m = 1.5.
When to use
Calculate cross-sectional area for common channel shapes.
P = b + 2y (rectangular); P = b + 2y√(1 + m²) (trapezoidal)
Meaning
Calculate wetted perimeter for rectangular and trapezoidal sections.
Watch out
√(1 + m²) is the length factor for sloped sides. For 1.5:1 slope: √(1 + 2.25) = 1.803.
When to use
Feed into R = A/P for Manning calculations.
Common values
Value
n = 0.012–0.013
Symbol
n
Quantity
Concrete channel (smooth)
Value
n = 0.015–0.020
Symbol
n
Quantity
Concrete with algae growth
Value
n = 0.025–0.035
Symbol
n
Quantity
Natural earth channel
Value
n = 0.040–0.060
Symbol
n
Quantity
Vegetated channel
Important facts
- Manning equation is empirical; valid for uniform flow only.
- Roughness n increases with vegetation, sediment, irregularities; decreases with smooth concrete/plastic.
- Slope S directly affects velocity (proportional to √S); doubling slope increases velocity by √2 ≈ 1.41.
- For design: increasing R (wider, shallower) increases capacity; increasing n (rougher surface) decreases capacity.
- Uniform flow assumed steady state and long enough channel for entrance effects to vanish.
Key definitions
Uniform flow
Example
A long, straight canal at steady state with consistent water depth.
Flow at constant depth; energy slope equals bed slope; velocity is constant along the channel.
Hydraulic radius (R)
Example
For 3 m wide × 1.2 m deep rectangular channel: A = 3.6 m², P = 5.4 m, R = 0.667 m.
Ratio of flow area to wetted perimeter; R = A/P; represents efficiency of cross-section.
Wetted perimeter (P)
Example
Rectangular: P = b + 2y. For b = 3, y = 1.2: P = 5.4 m (three sides of rectangle).
Total length of channel boundary in contact with water; excludes free surface.
Manning's n
Example
n = 0.013 for smooth concrete; n = 0.03 for vegetated banks.
Roughness coefficient accounting for friction; typical: n = 0.012–0.015 (concrete), 0.025–0.035 (natural channels).
Bed slope (S)
Example
1 m drop per 1000 m length: S = 0.001.
Longitudinal channel gradient (vertical drop per unit horizontal distance); entered as decimal.
Diagrams to know
- Cross-section geometry: rectangular, trapezoidal, triangular, circular (showing b, y, m, P labels).
- Friction slope ≈ bed slope in uniform flow (energy grade line parallel to bed).
- Roughness classification chart (n vs. channel type).
Most Efficient (Best Hydraulic) Section
Formulas
For rectangular: b = 2y (optimal width = 2 × optimal depth)
Meaning
Channel width should be twice the flow depth to minimize wetted perimeter for given area.
Watch out
This minimizes P for fixed A, maximizing R and hence Q for given n and S. Not all designs can achieve this ratio due to practical constraints.
When to use
Design problems asking for 'most economical' or 'most efficient' rectangular channel.
For rectangular optimal section: R_opt = y/2
Meaning
Hydraulic radius of the optimal rectangular section.
Watch out
This follows from A = 2y² and P = 4y, giving R = y/2. Only valid when b = 2y.
When to use
Quick check that you have the right depth when designing for efficiency.
For trapezoidal optimal section: sides at 60° from horizontal (half-hexagon)
Meaning
Angle provides minimum wetted perimeter for given area in trapezoidal form.
Watch out
This is theoretical optimum; practical channels use gentler slopes (1:1, 1.5:1) for stability.
When to use
Advanced design; rarely asked but worth knowing. Side slope m ≈ 0.577 (or 1:1.73).
Circular channel: max discharge at y ≈ 0.94D (94% full)
Meaning
Circular pipe carrying open flow has maximum capacity slightly below full.
Watch out
At full depth (y = D), wetted perimeter is maximum, reducing R; there is a depth where Q peaks.
When to use
Partly-full pipe flow design or hydraulic comparisons.
Common values
Value
b = 2y
Symbol
—
Quantity
Optimal rectangular slope ratio
Value
m = 0.577 (1:1.73)
Symbol
m
Quantity
Optimal trapezoidal side slope
Important facts
- Minimizing P maximizes R = A/P for fixed A, thereby maximizing v and Q.
- Rectangular optimum (b = 2y) is most common in practice; easy to construct and remember.
- Trapezoidal half-hexagon is theoretical ideal but rarely used (stability concerns at 60° slope).
- Natural channels rarely achieve hydraulic optimum due to bank stability, vegetation, and construction constraints.
- Most efficient section reduces excavation and concrete cost per unit discharge.
Key definitions
Most efficient section
Example
Rectangular channel with b = 2y carries more flow than b = 4y for same area.
Channel geometry that maximizes discharge (or carries given Q at minimum cost) for fixed area, slope, and roughness.
Hydraulic optimum
Example
Rectangular optimum: P_min when width = twice depth.
Condition where wetted perimeter is minimized for given cross-sectional area.
Diagrams to know
- Rectangular cross-section with b = 2y dimensions labeled.
- Trapezoidal half-hexagon (60° sides) with optimal geometry.
- Plot of discharge Q vs. width (for fixed area) showing a peak.
Specific Energy & Critical Flow
Formulas
E = y + v²/(2g)
Meaning
E = specific energy (m) relative to channel bottom; y = depth (m); v = velocity (m/s); g = 9.81 m/s².
Watch out
Energy per unit weight (not force); measured from channel bed, not surface. Do NOT use g = 10 unless explicitly told.
When to use
Analyze flow regimes, find critical depth, determine if jump occurs.
E = y + Q²/(2gA²)
Meaning
Alternative form using discharge Q and area A instead of velocity.
Watch out
Same energy; just rearranged for different input data.
When to use
When v is unknown or Q is given directly.
Fr = v / √(gy)
Meaning
Fr = Froude number (dimensionless); gy is velocity of shallow-water wave.
Watch out
For non-rectangular sections, use hydraulic depth D_h = A/T where T = top surface width. Then Fr = v/√(gD_h).
When to use
Classify flow regime. Fr < 1 subcritical; Fr = 1 critical; Fr > 1 supercritical.
y_c = (q²/g)^(1/3) (rectangular)
Meaning
y_c = critical depth (m); q = unit discharge = Q/b (m³/s per meter width).
Watch out
This is ONLY for rectangular sections. For trapezoidal or circular, use Fr = 1 condition iteratively or charts.
When to use
Find critical depth for rectangular channels.
E_min = (3/2) y_c (rectangular)
Meaning
Minimum specific energy at critical flow in rectangular channel.
Watch out
At critical depth, velocity v_c = √(g y_c); verify E_min = y_c + g×y_c/(2g) = 1.5 y_c.
When to use
Design; minimum energy to pass a given unit discharge.
v_c = √(g y_c)
Meaning
Velocity at critical flow; equals shallow-water wave speed.
Watch out
This is the transition velocity between subcritical and supercritical.
When to use
Quick check: at critical flow, Froude = 1, so v = √(gy).
Common values
Value
g = 9.81 m/s²
Symbol
g
Quantity
Gravitational acceleration
Value
Fr = 1.0
Symbol
Fr
Quantity
Froude classification boundary
Important facts
- E vs. y curve (for constant Q) is hyperbolic; has a minimum at y = y_c where Fr = 1.
- Two depths can carry same energy: one subcritical (left limb, lower energy term v²/2g), one supercritical (right limb, higher velocity).
- Critical depth is independent of slope and roughness; depends only on Q and channel geometry.
- At critical flow, the channel has maximum capacity (for fixed E); any obstacle triggers flow depth adjustment.
- Froude number < 1 means gravity dominates (inertial forces weak); > 1 means inertia dominates (gravity weak).
Key definitions
Specific energy (E)
Example
For y = 1 m, v = 2 m/s: E = 1 + 4/(2×9.81) = 1 + 0.204 = 1.204 m.
Total mechanical energy per unit weight relative to channel bottom; E = y + v²/(2g).
Critical flow
Example
Rectangular channel with y_c = 0.742 m carrying unit discharge q = 2 m³/s/m.
Flow at Froude number = 1; transition between subcritical (slow) and supercritical (fast); specific energy is minimum.
Froude number (Fr)
Example
Fr = 0.541 (subcritical), Fr = 1.0 (critical), Fr = 1.5 (supercritical).
Ratio of flow velocity to shallow-water wave speed; Fr = v/√(gy); dimensionless.
Subcritical flow (Fr < 1)
Example
Deep, slow-moving river; long shallow water surface waves propagate upstream.
Slow, tranquil flow controlled by downstream conditions; depth and pressure high, velocity low.
Supercritical flow (Fr > 1)
Example
Steep spillway; disturbances cannot propagate upstream (Mach cone effect).
Fast, rapid flow controlled by upstream conditions; depth and pressure low, velocity high.
Diagrams to know
- E–y diagram (specific energy curve): hyperbolic, two branches (subcritical left, supercritical right), minimum at y_c.
- Fr vs. y plot: Fr = 1 at critical depth; Fr < 1 (subcritical), Fr > 1 (supercritical).
- Velocity and energy profiles for different flow regimes.
Hydraulic Jump
Formulas
y₂/y₁ = (1/2) × [√(1 + 8Fr₁²) − 1]
Meaning
y₂ = sequent (conjugate) depth after jump (m); y₁ = depth before jump (m); Fr₁ = Froude number at entry.
Watch out
Fr₁ > 1 REQUIRED (supercritical entering). If Fr₁ < 1, no jump occurs. Most exams: Fr₁ is given or calculate from v₁ and y₁.
When to use
Find downstream depth after abrupt transition from supercritical to subcritical.
ΔE = E₁ − E₂ (energy loss in jump)
Meaning
Energy dissipated (per unit weight) during jump; always positive for supercritical entry.
Watch out
Jump is not isentropic; significant energy lost to turbulence, heat, sound. E₂ < E₁.
When to use
Quantify energy loss to check jump feasibility or size stilling basin.
ΔE = (y₂ − y₁)³ / [4y₁y₂]
Meaning
Compact form for energy loss in rectangular channel jump.
Watch out
Derived from momentum and energy equations; only valid for rectangular channels.
When to use
Quick calculation of energy dissipation.
L_jump ≈ 5 to 7 × y₂
Meaning
Approximate jump length (horizontal distance over which jump occurs).
Watch out
Empirical; varies with channel roughness and approach conditions. L_jump can be 5–12 y₂ depending on Fr.
When to use
Design stilling basin length.
Common values
Value
5 to 7 × y₂
Symbol
L
Quantity
Typical jump length
Value
Fr = 3 to 5
Symbol
Fr
Quantity
Jump classification: Froude range for steady jump
Important facts
- Jump occurs only if entering flow is supercritical (Fr₁ > 1); subcritical flow (Fr₁ < 1) cannot jump.
- Jump is highly dissipative: energy loss is (y₂ − y₁)³ / [4y₁y₂]. High Fr means large loss.
- Momentum is conserved across jump (in inviscid analysis); energy is NOT conserved (irreversible).
- Jump length 5–7 y₂ is typical; longer jumps more stable but require larger basin.
- Jump classification: weak (Fr 1–2), oscillating (Fr 2–3), steady (Fr 3–5), strong (Fr > 5).
Key definitions
Hydraulic jump
Example
Water jet hits basin floor, rises suddenly (jump), slows to tranquil downstream flow.
Abrupt, turbulent transition from supercritical to subcritical flow; dissipates large energy.
Conjugate (sequent) depths
Example
y₁ = 0.5 m (supercritical) → y₂ = 2.3 m (subcritical) after jump.
Pair of depths (y₁, y₂) connected by jump; satisfy momentum and continuity equations.
Stilling basin
Example
Spillway energy dissipator; baffle blocks and end sill reduce scour downstream.
Basin or structure below spillway designed to contain and dissipate hydraulic jump energy.
Diagrams to know
- Hydraulic jump profile: supercritical approach (shallow, fast) → sudden rise → subcritical exit (deep, slow).
- Jump length diagram with baffle and end-sill blocks.
- Energy curve showing E₁ > E₂ with vertical jump path.
Channel Design & Practical Applications
Formulas
Design approach: assume uniform flow, use Manning equation iteratively.
Meaning
Given Q, n, S, solve for y (and b for rectangular) such that Manning Q matches required discharge.
Watch out
Manning assumes uniform flow; real channels may have non-uniform reaches. Always verify Fr to confirm regime.
When to use
All practical channel design and capacity checks.
For most efficient rectangular: b = 2y, then solve Manning for y.
Meaning
Combines efficiency constraint with discharge requirement.
Watch out
Final design often adjusted for constructability, freeboard, maintenance access.
When to use
Economy or minimum-cost design.
Freeboard = 0.3 to 0.5 m (typical for small channels); 10–15% of depth for large ones.
Meaning
Safety margin above design water surface to prevent overtopping.
Watch out
Larger in steep or unlined channels. Account for wave action, surges, settlement.
When to use
Channel design (not formula-based; engineering judgment).
Common values
Value
0.3 to 0.5 m
Symbol
FB
Quantity
Freeboard for small channels
Value
10 to 15% of depth
Symbol
FB
Quantity
Freeboard for large channels
Important facts
- Channel design iterates Manning equation; no closed-form solution for rectangular section.
- Economic design: minimize excavation cost + lining cost; often not hydraulically optimal (b ≠ 2y).
- Side slopes depend on soil: cohesive (1:1 or steeper), sand (1.5:1), clay (0.5:1 to 1:1).
- Manning n varies with season (vegetation), maintenance, and sediment load.
- For trapezoidal or natural channels, use standard design tables or numerical solvers.
Key definitions
Channel capacity
Example
Rectangular channel 3 m × 1.5 m at S = 0.001, n = 0.013 carries Q ≈ 7 m³/s.
Maximum discharge the channel can carry safely; determined by Manning equation or allowable depth.
Design depth (normal depth)
Example
y_n = depth at which Q_Manning = Q_design.
Uniform-flow depth for given Q, n, S; satisfies Manning Q = required Q.
Diagrams to know
- Channel cross-section with freeboard allowance labeled.
- Manning Q vs. depth curve showing design intersection point.
- Typical side-slope stability envelope for different soils.
Non-Uniform Flow Basics
Formulas
y ≠ constant (non-uniform flow); S_f ≠ S_bed.
Meaning
Energy slope S_f (friction loss per unit distance) differs from bed slope S.
Watch out
Chapter focuses on UNIFORM flow; non-uniform flow (backwater, drawdown curves) is separate topic.
When to use
Recognize that uniform-flow assumptions break down (near controls, transitions, obstacles).
Important facts
- Non-uniform flow requires step-by-step integration (numerical methods).
- Backwater curve: depth increases upstream of an obstacle (M1, M2 curves, etc.); depends on Fr and bed slope class.
- Drawdown curve: depth decreases approaching a sudden drop (S2 curve on steep bed).
- Manning uniform-flow assumption valid away from controls; near controls, specific-energy / momentum approach needed.
Key definitions
Non-uniform flow
Example
Approach to a dam, exit from a spillway, flow over an obstacle.
Flow with varying depth along channel; arises when bed slope ≠ energy slope or geometry changes.
Diagrams to know
- Backwater profile upstream of dam showing M1 (mild slope), M2, M3 curves.
- Drawdown profile on steep slope approaching brink (S2 curve).
- Control point concept (critical depth, weir crest, gate lip).
Must remember
- 1. Manning SI equation: v = (1/n) R^(2/3) S^(1/2). Slope S is DECIMAL (not %, not ratio). This is NOT the 1.49 US version.
- 2. Hydraulic radius R = A/P (area divided by WETTED perimeter only—no free surface). Critical exam mistake: using wrong perimeter.
- 3. Most efficient rectangular section: b = 2y. Minimizes P for given A, maximizing discharge. Direct exam favorite.
- 4. Critical flow: Fr = 1. Use Fr = v/√(gy) for rectangular; for non-rectangular, use hydraulic depth D_h = A/T.
- 5. Critical depth (rectangular): y_c = (q²/g)^(1/3) where q = Q/b. Minimum specific energy at this depth.
- 6. Specific energy: E = y + v²/(2g). Plot has minimum at critical depth; two depths possible for same E.
- 7. Froude zones: Fr < 1 subcritical (downstream control), Fr = 1 critical (transition), Fr > 1 supercritical (upstream control).
- 8. Hydraulic jump formula: y₂/y₁ = (1/2)[√(1 + 8Fr₁²) − 1]. Requires Fr₁ > 1 (supercritical input). Energy dissipated.
- 9. Manning n values: 0.012–0.013 smooth concrete, 0.025–0.035 natural earth, 0.04–0.06 vegetated. Know typical ranges cold.
- 10. Uniform flow assumption requires constant depth, long channel, steady state. Breaks down near controls, gates, transitions.
Last minute tips
- Always verify units before plugging into Manning: S as decimal (0.001, not 1:1000), R in meters, v in m/s, n dimensionless. Single unit error ruins the answer.
- When designing for 'most efficient' section, solve for depth first using efficiency constraint (b = 2y for rectangular), then check Manning Q against required discharge. If mismatch, iterate or note that practical constraints override efficiency.
- In hydraulic jump problems, CHECK Fr₁ immediately. If Fr₁ < 1, no jump occurs—the question may be asking you to recognize this and state 'no jump' rather than calculate depths.
- For critical depth, ALWAYS use the rectangular formula y_c = (q²/g)^(1/3) only if the channel is rectangular. For trapezoidal or circular, you must use the Fr = 1 condition and solve iteratively or use charts—there is no shortcut formula.
- Energy dissipation in jump: ΔE = (y₂ − y₁)³ / [4y₁y₂]. Large y₂/y₁ ratios mean large energy loss. If exam asks 'how much energy is lost?', calculate this—common design check.
Comparison tables
Flow Regimes & Froude Number
| Regime | Froude (Fr) | Depth | Velocity | Wave Propagation | Controlled By |
|---|---|---|---|---|---|
| Subcritical | Fr < 1 | Large (slow) | Low | Upstream & downstream | Downstream conditions |
| Critical | Fr = 1 | y_c (optimum) | v_c = √(gy_c) | Stationary (transition) | Channel geometry only |
| Supercritical | Fr > 1 | Small (rapid) | High | Downstream only (V-cone) | Upstream conditions |
Manning Roughness Coefficient (n) — Typical Values
| Channel Type | Minimum n | Normal n | Maximum n | Notes |
|---|---|---|---|---|
| Smooth concrete | 0.011 | 0.012–0.013 | 0.015 | Best for design; algae increases n |
| Brick / stone masonry | 0.012 | 0.015 | 0.020 | Joints increase roughness |
| Natural earth channel | 0.020 | 0.025–0.030 | 0.035 | Depends on weed growth, sediment |
| Vegetated channel | 0.035 | 0.045–0.060 | 0.100 | Seasonal variation significant |
| Gravel bed river | 0.025 | 0.030 | 0.040 | Increases with stone size |
Most Efficient Channel Sections — Quick Reference
| Shape | Optimal Geometry | Optimal R | When Used | Stability |
|---|---|---|---|---|
| Rectangular | b = 2y | R = y/2 | Small lined channels, flumes | Excellent (concrete/steel) |
| Trapezoidal | Half-hexagon (60° sides, m ≈ 0.577) | R ≈ 0.5y | Theoretical optimum | Poor (bank slope too steep) |
| Practical trapezoidal | m = 1.0 to 1.5 | R depends on b, y | Earth/lined channels (typical) | Good (1:1 to 1.5:1 slopes) |
| Circular | Maximum Q at y ≈ 0.94D | Varies | Pipe sewers, culverts | Excellent (structural) |
Hydraulic Jump Parameters — Key Relationships
| Fr₁ Range | Jump Type | Depth Ratio (y₂/y₁) | Energy Loss | Stability |
|---|---|---|---|---|
| 1.0 < Fr₁ < 1.7 | Weak/undular | < 2 | Low (< 5%) | Weak, oscillating |
| 1.7 < Fr₁ < 2.5 | Oscillating | 2 to 3 | Moderate | Poor (waves upstream) |
| 2.5 < Fr₁ < 4.5 | Steady / strong | 3 to 6 | High (15–45%) | Good (best for stilling) |
| Fr₁ > 4.5 | Strong / rough | > 6 | Very high (45%+) | Excellent (intense dissipation) |
Diagrams
Open-Channel Flow — Concept Hierarchy
Hierarchical map showing how uniform flow, critical flow concepts, hydraulic jumps, and design applications connect in open-channel hydraulics.
Open-Channel Problem-Solving Decision Tree
Decision flowchart guiding students through open-channel analysis: discriminate between Manning capacity checks, flow regime classification, and hydraulic jump design.
Manning Equation Application Flow
Step-by-step Manning equation workflow emphasizing unit consistency, iterative solving, and post-calculation Froude verification.
Flow Regime State Transitions
State diagram showing flow regime transitions, key Froude boundaries, and the role of hydraulic jump in connecting supercritical to subcritical flow.