Given a stream function for a cylinder in a uniform flow with circulation: a) Sketch the flow pattern in terms of streamlines.

Transcription

1 Question Given a stream function for a cylinder in a uniform flow with circulation: R Γ r ψ = U r sinθ + ln r π R a) Sketch the flow pattern in terms of streamlines. b) Derive an expression for the angular position, θ, of the front and rear stagnation points in terms of the circulation, Γ, and the freestream velocity U. c) The circulation around a spinning cylinder causes the stagnation points to be displaced by 30 o downwards. i) Calculate the magnitudes of the flow velocity at the top and bottom of the cylinder, and the corresponding surface pressure coefficients. ii) Using the Kutta-Joukowski Theorem, evaluate the lift coefficient per unit span. Page of 7

2 Question A thin aerofoil has a circular arc camber line with a maximum camber of.5%. The camber line can be approximated by: y c ax x = c c where a is a constant. Given that the general loading on an aerofoil is given by: k + cosθ = 0 sin sinθ n= ( θ ) U A + A n nθ where: A 0 = α π π 0 dy dθ dx and: A n = π π 0 dy cos nθdθ dx a) Determine the parameters A 0 and A n (n= to infinity) [9 marks] b) Derive the equation for the lift coefficient C L in terms of A 0 and A. [9 marks] c) Calculate the magnitude of constant a, and the zero lift angle of attack, and then sketch the lift curve. [7 marks] Hint: Remember the transformation between Cartesian and polar coordinates. Page of 7

3 Question 3 A wing of span, b, and a planform area, S, with an elliptic lift distribution, can be described by the equation below: y Γ ( y) = Γ0 s where y is the spanwise ordinate and s is the span (maximum y). The wing is to be modelled by a horseshoe vortex of strength, Γ 0, and span, b 0. i) Show that for the horseshow vortex model: πb b 0 = 4 and b C 0 0 L = Γ V S ii) Use the Biot-Savart Law for a straight vortex segment filament, given below, to show that the vertical velocity, w, induced on the plane of symmetry of the horseshoe vortex is: Γ w( x) = 0 + πb0 x + ( b ) 0 x where x is the longitudinal distance from the wing quarter chord (positive downstream). iii) Sketch the variation of w with x. Page 3 of 7

4 Question 4 For a laminar boundary layer on a flat plate at zero angle of incidence, the velocity profile is assumed to have a profile: u U y y = δ δ a) Obtain the relationships for displacement thickness, δ *, and momentum thickness, θ, in terms of the boundary layer thickness, δ. b) Determine the expression for: i) how the boundary layer thickness, δ, varies with the distance from the plate leading edge, x. ii) the variation of momentum thickness, θ, with distance from the plate leading edge, x. Note that: dθ C f = dx Page 4 of 7

5 Question 5 a) Describe the physical characteristics of a turbulent boundary layer, including a description of each physical layer, and a plot of the variation between u + and y + within a turbulent boundary layer. [9 marks] b) For a turbulent boundary layer on a flat plate at zero angle of attack, the boundary layer velocity profile can be modelled as: u y 7 = U δ Given that the local skin friction coefficient, C f, is: C f = Re 4 x where Reynolds number based on distance, x, from the fictitious origin of the turbulent boundary layer thickness, determine an expression for the variation of boundary layer thickness, δ, with x. [6 marks] Page 5 of 7

6 Question 6 The diamond-wedge high speed aerofoil shown in Figure Q6 below is to be tested at an angle of attack of α=5 o to a Mach 3.0 free stream air flow. a) Sketch the flow structure around the aerofoil showing all shock waves and expansion fans. [6 marks] b) Using the tables/charts provided, use Shock-Expansion theory to calculate the pressure ratios; p /p, p 3 /p etc. for each surface. [ marks] c) Calculate the theoretical lift and drag coefficients per unit span. [7 marks] α 0 o M Figure Q6 Page 6 of 7

7 Question 7 a) An aircraft is flying at a velocity of 40m/s at low altitude, where standard sea level conditions can be assumed (.03bar, 88K). Calculate: i) The Mach number the aircraft is flying at, ii) The pressure measured by a nose mounted pitot tube, iii) The static pressure measured at a point on the upper wing surface where the local velocity is sonic (M=.0) iv) The local velocity at this point. [ marks] b) A model of the same aircraft is tested in a wind tunnel which works by inducing atmospheric air through a smooth contraction into the working section and then into a low pressure system. If the freestream Mach number in the tunnel working section is to match the real flight value, calculate: i) The corresponding static pressure and freestream velocity in the working section. ii) The pressure measured by the nose mounted pitot tube. c) Given the difference in freestream velocity and model scale between flight and wind tunnel test, comment on the likely accuracy of the wind tunnel measurements. [3 marks] Page 7 of 7