# Open Channel Flow

 Question 1
A hydraulic jump takes place in a 6 m wide rectangular channel at a point where the upstream depth is 0.5 m (just before the jump). If the discharge in the channel is 30 $m^3/s$ and the energy loss in the jump is 1.6 m, then the Froude number computed at the end of the jump is ___________. (round off to two decimal places) (Consider the acceleration due to gravity as 10 $m/s^2$.)
 A 0.4 B 0.85 C 0.65 D 0.75
GATE CE 2022 SET-2   Fluid Mechanics and Hydraulics
Question 1 Explanation: $Q=30m^3/sec$
$B=6m$
$y_1=0.5m$
$E_L=1.6m$
$q= \frac{Q}{B} =\frac{30}{6}=5m^3/sec/m$
We know that,
\begin{aligned} E_L=\frac{(y_2-y_1)^3}{4y_1y_2}&=1.6m\\ \frac{(y_2-0.5)^3}{4 \times 0.5 \times y_2}&=1.6\\ y_2^3-1.5y_2^2+0.75y_2-0.125&=3.2y_2\\ y_2=2.5m,-0.0527m,&-0.0947m \end{aligned}
Hence, $y_2=2.5m$
Post jump Froude's No.
$(F_2)=\frac{V_2}{\sqrt{\sqrt{gy_2}}}=\frac{\left ( \frac{30}{6 \times 2.5} \right )}{\sqrt{10 \times 2.5}}=0.4$

 Question 2
Water is flowing in a horizontal, frictionless, rectangular channel. A smooth hump is built on the channel floor at a section and its height is gradually increased to reach choked condition in the channel. The depth of water at this section is $y_2$ and that at its upstream section is $y_1$. The correct statement(s) for the choked and unchoked conditions in the channel is/are
 A In choked condition, $y_1$ decreases if the flow is supercritical and increases if the flow is subcritical. B In choked condition, $y_2$ is equal to the critical depth if the flow is supercritical or subcritical. C In unchoked condition, $y_1$ remains unaffected when the flow is supercritical or subcritical. D In choked condition, $y_1$ increases if the flow is supercritical and decreases if the flow is subcritical.
GATE CE 2022 SET-2   Fluid Mechanics and Hydraulics
Question 2 Explanation: Question 3
Depth of water flowing in a 3 m wide rectangular channel is 2 m. The channel carries a discharge of 12 $m^3/s$. Take g = 9.8 $m/s ^2$.
The bed width (in m) at contraction, which just causes the critical flow, is _________ without changing the upstream water level. (round off to two decimal places)
 A 2.85 B 4.25 C 2.15 D 1.55
GATE CE 2022 SET-1   Fluid Mechanics and Hydraulics
Question 3 Explanation:
Given: $B = 3m, y = 2m, Q = 12 m^3 /sec$
$Velocity(v)=\frac{Q}{A} =\frac{12}{2 \times 3}=2m/sec$ Specific energy at section (1-1)
$E=y+\frac{v^2}{2g}=2+\frac{2^2}{2 \times 9.8}=\frac{108}{49}m$
When channel section is contracted to minimum width and for constant discharge Q, the flow over contracted section will be critical flow and under the assumption that no energy loss has taken place.
$E=E_c=\frac{108}{49}$
We have that, $E_c=\frac{3}{2}y_c$ (for rectangular crosssection)
$y_c=\frac{2}{3}E_c=\frac{2}{3} \times \frac{108}{49}=\frac{72}{49}$
For critical flow condition,
\begin{aligned} \frac{Q^2T}{gA^3}&=1\\ \frac{Q^2B_{min}}{g(B_{min} \times y_c)^3}&=1\\ (B_{min})^2&=\frac{Q^2}{gy_c^3}\\ B_{min}&=\left ( \frac{(12)^2}{9.8 \times \left ( \frac{72}{49} \right )^3} \right )^{1/2}\\ &=2.152m \end{aligned}
 Question 4
A rectangular channel with Gradually Varied Flow (GVF) has a changing bed slope. If the change is from a steeper slope to a steep slope, the resulting GVF profile is
 A $S_3$ B $S_1$ C $S_2$ D either $S_1$ or $S_2$, depending on the magnitude of the slopes
GATE CE 2022 SET-1   Fluid Mechanics and Hydraulics
Question 4 Explanation: Question 5
A rectangular open channel of 6 m width is carrying a discharge of $20 \mathrm{~m}^{3} / \mathrm{s}$. Consider the acceleration due to gravity as $9.81 \mathrm{~m} / \mathrm{s}^{2}$ and assume water as incompressible and inviscid. The depth of flow in the channel at which the specific energy of the flowing water is minimum for the given discharge will then be
 A 0.82 m B 1.04 m C 2.56 m D 3.18 m
GATE CE 2021 SET-2   Fluid Mechanics and Hydraulics
Question 5 Explanation: Minimum specific energy will correspond to a critical flow condition.
The critical depth \begin{aligned} \left(Y_{C}\right) &=\left[\frac{q^{2}}{g}\right]^{1 / 3} \\ Y_{C} &=\left[\frac{(20 / 6)^{2}}{9.81}\right]^{1 / 3}=1.042 \mathrm{~m} \end{aligned}
 Question 6
If water is flowing at the same depth in most hydraulically efficient triangular and rectangular channel sections then the ratio of hydraulic radius of triangular section to that of rectangular section is
 A $\frac{1}{\sqrt{2}}$ B $\sqrt{2}$ C 1 D 2
GATE CE 2021 SET-1   Fluid Mechanics and Hydraulics
Question 6 Explanation:
Efficient channel section \begin{aligned} A & =y^{2} & & A=2 y^{2} \\ P & =2 \sqrt{2} y & P & =4 y \\ R_{I} & =\frac{y}{2 \sqrt{2}} & R_{I I}&=\frac{y}{2} \\ \therefore \qquad \qquad \qquad \frac{R_{I}}{R_{I I}} & =\frac{1}{\sqrt{2}} & \end{aligned}
 Question 7
A hydraulic jump occurs, in a triangular (V-shaped) channel with side slopes 1:1 (vertical to horizontal). The sequent depths are 0.5 m and 1.5 m. The flow rate (in $m^3/s$, round off to two decimal places) in the channel is _________.
 A 1.24 B 1.68 C 1.73 D 2.14
GATE CE 2020 SET-2   Fluid Mechanics and Hydraulics
Question 7 Explanation: \begin{aligned} A &=\frac{1}{2} \times 2Y \times Y =Y^2 \\ \bar{Y} &=\frac{Y}{3} \end{aligned}
For a horizontal and frictionless channel
Specific Force (F) $=A\bar{Y}+\frac{Q^2}{Ag}=constant$
$\Rightarrow \; Y^2\left ( \frac{Y}{3} \right )+\frac{Q^2}{(Y^2)g}=constant$
$\Rightarrow \; \frac{Y^3}{3}+\frac{Q^2}{gY^2}=constant$
If $Y_1$ and $Y_2$ are conjugate depth
\begin{aligned} \frac{Y_1^3}{3}+\frac{Q^2}{gY_1^2} &= \frac{Y_2^3}{3}+\frac{Q^2}{gY_2}\\ \frac{0.5^3}{3}+\frac{Q^2}{g \times 0.5^2}&=\frac{1.5^3}{3}+\frac{Q^2}{g \times 1.5^2} \\ \frac{1.5^3}{3}-\frac{0.5^3}{3}&=\frac{Q^2}{g}\left ( \frac{1}{0.5^2}-\frac{1}{1.5^2} \right ) \\ Q&=1.728 m^3/sec \end{aligned}
 Question 8
Water flows at the rate of 12 $m^3/s$ in a 6 m wide rectangular channel. A hydraulic jump is formed in the channel at a point where the upstream depth is 30 cm (just before the jump). Considering acceleration due to gravity as 9.81 $m/s^2$ and density of water as 1000 $kg/m^3$, the energy loss in the jump is
 A 114.2 kW B 114.2 MW C 141.2 h.p. D 141.2 J/s
GATE CE 2020 SET-1   Fluid Mechanics and Hydraulics
Question 8 Explanation:
Assuming channle bed to be horizontal and frictionless.
$q=\frac{12}{6}=2 m^3/s/m$ Initial Froude No.
\begin{aligned} (F_r) &=\left ( \frac{q^2}{gY_1^3} \right )^{1/2} \\ &= \left ( \frac{2^2}{9.81 \times 0.3^3} \right )^{1/2}\\ &=3.88 \end{aligned}
From Belenger's Momentum equation for a rectangular channel
\begin{aligned} \frac{Y_2}{Y_1}&=\frac{1}{2}(-1+\sqrt{1+8F_1^2}) \\ &=\frac{1}{2}(-1+\sqrt{1+8\times 3.88^2}) \\ &= 5.018\\ Y_2&=5.018 \times 0.3=1.505m \end{aligned}
Heal loss in the jump
\begin{aligned} (h_L) &=\frac{(Y_2-Y_1)^3}{4Y_1Y_2} \\ &= \frac{(1.505-0.3)^3}{4 \times 1.505 \times 0.3}\\ &= 0.968m \end{aligned}
Power lost in the jump
\begin{aligned} &=\gamma _wQh_L \\ &= (9.81 \times 12 \times 0.968)kW\\ &=114.04 kW \end{aligned}
 Question 9
A 4 m wide rectangular channel carries 6 $m^3/s$ of water. The Manning's 'n' of the open channel is 0.02. Considering g = 9.81 $m/s^2$, the critical velocity of flow (in m/s, round off to two decimal places) in the channel, is ________.
 A 1.32 B 5.65 C 2.45 D 4.25
GATE CE 2020 SET-1   Fluid Mechanics and Hydraulics
Question 9 Explanation:
\begin{aligned} \text{Critical depth} \; (Y_c)&=\left ( \frac{q^2}{g} \right )^{1/3} \\ &=\left ( \frac{1.5^2}{9.81} \right )^{1/3} \\ &= 0.612m\\ \text{Critical velocity}\; (V_c)&=\sqrt{gY_c}=\sqrt{9.81 \times 0.612} \\ &= 2.45 m/s \end{aligned}
 Question 10
A rectangular open channel has a width of 5 m and a bed slope of 0.001. For a uniform flow of depth 2 m, the velocity is 2 m/s. The Manning's roughness coefficient for the channel is
 A 0.002 B 0.017 C 0.033 D 0.05
GATE CE 2019 SET-1   Fluid Mechanics and Hydraulics
Question 10 Explanation:
Given,
\begin{aligned} V&= 2 m/sec, \; S=0.001\\ A&=5 \times 2=10m^2\\ P&=5+2+2=9m\\ & \text{Hydraulic mean radius}\\ (R)&=\frac{A}{P}=\frac{10}{9}\\ V&=\frac{1}{n}R^{2/3}S^{1/2}\\ 2&=\frac{1}{n}\left ( \frac{10}{9} \right )^{2/3}(0.001)^{1/2}\\ n&=0.0169\simeq 0.017 \end{aligned}
There are 10 questions to complete.