open-channel

name	open-channel
description	Open channel flow — Manning's equation, specific energy, critical flow, hydraulic jump, gradually varied flow, channel design, weirs, culverts.
metadata	{"priority":5,"promptSignals":{"phrases":["open channel","Manning","hydraulic jump","specific energy","critical flow","weir","culvert","channel flow"],"minScore":4}}

Open Channel Flow — Complete Skill

Manning's Equation (Uniform Flow)

Q = (1/n) × A × R^(2/3) × S^(1/2) [SI, Q in m³/s] Q = (1.486/n) × A × R^(2/3) × S^(1/2) [US customary, ft³/s]

Where:

• n = Manning's roughness coefficient
• A = cross-sectional area of flow [m²]
• R = A/P = hydraulic radius [m] (P = wetted perimeter)
• S = channel slope [m/m] = tan θ ≈ sin θ

Manning's n values:

Surface	n
Smooth concrete	0.012
Rough concrete	0.016
Clean excavated earth	0.020-0.025
Gravelly bottom	0.025-0.035
Natural stream, clean	0.025-0.035
Natural stream, heavy brush	0.050-0.100
PVC pipe	0.009-0.011
CMP (corrugated metal)	0.023-0.025

Normal Depth and Critical Depth

Normal Depth y_n (uniform flow)

Solve Manning's equation for y_n iteratively: Q = (1/n) × A(y_n) × R(y_n)^(2/3) × S^(1/2)

For wide rectangular channel (width b >> depth): A = by, R ≈ y y_n = (Qn/(bS^(1/2)))^(3/5)

Critical Depth y_c (Froude = 1)

Fr = V/√(gy) = 1 at critical flow Q²/g = A³/T (T = top width)

Wide rectangular: y_c = (Q²/(g×b²))^(1/3) Critical velocity: V_c = √(g×y_c)

Flow Classification

Fr < 1: subcritical (tranquil) — controlled from downstream Fr > 1: supercritical (rapid) — controlled from upstream Fr = 1: critical flow

Specific Energy

E = y + V²/(2g) = y + Q²/(2gA²)

E_min = (3/2) × y_c [rectangular channel] At given E: two alternate depths (one sub, one supercritical) y₁ and y₂ satisfy: E(y₁) = E(y₂)

Hydraulic Jump (Sequent Depths)

Jump occurs when Fr₁ > 1 → rapidly transitions to subcritical Energy dissipated in jump (turbulence)

Sequent depth ratio (rectangular): y₂/y₁ = ½(√(1 + 8Fr₁²) - 1)

Energy loss in jump: ΔE = (y₂ - y₁)³ / (4y₁y₂)

Efficiency: η = E₂/E₁ = 1 - ΔE/E₁

Length of jump: L_j ≈ 6(y₂ - y₁) [empirical]

Hydraulic jump types: Fr₁ = 1.0-1.7: undular jump (small loss) Fr₁ = 1.7-2.5: weak jump Fr₁ = 2.5-4.5: oscillating jump (instability) Fr₁ = 4.5-9.0: steady jump (stable, ~45% energy loss) — best for stilling basins Fr₁ > 9.0: strong jump (>70% energy loss)

Gradually Varied Flow (GVF)

dy/dx = (S₀ - S_f) / (1 - Fr²) S_f = friction slope (from Manning's, using actual y) S₀ = bed slope

Water surface profiles (12 types, M/S/C/H/A designation): M1: mild slope, subcritical, y > y_n — backwater curve (M1 upstream of dam) M2: mild slope, subcritical, y_n > y > y_c — drawdown (M2 at free overfall) M3: mild slope, supercritical, y < y_c — steep inlet S1: steep slope, subcritical (above y_c) S2: steep slope, supercritical, y_n < y < y_c S3: steep slope, supercritical, y < y_n

Numerical solution: step method (forward from known control section) Standard step: solve E equation iteratively for each spatial step Δx

Weirs

Sharp-Crested Rectangular Weir:

Q = (2/3) × C_d × L × √(2g) × H^(3/2) C_d = 0.611 + 0.075(H/P) [Francis formula, without end contractions] For contracted weir: L_eff = L - 0.2H (two end contractions)

Triangular (V-notch) Weir:

Q = (8/15) × C_d × tan(θ/2) × √(2g) × H^(5/2) C_d ≈ 0.60 (for θ = 90°) 90° V-notch: Q = 1.38H^(5/2) [SI, Q in m³/s, H in m]

Broad-crested Weir:

Q = 1.705 × C_d × L × H^(3/2) [approximate, SI] Approaches critical flow over crest

Culvert Hydraulics

Inlet control (weir flow): Q = C × A × (2gH)^(1/2) (orifice-like) Outlet control (pipe flow): use energy equation with losses HW = TW + V²/2g + h_f + K_e × V²/2g K_e = entrance loss coefficient (0.2 for well-rounded, 0.5 for square edge, 0.9 for projecting)

Channel Design (Best Hydraulic Section)

For given A, minimize P to maximize Q (or minimize n and S requirements)

Shape	Best hydraulic section	R_max
Rectangular	b = 2y (semi-square)	y/2
Trapezoidal	b = 2y/√3, z = 1/√3	y/2
Circular	y = D/2 (half-full)	D/4

Note: trapezoidal with z = 1/√3 (side slope 1:√3) is most efficient open section

Output

Provide: Q [m³/s], y_n [m], y_c [m], Fr, hydraulic jump sequent depth y₂ [m], energy loss ΔE [m], channel dimensions for best hydraulic section.