Open Channel Design: Normal Depth, Critical Flow and Lining Selection

A 2-PDH course on designing a roadside ditch or drainage channel: trapezoidal geometry, Manning normal depth by iteration, specific energy, critical depth and the Froude number, lining selection by permissible velocity and tractive force, riprap sizing and freeboard. Worked examples, graded quiz, certificate.

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Instructor: Michael Flynn, PE · Estimated time: 120 minutes including the quiz · Which states accept it?

Learning objectives

Compute area, wetted perimeter, top width and hydraulic radius for trapezoidal, triangular and rectangular sections.
Solve Manning's equation for normal depth by iteration and check the result.
Find critical depth and the Froude number, classify the flow, and explain why near-critical flow is avoided.
Check a lining by permissible velocity and by tractive force, including the side-slope factor.
Size a riprap lining with the Shields relation and set freeboard for the design and check flows.

1. Scope and criteria

Open channels in site and roadway work range from grassed swales and roadside ditches to riprap-lined outfall channels. The design questions are always the same: how deep will the design flow run, is the flow subcritical or supercritical, will the lining stay in place, and how much freeboard is left. This course covers uniform-flow design of prismatic channels with Manning's equation, critical-flow checks, and lining selection by permissible velocity and by tractive force, the method in FHWA HEC-15, Design of Roadside Channels with Flexible Linings.

Criteria to look up first: the design storm for the channel (often 10 years for roadside ditches and 25 to 100 years for larger channels), the check storm, the required freeboard, maximum side slopes (3:1 or flatter is common where mowing or vehicle recovery matters), allowable linings, and the Manning n values the agency accepts.

2. Section geometry

For a trapezoid with bottom width b, side slopes z:1 (horizontal:vertical) and flow depth y:

A = (b + z y) y P = b + 2y√(1 + z²) T = b + 2 z y R = A/P D_h = A/T

A triangle is the trapezoid with b = 0 and a rectangle the one with z = 0. R is the hydraulic radius, used in Manning's equation; D_h = A/T is the hydraulic depth, used in the Froude number. For wide shallow channels R and D_h both approach the depth; for narrow deep ones they differ a great deal.

3. Uniform flow and normal depth

Q = (1.486 / n) · A · R^2/3 · S^1/2

In a long prismatic channel of constant slope, flow settles to the depth at which gravity and friction balance, the normal depth y_n. Because A and R both depend on y, the equation cannot be solved directly for y. Iterate: guess y, compute Q, and adjust until the computed Q matches the design flow (or use a solver; the PE-Calc Manning calculator does this).

Example 1: capacity at a given depth. A grass-lined trapezoidal ditch has b = 4 ft, z = 3, n = 0.035, S = 0.004. At y = 1.50 ft:
A = (4 + 3 × 1.50) × 1.50 = 12.75 ft²; P = 4 + 2 × 1.50 × √(1 + 3²) = 13.49 ft; R = 0.945 ft
Q = (1.486/0.035) × 12.75 × 0.945^2/3 × 0.004^1/2 = 33.0 cfs

Example 2: normal depth for the design flow. The 10-year flow is 60 cfs. Iterating on y:

y (ft)	A (ft²)	P (ft)	R (ft)	Q (cfs)
1.50	12.75	13.49	0.945	33.0
2.00	20.00	16.65	1.201	60.7
1.80	16.92	15.38	1.100	48.4
1.99	19.84	16.59	1.196	60.0

y_n = 1.99 ft, A = 19.83 ft², V = Q/A = 3.03 ft/s, top width 15.9 ft. Note how much each trial moves Q: in this range Q grows roughly with y^2.1, so a 10% error in n changes the depth by only about 5%.

The n value deserves care. For grass it depends strongly on the grass height and on the flow depth (the "retardance" of the vegetation): short grass in deep flow can be 0.03 or less, tall grass in shallow flow 0.1 or more. HEC-15 gives a procedure that computes n from the grass properties and the shear; many manuals instead tabulate n by retardance class. For capacity, use the n of the grass at its tallest; for stability, at its shortest.

3.1 Composite roughness

A channel with a paved or riprap bed and grass sides has different n values around its perimeter. The Horton and Einstein method assumes equal velocity in each part and gives

n_c = [ Σ P_i n_i^1.5 / P ]^2/3

Composite n. The Example 1 section at y = 1.5 ft with a concrete bed (n = 0.015, P = 4.0 ft) and grass sides (n = 0.035, P = 9.49 ft): n_c = 0.030. A simple perimeter-weighted average gives 0.029, close here; the methods diverge more when the parts are very different.

4. Specific energy, critical depth and the Froude number

Specific energy is the energy above the channel bottom: E = y + V²/2g. For a given flow it is smallest at the critical depth y_c, where

Q² T / (g A³) = 1, equivalently Fr = V / √(g D_h) = 1

For a rectangle this reduces to y_c = (q²/g)^1/3, with q = Q/b; for a trapezoid, iterate. Flow deeper than critical (Fr < 1) is subcritical: tranquil, controlled from downstream. Flow shallower than critical (Fr > 1) is supercritical: fast, controlled from upstream, and it ends in a hydraulic jump where it meets slower water. A channel whose normal depth is above critical is on a mild slope; one whose normal depth is below critical is on a steep slope.

Near Fr = 1 the water surface is unstable: a small change in energy causes a large change in depth, and the surface carries standing waves. Designers avoid normal flow with a Froude number between roughly 0.8 and 1.2 (USBR guidance uses about 0.86 to 1.13), by changing the slope, width or lining.

Example 3: classify the flow in Example 2. Solving Q²T/(gA³) = 1 for 60 cfs gives y_c = 1.37 ft. The normal depth, 1.99 ft, is above it, so the slope is mild and the flow subcritical: Fr = 3.03 / √(32.2 × 19.83/15.9) = 0.48.
Specific energy at normal depth: E = 1.99 + 3.03²/64.4 = 2.13 ft; at critical depth E_c = 1.82 ft, the minimum for this flow.
The slope at which this flow would be critical in this channel, the critical slope, is S_c = [Q n / (1.486 A_c R_c^2/3)]² = 0.0193. Any reach steeper than that runs supercritical.

4.1 Water-surface profiles

Uniform flow is only reached in long, unobstructed reaches. Near a culvert, a slope break or a confluence the depth varies gradually between normal and critical depth, and the profile is classified by the slope (M for mild, S for steep) and by where the depth lies: zone 1 above both y_n and y_c, zone 2 between them, zone 3 below both. Backwater behind a culvert on a mild ditch is an M1 curve, rising above normal depth as it approaches the culvert; flow accelerating toward a free overfall or a steeper reach is an M2 curve, falling toward critical depth. On a steep reach the flow approaches its normal depth from critical depth along an S2 curve. Subcritical profiles are computed upstream from a downstream control; supercritical ones downstream from an upstream control. The standard-step method (or HEC-RAS) does the computation; for ditch design the point is to know when normal depth is not the answer, for instance where a culvert headwater backs up into the channel and the freeboard must be checked against the backwater profile.

5. Will the lining stay put?

5.1 Permissible velocity

The older method compares the mean velocity with a permissible value for the lining: Fortier and Scobey's values for unlined soils, and NRCS values for grasses (for Bermudagrass on easily eroded soil at slopes up to 5%, about 6 ft/s; on erosion-resistant soil about 8 ft/s). It is simple and still cited, but velocity is only a proxy for the force on the boundary, and the same velocity exerts more force in a shallow channel than in a deep one.

5.2 Tractive force

The current method, in HEC-15, compares the shear stress on the lining with the lining's permissible shear. The maximum shear on the bed in uniform flow is

τ_d = γ · d · S

with γ = 62.4 lb/ft³, d the maximum depth (not the hydraulic radius) and S the slope. The shear on the side slopes is K₁τ_d, with K₁ = 0.77 for z ≤ 1.5, 0.066z + 0.67 for 1.5 < z < 5 and 1.0 for z ≥ 5; in bends the shear rises by a factor K_b of up to 2.0 for tight bends. The design check is τ_p ≥ SF · τ_d, with a safety factor SF of 1.0 to 1.5.

Example 4: lining checks for Example 2. V = 3.03 ft/s is below the 6 ft/s permissible velocity for Bermudagrass on easily eroded soil, so a good grass stand passes the velocity screen.
Bed shear: τ_d = 62.4 × 1.99 × 0.004 = 0.50 lb/ft²; side slopes: K₁ = 0.868, τ_s = 0.43 lb/ft².
Bare soil before the grass is established (Fortier and Scobey, clear water): Ordinary firm loam, τ_p = 0.15 lb/ft²: fails; Stiff clay, very colloidal, τ_p = 0.26 lb/ft²: fails; Coarse gravel, noncolloidal, τ_p = 0.3 lb/ft²: fails.
The channel is stable only once vegetated, so it needs a temporary rolled erosion control product (RECP) rated for at least 0.50 lb/ft² until the grass is established. That is a common outcome, and a plan that shows "seed and mulch" alone on a channel like this one is a frequent review comment.

5.3 Riprap

For rock, the permissible shear comes from the Shields relation:

τ_p = F_* (γ_s − γ) D₅₀ ⇒ D₅₀ ≥ SF · τ_d / [F_* (γ_s − γ)]

with γ_s the unit weight of the stone (about 165 lb/ft³) and F_* the Shields parameter. HEC-15 uses F_* = 0.047 at lower particle Reynolds numbers, rising to 0.15 for large rock in steep, rough flow, and it adjusts the safety factor on steep slopes; 0.047 is the conservative value used here. Rock riprap also needs a filter (granular or geotextile) between the rock and the soil, and a thickness of at least about 1.5 D₅₀ or D₁₀₀.

Example 5: a steep reach. Below a road crossing the same section drops at S = 0.05, with n = 0.04 for the rock lining. Normal depth for 60 cfs: y_n = 1.16 ft, V = 6.94 ft/s, Fr = 1.38, so this reach is supercritical (and a hydraulic jump will form where it meets the milder reach downstream; the lining must extend through it).
τ_d = 62.4 × 1.16 × 0.05 = 3.61 lb/ft².
D₅₀ ≥ 1.5 × 3.61 / [0.047 × (165 − 62.4)] = 1.123 ft = 13.5 in on the bed; the side slopes need 11.7 in. Use a 15-in D₅₀ gradation over a filter. The n value assumed for the rock should then be checked against the rock chosen and the flow depth, since rock n rises as the depth falls toward the rock size.

5.4 Bends

In a bend the water surface rises on the outside. For subcritical flow the superelevation is about Δy = V² T / (g R_c), with R_c the centerline radius. The outside bank also carries higher shear, by the factor K_b given earlier, which extends some distance downstream of the bend.

A bend in the Example 2 channel. R_c = 40 ft, T = 15.9 ft, so R_c/T = 2.51. Superelevation: 3.03² × 15.9 / (32.2 × 40) = 0.11 ft, which adds to the freeboard requirement on the outside bank. K_b = 2.38 − 0.206(2.51) + 0.0073(2.51)² = 1.91, so the bend shear is 0.95 lb/ft², which still requires the RECP and established grass of Example 4 to be checked at this higher value.

5.5 The hydraulic jump

Where supercritical flow meets subcritical flow it jumps to the sequent depth. For a rectangular section

y₂ = (y₁/2) [ √(1 + 8 Fr₁²) − 1 ], ΔE = (y₂ − y₁)³ / (4 y₁ y₂)

The jump is about 6 y₂ long on a flat apron, dissipates energy violently and is where a lining is most likely to fail. If the tailwater is lower than y₂, the jump moves downstream; if higher, it moves upstream and may drown.

Jump at the foot of a chute. A 6-ft wide rectangular concrete chute carries 60 cfs at y₁ = 0.55 ft: V₁ = 18.2 ft/s, Fr₁ = 4.32 (critical depth is 1.46 ft). Sequent depth y₂ = 0.55/2 × [√(1 + 8 × 4.32²) − 1] = 3.10 ft; energy lost ΔE = 2.42 ft; length about 19 ft. The apron or stilling basin must hold the jump, and the channel below it needs a tailwater near 3.10 ft or the jump will sweep downstream.

6. Freeboard and the check flow

Freeboard is the vertical distance from the design water surface to the top of the channel. It covers waves, debris, settlement and the uncertainty in n. Requirements vary: 0.5 ft for small roadside ditches is common, more for larger channels, superelevation in bends, and supercritical reaches with standing waves. Many agencies also require that a larger check flow stay in the channel, with or without freeboard.

Example 6: depth of channel. For the mild reach, the 100-year check flow is 95 cfs, with a normal depth of 2.45 ft. With 0.5 ft of freeboard on the check flow, the channel needs to be at least 2.95 ft deep; specify 3.00 ft. At that depth the 10-year flow has 1.01 ft of freeboard.

7. Common mistakes

Using the hydraulic radius instead of the maximum depth in τ = γ d S.
Using one n value for both capacity and stability on a grass channel.
Checking the lining only on the bed, not the side slopes and bends.
Designing a channel whose normal flow sits near Fr = 1.
Ending riprap at the slope break where a hydraulic jump forms downstream of a steep reach.
Relying on grass that does not yet exist: the bare-soil condition governs until the stand is established.
Omitting the filter under riprap.

8. Summary

Channel design is a short sequence of checks on a trial section: normal depth by Manning, the Froude number against critical flow, the lining against shear (bed, sides, bends), and the channel depth against freeboard and the check flow. Change the section, slope or lining and repeat until each check passes. The judgment lies in the n value, the lining data and the condition of the lining when the design storm actually arrives.

References: FHWA HEC-15, Design of Roadside Channels with Flexible Linings, 3rd ed. (2005); Chow, Open-Channel Hydraulics (1959); USDA NRCS, National Engineering Handbook Part 654; Fortier and Scobey (1926), Permissible Canal Velocities; see also the PE-Calc permissible velocity and shear tables.

Final quiz

24 questions. You need 70% to pass. You can retake it as many times as you need.

1. A trapezoid with b = 6 ft, z = 2 flows 2.0 ft deep. The flow area (ft²) is:

(A) 12.0 ft² (B) 20.0 ft² (C) 28.0 ft² (D) 16.0 ft²

2. Same section: the hydraulic radius (ft) is most nearly:

(A) 1.43 ft (B) 2.00 ft (C) 0.91 ft (D) 1.34 ft

3. b = 3 ft, z = 2, n = 0.030, S = 0.006, y = 1.2 ft. Q by Manning (cfs) is most nearly:

(A) 21.0 cfs (B) 14.1 cfs (C) 66.3 cfs (D) 33.5 cfs

4. Normal depth in a channel is found by iteration because:

(A) Manning n is unknown (B) Both A and R depend on the depth (C) The slope changes with depth (D) Critical depth must be found first

5. A rectangular channel carries q = 10 cfs per foot of width. The critical depth (ft) is most nearly:

(A) 0.31 ft (B) 0.56 ft (C) 1.46 ft (D) 2.19 ft

6. Flow at 5.0 ft/s with a hydraulic depth of 1.5 ft has a Froude number of most nearly:

(A) 0.10 (B) 4.08 (C) 1.44 (D) 0.72

7. Specific energy for y = 1.5 ft and V = 5.0 ft/s (ft) is most nearly:

(A) 1.89 ft (B) 1.58 ft (C) 2.28 ft (D) 0.39 ft

8. Normal depth below critical depth means the channel slope is:

(A) Mild, and the flow subcritical (B) Steep, and the flow supercritical (C) Horizontal (D) Adverse

9. Designers avoid normal flow with a Froude number near 1 because:

(A) Manning's equation is invalid there (B) The lining shear is zero (C) The water surface is unstable and carries standing waves (D) Critical depth cannot be computed

10. Bed shear in uniform flow is computed as γ × S × :

(A) The hydraulic radius (B) The mean velocity (C) The top width (D) The maximum flow depth

11. Flow 1.8 ft deep on a 1% slope. The bed shear (lb/ft²) is most nearly:

(A) 1.12 lb/ft² (B) 11.23 lb/ft² (C) 0.75 lb/ft² (D) 0.62 lb/ft²

12. With 3:1 side slopes, the side-slope shear for the flow in the previous question (lb/ft²) is most nearly:

(A) 0.86 lb/ft² (B) 1.68 lb/ft² (C) 0.97 lb/ft² (D) 0.37 lb/ft²

13. A riprap bed carries 1.25 lb/ft² of design shear. With SF = 1.5, F_* = 0.047 and γ_s = 165 lb/ft³, the minimum D₅₀ (in) is most nearly:

(A) 0.4 in (B) 4.7 in (C) 1.9 in (D) 3.1 in

14. A new grass channel has τ_d above the permissible shear of the bare soil but below that of established grass. It needs:

(A) Nothing, the grass governs (B) A temporary rolled erosion control product until the grass establishes (C) A larger n value (D) A steeper slope

15. For a grass channel, the n value for capacity should reflect:

(A) The grass at its shortest (B) The bare soil (C) The grass at its tallest (D) Any average value

16. Riprap on soil normally also needs:

(A) A filter of granular material or geotextile beneath it (B) A concrete cap (C) A grass cover (D) Nothing else

17. A steep supercritical reach discharges into a mild reach. The designer should expect:

(A) A drawdown curve (B) A hydraulic jump near the slope break, which the lining must cover (C) Critical depth everywhere (D) No change in depth

18. A triangular (b = 0) ditch with 3:1 sides, n = 0.030 and S = 0.01 carries 20 cfs. Normal depth (ft) is most nearly:

(A) 1.75 ft (B) 1.04 ft (C) 0.87 ft (D) 1.35 ft

19. Freeboard in an open channel is:

(A) The distance from the design water surface to the top of the channel (B) The depth of flow above critical depth (C) The channel bottom width (D) The difference between the 10- and 100-year depths

20. In a tight bend (R_c/T ≤ 2), HEC-15 increases the design shear by a factor of:

(A) 1.05 (B) 1.5 (C) 2.0 (D) 0.77

21. Rectangular flow at y₁ = 0.6 ft and V₁ = 12 ft/s jumps. The sequent depth (ft) is most nearly:

(A) 3.05 ft (B) 1.64 ft (C) 1.36 ft (D) 2.04 ft

22. Subcritical flow at 4 ft/s, top width 20 ft, rounds a bend of 60-ft radius. The superelevation (ft) is most nearly:

(A) 0.17 ft (B) 0.33 ft (C) 0.08 ft (D) 0.25 ft

23. A channel has 5 ft of perimeter at n = 0.013 and 10 ft at n = 0.030. The Horton-Einstein composite n is most nearly:

(A) 0.013 (B) 0.030 (C) 0.025 (D) 0.020

24. Backwater rising toward a culvert on a mild ditch is classified as:

(A) An S2 profile (B) An M1 profile (C) An M2 profile (D) Uniform flow

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