Variation of hydraulic radius with depth

Hydraulic radius $R=\dfrac{A}{P}$ is the ratio of flow area to wetted perimeter. For a rectangle of bed width $b$ and depth $y$:

$$A=by,\qquad P=b+2y,\qquad R=\frac{by}{b+2y}$$

Variation

• For a given $b$, $R$ increases with $y$ from zero (at $y=0$) and approaches the limit $b/2$ for very large $y$.
• For a given $y$, $R$ increases with $b$ and approaches $y$ for very large $b$.

Wide rectangular channel

Divide numerator and denominator by $b$:

$$R=\frac{y}{1+2y/b}$$

For $b\gg y$, $2y/b\to0$, so

$$R\approx y$$

The side walls contribute negligible wetted perimeter, so $P\approx b$ and $R=by/b=y$.

Deep narrow channel (gorge)

Divide numerator and denominator by $y$:

$$R=\frac{b}{b/y+2}$$

For $y\gg b$, $b/y\to0$, so

$$R\approx\frac{b}{2}$$

Here the bed is small compared with the two deep walls, so $P\approx2y$ and $R=by/2y=b/2$.

 wide: R ~ y            gorge: R ~ b/2
 ~~~~~~~~~~~~~~~        |~~|
 |_____________|        |  |  y >> b