--- title: "Shield's diagram and its application" subject: "Hydraulics" chapter: 10 key: "semester_4/hydraulics:shield-diagram-application-design" source: https://www.bctengineer.com/q/semester_4/hydraulics/shield-diagram-application-design site: BCT Engineering --- # Shield's diagram and its application Hydraulics · Chapter 10: Flow in mobile boundary channel - Asked in: 2081 Chaitra, 4 marks; 2079 Chaitra, 4 marks; 2076 Bhadra, 6 marks; 2074 Bhadra, 3 marks; 2070 Magh, 6 marks (asked 5 times) - Video: [Sediment Transport Mechanism Shield's Theory - Design of Irrigation Channels & Spillways - GATE](https://www.youtube.com/watch?v=lhRbjNhBroE) — Ekeeda GATE & ESE - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/shield-diagram-application-design - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-10 ## Question Explain Shield's diagram (with sketch, including the important points based on critical Reynolds number) and its application in designing / predicting critical tractive force for a mobile boundary channel. ## Short answer Shields' diagram relates dimensionless critical shear stress τ* = τc/((γs−γ)d) to boundary Reynolds number Re* = u*d/ν. For Re* > 400 (fully rough) τ* ≈ 0.056; a minimum of about 0.03 occurs near Re* ≈ 10. Design iterates: assume τ*, compute τc and Re*, refine τ* from the diagram, then check that actual bed shear γRS0 ≤ τc. ## Answer #### Shields diagram Shields (1936) showed by experiments that the start of motion of non-cohesive sediment depends on two dimensionless numbers: - **Shields parameter (dimensionless critical shear stress):** $\tau_* = \dfrac{\tau_c}{(\gamma_s - \gamma)\,d}$ - **Boundary (shear) Reynolds number:** $Re_* = \dfrac{u_*\,d}{\nu}$, with $u_* = \sqrt{\tau_0/\rho}$ the shear velocity where $\tau_c$ is the critical bed shear stress, $\gamma_s$ and $\gamma$ the specific weights of sediment and water, $d$ the grain size and $\nu$ the kinematic viscosity. ```text tau* 0.1 |\ | \ | \ ___________ 0.056 0.05| '. _.-' | '. _.-' 0.03| '-..__..-' <- minimum | (smooth) (transition) (rough) +------+--------+---------+------------> Re* 2 10 400 above curve: bed moves ; below curve: no motion ``` **Important points (based on $Re_*$):** | Range of $Re_*$ | Boundary type | Behaviour of $\tau_*$ | |---|---|---| | $Re_* < 2$ (about) | hydraulically smooth; grains lie inside the viscous sublayer | $\tau_*$ falls as $Re_*$ rises (about 0.1 at $Re_* \approx 1$) | | $2 < Re_* < 400$ | transition | $\tau_*$ passes through a **minimum of about 0.03** near $Re_* \approx 10$ and rises again | | $Re_* > 400$ | fully rough turbulent | $\tau_*$ becomes constant, about **0.056 (taken as 0.06)**, independent of $Re_*$ | The curve separates the region of no motion (below) from the region of sediment motion (above). #### Application in design **To find the critical shear stress for a given grain size $d$:** 1. Assume $\tau_*$ (for coarse material 0.056 to 0.06). 2. Find $\tau_c = \tau_*(\gamma_s - \gamma)d$ and $u_* = \sqrt{\tau_c/\rho}$. 3. Compute $Re_* = u_*d/\nu$, read the correct $\tau_*$ from the diagram and repeat until it agrees. **To check or design a channel:** 4. Compute the actual bed shear stress $\tau_0 = \gamma R S_0$ (or $\gamma y S_0$). 5. If $\tau_0 < \tau_c$ the bed is stable; if $\tau_0 > \tau_c$ the bed material moves. 6. For design, choose the bed material (or a stable size $d$) so that $\tau_c \ge \tau_0$. For fully rough flow, the size that just stays stable is $d_{cr} \approx 10\,R S_0$ (with $\gamma_s = 2.65\gamma$). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which two dimensionless numbers does the Shields diagram relate? A. Froude number and Reynolds number B. Shields parameter tau* and boundary Reynolds number Re* C. Shields parameter and Froude number D. Re* and relative roughness only Answer: B. The diagram plots tau* = tau_c/((gamma_s - gamma)d) against Re* = u* d / nu. (Key point: Shields variables) 2. What is the behaviour of tau* for Re* > 400? A. It rises steadily with Re* B. It becomes constant at about 0.056 (taken as 0.06), independent of Re* C. It falls to zero D. It reaches its minimum of 0.03 Answer: B. In fully rough turbulent flow the viscous sublayer is negligible, so tau* is constant. (Key point: Fully rough range) 3. Where does the curve show its minimum and what is its approximate value? A. tau* about 0.06 at Re* about 400 B. tau* about 0.1 at Re* about 1 C. tau* about 0.03 near Re* about 10, in the transition range D. tau* about 0.03 at Re* greater than 1000 Answer: C. The curve falls from the smooth range, passes a minimum of about 0.03 near Re* = 10 and rises again to 0.056. (Key point: Minimum of the curve) 4. In design, how is the bed judged stable using the critical shear stress? A. Bed is stable if tau_0 is greater than tau_c B. Bed is stable only if Re* is below 2 C. Bed is stable if tau* equals 0.03 D. Bed is stable if actual tau_0 = gamma R S0 is less than tau_c Answer: D. If the actual bed shear stress is below the critical value the bed does not move; if above, material moves. 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