HW3_soln

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Jan 9, 2024

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Name: HW3 Problem 1 Page 1 / 7 ENAE283 Homework #3 Due Wednesday, June 15 at 11:59PM For all homework assignments in this course, you are required to submit fully- explained solutions, indicating the sources for any numbers and equations used. The boxed areas should be used for your final answers. If you cannot use this .pdf directly as your worksheets, please work neatly on your own paper and include a header on each page like the ones shown, write the problem statement at the top and box your final answer at the lower right corner. Pay attention to appropriate significant figures, usually 2 places past the decimal. 1. An airfoil has a critical Mach number of 0.7. What is the minimum pressure coefficient on the surface of this airfoil at low speed. At 𝑀𝑀 ∞ = 0.7 , the critical pressure coefficient is given by eq 5.45 𝐶𝐶 𝑃𝑃 , 𝑐𝑐𝑐𝑐 = 2 𝛾𝛾𝑀𝑀 ∞ 2 �� 2 + ( 𝛾𝛾 − 1) 𝑀𝑀 ∞ 2 𝛾𝛾 + 1 � 𝛾𝛾 𝛾𝛾−1 − 1 � = − 0.7791 Reversing the Prandtl-Glauert correction factor 𝐶𝐶 𝑝𝑝 = 𝐶𝐶 𝑝𝑝 , 0 � 1 − 𝑀𝑀 ∞ 2 → 𝐶𝐶 𝑝𝑝 , 0 = 𝐶𝐶 𝑝𝑝 � 1 − 𝑀𝑀 ∞ 2 = ( − 0.7791) � 1 − (0.7) 2 = − 0.5564

Name: HW3 Problem 2 Page 2 / 7 2. A thin, low-camber airfoil is mounted in a low-speed wind tunnel test section at an angle of attack 𝛼𝛼 =5°. Free stream static and total pressure are measured to be 100 and 105 kPa, respectively (where 1 kPa = 1,000 N/m 2 ). The pressure distribution along the upper and lower surface are given by the following expressions: 𝑝𝑝 𝑢𝑢 [ 𝑘𝑘𝑘𝑘𝑘𝑘 ] = 107 � 𝑠𝑠 𝑐𝑐 � 4 − 256 � 𝑠𝑠 𝑐𝑐 � 3 + 225 � 𝑠𝑠 𝑐𝑐 � 2 − 78 � 𝑠𝑠 𝑐𝑐 � + 105 𝑝𝑝 𝑙𝑙 [ 𝑘𝑘𝑘𝑘𝑘𝑘 ] = 43 � 𝑠𝑠 𝑐𝑐 � 4 − 107 � 𝑠𝑠 𝑐𝑐 � 3 + 101 � 𝑠𝑠 𝑐𝑐 � 2 − 39 � 𝑠𝑠 𝑐𝑐 � + 105 A) Determine expressions for the pressure coefficients along both the upper and lower surfaces, and plot these curves of 𝐶𝐶 𝑝𝑝 vs. 𝑥𝑥 / 𝑐𝑐 on a single plot, using the standard plotting convention that (-) is up. B) Find the total lift coefficient, 𝑐𝑐 𝑙𝑙 , for this airfoil. (You are encouraged to use a computing tool such as Excel or MATLAB to generate your plot and calculate the lift coefficient.) b) Total lift force is the area between the curves, from numerical integration (easiest way since the profiles are already in MATLAB/Excel) 𝑐𝑐 𝑙𝑙 = 0.523

Name: HW3 Problem 3 Page 3 / 7 3. A supersonic jet with thin, straight wings is in steady, level flight at Mach 2.5 at a standard altitude of 8 km. If this vehicle has a planform area of 5 m 2 and experiences a wave drag of 1,200 N, what is its total weight? (Neglect any lift due to the body of the aircraft.)

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Name: HW3 Problem 5 Page 4 / 7 4. A wing with planform area 3 ft 2 spans the full width of a wind tunnel test section having a flow velocity of 95 ft/s at standard sea-level conditions. At an angle of attack of -2°, the lift and drag forces are measured to be 0 and 0.27 pounds, respectively. When the wing is pitched to α = 3°, the lift and drag are then measured as 4 and 0.3 pounds, respectively. If we implemented this same airfoil on a real aircraft with an aspect ratio of 6, what lift slope would we expect? Assume a span efficiency factor e = 0.8767.

Name: HW3 Problem 5 Page 5 / 7 5. Airfoil Choice for an Aerobatic Airplane US National Aerobatics Champion, Patty Wagstaff, spends a lot of her flying time upside- down. The wing of her aerobatic airplane has a chord length of 3 ft, a tip to tip span length of 20 ft, a planform area of roughly 60 ft 2 , and she flies at an average speed of 120 ft/sec with a Reynolds number of 6 x 10 6 . Assume sea level conditions. Your task is to choose between a NACA 4412 and a NACA 0009 airfoil (see wing section data charts in Appendix D of Dr. Anderson’s Introduction to Flight textbook). A) For each airfoil, determine the lift coefficient upright at 𝛼𝛼 = 6° and inverted at 𝛼𝛼 = -6°, and calculate the resulting lift force on the plane.

Name: HW3 Problem 5 Page 6 / 7 B) For each airfoil, determine the drag coefficient upright at 𝛼𝛼 = 6° and inverted at 𝛼𝛼 = - 6°, and calculate the resulting drag force on the plane.

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Name: HW3 Problem 5 Page 7 / 7 C) Calculate the moment about the quarter-chord point at 𝛼𝛼 = 6° with and without flap deflected 60° for each airfoil shape. D) Based on the numbers calculated above, which airfoil do you think would be better for this aerobatic airplane?