a) Derive general expressions for the stream function Ψ and the velocity potential function φ for the combined flow. [12 Marks]

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1 Question 1 A horizontal irrotational flow system results from the combination of a free vortex, rotating anticlockwise, of strength K=πv θ r, located with its centre at the origin, with a uniform flow of velocity U in the x-direction. a) Derive general exressions for the stream function Ψ and the velocity otential function φ for the combined flow. [1 Marks] b) Sketch the streamline attern for the combined flow, and state what kind of flow might be redicted by this model. c) In a secific instance, the uniform velocity comonent U=3.0m/s, the strength of the vortex K=7.0m /s and the density of the fluid is ρ =1000kg/m 3. Calculate the ressure difference between a oint on the ositive x-axis and a oint on the ositive y-axis, each 0.5m from the origin. (Assume gravity acts in the z-axis). [8 Marks] Page 1 of 7 AE303 Aerodynamics May 05.doc

2 Question Thin aerofoil theory shows that the vortex strength distribution reresenting a symmetrical aerofoil is: γ = α U ( 1 cosθ ) sinθ where θ is related to x by x=(c/)(1-cosθ ), and non-dimensional load distribution is given by C P = γ/u. a) Integrate the load distribution to show that the lift coefficient, C L, is given by C L =πα, and the itching moment about the leading edge, Cm LE, is given by Cm LE =-πα/. [15 Marks] b) Obtain an exression for C P in terms of x, and sketch the load distribution commenting on the values at the leading and trailing edges. [8 Marks] c) Comment on the alicability of thin aerofoil theory, and the limitations of the method. [ Marks] Page of 7 AE303 Aerodynamics May 05.doc

3 Question 3 For a finite san wing: a) Describe how the lift and drag forces generated by a finite san wing differ from the theoretical case of an infinite san wing, and the hysical mechanism for these differences. b) A wing can be reresented by two horseshoe vortices as shown below in figure Q3. 0.b Γ/4 b 0.3b 3Γ/4 c By calculating the downwash at the oint and relating the geometric and effective incidences at that oint, show that the lift-curve sloe is given by: dc L dα πar = AR where AR is the asect ratio. [18 Marks] c) Comment on the significance of the value of dc L /dα when AR. [ Marks] Page 3 of 7 AE303 Aerodynamics May 05.doc

4 Question 4 An air flow at Mach number.5 and ressure 50 kn/m enters a divergent section of a duct having an inlet area of 50mm. A normal shock occurs at the exit where the Mach number is 4.0 and the temerature immediately ustream of the shock is 10K. a) Determine the mass flow rate, the exit area of the duct and the air ressure immediately downstream of the shock. [18 Marks] b) Sketch the general trends of ressure, temerature, density, velocity and Mach number, total ressure and total temerature through the divergent duct. Assume reversible adiabatic flow, other than through the shock. [9 Marks] The following equations may be used, where aroriate, and the tye of each equation used must be stated, e.g: steady flow continuity equation. 1 T = T 1 γ ( γ 1) γ 1 T 1 + M = constant ( 1 + γm ) = constant M = + γm1 ( γ 1) M1 ( γ 1) Page 4 of 7 AE303 Aerodynamics May 05.doc

5 Question 5 The suersonic flow ast a wedge bilane wing is shown in figure Q5 below. A l α α B M h C α D α E a) Comlete the sketch the suersonic flow field (drawing in your exam work book) by adding all the other waves emanating from oints ABCD and E and label them as shock or exansion waves. Also draw some streamlines above, between and below the wings. b) Using linearised (Ackeret) theory, find the lift and drag coefficients (based on the area of one wing) and the lift to drag ratio, in terms of wedge angle, α, and free stream Mach number, M, for the bilane wing. State all assumtions in Ackeret s theory. [8 Marks] c) Calculate the correct sacing, h, for the wings in terms of the free stream Mach number and the dimension, l. [ Marks] d) Assume the wings are now moved further aart so that they do not interfere. Calculate the sacing required to achieve this and the corresonding values of C L and C D. Comment on the merits of the suersonic wedge bilane arrangement. [10 Marks] Page 5 of 7 AE303 Aerodynamics May 05.doc

6 Question 6 Laminar flow occurs ast a flat solid boundary. The velocity distribution may be aroximated by the exression: u U y y = δ δ where u is the velocity at a distance y from the boundary, U is the free stream velocity and δ is the boundary layer thickness. a) Show that this distribution satisfies the boundary requirements for a boundary layer. b) Show that the boundary layer thickness varies with the distance x from the leading edge of the boundary and the Reynolds number Re x =ρux/µ according to the exression δ = 5.48x(Re x ) -1/ [0 Marks] The following relationshis may be assumed without roof. Local wall shear stress: τ w = ρu dθ dx where the momentum thickness: u θ = δ 1 0 U u dy U Page 6 of 7 AE303 Aerodynamics May 05.doc

7 Question 7 For a turbulent boundary layer flow: a) exlain the following terms, including the equations relating u + and y + : i) laminar sublayer ii) inner layer iii) outer layer Include in your answer a sketch the grah relating u + and y + through a turbulent boundary layer [15 Marks] b) Assuming that for flow through a long engine intake the friction coefficient is given by C F = 4C f = 0.316(Re) -1/4, that the wall shear stress is indeendent of ie radius R and that the ratio of local velocity u to freestream velocity U does not change with freestream velocity, show that: u U 1 y 7 R [10 Marks] Note that: + u u =, u τ y ρu µ + τ y = where wall friction velocity is u τ = τw ρ Examiner : Dr. S. Prince External Examiner : Professor D.I.A. Poll Page 7 of 7 AE303 Aerodynamics May 05.doc