Design by permissible velocity approach

The maximum permissible velocity is the greatest mean velocity that will not erode the channel boundary; the minimum permissible velocity is the smallest that prevents silting and weed growth. The velocity is chosen from tables or empirical formulas for the boundary material.

Rigid boundary (minimum velocity approach)

1. Choose a minimum velocity $V_{min}$ (typically 0.6 to 0.9 m/s for lined canals) to avoid sedimentation and weeds.
2. Choose side slope $z$, Manning's $n$ (from the lining) and the bed slope $S_0$.
3. Required area: $A = Q/V$ with $V \ge V_{min}$.
4. Manning's equation gives the hydraulic radius: $R = \left(\dfrac{nV}{\sqrt{S_0}}\right)^{3/2}$, and the wetted perimeter $P = A/R$.
5. Solve for depth and width from $A = (b + zy)y$ and $P = b + 2y\sqrt{1 + z^2}$.
6. Check that the velocity is below the permissible limit of the lining, and add freeboard.

Mobile boundary (maximum velocity approach)

1. Select the maximum permissible velocity $V_{max}$ for the material (for example Fortier and Scobey's values: fine sand about 0.5 m/s, coarse gravel about 1.2 to 1.5 m/s, hard clay higher) or use Kennedy's critical velocity

$$V_0 = 0.55\,m\,y^{0.64}\ \text{(m/s, } y \text{ in m)}$$

where $m$ is the critical velocity ratio. 2. Choose the side slope $z$ (as steep as the soil permits) and Manning's $n$. 3. Area: $A = Q/V_{max}$. 4. Hydraulic radius from Manning's equation: $R = \left(\dfrac{nV_{max}}{\sqrt{S_0}}\right)^{3/2}$ (if $S_0$ is not fixed, it is found from $S_0 = \dfrac{n^2V^2}{R^{4/3}}$ for a chosen $R$). 5. Solve the geometry for $b$ and $y$: $P = A/R$, $A = (b + zy)y$, $P = b + 2y\sqrt{1 + z^2}$. 6. Check that $V \le V_{max}$ and $V \ge V_{min}$, and adjust $z$, $b$ and $y$ if not.

Limitation: the method uses only the mean velocity and ignores the side-slope effect and the shear stress distribution, so the tractive force method is preferred for important designs.