Gradually Varied Flow

Water-surface profiles and the phase portrait of

Channel

Bottom width

Manning

In SI units, , the inverse of the Strickler coefficient.

Flow & bed slope

Discharge (log scale) Bed slope (signed log)

Centre = 0 (horizontal, H); left of centre = adverse (A). Both have no normal depth.

Control section

Control depth

Jump to a profile type

Same channel () every time — only the bed slope and the control depth change.

Direct-step method

Number of steps Show the direct-step staircase

Show

Water particles (local ) Zone shading (1 / 2 / 3) Energy grade line Wave speeds

Aha demo: press March from the control. The probe integrates the profile away from the control section while the same point rides the curve on the right: watch it slide toward the zero of the numerator (the NDL — every profile is attracted to ) or toward the pole of the denominator (the CDL — where becomes unbounded and GVF theory no longer applies, so a jump or a free overfall takes over). Then press Sweep : the zero and the pole swap places as crosses , and the whole family changes M → C → S.

Classification

Normal depth · NDL
Critical depth · CDL
Critical slope
Slope class
Zone of

M1 — BACKWATER

READY

The equation at the control

Froude number
Friction slope
Numerator
Denominator

Control acts from

Integrated profile (RK4)

Reach length to 1% of
End condition
Particle time compression

Direct step (quantitative)

Single step (classic)
N-step over the same range
N-step ÷ RK4 length

Self-checks

Subcritical flow () carries information upstream, so its control sits downstream and the profile is integrated upstream from it; supercritical flow is controlled from upstream. The profile flattens onto the NDL asymptotically and meets the CDL vertically — there the gradually-varied assumption fails and a jump, a gate or an overfall takes over.