Gradually Varied Flow Calculator

Classify the water surface profile (M1–M3, S1–S3, C1/C3) and compute dy/dx for GVF in a rectangular channel using the standard GVF equation.

Open Channel Flow · GVF

Inputs

Flow rate Q (m³/s)

Channel width (rectangular) b (m)

Bed slope S₀ (m/m)

e.g. 0.001 = 1 m drop per 1000 m

Manning's roughness n (dimensionless)

Current flow depth y (m)

Depth at the section of interest — determines profile zone

Theory

GVF equation:

dy/dx = (S₀ − S_f) / (1 − Fr²)

S_f = (nV / R^2/3)² [Manning's friction slope]

S_c = (nV_c / R_c^2/3)² [critical slope at y = y_c]

Where:

S₀ = channel bed slope [m/m]
S_f = friction slope from Manning's equation at depth y [m/m]
S_c = critical slope — S₀ at which y_n = y_c [m/m]
Fr = V/√(gy) — Froude number at depth y (rectangular channel)
y_n = normal depth — uniform flow depth from Manning's equation [m]
y_c = critical depth — depth at Fr = 1 [m]

Profile classification:

Slope	Condition	Zone 1	Zone 2	Zone 3
Mild	S₀ < S_c (y_n > y_c)	M1 — y > y_n	M2 — y_c < y < y_n	M3 — y < y_c
Steep	S₀ > S_c (y_n < y_c)	S1 — y > y_c	S2 — y_n < y < y_c	S3 — y < y_n
Critical	S₀ = S_c (y_n = y_c)	C1 — y > y_c	—	C3 — y < y_c

GVF occurs when depth changes gradually along the channel. The profile type depends on the relative position of y to y_n and y_c. M1 (backwater) and M2 (drawdown) are most common in practice. Near Fr = 1 the GVF equation has a singularity — a hydraulic jump or free overfall occurs instead of a smooth profile.