Uniform flows and gradually varied flows
Part 1 Revision exercises
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Fig. R.2 Sketch of a steppedtransition.
Compute and give the values (and units) of the specified quantities in the following list: (a) Velocity of flow at section 1. (b) Specific energy at section 1. (c) Specific energy at section 2.
(d) Assumption used in answer (c). (e) Depth of flow at section 2. (f) Velocity of flow at sec- tion 2. (g) The force acting on the sluice gate, (h) The direction of the force in (g): i.e.
upstream or downstream, (i) The critical depth for the flow in Fig. R. 1. (j) What is maximum possible discharge per unit width for a flow with specific energy entered at (b). (This ques- tion is a general question, not related to the sluice gate.)
REVISION EXERCISE NO. 2
A channel step, sketched in Fig. R.2, is in a 12 m wide channel and the discharge, g, is 46 m^/s.
The flow depth at section 1 is 1.6 m and the step height is 0.4 m. The bed of the channel, upstream and downstream of the step, is horizontal and smooth.
• Sketch on Fig. R.2 the fi*ee-surface profile between sections 1 and 2. (You might have to modify it after your calculations in the following questions.)
• Sketch on Fig. R.2 the variation of the pressure with depth at sections 1 and 2, and on the verti- cal face of the step. Sections 1 and 2 are located far enough fi*om the step for the velocity to be essentially horizontal and uniform.
• Show on Fig. R.2 the forces acting on the control volume contained between sections 1 and 2. Show also your choice for the positive direction of distance and of force.
• Write the momentum equation as applied to the control volume between sections 1 and 2, using the sign convention you have chosen. Show on Fig. R.2 the forces and velocities used in the momentum equation.
• Compute and give the values (and units) of the specified quantities in the following list:
(a) Velocity of flow at section 1. (b) Specific energy at section 1. (c) Specific energy at sec- tion 2. (d) Assumption used in answer (c). (e) Depth of flow at section 2. (f) Plot the correct fi-ee-surface profile on Fig. R.2 using (e). (g) Velocity of flow at section 2. (h) The force acting on the step, (i) The direction of the force in (h): i.e. upstream or downstream, (j) The critical depth for the flow in Fig. R.2. (k) What is maximum possible discharge per unit width for a flow with specific energy entered at (b). (This question is a general question, not related to the stepped channel.)
Revision exercise No. 4 113
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Fig. R.3 Sketch of an upward-step.REVISION EXERCISE NO. 3
A channel step, sketched in Fig. R.3, is in a 5 m wide channel of rectangular cross-section. The total discharge is 64000kg/s. The fluid is slurry (density 1200 kg/m^). The downstream flow depth (i.e. at section 2) is 1.5 m and the step height is 0.8 m. The bed of the channel, upstream and downstream of the step, is horizontal and smooth. The flow direction is from section 1 to section 2.
• Compute and give the values (and units) of the following quantities: (a) Velocity of flow at section 2. (b) Froude number at section 2. (c) Specific energy at section 2.
• For the flow at section 2, where would you look for the hydraulic control?
• Sketch on Fig. R.3 the fi"ee-surface profile between sections 1 and 2. You might have to mod- ify it after your calculations in the following questions.
• Show on Fig. R.3 all the forces acting on the control volume contained between sections 1 and 2. Show also your choice for the positive direction of distance and of force.
• Compute and give the values (and units) of the specified quantities in the following list: (a) Specific energy at section 1. (b) Depth of flow at section 1. (c) Velocity of flow at section 1.
(d) Froude number at section 1. (e) Plot the correct free-surface profile on Fig. R.3 using (b). (f) The force acting on the step, (g) The direction of the force in (f): i.e. upstream or downstream, (h) The critical depth for the flow in Fig. R.3.
REVISION EXERCISE NO. 4
The horizontal rectangular channel sketched in Fig. R.4 is used as a stilling basin in which energy dissipation takes place in a hydraulic jump. The channel is equipped with four bafile blocks. The width of the channel is 20 m. The bed of the channel is horizontal and smooth. The inflow conditions are dx =4.1 m, V\ = 22 m/s, and the observed downstream flow depth is d2= 19.7m.
• Sketch on Fig. R.4 an appropriate control volume between the upstream and downstream flow locations (on the cross-sectional view).
• Sketch on Fig. R.4 (cross-sectional view) the variation of the pressure with depth at sections 1 and 2. Sections 1 and 2 are located far enough from the hydraulic jump for the velocity to be essentially horizontal and uniform.
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Fig. R.4 Sketch of a stilling basin.
• Show on Fig. R.4 all the forces acting on the control volume contained between sections 1 and 2. Show also your choice for the positive direction of distance and of force.
• Write the momentum equation as applied to the control volume between sections 1 and 2, using the sign convention that you have chosen.
• Compute and give the values (and units) of the specified quantities in the following list: (a) Total flow rate at section 1. (b) Specific energy at section 1. (c) Froude number at section 1. (d) Total force acting on the baffle blocks, (e) Velocity of flow at section 2. (f) Specific energy at section 2. (g) Energy loss between section 1 and 2. (h) Force acting on a single block, (i) The direction of the force in (h) and (g): i.e. upstream or downstream.
REVISION EXERCISE NO. 5
The 'Venturi' flume sketched on Fig. R.5 is in a rectangular channel and the discharge is 2m^/s.
The channel bed upstream and downstream of the throat is horizontal and smooth. The channel width is ^1 = ^3 = 5 m. The throat characteristics are AZQ = 0.5 m and B2 = 2.5 m. The upstream flow depth is (ii = 1.4 m.
• Sketch on Fig. R.5 a fi-ee-surface profile in the Venturi flume between sections 1 and 3. (You might have to modify it after your calculations in the following questions.)
• Sketch on Fig. R.5 the variation of the pressure with depth at sections 1 and 3. Sections 1 and 2 are located far enough fi*om the throat for the velocity to be essentially horizontal and uniform.
Revision exercise No. 6 115
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Fig. R.5 Sketch of a Venturi flume.
Compute and give the values (and units) of the specified quantities in the foUov^ing Ust: (a) Velocity of flow at section 1. (b) Specific energy at section 1. (c) Specific energy at section 2 (i.e.
at throat), (d) Assumption used in answer (c). (e) Depth of flow at section 2 (i.e. at throat), (f) Velocity of flow at section 2 (i.e. at throat), (g) Depth of flow at section 3. (h) Velocity of flow at section 3. (i) Specific energy at section 3. (j) The critical depth for the flow at section 2 (i.e. at throat), (k) What is maximum possible discharge per unit width for a flow with specific energy entered at (c) (i.e. at throat)? (This question is a general question, not related to the sluice gate.) REVISION EXERCISE NO. 6
An (undershot) sluice gate is in a rectangular channel 0.8 m wide. The flow depth upstream of the gate is 0.45 m and, downstream of the sluice gate, the free-surface elevation is 0.118 m above the channel bed. The bed of the channel is horizontal and smooth.
• Sketch the sluice gate and the fi'ee-surface profiles upstream and downstream of the sluice gate.
• Sketch on your figure the variation of the pressure with depth at a section 1 (upstream of the gate) and a section 2 (dovmstream of the gate), and on the upstream face of the sluice gate.
Sections 1 and 2 are located far enough from the sluice gate for the velocity to be essentially horizontal and uniform.
• Compute and give the values (and units) of the following quantities:
(a) Flow rate, (b) Assumption used in answer (a), (c) Velocity of flow at section 1. (d) Specific energy at section 1. (e) Specific energy at section 2. (f) Velocity of flow at section 2.
• Write the momentum equation as applied to the control volume between sections 1 and 2, using a consistent sign convention. Show on your figure the forces and velocities used in the momentum equation.
• Compute and give the values (and units) of the specified quantities in the following list (a) The force acting on the sluice gate, (b) The direction of the force in (a): i.e. upstream or downstream, (c) The critical depth for the flow
A broad-crested weir is to be constructed downstream of the sluice gate. The purpose of the gate is to force the dissipation of the kinetic energy of the flow, downstream of the sluice gate, between the gate and the weir: i.e. to induce a (fully developed) hydraulic jump between the gate and the weir. The channel between the gate and the weir is horizontal and smooth. The weir crest will be set at a vertical elevation Az above the channel bed, to be determined for the flow conditions in the above questions.
• Sketch the channel, the sluice gate, the hydraulic jump and the broad-crested weir.
• Write the momentum equation between section 3 (located immediately upstream of the jump and downstream of the gate) and section 4 (located immediately downstream of the jump and upstream of the weir).
• For the above flow conditions, compute and give the values (and units) of the specified quan- tities in the following list: (a) Flow depth upstream of the hydraulic jump (section 3). (b) Specific energy upstream of the jump (section 3). (c) Flow depth downstream of the jump (section 4). (d) Specific energy downstream of the jump (section 4). (e) Flow depth above the crest, (f) Specific energy of the flow at the weir crest, (g) Weir crest elevation Az. (h) Assump- tions used in answer (g).
REVISION EXERCISE NO. 7
A settling basin is a deep and wide channel in which heavy sediment particles may settle, to pre- vent siltation of the downstream canal. Settling basins are commonly located at the upstream end of an irrigation canal. In this section we shall investigate the flow characteristics of the out- let of the settling basin (Fig. R.6).
The outlet section is a rectangular channel sketched in Fig. R.6. The channel bed is smooth and, at sections 1, 2 and 3, it is horizontal. The channel width equals B^ = 12 m, ^2 ~ ^3 — 4.2 m.
The change in bed elevation is Az^ = 0.4 m. The total flow rate is 48 200 kg/s. The fluid is slurry (density 1405 kg/m^). The inflow conditions are rfj = 4 m.
(a) Compute and give the value of velocity of flow at section 1. (b) Compute the specific energy at section 1. (c) Calculate the specific energy at section 2. (d) In answer (c), what basic principle did you not use? (e) Compute the depth of flow at section 2. (f) Calculate the velocity of flow at section 2. (g) Compute the depth of flow at section 3. (h) Compute the velocity of flow at section 3. (i) Calculate the specific energy at section 3. (j) Compute the critical depth for the flow at section 2. (k) What is maximum possible discharge per unit width for a flow with specific energy entered at question (i). (This question is a general question, not related to the settling basin.)
REVISION EXERCISE NO. 8
A broad-crested weir is to be designed to operate with critical flow above its crest and no down- stream control. The maximum discharge capacity will be 135 m^/s. The weir crest will be 2.5 m above the natural channel bed and its width will be 15 m. (Assume the channel to be prismatic, smooth and rectangular.)
Revision exercise No. 9 117
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Fig. R.6 Sketch of the set- tling basin outlet.
Sketch the free-surface profile between sections 1 and 3 located respectively upstream and downstream of the weir. (You might have to modify it after your calculations in the following questions.) Sketch also the variation of the pressure with depth at sections 1 and 3. Sections 1 and 3 are located far enough from the throat for the velocity to be essentially horizontal and uniform.
Calculate (a) the upstream flow depth at maximum flow rate, (b) the upstream specific energy at maximum flow rate, (c) the specific energy at the crest at maximum flow rate, (d) the down- stream specific energy at maximum flow rate, (e) the downstream flow depth at maximum flow rate, (f) At maximum flow rate, what will be the longitudinal free-surface profile? (g) Calculate the horizontal component of the sliding force acting on the broad-crested weir at maximum flow rate, (h) What is maximum possible discharge per unit width for a flow with specific energy entered at (b) (i.e. upstream). (This question is a general question, not related to the broad-crested weir.) REVISION EXERCISE NO. 9
The backward-facing step sketched in Fig. R.7 is in a 1.3 m wide channel and the discharge is 1901/s. The bed of the channel is horizontal and smooth (both upstream and downstream of the step). The flow direction is from section 1 to section 2.
(a) Show in Fig. R.7 all the forces acting on the control volume contained between sections 1 and 2. Show also your choice for the positive direction of distance, of velocity and of force.
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Fig. R.7 Sketch of stepped channel and overfall.(b) Compute and give the values (and units) of the specified quantities in the following list: spe- cific energy at sections 1 and 2, assumption used in the answer, depth of flow at section 2, the force acting on the step and the direction of the force (i.e. upstream or downstream).
Then plot the correct fi-ee-surface profile in Fig. R.7.
(c) A fi-ee overfall is located at a short distance downstream of the backward-facing step.
Considering now the overfall at the downstream end of the channel (Fig. R.7). What is the specific energy at section 3? What is the flow depth at section 3? Plot in Fig. R.7 the fi*ee- surface profile downstream of the step (i.e. between sections 2 and 3) AND downstream of the overfall (i.e. downstream of section 3). Sketch in Fig. R.7 the variation of the pressure with depth at section 3.