1. As part of your role at a heavy-goods vehicle manufacturer, you have been asked to determine the drag caused by the boundary layer developing along a section of the trailer roof, which can be treated as having constant pressure, as the truck travels at 10 m/s. The velocity profile in this region is given by:
u- us
2/7 +A ++1)+B
where us is the velocity at the edge of the boundary layer. A and B are constants.
a) How many primary and auxiliary boundary conditions need to be satisfied? What values must A and B take such that these boundary conditions are satisfied?
[8 marks]
b) By evaluating the ratio between the displacement thickness (8*) and the momentum thickness (0) of the boundary layer, determine whether the boundary layer on the trailer roof is laminar or turbulent.
[7 marks]
c) The thickness of the boundary layer at locations x = 0.1 m and 0.5 m is measured using Pitot probe data to be 8 = 4 mm and 8 = 14.5 mm, respectively. Calculate the displacement and momentum thicknesses at these locations.
[4 marks]
d) The trailer roof has width 2 m. Use the von Karman integral momentum equation to determine the total drag contribution due to the boundary layer between x = 0.1 m and 0.5 m. Assume p = 1.3 kg/m3.
[6 marks]
2. You work for an aerospace company which is due to start testing a design in a supersonic wind tunnel. It is a two-dimensional, blow down wind tunnel with rectangular cross-section. The width of the wind tunnel is 100mm and the height at key points is provided in Figure Q2a. At x = 0 mm, the stagnation pressure is always 175 kPa and the stagnation temperature is 300 K. Prior to your test entry, you are given calibration data measured using a Pitot probe mounted in the centre of the test section during wind tunnel start up when the wind tunnel is empty. Your task is to evaluate whether the readings provided by the Pitot probe are sensible to check whether it is working correctly. You can neglect boundary-layer effects throughout this question.
a) During supersonic wind tunnel operation, the Mach number in the test section of the tunnel is 2.5. By considering the area ratio between the test section and the throat. determine the height, h, at the throat (x = 50 mm).
b) When the peak Mach number in the wind tunnel is 1, what is the stagnation pressure at the Pitot probe location, x = 600 mm?
c) When a normal shock wave is in the test section at x = 400 mm, what is the Mach number of the flow ahead of this shock wave? What is the stagnation pressure before and after the shock?
d) The wind tunnel starts at time t = 2 s and shuts down at time, t = 15 s. The normal shock wave crosses the Pitot probe at t = 10 s. You are given Figure Q2b, which shows the pressure measured by the Pitot probe as a function of time. Annotate the key timings on a copy of this graph and describe what these correspond to in terms of the tunnel flow. Also explain (with the help of sketches) whether the Pitot probe is functioning correctly.
[7 marks]
[4 marks]
[6 marks]
[8 marks]
3. You are on a placement in a chemical plant and have been asked to calculate the force on the walls of a circular pipe (radius, R = 0.02 m) using pressure tappings at the two stations (A and B) shown in Figure Q3. The chemical in the pipe has viscosity, u = 5 x 10-3 kg / m s, and density, p = 660 kg/m3. The pressure tappings are separated by a distance, L = 20 m, and are connected to a manometer filled with water (p = 1000 kg/m3) with an indication, Ah = 40 mm.
a) Is the pressure at B higher or lower than the pressure at A? For g = 9.8 m/s2, calculate the magnitude of this pressure change.
[7 marks]
b) The flow in the pipe is laminar and fully developed, so can be assumed to follow a Hagen-Poiseuille distribution:
u = 4 4u dx 1 ªP -(r2 - R2) What is the velocity at point 1 in section B, which is located 0.01 m away from the pipe axis? [5 marks]
c) What is the maximum velocity across the cross-section at station B? What is the maximum velocity at station A?
[7 marks]
d) Recall that the critical Reynolds number for transition in pipes occurs at Reynolds numbers based on the diameter in the range Rep = 2300 - 3500. Explain whether our assumption of laminar flow is reasonable.
[6 marks]
A
L
B
0.01 m
chemical
Y
Ah
water
Figure Q3: Chemical flow through a pipe
1
R
4. You are responsible for the design of the convergent nozzle for a turbojet engine fitted to a civil aircraft, as shown in Figure Q4. During operation at an altitude of 10,000 m, the aircraft flies at 300 m/s and the flow exhausts to atmosphere, where the pressure is 26.5 kPa and the temperature is 223 K. In this question, you should treat the flow as quasi-one-dimensional and inviscid.
a) The radius of the nozzle satisfies the equation R(x) = 2-0.44 x2 where both x and R are expressed in metres. What is the area ratio between the nozzle inlet (x = 0.0 m) and the nozzle exit (x = 1.5 m)?
b) The turbine exit temperature is the stagnation temperature, To = 400 K, of the flow in the nozzle. Similarly, the turbine exit pressure is the stagnation pressure, Po, of the flow in the nozzle. When the turbine exit pressure is 35 kPa, what is the Mach number at the nozzle exit?
c) In a different scenario, the turbine exit pressure is 70 kPa. What is the Mach number of the flow at the nozzle exit? Also determine the Mach number at the nozzle inlet.
d) The turbo-machinery team within the company have asked you to specify what exit pressure of the turbines results in perfectly expanded flow, with minimal expansion and compression waves outside the nozzle. What is this turbine exit pressure?
[6 marks]
[6 marks]
[7 marks]
[6 marks]
combustion
compressor
chamber
turbine
nozzle
p = 26.5 kPa T = 223 K
capture streamtube
p = 26.5 kPa T = 223 K
Ro
nozzle inlet,
nozzle exit,
x = 0.0 m
x = 1.5 m
Figure Q4: Diagram of turbojet engine
100 m
