Non-Uniform (Gradually Varied) Flow Notes

Non-Uniform (Gradually Varied) Flow

Assumptions of gradually varied flow; dynamic equation of GVF and its forms for wide rectangular channels; channel slope classification (mild, steep, critical, horizontal, adverse); zones and the 12 water surface profiles (M1–M3, S1–S3, C1, C3, H2, H3, A2, A3) with practical examples; control sections; backwater and afflux; computation of profiles by direct step, standard step and graphical integration — with solved numericals.

Last reviewed 16 Sept 2026 · 7 min read

Gradually varied flow

Gradually varied flow (GVF) is steady non-uniform flow in which the depth changes gradually over a long distance — for example the backwater curve upstream of a dam or weir.

Assumptions

Flow is steady; streamlines are practically parallel, so pressure distribution is hydrostatic.
The head loss at a section equals that of uniform flow with the same depth and velocity — Manning or Chezy formulas apply using the local depth.
Channel slope is small; the channel is prismatic.
Velocity distribution is fixed (α constant); roughness is constant along the reach.

Dynamic equation of GVF

FormulaGVF equation

\frac{d y}{d x} = \frac{S _{0} - S _{f}}{1 - F r ^{2}} = \frac{S _{0} - S _{f}}{1 - \frac{Q ^{2} T}{g A ^{3}}}

$S_{0}$ = bed slope; $S_{f}$ = energy (friction) slope at depth $y$ ; $x$ measured along the flow.

Wide rectangular channel:

Manning: $\frac{d y}{d x} = S_{0} \frac{1 - ( y _{n} / y ) ^{10/3}}{1 - ( y _{c} / y ) ^{3}}$
Chezy: $\frac{d y}{d x} = S_{0} \frac{1 - ( y _{n} / y ) ^{3}}{1 - ( y _{c} / y ) ^{3}}$

$d y / d x > 0$ : depth increases in the flow direction (backwater / rising curve).
$d y / d x < 0$ : depth decreases (drawdown / falling curve).
$d y / d x$ = 0: uniform flow ( $y = y_{n}$ ).
$d y / d x \to \infty$ as $y \to y_{c}$ : the surface meets the critical depth line vertically in theory — in reality a hydraulic jump (rising) or a free fall/rapid drawdown occurs; GVF assumptions break down there.

Classification of channel slopes

Slope	Condition	Relation of depths
Mild (M)	$0 < S_{0} < S_{c}$	$y_{n} > y_{c}$
Steep (S)	$S_{0} > S_{c}$	$y_{n} < y_{c}$
Critical (C)	$S_{0} = S_{c}$	$y_{n} = y_{c}$
Horizontal (H)	$S_{0} = 0$	$y_{n} = \infty$
Adverse (A)	$S_{0} < 0$	$y_{n}$ imaginary

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Non-Uniform (Gradually Varied) Flow

Gradually varied flow

Assumptions

Dynamic equation of GVF

Classification of channel slopes

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