MP4005

NANYANG TECHNOLOGICAL UNIVERSITY

SEMESTER 2 EXAMINATION 2013-2014

MP4005 – FLUID DYNAMICS

April/May 2014 Time Allowed: 2 ½ hours INSTRUCTIONS 1. This paper contains FOUR (4) questions and comprises FIVE (5) pages. 2. Answer ALL FOUR questions. 3. All questions carry equal marks. 4. This is a CLOSED-BOOK examination.

5. The Compressible Gas Tables comprising of ELEVEN (11) pages are enclosed. 1(a) A convergent-divergent nozzle has a throat area of 0.02 m2 and an exit area of 0.05

m2. As the back pressure is reduced progressively from the atmospheric value, air starts to flow from the atmosphere into the nozzle. The pressure and temperature of air in the atmosphere are 101 kPa and 300 K respectively.

For air, 1.4γ = and 287 J/kg KR = ⋅ .

(i) When the back pressure is 99 kPa, the flow is not choked and is fully subsonic and isentropic. Determine the Mach number at the exit, the Mach number at the throat and the mass flow rate of air.

(ii) When the back pressure is 85 kPa, determine whether the flow is choked and the

Mach number at the throat of the nozzle. Calculate the mass flow rate of air.

(17 marks)

(b) Consider the Fanno line flow through a circular duct of constant diameter. The flow at the duct inlet is subsonic. For a certain duct length, the flow at the duct outlet is sonic. What would happen to the flow if the duct length is made longer? Give physical reasons to justify your answer.

If the duct diameter is 0.06 m, the duct length is 2.93 m, the duct friction factor is 0.013 and the Mach number at the duct inlet is 0.37, determine the Mach number at the duct outlet. What would be the duct length if the Mach number at the outlet is 1 while the Mach number at the inlet remains at 0.37?

(8 marks)

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0.2 m

0.2 m

concave, radius = 0.1 m

convex, radius = 0.1 m

U = 15 m/s

2 (a) The velocity profile in a turbulent boundary layer with zero pressure gradient may be

expressed as 1/9u y

U = δ

where u is the boundary layer velocity, U is the free-stream

velocity, y is the normal distance from the surface and δ is the boundary layer

thickness. The wall shear stress wτ is given by 1/4

20.0233wUU

−δ τ = ρ ν where ρ is

the fluid density and ν is the kinematic viscosity. Determine an expression for the local skin friction coefficient in terms of the Reynolds number using the momentum integral equation. State any assumptions made.

(11 marks)

(b) For flow over a flat plate aligned with the stream-wise direction, use the momentum equation to show that the skin friction per unit width of the plate, fD , is given by the expression 2

f LD U= ρ θ where ρ is the fluid density, U is the free-stream velocity and

Lθ is the momentum thickness at the trailing edge of the plate. Does this expression apply to both laminar and turbulent boundary layers?

(6 marks)

(c) The anemometer as shown in Figure 1 (top view) consists of two identical hemispherical cups connected by a bar. The anemometer is hinged at the midpoint of the bar. The radius of the cups is 0.1 m and the length of the bar is 0.4 m. The anemometer is placed in the wind blowing at a velocity of 15 m/s. The total drag coefficients, DC , of the convex surface and concave surface of the cups are 0.34 and 1.33 respectively. Determine the turning moment of the anemometer. Take air density to be 1.2 kg/m3. State any necessary assumptions.

(8 marks)

Figure 1

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3 (a) A velocity field is 4 2 2 4 3 3( 6 ) ( )V A x x y y i B x y xy j= − + + −

where 3 12m sA − −= , B is a constant and the coordinates are measured in metres, such that the velocity is in m/s.

(i) Find B for this flow field to be an incompressible flow field. (ii) Obtain the stream function ( , )x yy of this flow field. (iii) Find ω . Is this an irrotational flow field? (iv) If the answer to part 3(a)(iii) is yes, determine the potential function ( , )x yφ , if

the answer is no, explain why. (v) For this flow field, determine the flow rate between the two streamlines passing

through the points (1,1)C and (2,2)D . What is the direction of the flow?

Note: ( )0.5 ,Vω = ∇×

uyy∂

=∂

, and vxy∂

= −∂

(13 marks)

(b) Oil of density ρ and dynamic viscosityµ , drains steadily down the side of a vertical plate, as shown in Figure 2. After a development region near the top of the plate, the oil film becomes independent of z and has a constant thickness .δ Assume that

( )w w x= only and that the atmosphere offers no shear resistance to the surface of the

film, i.e., 0xzw ux z

τ µ ∂ ∂ = + = ∂ ∂ at .x δ=

(i) Solve the Navier-Stokes equation and show that ( )2( ) 2 .2

gw x x xρ δµ

= −

(ii) Find the flow rate per unit width, 0

.q wdxδ

= ∫

Figure 2

Note: 2 2 2

2 2 2zw w w w p w w wu v w gt x y z z x y z

ρ ρ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + + = − + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (6 marks)

Note: Question 3 continues on page 4.

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(c) Air flowing past a wall encounters a step of height h = 10 mm and the leading edge of

the wall is profiled to avoid sharp changes in flow direction as shown in Figure 3. The flow may be represented by the upper half-plane of a uniform flow of velocity U parallel to the axis and a source of strength m, i.e., a half-body. If the velocity of the airstream is 40 m/s, determine

(i) the distance b that the source is located behind the leading edge of the step, (ii) the strength of the source, and (iii) the horizontal and vertical velocity components (i.e. u and v) at a point

(0,5 mm)A on the step.

Figure 3

Note: For a half body, ( )1/22 2cos ln or ln ,2 2m mUr r Ux x yφ θ φπ π

= + = + +

( ) .sin

br π θθ−

= For a source, and 0,2rmV V

r θπ= = where m > 0 for source, and

( ) 1ln .d xdx x

=

(6 marks)

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4 (a) A vertical axis tank is conical in shape whereby the diameter increases uniformly

from 1.5 m at the base to 3.5 m at a height of 5 m. The water in the tank can be reduced by means of a 100 mm diameter pipe of length 5 m and friction factor 0.008, with the height of the pipe 4 m below the base of the tank. Assume the effect of the orifice to pipe connection can be represented by a separation loss with a k value of 2. Determine the time required to reduce the water in the tank by 3 m if the tank is full initially.

Note that 2 2 2 2

1 1 2 21 22 2 2 2

p V p V L V Vz z f kg g g g D g gρ ρ+ + = + + + +

and

2

1

zs

i oz

At dzQ Q

=−∫

(13 marks)

(b) A centrifugal pump is designed to deliver water at 30 l/s and the inlet velocity has no tangential component. The pump has the following dimensions: Parameter Inlet Outlet Radius, r (mm) 75 150 Blade width, b (mm) 7.5 6.25 Blade angle, β (deg) 25 40 (i) Draw the inlet and outlet velocity diagrams. (ii) Determine the rotating speed of the pump. (iii) Determine the outlet absolute flow angle measured relative to the normal

direction. (iv) Evaluate the theoretical head developed by the pump.

(12 marks)

End of Paper

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