Chapter Questions
A large tank of water having depth of $5 \mathrm{~m}$ is kept on a descending elevator. Determine the speed of a wave created on its surface if the rate of descent is (a) constant at $10 \mathrm{~m} / \mathrm{s},(\mathrm{b})$ accelerated at $5 \mathrm{~m} / \mathrm{s}^{2},(\mathrm{c})$ accelerated at $9.81 \mathrm{~m} / \mathrm{s}^{2}$.
The flow through a rectangular channel of width $3 \mathrm{~m}$ is given as $8 \mathrm{~m}^{3} / \mathrm{s}$, determine the Froude number when the water depth is $0.6 \mathrm{~m}$. At this depth, is the flow subcritical or supercritical? Also, what is the critical speed of the flow?
A rectangular channel of width $4 \mathrm{~m}$ transports water at $12 \mathrm{~m}^{3} / \mathrm{s}$. If the water depth is $3 \mathrm{~m}$, is the flow subcritical or supercritical?
A river having $6 \mathrm{~m}$ depth flows at an average speed of $4 \mathrm{~m} / \mathrm{s}$. If a stone is thrown into it, determine how fast the waves will travel upstream and downstream.
The flow over a $2.5 \mathrm{~m}$ wide rectangular channel is 6 $\mathrm{m}^{3} / \mathrm{s}$, determine the Froude number when the water depth is $2 \mathrm{~m}$. At this depth, is the flow subcritical or supercritical? Also, what is the critical speed of the flow?
Water flows in a rectangular channel with a speed of $3 \mathrm{~m} / \mathrm{s}$ and depth of $1.25 \mathrm{~m}$. What other possible depth of flow provides the same specific energy?
A rectangular channel of a width of $4 \mathrm{~m}$ is required to transport $50 \mathrm{~m}^{3} / \mathrm{s}$ of water. Determine the critical depth and critical velocity of the flow. Also, find the specific energy at the critical depth, and also when the depth is $3 \mathrm{~m}$ ?
Water flows in a rectangular channel with a mean speed of $6 \mathrm{~m} / \mathrm{s}$ and depth of $4 \mathrm{~m}$. What other possible average velocity of flow provides the same specific energy?
Water flows within the rectangular channel with a flow of $8 \mathrm{~m}^{3} / \mathrm{s}$. Determine the two possible flow depths, and identify the flow as supercritical or subcritical, if the specific energy is $2 \mathrm{~m}$. Also, plot the specific energy diagram.
The rectangular channel transports water at $8 \mathrm{~m}^{3} / \mathrm{s}$. Determine the critical depth $y_{c}$ and plot the specific energy diagram for the flow. Indicate $y$ for $E=2 \mathrm{~m}$.
The channel transports water at $8 \mathrm{~m}^{3} / \mathrm{s}$. If the depth of flow is $y=1.5 \mathrm{~m},$ determine if the flow is subcritical or supercritical. What is the critical depth of flow? Compare the specific energy of the flow with its minimum specific energy.
The rectangular channel transports water at $4 \mathrm{~m}^{3} / \mathrm{s}$. Determine the critical depth $y_{c}$ and plot the specific energy diagram for the flow. Indicate $y$ for $E=1.25 \mathrm{~m}$.
The rectangular channel becomes narrow to $2 \mathrm{~m}$ as shown. If the flow is $6 \mathrm{~m}^{3} / \mathrm{s}$ and $y_{A}=4 \mathrm{~m}$, determine the depth of flow at $B$.
The rectangular channel becomes narrow to $2 \mathrm{~m}$ as shown. If the flow is $6 \mathrm{~m}^{3} / \mathrm{s}$ and $y_{A}=6 \mathrm{~m}$, determine the depth of flow at $B$.
A venturi is used to measure the volumetric flow in a channel. If the depth of flow at $A$ is $y_{A}=3 \mathrm{~m}$ and at the throat $B$ is $y_{B}=2.5 \mathrm{~m}$, determine the flow through the channel.
Water flows within the rectangular channel such that the flow is $4 \mathrm{~m}^{3} / \mathrm{s}$. Determine the critical depth of flow and the minimum specific energy. If the specific energy is $8 \mathrm{~m},$ what are the two possible flow depths?
The rectangular channel transports water at a flow of $8 \mathrm{~m}^{3} / \mathrm{s}$. Plot the specific energy diagram for the flow and indicate $y$ for $E=3 \mathrm{~m}$.
The channel has a width of $3 \mathrm{~m}$ and used to transport water at a flow rate of $25 \mathrm{~m}^{3} / \mathrm{s}$. If the elevation of the bed is lowered by $0.2 \mathrm{~m},$ determine the new depth $y_{2}$ of the water.
The channel is $2 \mathrm{~m}$ wide and transports water at $18 \mathrm{~m}^{3} / \mathrm{s}$. If the elevation of the bed is raised $0.25 \mathrm{~m}$, determine the new depth $y_{2}$ of the water and the speed of the flow. Is the new flow subcritical or supercritical?
Water flows within the 4 -m-wide rectangular channel at $20 \mathrm{~m}^{3} / \mathrm{s}$. Determine the depth of flow $y_{B}$ at the downstream end and the velocity of flow at $A$ and $B$. Take $y_{A}=5 \mathrm{~m}$.
Water flows within the 4 -m-wide rectangular channel at $20 \mathrm{~m}^{3} / \mathrm{s}$. Determine the depth of flow $y_{B}$ at the downstream end and the velocity of flow at $A$ and $B$. Take $y_{A}=0.5 \mathrm{~m}$.
The rectangular channel is $2 \mathrm{~m}$ wide, and the depth of the water is $1.5 \mathrm{~m}$ as it flows with an average velocity of $0.5 \mathrm{~m} / \mathrm{s}$. Show that the flow is tranquil, and determine the required height $h$ of the bump so that the flow can change to rapid flow after it passes over the bump. What is the new depth $y_{2}$ for rapid flow?
The rectangular channel is $2 \mathrm{~m}$ wide, and the depth of the water is $0.75 \mathrm{~m}$ as it flows with an average velocity of $4 \mathrm{~m} / \mathrm{s}$. Show that the flow is rapid, and determine the required height $h$ of the bump so that the flow can change to tranquil flow after it passes over the bump. What is the new depth $y_{2}$ for tranquil flow?
The 2 -m-wide sluice gate is used to control the flow of water from a reservoir. If the depths $y_{1}=4 \mathrm{~m}$ and $y_{2}=0.75 \mathrm{~m},$ determine the volumetric flow through the gate and the depth $y_{3}$ just before the gate.
The 2 -m-wide sluice gate is used to control the flow of water from a reservoir. If the flow is $10 \mathrm{~m}^{3} / \mathrm{s}$ and $y_{1}=4 \mathrm{~m}$ determine the depth $y_{2}$, and depth $y_{3}$ just before the gate.
The sluice gate and channel both have a width of $2 \mathrm{~m}$. If the depth of flow at $A$ is $y_{1}=3 \mathrm{~m}$, determine the volumetric flow through the channel as a function of depth(a) $1 \mathrm{~m}$ $y_{2}$ and specify $Q$ when the depth $y_{2}$ is(b) $1.5 \mathrm{~m}$
Determine the hydraulic radius for each channel cross section.
The channel has a triangular cross section. Determine the critical depth $y=y_{c}$ in terms of $\theta$ and the flow $Q$.
A rectangular channel has a width of $2 \mathrm{~m}$, is made of unfinished concrete, and is inclined at a slope of $0.0014 .$ Determine the volumetric flow when the depth of flow of the water is $1.5 \mathrm{~m}$.
Water flows uniformly down the triangular channel having a downward slope of 0.0083 . If the walls are made of finished concrete, determine the volumetric flow when $y=1.5 \mathrm{~m}$.
The channel is made of unfinished concrete and has a downward slope of 0.003 . Determine the volumetric flow if the depth is $y=2 \mathrm{~m}$. Is the flow subcritical or supercritical?
The channel is made of unfinished concrete and has a downward slope of 0.003 . Determine the volumetric flow if the depth is $y=3 \mathrm{~m}$. Is the flow subcritical or supercritical?
The culvert carries water and is at a downward slope $S_{0} .$ Determine the depth $y$ that will produce the maximum volumetric flow.
The culvert carries water and is at a downward slope $S_{0}$. Determine the depth $y$ that will produce maximum velocity for the flow.
The drainage canal has a downward slope of 0.002 . If its bottom and sides have weed growth, determine the volumetric flow of water when the depth of flow is $2.5 \mathrm{~m}$.
A rectangular channel has a downward slope of 0.006 and a width of $3 \mathrm{~m}$. The depth of the water is $4 \mathrm{~m}$. If the volumetric flow through the channel is $30 \mathrm{~m}^{3} / \mathrm{s}$, determine the value of $n$ in the Manning formula. The channel is made of unfinished concrete and has the cross section shown. If the downward slope is 0.0008 , determine the flow of water through the channel when $y=4 \mathrm{~m}$
The channel is made of unfinished concrete and has the cross section shown. If the downward slope is 0.0008 , determine the flow of water through the channel when $y=6 \mathrm{~m}$.
The channel is made of finished concrete and has a trapezoidal cross section. If the average velocity of the flow is to be $6 \mathrm{~m} / \mathrm{s}$ when the water depth is $2 \mathrm{~m}$, determine the required slope.
The unfinished concrete channel is intended to have a downward slope of 0.002 and sloping sides at $60^{\circ} .$ If the flow is estimated to be $100 \mathrm{~m}^{3} / \mathrm{s}$, determine the base dimension $b$ of the channel bottom.
A rectangular channel has a width of $2.5 \mathrm{~m}$ and is made of unfinished concrete. If it is inclined downward at a slope of $0.0014,$ what depth of water will produce a discharge of $12 \mathrm{~m}^{3} / \mathrm{s} ?$
Determine the length of the sides $a$ of the channel in terms of its base $b$, so that for the flow at full depth it provides the best hydraulic cross section that uses the minimum amount of material for a given discharge.
Determine the volumetric flow of water through the channel if the depth of flow is $y=1.25 \mathrm{~m}$ and the downward slope of the channel is $0.005 .$ The sides of the channel are finished concrete.
Determine the normal depth of water in the channel if the flow is $Q=15 \mathrm{~m}^{3} / \mathrm{s}$. The sides of the channel are finished concrete, and the downward slope is 0.005 .
Determine the angle $\theta$ of the channel so that it has the best hydraulic triangular cross section that uses the minimum amount of material for a given discharge.
Show that the width $b=2 h(\csc \theta-\cot \theta)$ in order to minimize the wetted perimeter for a given crosssectional area and angle $\theta$. At what angle $\theta$ will the wetted perimeter be the smallest for a given cross-sectional area and depth $h$ ?
Show that when the depth of flow $y=R$, the semicircular channel provides the best hydraulic cross section.
Determine the angle $\theta$ and the length $l$ of its sides so that the channel has the best hydraulic trapezoidal cross section of base $b$.
A rectangular channel is made of unfinished concrete, and it has a width of $1.25 \mathrm{~m}$ and an upward slope of 0.01 . Determine the surface profile for the flow if it is $0.8 \mathrm{~m}^{3} / \mathrm{s}$ and the depth of the water at a specific location is $0.5 \mathrm{~m}$. Sketch this profile.
A rectangular channel is made of finished concrete, and it has a width of $1.25 \mathrm{~m}$ and a downward slope of 0.01 . Determine the surface profile for the flow if it is $0.8 \mathrm{~m}^{3} / \mathrm{s}$ and the depth of the water at a specific location is $0.6 \mathrm{~m}$. Sketch this profile.
A rectangular channel is made of finished concrete, and it has a width of $1.25 \mathrm{~m}$ and an upward slope of 0.01 . Determine the surface profile for the flow if it is $0.8 \mathrm{~m}^{3} / \mathrm{s}$ and the depth of the water at a specific location is $0.2 \mathrm{~m}$. Sketch this profile.
Water flows at $4 \mathrm{~m}^{3} / \mathrm{s}$ along a horizontal channel made of unfinished concrete. If the channel has a width of $2 \mathrm{~m},$ and the water depth at a control section $A$ is $0.9 \mathrm{~m}$, approximate the depth at the section where $x=2 \mathrm{~m}$ from the control section. Use increments of $\Delta y=0.004 \mathrm{~m}$ and plot the profile for $0.884 \mathrm{~m} \leq y \leq 0.9 \mathrm{~m}$
A rectangular channel is made of finished concrete, and it has a width of $1.25 \mathrm{~m}$ and a downward slope of 0.01 . Determine the surface profile for the flow if it is $0.8 \mathrm{~m}^{3} / \mathrm{s}$ and the depth of the water at a specific location is $0.3 \mathrm{~m}$. Sketch this profile.
Water flows at $12 \mathrm{~m}^{3} / \mathrm{s}$ down a rectangular channel made of unfinished concrete. The channel has a width of $4 \mathrm{~m}$ and a downward slope of 0.008 , and the water depth is $2 \mathrm{~m}$ at the control section $A$. Determine the distance $x$ from $A$ to where the depth is $2.4 \mathrm{~m}$. Use increments of $\Delta y=0.1 \mathrm{~m}$ and plot the profile for $2 \mathrm{~m} \leq y \leq 2.4 \mathrm{~m}$
Water flows at $4 \mathrm{~m}^{3} / \mathrm{s}$ along a horizontal channel made of unfinished concrete. If the channel has a width of $2 \mathrm{~m},$ and the water depth at a control section $A$ is $0.9 \mathrm{~m}$ determine the approximate distance $x$ from $A$ to where the depth is $0.8 \mathrm{~m}$. Use increments of $\Delta y=0.025 \mathrm{~m}$ and plot the profile for $0.8 \mathrm{~m} \leq y \leq 0.9 \mathrm{~m}$.
Water flows under the partially opened sluice gate, which is in a rectangular channel. If the water has the depth shown, determine if a hydraulic jump forms, and if so, find the depth $y_{C}$ at the downstream end of the jump.
Water runs from a sloping channel with a flow of $8 \mathrm{~m}^{3} / \mathrm{s}$ onto a horizontal channel, forming a hydraulic jump. If the channel is $2 \mathrm{~m}$ wide, and the water is $0.25 \mathrm{~m}$ deep before the jump, determine the depth of water after the jump. What energy is lost during the jump?
The hydraulic jump has a depth of $5 \mathrm{~m}$ at the downstream end, and the velocity is $1.25 \mathrm{~m} / \mathrm{s}$. If the channel is $2 \mathrm{~m}$ wide, determine the depth $y_{1}$ of the water before the jump and the energy head lost during the jump.
Water flows at $18 \mathrm{~m}^{3} / \mathrm{s}$ over the 4 -m-wide spillway of the dam. If the depth of the water at the bottom apron is $0.5 \mathrm{~m},$ determine the depth $y_{2}$ of the water after the hydraulic jump.
The sill at $A$ causes a hydraulic jump to form in the channel. If the channel width is $1.5 \mathrm{~m},$ determine the average upstream speed and downstream speed of the water. What amount of energy head is lost in the jump?
The flow of water over the broad-crested weir is $15 \mathrm{~m}^{3} / \mathrm{s}$. If the weir and the channel have a width of $3 \mathrm{~m}$ determine the depth of water $y$ within the channel. Take $C_{w}=0.80$.
The rectangular channel has a width of $3 \mathrm{~m}$ and the depth of flow is $1.5 \mathrm{~m}$. Determine the volumetric flow of water over the rectangular sharp-crested weir. Take $C_{d}=0.83$.