Gradually Varied Flow (dy/dx)

\frac{d y}{d x} = \frac{S _{0} - S _{f}}{1 - Fr ^{2}}

The water-surface slope relative to the bed follows from energy. Sf is the friction slope (Manning). On a mild slope (yn > yc): M1 is above yn (backwater), M2 between yn and yc, M3 below yc. Sign of dy/dx flips across Fr = 1.

$S_{0}$: Bed slope
$S_{f}$: Friction slope
$Fr$: Froude number

Step-by-step solved example

Mild slope, y > yn > yc so Fr < 1 and Sf < S0. Is dy/dx positive?

Gradually varied flow: dy/dx = (S0−Sf)/(1−Fr²).

1. Numerator
S0 − Sf > 0.
2. Denominator
Fr < 1 ⇒ 1−Fr² > 0. So dy/dx > 0 (depth increases downstream: M1).

Answer: Yes — M1 profile, depth increases downstream

10 practice questions

0/10 correct

1.Critical flow (Fr = 1) makes dy/dx

2.Uniform flow means

3.Mild slope is defined by

4.Steep slope has

5.M2 profile sits

6.An S1 profile is

7.If Fr > 1 and S0 > Sf, dy/dx is

8.Control for subcritical flow is

9.Control for supercritical flow is

10.Standard-step GVF computation marches using