Mobile boundary design: three approaches

1. Maximum permissible velocity method

The channel is stable if the mean velocity is below the velocity that would erode the boundary.

1. Select the maximum permissible velocity $V_{max}$ for the soil (tables, or Kennedy's $V_0 = 0.55\,m\,y^{0.64}$).
2. Select side slope $z$, bed slope $S_0$ and Manning's $n$.
3. Area $A = Q/V_{max}$; hydraulic radius $R = \left(\dfrac{nV_{max}}{\sqrt{S_0}}\right)^{3/2}$; perimeter $P = A/R$.
4. Solve $A = (b + zy)y$ and $P = b + 2y\sqrt{1 + z^2}$ for $b$ and $y$.

2. Tractive force method

The channel is stable if the shear stress on the boundary is below the critical shear stress of the soil.

1. Find $\tau_c$ for the grain size: $\tau_c = \tau_*(\gamma_s - \gamma)d$ ($\tau_* \approx 0.06$ for coarse material).
2. Select $z$, compute $K = \sqrt{1 - \sin^2\theta/\sin^2\phi}$ with $\tan\theta = 1/z$.
3. Bed: $\gamma y S_0 \le \tau_c$. Sides: $0.75\,\gamma y S_0 \le K\tau_c$. Take the smaller permitted depth $y$.
4. Find $b$ from Manning's equation: $Q = \dfrac{1}{n}AR^{2/3}S_0^{1/2}$.
5. Check the Froude number and add freeboard.

3. Regime theory (Lacey)

For a channel in alluvium that carries sediment, the dimensions follow empirical regime relations. With silt factor $f = 1.76\sqrt{d_{mm}}$:

$$V = \left(\frac{Qf^2}{140}\right)^{1/6},\quad A = \frac{Q}{V},\quad P = 4.75\sqrt{Q},\quad R = \frac{5V^2}{2f},\quad S_0 = \frac{f^{5/3}}{3340\,Q^{1/6}}$$

Then find $b$ and $y$ from the geometry (side slope 0.5H : 1V).