--- title: "Mobile boundary design: three approaches" subject: "Hydraulics" chapter: 10 key: "semester_4/hydraulics:design-three-methods-mobile-boundary" source: https://www.bctengineer.com/q/semester_4/hydraulics/design-three-methods-mobile-boundary site: BCT Engineering --- # Mobile boundary design: three approaches Hydraulics · Chapter 10: Flow in mobile boundary channel - Asked in: 2070 Bhadra, 6 marks - Video: [16 Hydraulics of mobile bed channel Part3](https://www.youtube.com/watch?v=aeKxpLvg5_E) — Tanveer Ahmad - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/design-three-methods-mobile-boundary - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-10 ## Question Write down the design procedures of mobile boundary channel using maximum permissible velocity method, tractive force method and regime theory approaches with appropriate expressions. ## Short answer Permissible velocity method sizes so mean velocity ≤ Vmax. Tractive force method limits bed shear to τc and side shear to Kτc (K = sqrt(1 - sin²θ/sin²φ)). Regime theory uses Lacey's relations: V=(Qf²/140)^(1/6), P=4.75√Q, f=1.76√d_mm. ## Answer #### 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). | Method | Controlling quantity | |---|---| | Permissible velocity | mean velocity $V \le V_{max}$ | | Tractive force | boundary shear $\tau_0 \le \tau_c$ (bed and sides) | | Regime theory | empirical relations of $Q$ and $f$ | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What controls design in the maximum permissible velocity method? A. Mean velocity V <= V_max for the soil, with A = Q/V_max B. Boundary shear stress C. Silt factor D. Froude number Answer: A. The channel is sized so the mean velocity stays below the erosion velocity. (Key point: Velocity method) 2. In the tractive force method, how is the side slope condition written? A. gamma y S0 <= K tau_c B. 0.75 gamma y S0 <= K tau_c C. 0.75 gamma y S0 <= tau_c / K D. gamma y S0 <= 0.75 tau_c Answer: B. Maximum side shear is 0.75 gamma y S0 and must not exceed the reduced critical stress K tau_c. (Key point: Tractive method) 3. In Lacey's regime theory, how is the wetted perimeter obtained? A. P = Q/V B. P = 4.75 Q C. P = 4.75 sqrt(Q) D. P = 2.5 sqrt(Q) Answer: C. Lacey gives P = 4.75 sqrt(Q), together with V from Q and f. (Key point: Lacey perimeter) 4. How is Lacey's silt factor found? A. f = 1.76 d^2 B. f = d / 1.76 C. f = 140 d D. f = 1.76 sqrt(d in mm) Answer: D. f = 1.76 sqrt(d_mm), used to find V, R and S. 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