ENGG S202F – Fluid Mechanics Autumn 2023
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ENGG S207F - Fluid Mechanics
Assignment 01
Instructions
. Answer ALL questions.
. You must clearly state any assumption you made in the solution.
. Submit a hardcopy of solutions to the lecturer on or before 23rd November, 2023.
. Submit asoftcopy of solutions to OLE onor before 23rd November, 2023.
. Plagiarism is a serious academic dishonesty. DO NOT copy solutions from other students, textbooks, or internet.
2. (a) The cross section of a dam is shown in Figure 2a.
i. Draw the pressure diagram for the upstream face of the dam
ii. Calculate the hydrostatic thrust per unit length of the dam
Figure 2a
[9 Marks]
(b) A rectangular opening in the vertical side of the dam of a reservoir containing water is 0.3 m × 0.6 m,with the 0.6 m side vertical. It is closed by means of a gate as shown in Figure 2b. The gate is hinged along its top edge and kept closed by a counterweight on a lever arm attached to the gate. The weight of the level arm can be neglected. The
water level in the reservoir is ‘h’ (in m) above the hinge of the gate.
i. What is the hydrostatic thrust on the gate
ii. The distance to the center of pressure from the hinge of the gate
iii. If the gate is about to open when the water level ‘h’ above the hinge of the gate is 0.3 m, what is the mass of the counterweight?
Figure 2b
[9 Marks]
(c) For a gate width of 2 m into the paper (Figure 2c), determine the force F required to hold the gate ABC at its location.
Figure 2c
[7 Marks]
3. (a) A metal sphere of volume Vm = 1 m3. Relative density Sm = 2 and fully immersed in water is attached by a flexible wire to a buoy of volume Vb = 1m3 and relative density Sb = 0.1 (Figure 3a). Calculate the tension T in the wire and volume of the buoy that is submerged.
Figure 3a
[7 Marks]
(b) It is said that Archimedes discovered his principle during a bath while thinking about how he could determine if King Hiero’s crown was actually made of pure gold. While in the bathtub, he conceived the idea that he could determine the average density of an irregularly shaped object by weighing it in air and alos in water. If the crown weighed 34.8
N in air and 31.9 N in water.
i. Determine if the crown is made of pure gold (the density of gold = 19300 kg/m3)
ii. Discuss how you can solve this problem withoutweighing the crown in water but
by using an ordinary bucket with no calibration for volume. You may weigh
anything in air.
[9 Marks]
(c) A closed tank 6 mlong, 2 mwide and 1.8 m deep initially contains water to a depth of 1.2 m. the top has an opening in the front part to have air space at the atmospheric pressure (Figure 3b). If the tank is given a horizontal acceleration at a constant value of 2.4 m/s2 along its length, calculate the total pressure force on the top of the tank.
Figure 3b
[9 Marks]
4. (a) the velocity vector in an incompressible flow is given by
V = (6xt + yz2 )i + (3t + xy2 )j + (xy 
2xyz 
6tz)k
i. Verify whether the continuity equation is satisfied
ii. Determine the acceleration vector at point A (1, 1, 1) at t = 1.0.
[9 Marks]
(b) Consider steady flow of air through the diffuser portion of a wind tunnel (Figure 4a). along the centerline of the diffuser, the air speed decreases from uenterance to uexit as sketched. Measurements reveal that the centerline air speed decreases parabolically through the diffuser.
i. Write an equation for centerline speed u(x), based on the parameters given
here, from x = 0 to x =L
Figure 4a
ii. Calculate the fluid acceleration along the diffuser centerline as a function of x and
the given parameters. ForL = 1.56 m, uenternace = 22.6 m/s, and uexit = 17.5 m/s,
calculate the acceleration atx = 0 and x = 1.0 m.
[6 Marks]
(c) Mass conservation is one of the main principles in fluid mechanics.
i. Define mass and volume flow rates. How are they related to each other
ii. Consider a device with one inlet and one outlet. If the volume flow rates at the inlet and at the outlet are the same, is the flow through this device necessarily
steady? Why?
iii. A garden hose attached with a nozzle is used to fill a 75-liter bucket. The inner diameter of the hose is 2.5 cm and it is reduced to 6 mm at the nozzle exit. If
the average velocity in the hose is 1.8 m/s, determine
I. The volume and mass flow rates of water through the hose
II. How long it will take to fill the bucket with water
III. The average velocity of water at the nozzle exit
[10 Marks]
2023-11-30