MP4A01

NANYANG TECHNOLOGICAL UNIVERSITY

SEMESTER 1 EXAMINATION 2009-2010

MP4A01 - AERODYNAMICS November/December 2009 Time Allowed: 2 ½ hours INSTRUCTIONS 1. This paper contains FIVE (5) questions and comprises FOUR (4) pages.

2. Answer ALL FIVE (5) questions.

3. All questions carry equal marks. 4. Relevant formulae are provided in the Appendix on page 4. 5. This is a CLOSED BOOK examination. 1 (a) A two-dimensional source and a sink of same strength Λ are placed at (a,0) and (-a,0)

relative to a Cartesian coordinate system centered at (0,0). Assuming incompressible, inviscid and irrotational flow, show that the streamlines corresponding to the resulting flow pattern in the x-y plane are coaxial circles passing through the locations of the source and sink and for a given value of streamfunction ψ , state the radius of the circle and the location of its centre.

(10 marks)

(b) The streamlines of a two-dimensional incompressible inviscid flow in a two-dimensional Cartesian x-y plane are defined by the equation 2 ( 1)y x x C= + + where C is an arbitrary constant.

(i) If p(x,y) is the static pressure field and ρ is the density of the fluid, calculate the

value of p

ρ

∇r

at the point (2,3).

(6 marks)

(ii) Is the flowfield irrotational? If so, find the velocity potential. (4 marks)

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MP4A01

2 (a) What is the difference between the centre of pressure and the aerodynamic centre of an airfoil? On the basis of incompressible thin airfoil theory, explain the conditions under which the locations of the centre of pressure and the aerodynamic centre coincide.

(5 marks) (b) The mean camber of an airfoil of chord length c inclined at an angle α to a low speed

uniform airflow at V is defined in the Cartesian (x-y) system as ∞3 21 4 7 3 , 0 1

3y x x x xc c c c c

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + ≤⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠≤

Using incompressible thin airfoil theory, calculate

(i) the rate of change of the coefficient of moment about the leading edge with

angle of attack, (5 marks)

(ii) the rate of change of the position of the centre of pressure along the chord with

angle of attack, and (5 marks)

(iii) the coefficient of moment about the aerodynamic centre.

(5 marks)

3 (a) If the flow velocities upstream and downstream of a normal shock wave in adiabatic

compressible flow are respectively and , show that 1U 2U *1 2U U a= where is the

critical speed of sound.

*a

(8 marks)

(b) The upper and lower surfaces of an airfoil of chord length c immersed in a uniform supersonic free stream flow at Mach number M∞ and at an angle of attack α are

defined in the Cartesian coordinate system by the equations U

yc

⎛ ⎞⎜ ⎟⎝ ⎠

and L

yc

⎛⎜

⎞⎟

⎝ ⎠

respectively as

8 1 , 0 1

4 1 , 0 1

U

L

y x x xc c c c

y x xc c c

κ

κ

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

xc

= − ≤⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞

≤

= − − ≤⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠≤

where is a design constant. Using Ackeret’s supersonic thin airfoil theory, estimate κ (i) the coefficient of moment about the aerodynamic centre, and

(6 marks)

(ii) the wave drag polar of the airfoil. (6 marks)

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MP4A01

4 (a) Three identical airplanes of equal weight are flying abreast and aligned to each other

straight and level at the same altitude in formation flying at constant speed V . Each airplane has a wing span of length b and the wing tips of each airplane wing are separated by a distance s. By representing each airplane wing by a single horse shoe vortex system as in Prandtl’s lifting line theory,

∞

(i) obtain an expression for the total induced drag experienced by the middle

airplane as a result of this formation flight, and (4 marks)

(ii) for the case when the wing tip separation s= b/2 ,show that the power to

overcome the total induced drag experienced by the middle airplane is reduced by 25% compared to the power to overcome the induced drag if it was flying alone and not in formation flight.

(4 marks) (b) A three-dimensional wing of span b is immersed in an incompressible airflow at

uniform speed V and angle of attack ∞ α . Assuming a general distribution of circulation (( 2 0.05sin 0.01 2y bV )) sinθ θ∞ +Γ = and using Prandtl’s lifting line theory, estimate

(i) the spanwise variation of the induced angle of attack at the lifting line,

(4 marks) (ii) the planform efficiency corresponding to the wing, and

(4 marks) (iii) the coefficient of rolling moment in terms of wing aspect ratio.

(4 marks) 5 (a) Explain, with relevant sketches, the mechanism and cause of tip stalling in swept back

wings and briefly outline various methods and the rationale for each of these methods used by aircraft designers to mitigate the adverse effects of swept back wing tip stalling.

(6 marks) (b) Explain the term Mach number of drag divergence and state its significance in

transonic aircraft aerodynamic design considerations. Explain the rationale of all possible aerodynamic technologies available to an aircraft designer to delay the Mach number of drag divergence of transonic aerodynamic configurations.

(7 marks)

(c) Explain briefly with sketches, the differences in the aerodynamic effects of activating a split flap, a Fowler flap and a jet flap. Also show the aerodynamics effects on typical plots of the wing lift characteristics and wing drag polars corresponding to a plain cambered aircraft wing installed with these high lift devices.

(7 marks)

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MP4A01

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APPENDIX A1. INCOMPRESSIBLE THIN AIRFOIL THEORY You may assume that for the camber problem defined by the incompressible thin airfoil theory, the distribution of vorticity, ( )γ θ , along the chord of a thin airfoil is given by

01

1 cos( ) 2 sin( )nsin n

n

V A Aθγ θ θ⎨ ⎬⎩ ⎭θ

∞

∞=

⎧ + ⎫⎡ ⎤= +⎢ ⎥⎣ ⎦∑ where

0 0 0

1 dyA ddx

π

α θπ

= − ∫ and

0 0 0

2 cosndyAdx

π

n dθ θπ

= ∫ for n = 1,2,….; 0(1 cos )2cx θ= − , V∞ is the free stream

velocity,α is the angle of attack and c is the airfoil chord length. A2. SUPERSONIC THIN AIRFOIL THEORY The expression for the coefficient of pressure C at a point on an airfoil surface immersed in

a uniform free stream Mach number

p

M∞ which has a slope dy at that point, within the

framework of supersonic thin airfoil theory is

dx

( ) 1/222 1pdyC Mdx

−

∞⎛ ⎞= ± −⎜ ⎟⎝ ⎠

.

A3. PRANDTL’S LIFTING LINE THEORY You may assume that the general circulation distribution along the wing span of an arbitrary

finite wing is given as 1

( ) 2 sinn

nn

bV A nθ θ=∞

∞=∑Γ = corresponding to the transformation

cos2by θ= where y is the spanwise direction, b is the wing span and V∞ is the uniform free-

stream velocity. The expression for downwash estimated from Prandtl’s lifting line theory at

the spanwise location y = 0y is ( )( )2

0 20

/14

b wing

b

d dyw y dy

y yπ −

Γ⎡ ⎤= − ⎢ ⎥

−⎢ ⎥⎣ ⎦∫

A4: RELEVANT INTEGRALS FOR USE

000 0

0 0 0

0

sincos sin sin and coscos cos sin cos cos

for 1sin sin 2

0 for 1

nn nd d

nn d

n

π π

π

nπ θθ θ θθ θ π θθ θ θ θ θ

πθ θ θ

= =− −

⎧ =⎪=⎨⎪ ≠⎩

∫ ∫

∫

−

End of Paper

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