Name: Matric Number:________________ Number:________________
Department of Mechanical Engineering National University of Singapore
ME 2135E – Fluid Mechanics II – Take Home Test . . . . .day, . . th April 2011
Time Allocated: 2 Hours to 2 Hours and 30 Minutes
Instructions to Candidates: Candidates: 1. 2. 3. 4.
Answer ALL Questions. This Test is to gauge your understanding understanding on the Viscous Flow Part Part of the module. You may write your name and matric number on the top right right corner of this page. You can refer to the the Lecture Notes and any other other materials but try to complete within within the time allocated. You can still work on the Questions beyond the specified time but please record the total time.
Question 1
A hydrostatic circular step thrust bearing is to support a rotating load shaft of 50 kN. The bearing consists of a flat circular pad of 100 mm diameter and a deep central recess of 50 mm diameter. The shaft rotates at 1200 rpm and is lubricated by a lubricant with a viscosity of 2 0.02 Ns/m which is pumped at a constant flow rate into the recess. If the lubricant film thickness is to be maintained at 0.1 mm, determine (i) the recess pressure, (ii) the lubricant flowrate, and (iii) the total power loss of the bearing. (10 marks)
Question 2 (a)
Two step bearing designs are shown in the figure below. Based on the information available from the figure and using “infinitely wide” bearing theory, which design will give higher load capacity ? The pressure distributions in the lubricant film may be assumed to be triangular and the lubricant flow rate per unit width qx is given by: qx
Uh 2
h 3 dP 12μ dx
Uh 2
where the symbols used have their usual meaning. (9 marks)
Question 2 (b)
Two immiscible, incompressible, viscous liquids having the same densities (ρ) but different viscosities (μ1 and μ2) are contained between two infinite, horizontal, parallel plates as shown in the figure below. The bottom plate is fixed and the upper plate moves with a constant velocity U as shown. Assuming parallel, laminar flow, and by applying the equations: DV Dt
and
f
V
1 ρ
P
ν 2V
0
where f is the body force vector per unit mass and other symbols have their usual meaning, express the velocity at the interface in terms of U , μ1 and μ2 by using the co-ordinate system shown in the figure. Neglect body force due to the liquid weight, and the liquid motion is caused entirely by the movement of the upper plate, that is, there is no pressure gradient in the direction of velocity. Note: the liquid velocities are the same at the interface, and so are the shear stresses.
(11 marks)
Question 3
(20 marks)
Question 4 (a)
The velocity profile for a laminar boundary layer is to be approximated by the expression:
u U
a
bη
c η m where η
y δ
State three boundary conditions applicable to the velocity profile and evaluate a, b and c for the situation where m = 1.5. (10 marks)
Question 4 (b) 3
-5
In the flow of air (of density ρ = 1.2 kg/ m and dynamic viscosity μ = 1.8 x 10 kg/ms) past a flat plate as shown below, the wall shear stress is to be determined at position x from the plate leading edge by a floating element (a small area connected to a strain gauge force measurement). At x = 2 m, the element indicates a shear stress of 2.0 Pa. Assuming turbulent boundary layer flow with one-seventh power law velocity profile from the leading edge, estimate: (a) the free-stream velocity U , (b) the boundary layer thickness δ at the element, and (c) the flow velocity at 5 mm and at 5 cm above the element. Please check on the validity of using the assumption: one-seventh power law velocity profile ! (20 marks)
Question 5 (a)
Compare the differences between a laminar and turbulent boundary layer with the help of sketches of velocity profiles. Then explain which boundary layer is more resistant to flow separation and state the condition for flow separation to occur. Based on your answer, explain why golf balls have dimples. (8 marks) Question 5 (b) -5
2
Air (of kinematic viscosity = 1.5 x 10 m /s) flows over a smooth flat plate of 2.0 m length and 1 m width. If the transition from laminar to turbulent boundary layer takes place at Rex,tr = 5 5 x 10 , what will be the velocity of the air so that the boundary layer along the flat plate to remain laminar up to half of its length ? Then estimate the total friction drag on one side of the plate only, for this condition. D D/unit width Note: Drag coefficient C D 2 2 1 1 2 ρU A 2 ρU x For laminar boundary layer: CDl = 1.32 Rex For turbulent boundary layer: CDt = 0.072 Rex All symbols have their usual meaning. (12 marks) – END OF PAPER –
