ME2135
NATIONAL UNIVERSITY OF SINGAPORE
ME2135 – FLUID MECHANICS II
(Semester II : AY2009/2010)
Time Allowed : 2.5 Hours
INSTRUCTIONS TO CANDIDATES:
1.
This examination paper contains FIVE (5) questions and comprises SIX (6) printed pages.
2.
Answer ALL FIVE (5) questions.
3.
Programmable calculators are NOT allowed for this examination.
4.
This is a CLOSED-BOOK EXAMINATION with authorized materials: Students are allowed to bring TWO (2) A4 size sheets of notes/formulae written on both sides.
PAGE 2
ME2135
QUESTION 1
Consider a centrifugal pump A of diameter D A = 20 cm, used for pumping water, having the following characteristic curve, when it is run at N = 100 rad/sec.
H P = H 0 − CQ where H0 = 5 m and C = 10 s 2/m5. 2
The efficiency is given by
η = 0.9 cos[2π (Q − 0.4)].
Consider a geometrically similar pump B with DB = 10 cm running at N = 200 rad/sec. (a)
Find the non-dimensional form of the characteristic curve. (4 marks)
(b)
What will be the volume flow rate Q B against a head of H = 3m? (5 marks)
(c)
If this pump is run at highest efficiency, what is the head developed by this smaller pump? (6 marks)
(d)
How much shaft power is required to run the pump B at these conditions? (5 marks)
3
You may take density of water ρ water = 1000 kg/m and the standard acceleration of gravity on 2 Earth to be g = 9.81 m/s .
PAGE 3
ME2135
QUESTION 2
Consider an ideal fluid flowing past a stationary cylinder with its axis perpendicular to the direction of flow and located at the origin of a coordinate system. The stream function (in polar coordinates) is given by:
a 2 ψ = −U ∞ r sin θ 1 − 2 r (a)
, where a is the radius of the cylinder.
Determine the pressure distribution p(θ) along the surface of the cylinder. The free stream pressure can be taken to be p∞. Angle θ and the coordinate system is as shown in the FIGURE 1 below: (6 marks)
Figure 1
(b)
In the actual (real) flow past a cylind er, separation occurs near the shoulder or θ ~ + o 90 as shown in the Figure 1 above, leaving a constant pressure wake on the back side of the cylinder. Calculate the drag force, by assuming the stati c pressure in the wake is constant at the potential flow value where it separates. Compare your calculated drag coefficient C D value with the experimentally measured value of about C D ~ 1.0 in the moderate Reynolds number range and comment on the deviation, if any. The following formulae may be useful: 2 ∫ sin θ cosθ d θ =
sin θ 4
−
1 12
sin 3θ .
The drag coefficient CD per unit span is defined as : C D
=
Drag force 1 2
ρ U ∞2 a (14 marks)
PAGE 4
ME2135
QUESTION 3
(a)
2
2
Consider the flow field which has the velocity potential φ = (x - y ). (i)
Find the velocity components. (1 marks)
(ii)
Find the velocity vector at (x,y) = (2,2). (1 marks)
(iii)
Find the stream function ψ(x,y) for the flow. (2 marks)
(iv)
Determine the volume flow rate between points (1,1) and (2,2)? (2 marks)
(b)
(v)
Find the equation describing the streamline passing through (x,y) = (2,3). (2 marks)
(vi)
How long will it take a fluid element to move from (x,y) = (0,2) to (x,y) = (0,1) ? (2 marks)
3
-5
2
Air (of density = 1.20 kg/m and kinematic viscosity = 1.5 x 10 m /s) flows over a smooth flat plate of 2.0 m length and 1.5 m width, in the long side direction. If the 5 transition from laminar to turbulent boundary layer takes place at Rex,tr = 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 both sides of the plate for this condition. Note: Drag coefficient C D
=
D 1 2
2
ρU A
=
D/unit width 1 2
2
ρU x −1/2
For laminar boundary layer: CDl = 1.328 Rex
− 1/5
For turbulent boundary layer: CDt = 0.072 Rex All symbols have their usual meaning.
(10 marks)
PAGE 5
ME2135
QUESTION 4
(a)
Two step bearing designs are shown in Figure 2 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 q x is given by:
q x =
Uh 2
−
h 3 dP 12μ dx
where the symbols used have their usual meaning. (9 marks)
Figure 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 Figure 3 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 ∇ 2 V
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 coordinate 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)
PAGE 6
ME2135
Figure 3
QUESTION 5
(a)
The velocity profile of a boundary layer flow for y ≤ δ and u = U for y >
u described by :
U
= 2
y δ
y − 2 δ
δ is
3
Investigate whether the boundary layer flow is attached, separated, or on the point of separation, and comment whether it is laminar or turbulent. Then, by using the von Karman integral equation, obtain the expression of the boundary layer thickness development δ/x in terms of the local Reynolds number Re x. All the symbols have their usual meaning. (12 marks)
(b)
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)
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