Froude and Reynolds Numbers in a Rectangular Channel: 22-Agric-A4 Fluid Flow, December 2019, Q3

Problem 3: Froude and Reynolds Numbers in a Rectangular Channel (25 points)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Steady flow of 20°C water in an open rectangular channel of known width and depth.

Quantity	Value
Channel width, $b$	30 cm $=0.30$ m
Flow depth, $y$	10 cm $=0.10$ m
Flow rate, $Q$	$80{,}000\ \text{cm}^3/\text{s}=0.0800\ \text{m}^3/\text{s}$
Water, 20°C	$\rho=998\ \text{kg/m}^3$, $\mu=0.001\ \text{kg/m}\cdot\text{s}$

Find. (a) The Froude number $Fr$; (b) the (channel) Reynolds number $\text{Re}$.

Figure 3 — Cross-section of the rectangular channel: width $b=0.30$ m, flow depth $y=0.10$ m, carrying $Q=0.0800\ \text{m}^3/\text{s}$.

Approach. Get the mean velocity from continuity, then form the Froude number from $V$ and the depth, and the channel Reynolds number from $V$ and the hydraulic radius $R_h=A/P$ (White's open-channel convention, distinct from the pipe-flow hydraulic-diameter definition).

Cross-sectional area and mean velocity. $$A = by = (0.30)(0.10) = 0.0300\ \text{m}^2,\qquad V=\frac{Q}{A}=\frac{0.0800}{0.0300} = \boxed{V\approx2.667\ \text{m/s}}.$$
(a) Froude number. $$Fr = \frac{V}{\sqrt{gy}} = \frac{2.667}{\sqrt{(9.81)(0.10)}} = \boxed{Fr\approx2.69}.$$ Since $Fr>1$, the flow is supercritical.
Hydraulic radius. Wetted perimeter $P=b+2y=0.30+2(0.10)=0.50$ m, so $$R_h = \frac{A}{P} = \frac{0.0300}{0.50} = \boxed{R_h\approx0.0600\ \text{m}}.$$
(b) Channel Reynolds number. $$\text{Re} = \frac{\rho V R_h}{\mu} = \frac{(998)(2.667)(0.0600)}{0.001} = \boxed{\text{Re}\approx1.60\times10^5}.$$ This is far above the open-channel laminar limit ($\text{Re}\lesssim500$ on this $R_h$-based definition), confirming fully turbulent flow, consistent with the high Froude number.

Quantity	Result
Mean velocity, $V$	2.67 m/s
(a) Froude number, $Fr$	2.69 (supercritical)
Hydraulic radius, $R_h$	0.0600 m
(b) Reynolds number, $\text{Re}$	≈1.60×10⁵ (turbulent)

Concept & Theory

Topic. The two governing dimensionless groups of open-channel flow: the Froude number (gravity vs. inertia, wave/critical-depth behaviour) and the Reynolds number (viscous vs. inertial forces, laminar/turbulent state).

Key relations. $Fr=V/\sqrt{gy}$ for a rectangular channel; hydraulic radius $R_h=A/P$; channel Reynolds number $\text{Re}=\rho V R_h/\mu$ (White's convention — not the pipe-flow hydraulic diameter $D_h=4R_h$).

Why this works. Continuity gives the one velocity scale that both dimensionless groups need; $Fr$ compares that velocity to the shallow-water wave speed $\sqrt{gy}$, telling us whether disturbances can propagate upstream ($Fr<1$) or not ($Fr>1$), while $\text{Re}$ (built on the channel's own characteristic length, $R_h$) tells us whether viscosity meaningfully damps the flow or is negligible next to inertia — almost always the latter in open channels of this scale.

Common pitfall. Using the pipe-flow hydraulic diameter $4R_h$ in the open-channel Reynolds number (a 4× error) instead of $R_h$ directly; also forgetting the $\text{cm}^3\to\text{m}^3$ conversion on the given flow rate.

Source. White, Fluid Mechanics, Ch. 10 (Open-Channel Flow) — Froude Number and Channel Reynolds Number.

Question 3 of 4: Froude and Reynolds Numbers in a Rectangular Channel

Problem 3: Froude and Reynolds Numbers in a Rectangular Channel (25 points)