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4.6 EXERCISES Momentum equation
The backward-facing step, sketched in Fig. E.4.1, is in a 5 m wide channel of rectangular cross- section. The total discharge is 55 m^/s. The fluid is water at 20°C. The bed of the channel, upstream and downstream of the step, is horizontal and smooth. Compute the pressure force acting on the vertical face of the step and indicate the direction of the force (i.e. upstream or downstream).
1.35 m/s
0.1m
c
~^® (D
Fig. E.4.1 Sketch of a negative step.
Solution: Applying the continuity equation, the upstream flow depth equals dx = 8.15 m.
Assuming a hydrostatic pressure distribution, the upstream specific energy equals Ei = 8.24 m.
Note that the upstream flow is subcritical (Frj = 0.15).
The downstream specific energy is deduced fi*om the Bernoulli principle assuming a smooth transition:
Ex + AZQ = £'2 Bernoulli equation
where Az^ is the drop height (0.1 m). It gives £"2 = 8.34 m. The downstream flow depth is deduced fi"om the definition of the specific energy:
£". = J. + ^
2gB^4
where B is the channel width. The solution is: d2 = 8.25 m.
Considering the forces acting on the control volume contained between sections 1 and 2, the momentum equation as applied to the control volume between sections 1 and 2 is:
1 2 1 2
PQV2 — PQVY= —pgdi B — —pgd2B + F Momentum principle in the horizontal direction. Note that the friction is zero as the bed is assumed smooth.
The solution of the momentum equation is F = +40.3 kN (Remember: F is the force exerted by the step onto the fluid). In other words, the pressure force exerted by the fluid onto the verti- cal face of the step acts in the upstream direction and equals 40.3 kN.
Considering a broad-crested weir laboratory model (0.25 m wide rectangular channel), the upstream flow conditions (measured upstream of the weir) are di = 0.18 m and Vi = 0.36 m/s.
The bed of the channel is horizontal and smooth (both upstream and downstream of the weir).
Assuming critical flow conditions at the crest and supercritical flow downstream of the weir, compute the following quantities: (a) flow depth downstream of the weir and (b) the horizontal force acting on the weir (i.e. sliding force).
Considering a sluice gate in a rectangular channel (B = 0.80 m), the observed upstream and down- stream flow depth are respectively: di — 0.450 m and ^2 = 0.118 m. (a) Derive the expression of the flow rate in terms of the upstream and downstream depths, and of the channel width; (b) derive the expression of the force on the gate in terms of the upstream and downstream depths, the chan- nel width and the flow rate; (c) compute the flow rate and (d) compute the force on the gate.
Assume of smooth horizontal channel.
Hydraulic jump
Considering a hydraulic jump, answer the following questions:
Question Yes No Correct answer The upstream Froude number is less than 1
The downstream velocity is less than the upstream velocity The energy loss term is always positive
The upstream depth is larger than the downstream flow depth Hydraulic jump can occur in closed-conduit flow
Air entrainment occurs in a hydraulic jump Hydraulic jump is dangerous
If the upstream flow is such as Fri = 1 can hydraulic jump occur?
What is then the downstream Froude number?
4.6 Exercises 89
What is the definition of alternate depths, conjugate depths and sequent depths? In each case, explain your answer in words and use appropriate example(s) if necessary.
Considering a hydraulic jump in a rectangular horizontal channel, sketch the flow and write the following basic equations across the jump: (a) continuity equation, (b) momentum equation and (c) energy equation. Deduce the expression of the downstream flow depth as a function of the upstream flow depth and upstream Froude number only.
Application: The upstream flow depth is 1.95 m. The channel width is 2 m. The flow rate is 70m^/s. Compute the downstream flow depth and the head loss across the jump.
Considering a hydraulic jump in a horizontal channel of trapezoidal cross-section. The channel cross-section is symmetrical around the channel centreline, the angle between the sidewall and the horizontal being 5. Write the continuity and momentum equations for the hydraulic jump. Use the subscript 1 to refer to the upstream flow conditions and the subscript 2 for the downstream flow conditions. Neglect the bottom fiiction.
A hydraulic jump flow takes place in a horizontal rectangular channel. The upstream flow conditions axQ d = 1 m and q = 11.2m^/s. Calculate the downstream flow depth, downstream Froude number, head loss in the jump and roller length.
Considering the upstream flow Q = 160m^/s and 5 = 40 m, design a hydraulic jump dissipa- tion structure. Select an appropriate upstream flow depth for an optimum energy dissipation.
What would be the upstream and downstream Froude numbers?
Considering a hydraulic jump in a horizontal rectangular channel of constant width and neglect- ing the friction force, apply the continuity, momentum and energy equations to the following case: Q = 1500m^/s, ^ = 50 m. The energy dissipation in the hydraulic jump is A/f = 5 m.
Calculate the upstream and downstream flow conditions (di, d2, Fri, Fr2). Discuss the type of hydraulic jump.
Considering a rectangular horizontal channel downstream of a radial gate, a stilling basin (i.e.
energy dissipation basin) is to be designed immediately downstream of the gate. Assume no flow contraction at the gate. The design conditions are Q = 65 w?/s andB = 5m. Select an appropriate gate opening for an optimum design of a dissipation basin. Explain in words what gate opening would you choose while designing this hydraulic structure to dissipate the energy in a hydraulic jump. Calculate the downstream Froude number. What is the minimum length of the dissipation basin? What hydraulic power would be dissipated in the jump at design flow conditions? How many
100 W bulbs could be powered (ideally) with the hydraulic power dissipated in the jump?
Surges and bores
What is the difference between a surge, a bore and a wave? Give examples of bore.
Considering a flow downstream of a gate in an horizontal and rectangular channel, the initial flow conditions are Q = 500m^/s, B = I5m and d = 7.5 m. The gate suddenly opens and pro- vides the new flow conditions Q = 700m^/s. (a) Can the flow situation be described using the quasi-steady flow analogy? (Sketch the flow and justify in words your answer.) (b) Assuming a frictionless flow, compute the surge velocity and the new flow depth, (c) Are the new flow con- ditions supercritical? (d) Is the wave stable? (If a small disturbance start from the gate (i.e. gate vibrations) after the surge wave what will happen? Would the wave become unstable?)
Considering a flow upstream of a gate in an horizontal and rectangular channel, the initial flow conditions are Q = \00m^/s, B = 9m and d = 7.5 m. The gate suddenly opens and provides the new flow conditions Q = 270m^/s. Characterize the surge: i.e. positive or negative. Is the wave stable?
Considering the flow upstream of a gate, the gate suddenly closes. The initial flow conditions were Q = 5000 mVs, d = 5 m and 5 = 100 m. The new discharge is Q = 3000 mVs. Compute the new flow depth and flow velocity.
Solution: The surge is an advancing wave front (i.e. positive surge). Using the quasi-steady flow assumption, the initial flow conditions are the flow conditions upstream of the surge fi*ont di = 5m,Vi = lOm/s andFr = 1.43. To start the calculations, it may be assumed Kg = 0 (i.e.
stationary surge also called hydraulic jump). In this particular case (i.e. V^^g = 0), the continu- ity equation becomes Vidi = ^2^/2.
Notation equation 1st iteration 2nd iteration Solution
y
0.0 2 3.26
Fri 1.43 1.71 1.89
[C]
23.4 15 11.1
Fr2 [M]
0.72 0.62 0.57
V2 Def. Fr2 10.9 5.5 2.70
V2d2B Qi 2540 8229 3000
Notes
1. For an initialization step (where V^^^ = 0), the continuity equation can be rewritten as Fr^Jg^i = Fr2Vg^ using the definition of the Froude number. The above equation is more practical than the general continuity equation for surge.
2. It must be noted that the initial flow conditions are supercritical. The surge is a large distur- bance travelling upstream against a supercritical flow.
3. The surge can be classified as a hydraulic jump using the (surge) upstream Froude number Fri. For this example, the surge is a weak surge.
Flow resistance
Considering an uniform equilibrium flow down a rectangular channel, develop the momentum equation. For a wide rectangular channel, deduce the expression of the normal depth as a func- tion of the Darcy friction factor, discharge per unit width and bed slope.
For an uniform equilibrium flow down an open channel: (a) Write the Chezy equation. Define clearly all your symbols, (b) What are the SI units of the Chezy coefficient? (c) Give the expres- sion of the Chezy coefficient as a fimction of the Darcy-Weisbach fiiction factor, (d) Write the Gauckler-Manning equation. Define clearly your symbols, (e) What is the SI units of the Gauckler-Manning coefficient?
Considering a gradually varied flow in a rectangular open channel, the total discharge is 2.4 m^/s and the flow depth is 3.1 m. The channel width is 5 m, and the bottom and sidewalls are made of smooth concrete. Estimate: (a) the Darcy firiction factor and (b) the average boundary shear stress. The fluid is water at 20°C.
A rectangular (5.5 m width) concrete channel carries a discharge of 6m^/s. The longitudinal bed slope is 1.2m/km. (a) What is the normal depth at uniform equilibrium? (b) At uniform
4.6 Exercises 91
equilibrium what is the average boundary shear stress? (c) At normal flow conditions, is the flow subcritical, supercritical or critical? Would you characterize the channel as mild, critical or steep?
For man-made channels, perform flow resistance calculations based upon the Darcy-Weisbach fl'iction factor
Solution: (a) d = 0.64m, (b) To = 6.1 Pa and (c) Fr = 0.68: near-critical flow, although subcritical (hence mild slope) (see discussion on near critical flow in Section 5.1.2).
In a rectangular open channel (boundary roughness: PVC), the uniform equilibrium flow depth equals 0.9 m. The channel is 10 m wide and the bed slope is 0.0015°. The fluid is water at 20°C. Calculate (a) the flow rate, (b) Froude number, (c) Reynolds number, (d) relative roughness, (e) Darcy fiiction factor and (f) mean boundary shear stress.
Considering an uniform equilibrium flow in a trapezoidal grass waterway (bottom width: 15 m, sidewall slope: 1 V:5H), the flow depth is 5 m and the longitudinal bed slope is 3 m/km. Assume a Gauckler-Manning coefficient of 0.05 s/m^^^ (flood plain and light brush). The fluid is water at 20°C. Calculate: (a) discharge, (b) critical depth, (c) Froude number, (d) Reynolds number and (e) Chezy coefficient.
Norman Creek, in Southern Brisbane, has the following channel characteristics during a flood event: water depth: 1.16 m, width: 55 m, bed slope: 0.002 and short grass: k^ = 3 mm. Assume uniform equilibrium flow conditions in a rectangular channel. Calculate the hydraulic charac- teristics of the stream in flood.
Solution: Q = 200mVs and V = 3.1 m/s.
During a flood, measurement in Oxley Creek, in Brisbane, gave: water depth: 1.16 m, width:
55 m, bed slope: 0.0002 and short grass: k^ = 0.003 m. Assuming uniform equilibrium flow conditions in a quasi-rectangular channel, compute the flow rate.
Solution: V= 0.99m/s and V* = 0.047 m/s.
Considering an uniform equilibrium flow in a trapezoidal concrete channel (bottom width: 2 m and sidewall slope: 30°), the total discharge is lOm^/s. The channel slope is 0.02°. The fluid is water at 20°C. Estimate: (a) critical depth, (b) normal depth, (c) Froude number, (d) Reynolds number and (e) Darcy friction factor.
Considering the flood plain, sketched in Fig. E.4.2, the mean channel slope is 0.05°. The river channel is lined with concrete and the flood plain is riprap material (equivalent roughness height: 8 cm). The fluid is water with a heavy load of suspended sediment (fluid density:
1080kg/m^). The flow is assumed to be uniform equilibrium. Compute and give the values (and units) of the following quantities: (a) Volume discharge in the river channel, (b) Volume dis- charge in the flood plain, (c) Total volume discharge (river channel + flood plain), (d) Total mass flow rate (river channel + flood plain), (e) Is the flow subcritical or supercritical? Justify your answer clearly. (Assume no friction (and energy loss) at the interface between the river channel flow and the flood plain flow.)
Considering a river channel with a flood plain in each side (Fig. E.4.3), the river channel is lined with finished concrete. The lowest flood plain is liable to flooding (it is land used as flood water retention system) and its bed consists of gravel (k^ = 20 mm). The right bank plain is a grassed area (centipede grass, ^Manning ~ 0-06 SI units). The longitudinal bed slope of the river is 2.5 km.
River channel Flood plain
Fig. E.4.2 Sketch of a flood plain.
Flood retention channel
^\4-
River
channel Flood plain
51m
^ 5.3 m 3.2 m
^ ^ 22 m
// //
200 m
Fig. E.4.3 Sketch of a flood plain.
For the l-in-50-years flood (Q = 500 m^/s), compute (a) the flow depth in the main channel, (b) the flow depth in the right flood plain, (c) the flow rate in the main channel, (d) the flow rate in the flood water retention system and (e) the flow rate in the right flood plain. (Assume no fiiction (and energy loss) at the interface between the river channel flow and the flood plain flows.)
Considering the river channel and flood plain sketched below (Fig. E.4.4), the main dimensions of the channel and flood plain are: Wi = 2.5 m, PF2 — 25 m and Azi2 = 1.2 m. The main channel is lined with finished concrete. The flood plain is liable to flooding. (It is used as a flood water retention system.) The flood retention plain consists of light bush. (Field observations suggested
^Manning ^ 0.04 s/m ^^^.) The longitudinal bed slope of the river is 1.5 m/km.
During a storm event, the observed water depth in the main (deeper) channel is 1.9 m. Assuming uniform equilibrium flow conditions, compute (a) the flow rate in the main channel, (b) the flow rate in the flood water retention system, (c) the total flow rate and (d) What the flow Froude number in the deep channel and in the retention system? (Assume energy loss at the interface between the river channel flow and the flood plain flows.)
4.6 Exercises 93
^ Main channel
^
r
H^
1^^ P : ^
/ -
T *
Flood plain
1/1/2
^
w
w
w.
Fig. E.4.4 Sketch of flood plain.
Flood plain 2
4^ ^ -^
1 ^ P:"^
^ JM
1/1/3 r^
A Z i 3
Main channel w A
d^
^
AZ12
f 1 .
Flood plain 2
^ w
^ ^ 1/1/2
1/1/1 Fig. E.4.5 Sketch of flood plain.
For man-made channels, DO perform flow resistance calculations based upon the Darcy- Weisbach fr-iction factor.
Solution: (a) 18.4m^/s, (b) 13.1 mVs, (c) 31.5 mVs and (d) Fr = 0.7 and 0.3, respectively (see Section 3.4).
Considering a river channel with a flood plain in each side (Fig. E.4.5), the cross-sectional characteristics are Wx = 5m.,W2= 14m, Azi2 = 0.95m, Wi, = 58m, Azi3 = 1.35m. The river channel is lined with finished concrete. The lowest flood plain (Flood plain No. 2) is liable to flooding and its bed consists of gravel (k^ = 20 mm). The left bank plain (Flood plain No. 2) is a grassed area (centipede grass, ^Manning ~ 0-06 SI units).The longitudinal bed slope of the river is 3.2 m/km. For the l-in-50-years flood (Q = 150 m^/s), compute (a) the flow depth in the main channel, (b) the flow depth in the right flood plain, (c) the flow rate in the main channel, (d) the flow rate in the flood water retention system and (e) the flow rate in the right flood plain.
Assume no friction (and no energy loss) at the interface between the river channel flow and the flood plain flows. For man-made channels, DO perform flow resistance calculations based upon the Darcy-Weisbach friction factor.
Solution: (a) d^ = 2.17m, (d) Q2 = 53.2mVs and (e) ft = 39.2 m^/s.