---
title: "Critical stress for Re > 400 and critical diameter"
subject: "Hydraulics"
chapter: 10
key: "semester_4/hydraulics:critical-shear-stress-re-400-critical-diameter"
source: https://www.bctengineer.com/q/semester_4/hydraulics/critical-shear-stress-re-400-critical-diameter
site: BCT Engineering
---

# Critical stress for Re > 400 and critical diameter

Hydraulics · Chapter 10: Flow in mobile boundary channel

- Asked in: 2079 Chaitra, 2+2 marks; 2069 Poush, 2 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/critical-shear-stress-re-400-critical-diameter
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-10

## Question

Derive an expression for critical stress for Re* > 400 (fully developed turbulent flow). Hence, finally express critical diameter of sediment in terms of geometric properties of channel (i.e. prove $d_{cr} = 10 R S_0$, where R is hydraulic radius and $S_0$ is bed slope).

## Short answer

For Re* > 400, τ* ≈ 0.06, so τc = 0.06(γs−γ)d. Equating τc to actual bed shear γRS0 yields d_cr = γRS0 / [0.06(γs−γ)]. For quartz sediment (specific gravity 2.65), this simplifies to d_cr ≈ 10 R S0.

## Answer

#### Critical shear stress for $Re_* > 400$

From the Shields diagram, the dimensionless critical shear stress is

$$
\tau_* = \frac{\tau_c}{(\gamma_s - \gamma)\,d}
$$

For $Re_* = \dfrac{u_*d}{\nu} > 400$ (fully rough, fully developed turbulent flow) the viscous sublayer is negligible and $\tau_*$ becomes independent of $Re_*$ and equal to a constant:

$$
\tau_* \approx 0.056 \approx 0.06
$$

Therefore the critical shear stress is

$$
\boxed{\tau_c = 0.06\,(\gamma_s - \gamma)\,d}
$$

#### Critical diameter

The average boundary shear stress in uniform flow is

$$
\tau_0 = \gamma R S_0
$$

At the limit of stability $\tau_0 = \tau_c$ for the critical diameter $d_{cr}$:

$$
\gamma R S_0 = 0.06\,(\gamma_s - \gamma)\,d_{cr}
$$

$$
d_{cr} = \frac{\gamma R S_0}{0.06\,(\gamma_s - \gamma)} = \frac{R S_0}{0.06\,(s - 1)}
$$

where $s = \gamma_s/\gamma$ is the specific gravity of the sediment. For quartz sand and gravel, $s = 2.65$, so $s - 1 = 1.65$:

$$
d_{cr} = \frac{R S_0}{0.06 \times 1.65} = 10.1\,R S_0
$$

$$
\boxed{d_{cr} \approx 10\,R\,S_0}
$$

with $d_{cr}$, $R$ in the same units (for example metres). Grains larger than $d_{cr}$ remain at rest; smaller grains move.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is tau* a constant for Re* > 400?
   A. The grains are smooth
   B. The flow is laminar
   C. The grains are cohesive
   D. Flow is fully rough turbulent and the viscous sublayer is negligible, so tau* is about 0.056 to 0.06
   Answer: D. In fully rough flow tau* no longer depends on Re*. (Key point: Constant tau*)

2. Which expression gives tau_c for Re* > 400?
   A. tau_c = 0.06 (gamma_s - gamma) d
   B. tau_c = 0.06 gamma R S0
   C. tau_c = 0.06 rho V^2
   D. tau_c = 0.06 d / (gamma_s - gamma)
   Answer: A. From tau* = tau_c / ((gamma_s - gamma) d) with tau* = 0.06. (Key point: Critical shear stress)

3. How is d_cr = 10 R S0 obtained?
   A. Equate V = sqrt(g R S0) with the settling velocity
   B. Equate gamma R S0 = 0.06 (gamma_s - gamma) d and use s - 1 = 1.65, giving 1/(0.06 x 1.65) about 10
   C. Divide 0.06 by 1.65 R S0
   D. Use d = R / S0
   Answer: B. Setting tau_0 = tau_c gives d_cr = R S0 / (0.06 (s - 1)), about 10.1 R S0 for s = 2.65. (Key point: Critical diameter)

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