AER416-MT-1-W24-solution

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1 TORONTO METROPOLITAN UNIVERSITY Department of Aerospace Engineering AER 416 Flight Mechanics Mid-Term Examination - 1 March 2024 Instructions: 1. A Calculator and one (8.5in  11in) formula sheet (both sides) are permitted. 2. All questions have similar value, you are to attempt all problems. 3. Read all questions carefully. 4. You must show All of your work to receive credit. 5. You can write on the back of a page if you need more space. 6. No communication of any form is permitted during the exam. 7. The length of time given for the exam is 110 minutes. Question 1: __________/10 Question 2: __________/10 Question 3: __________/10 Total:_________/30 Name : _ Solution _________________________________ Student Number :______________________________ Section: ___________________________________

2  The first section is multiple choice. Select the response that best answers the given question. Record your response CLEARLY in the location provided. Ambiguous responses will not be accepted. Note, there is no penalty for an incorrect response. Note,  = 1.4, and the Universal Gas Constant for air is R = 287 J/kg K, and g o = 9.81 m/s 2 . 1) Air flowing over an aircraft wing has pressure. The pressure of the air exerts a force on the airfoil surface. The direction of this force is: a) Tangent to the airfoil surface b) Normal to the airfoil surface c) Parallel to the wing span d) None of the above 1)__B_____ 2) A pilot uses an aircraft's ailerons to cause rotation about which axis? a) the Lateral axis b) the Yaw axis c) the Longitudinal axis d) the Elevator axis 2)__C_____ 3) The determination of a Geopotential altitude assumes that, a) The air temperature is fixed at sea level b) The air density is variable c) The gravitational acceleration is variable d) The gravitational acceleration is constant at 9.81 m/s 2 3)___D____ 4) The Aerodynamic Centre of an airfoil is where: a) the moment is positive for all angles of attack in attached flow b) the moment is negative for all angles of attack in attached flow c) the moment is constant for all angles of attack in attached flow d) the moment is zero for all angles of attack in attached flow 4)___C_____ 5) An aircraft is flying at 70 m/s. At one point on the surface of its wing the velocity is measured to be 90 m/s. The flow is incompressible. What is the pressure coefficient ( C P ) at this point? a) C P = -0.653 b) C P = 0.934 c) C P = 1.28 d) C P = 0.395 5)___A_____ 6) A blow-down supersonic wind tunnel has a number of components that must function effectively for efficient operation. The component of the tunnel that allows the test section to operate below ambient air pressure is, a) The settling section b) The diffuser c) A properly designed nozzle d) A constant cross-section test area 6)____B____

3 7) Which dimensionless parameter is related to air viscosity? a) the Lift coefficient b) the Mach number c) the Reynolds number d) the Aspect ratio 7)___C_____ 8) An aircraft flies through air moving at 450 m/s. The static temperature and pressure of the free stream air is 255 K and 65 kPa respectively. What is the stagnation temperature of the air? a) 176 K b) 255 K c) 356 K d) 435 K 8)___C____ 9) On an airfoil, the lift coefficient C L = 0.2, and moment coefficient about the leading edge C MLE = - 0.1, what is the moment coefficient about the quarter chord point C Mc/4 ? a) C Mc/4 = - 0.05 b) C Mc/4 = - 0.25 c) C Mc/4 = - 0.15 d) C Mc/4 = 0.05 9)____ A ___ 10) For most airfoil profiles, the lift coefficient will reach a maximum value at the stall angle. The angle of attack at which stall occurs is a function of, a) Static pressure b) Mach number c) Reynolds number d) Aspect ratio 10)___C___

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4  An aircraft flies in steady, level flight at a speed of 250 m/s at an altitude such that the ambient air pressure and temperature are 60 kPa and 270 K respectively. The aircraft has a mass of 6000 kg, has an Oswald efficiency factor of 0.7, an aspect ratio of 6, a profile drag C Dprofile of 0.006, and a wing area of 15 m 2 . Determine: (note for air, γ = 1.4, R = 287 J/(kg K) , specific heat at constant pressure C p = 1005 J/(kg K) ) a) The Mach number at which the aircraft is flying (2 marks) b) The lift coefficient C L acting on the aircraft (4 marks) c) The total drag coefficient (3 marks) d) The total drag force acting on the aircraft (1 mark) a) Speed of sound, 𝑎 ∞ = √𝛾?? ∞ = √(1.4)(287)(270) = 329 ? ? Mach number ? ∞ = 𝑉 ∞ 𝑎 ∞ = 250 329 = 0.76 b) Air density, ? ∞ = ? ∞ ?? ∞ = (60000) (287)(270) = 0.774 𝑘𝑔/𝑚 3 Dynamic pressure, ? ∞ = 1 2 ? ∞ ? ∞ 2 = 1 2 (0.774)(250) 2 = 24188 𝑃𝑎 Aircraft weight, W = (mass)(9.81) = (6000)(9.81) = 58860 N Steady, level flight, Lift = weight, L = W Lift coefficient, ? 𝐿 = ? ? ∞ ? = (58860) (24188)(15) = 0.162 c) Total drag, ? 𝐷 = ? 𝐷????𝑖?? + ? 𝐷𝑖 = ? 𝐷????𝑖?? + (? 𝐿 ) 2 ?𝑒 𝐴? = 0.006 + (0.162) 2 ?(0.7)(6) = 0.008 d) Total drag force, ? = ? 𝐷 ? ∞ ? = (0.008)(24188)(15) = 2902.6 ?

5  Air at standard temperature (288 K) and pressure (101.325 kPa) flows over a rectangular flat plate with a velocity of V  = 25 m/s as shown. The plate has a chord of C = 1 m and a span of b = 2 m. A boundary layer develops on the surface of the plate. The viscosity of air is  = 1.79  10 -5 kg/(m s) . . The distance x is measured from the leading edge of the plate as shown. Find: a) Assume transition from laminar to turbulent flow occurs at Re x = 500,000. Find the distance x cr from the leading edge where transition occurs. (4 marks) b) Assume that the boundary layer is completely turbulent (from x = 0 onward). Consider only one side of the plate, find the total drag force due to shear on the plate (3 marks) c) Consider the laminar-then-turbulent arrangement as shown, find the total drag force on one side of the plate (3 marks) a) ? = 𝑃 𝑅𝑇 = 101325 (287)(288) = 1.225 𝑘𝑔/𝑚 3 ?𝑒 𝑐? = 𝜌𝑉𝑥 𝑐𝑟 𝜇 = 500000 𝑥 𝑐? = (500000)(1.79×10 −5 ) [(1.225)(25)] = 0.292 𝑚 b) Total drag on plate if it was completely turbulent ? ∞ = 1 2 ? ∞ ? ∞ 2 = (0.5)(1.225)(25) 2 = 382.8 ?𝑒 𝑐 = 𝜌𝑉 ∞ 𝑐 𝜇 = (1.225)(25)(1.0) 1.79×10 −5 = 1.71 × 10 6 ? ? = 0.074 𝑅? 𝐿 0.2 = (0.074) (1.71×10 6 ) 0.2 = 0.0042 ? ?𝑇 = ? ? ? ∞ ? = (0.0042)(382.8)(2)(1) = 3.21 ? c) Drag on a turbulent plate up to x cr , (area A) ? ? = 0.074 ?𝑒 𝑐 1/5 = 0.074 (5.00 × 10 5 ) 1/5 = 0.00536 Top view Side view

6  workspace D fA = q  SC f = (382.8)(0.292  2)(0.00536) = 1.2 N Drag on area B, turbulent, D fB = D fT – D fA = 3.21 – 1.2 = 2.0 N Laminar drag on A only, ? ? = 1.328 √?𝑒 𝐿 = 1.328 √500,000 = 0.001878 D fA = q  SC f = (382.8)( 0.292  2)(0.001878) = 0.42 N Total drag, D f = D f,lamA + D f,turbB = 0.42 + 2.0 = 2.42 N

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