Manning Uniform Flow

OpenHydroLab · Normal depth as a force balance · Slope classification

Channel

Roughness (Manning n)

Flow & slope

Find normal depth (experiment)

Release the reach away from equilibrium and watch gravity vs friction drive the depth onto yₙ — normal depth is the attractor of the momentum balance, the open-channel "terminal velocity".

Show

Aha demo: press Release too shallow — the fast, shallow flow has friction < gravity? No: watch the arrows. The reach fills (outflow < inflow), friction and gravity re-balance, and depth locks onto yₙ with τ₀P = ρgAS₀ exactly ✓. Then press Sweep slope: yₙ falls as the channel steepens while y꜀ stays put — their crossing is the critical slope S꜀ where the classification flips from mild to steep.

Depths (solved)

Normal depth yₙ (Manning)
Critical depth y꜀ (Fr = 1)
Critical slope S꜀
MILD SLOPE
UNIFORM FLOW at yₙ

Force balance (per m of channel)

Gravity drive ρgAS₀
Friction resist τ₀P
Ratio τ₀P / ρgAS₀
Experiment y → yₙ check—

Section at current depth

Flow area A
Wetted perimeter P
Hydraulic radius R = A/P
Top width T
Velocity V
Froude number Fr
Bed shear τ₀ = ρgRS_f
Chezy C = R¹ᐟ⁶/n
Best-section b/y (min P)
V = (1/n)·R^⅔·S₀^½  ·  Q = V·A  ·  R = A/P
S_f = n²V²/R^{4/3}  ·  τ₀ = ρgRS_f
y꜀ : Q²T/(gA³) = 1  ·  S꜀ = [nQ/(A꜀R꜀^⅔)]²
experiment: dA/dt = (Q − VA)/L,   dV/dt = g(S₀ − S_f)
Uniform flow: the gravity component along the bed exactly balances boundary friction, so depth and velocity are constant — bed, water surface and energy line are all parallel (S_f = S_w = S₀). Manning's equation IS this balance with the empirical closure τ₀ ∝ V²/R^⅓; solving it for depth gives yₙ.