AE 311 Incompressible Flow Fall 2016 Homework Set # 7 Due: In class on Wednesday 2 November Show all your equations, data, and work to receive full credit. Clearly indicate your answers. 1. (20%) Assume potential flow. A circular cylinder produces lift in a freestream flow of 30 m/s. The diameter is 0.5 m, and the amount of circulation is 4 m 2 /s. For this case, sketch the streamlines around the cylinder, clearly showing the stagnation points. Also, plot the C p distribution around the circular cylinder. Integrate this C p distribution (show all equations) and calculate the lift produced. Assume standard sea level conditions for the density. 2. (10%) Use the Kutta-Joukowski Theorem to determine the circulation around an airfoil given: c = 2 m, V = 90 m/s, h = 3000 m, and C l = 0.25. Determine the air density from the standard atmosphere. 3. (20%) Two airfoils operate in a biplane arrangement with one airfoil directly above another. Both airfoils have a chord length of 4.5 ft. The spacing is 5.5 ft. Sketch the arrangement. The top airfoil has a circulation of 320 ft 2 /s, and the bottom airfoil has a circulation of 345 ft 2 /s. Take the freestream flow speed to be 100 mph. Assume standard sea level conditions. Draw to scale the total flow velocity (showing the components) on each airfoil. Draw the resulting force vector to scale for each airfoil. Draw to scale and compute the lift and drag on each airfoil. What is the total lift and total drag on the arrangement? What is the lift coefficient for each airfoil? Normalize your lift forces using the airfoil chord of 4.5 ft. 4. (20%) For the prior homework involving the website http://www.profoil.org, you designed an airfoil and plotted the C p distribution for a range of angles of attack. For your airfoil, calculate the lift coefficient for each angle of attack given by the website. Determine these lift coefficient values by integrating the C p distribution. Show your work and tabulate your results. Upload into Compass 2G your spreadsheet (or Matlab code) that is used to determine the lift coefficient from the pressure coefficient distribution. [Note that sharing electronic files is considered copying. Do not share files; see policy in syllabus.] 5. (30%) A circular cylinder has circulation and produces lift in a potential flow. The stagnation points are located at 20 deg and 200 deg on the circumference of the circle. Use Matlab to write a code and plot the flow vectors (see the quiver Matlab function) around the surface of the circular cylinder and at finely space points on a square grid that surrounds the circular cylinder. Also, create a second plot showing the same flow vectors and overlay on top of this your sketch of the corresponding streamlines (or alternatively compute the streamlines and plot). Upload your Matlab code into Compass 2G. See the following page about including units on parameters in code (highlighted section). [Again: Note that sharing electronic files is considered copying. Do not share files; see policy in syllabus.] 1 Copyright c 2016 Michael Selig. All Rights Reserved.
As it may apply: For problems that involve sketching, make your answers neat and clear. Assume sea level conditions except where otherwise noted. Constants for properties of air were provided in class. All parameters in code that have units must have units included as a suffix in the variable name, and all equations should have consistent units (e.g. English or SI). See the example naming style below. In the table, the unit syntax LbpFt2 means lb per ft 2. Parameter Variable Units Naming in code Wing span b ft b_ft Wing area S ft^2 Sref_Ft2 Static pressure p lb/ft^2 pstatic_lbpft2 Temperature T deg F T_DegF Various: For problems in English units, lb will always means lb force. Do not write lbf or lb f because if you do points will be taken off your homework. We will not use lbm in AE 311. We will leave lbm for the mechanical engineers. Another point is that slug is mass and lb is force. This is exactly the same as would be used in SI units where the mass is in kilograms and the weight is in Newtons. Do not write the unit slugs for slug,... or else points will be taken off your homework. These course notes and materials Copyright c 2016 Michael Selig, TA s, and, UIUC. All rights reserved. Permission is granted to use these materials for your personal use. No other permission is granted, e.g. no permission is granted for sharing online. 2 Copyright c 2016 Michael Selig. All Rights Reserved.
3.5 3 2.5 2 1.5 c P 1 0.5 0 0.5 1 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π θ 0.2 0.1 y (m) 0 0.1 0.2 0.6 0.4 0.2 0 0.2 0.4 0.6 x (m)
Contents Probem 4 Probem 4 Input data copied from Profoil Calculate lift Input data copied from Profoil
Calculate lift
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2 Flow Vectors 1.5 1 0.5 y 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x
2 Flow Vectors and Streamlines 1.5 1 0.5 y 0 0.5 1 1.5 2 2 1.5 1 0.5 0 0.5 1 1.5 2 x