MAE 3241 Aerodynamics and Flight MechanicsAssigned: Feb 9, 2015 Homework #3 Due: Feb 18, 2015 Submit your answers to all questions below. To ensure full credit, show your working steps in sufficient details and plot your graphs properly. No late submission is accepted for this homework. 1. (10 pts.) The components of velocity of an incompressible flow are described by u = A and v = By, where A and B are constants. a. Find its stream function if it exists. b. Find its velocity potential if it exists. c. Is this flow physically possible? Briefly explain the reason for your answer. 2. (15 pts.) The stream function of an incompressible, irrotational two-dimensional flow is given by 2xy a. Determine the velocity field of this flow. Also calculate the magnitude and direction of the velocity at (1, 1) and at (2, 0.5). b. Is this flow rotational? Give clear justification for your answer. c. Determine the velocity potential for this flow if it exists. d. Plot some streamlines and equipotential lines of this flow in the region where x and y are positive on a graph paper (alternatively, you may use software like MATLAB to generate the plot [submit also your command lines/code]). 3. (15 pts.) The velocity field of a two-dimensional, incompressible, steady flow is given by: y3 2 2 V x y xy i xy2 j 3 a. If it exists, find the stream function (x,y) for this flow. If it does not exist, explain why. b. If it exists, find the velocity potential (x,y) for this flow. If it does not exist, explain why. c. What is the circulation of the flow on a triangular region bounded by the points (0,0), (1,0) and (1,1) as shown in the figure below. 1 as shown in the figure below. The freestream velocity of the air in the test section is 60 m/s. a. b. Gage 1 indicates the pressure of 1550 N/m2 above the ambient pressure. Determine h. Determine the expression for the height of the cliff (y) as a function of V∞. c. while gage 2 shows the pressure of 3875 N/m2 below the ambient pressure. Determine the vertical wind speed profile on the surface of the cliff as a function of V∞ and θ. a.4.) During a low-speed flight test. (10 pts. b. What are the pressure coefficients at gage 1 and gage 2? 5. θ. determine the airspeed and Mach number of the flight. If the velocity of the air right at the inlet of the tunnel is 1 m/s.) Horizontal wind field past a cliff can be represented as air flow over semi-infinite body using the combination of a uniform horizontal flow with speed V∞ and a line source flow with strength Λ. a. Hint: Vertical flow speed v Vr sin V cos . Determine the speed of the air near gage 2 relative to the airplane and relative to the ground. which represents the limit height of the cliff at the faraway distance. an airplane is equipped with some pressure gages in several locations. and Λ. (10 pts. What is the value of the pressure coefficient at the stagnation point on a model tested in the tunnel? c. If gage 1 is known to be at the stagnation point of the air flow. assuming no wind in the atmosphere.) An in-draft wind tunnel with circular cross section at sea level takes air from the stationary atmosphere outside of the tunnel and accelerates it in the converging section. 2 . Hint: y r sin in polar coordinates. b. (20 pts. The test altitude is 8 km. c. The upper part of the dividing/stagnation streamline from this combination can be considered as the surface of the cliff. what is the ratio of the diameter of the test section and the diameter of the inlet to achieve the freestream velocity above at the test section? 6. Determine the freestream static pressure inside the test section. 61E-05 1.5 5 268.5 1.51E-05 1.6 324.7 299.98 229.49575 0.95696 330.6 326.53E-05 1.5 2 2.66 89876 1.16 79501 1. h km 0 0.41 84560 1.9 301.90926 0. Determine the general forms of velocity potential and stream function for this tornado in polar coordinates.4 278.5 271.6 322.6 1.2 1. a m/s 288.5 Characteristics of the International Standard Atmosphere.93 255.41351 308.74 226. Radial lines from the tornado centerline will intersect with the streamlines.5 10 236.76E-05 1.5 6 6. Determine the expression for the angle between the streamline and the radial line at any intersection point.3 310. SI Units Temperature.5 223.44 249.69 70121 65780 61660 57752 54048 0.68E-05 1.4 281.0581 334.74E-05 1.79E-05 1.73E-05 1.18 258. what is the local pressure and velocity at a radial distance of 50 m from the centerline of the tornado? Appendix Altitude.2 245. What can you say about the dependency of this angle to r and θ coordinates? c.5 314.63E-05 5.16 101325 1.1 306 303.4 312.1673 338.43966 0.55719 318.6 Viscosity.66E-05 1.) A tornado is simulated two dimensionally by a line sink with strength of 3000 m2/s plus a line vortex with strength of 5200 m2/s that coincide at the tornado centerline.62431 0.5 316.65E-05 1.3 284. T Pressure. μ kg/m s 1.56E-05 1.48E-05 1.225 340.1117 336.5 7 7.69747 0.69E-05 1.66011 0. P Density.92 74692 0.6E-05 1.46706 0.67 265.23 232.59002 0.6 320.7.5 275. a.86341 0. Plot some streamlines of this tornado on a graph paper (hint: you can use a numerical solver like MATLAB to generate an accurate plot [include your codes/command lines in your submission]).77E-05 1.47 50539 47217 44075 41105 38299 0.95 242.77704 0.5 1 1. (20 pts.46E-05 3 .58E-05 1.71 239.5 4 4. ρ Speed of K N/m2 kg/m3 Sound.91 95461 1.5 9 9. b.54E-05 8 8.26 35651 33154 30800 28584 26500 0.5 252.42 262.73643 328.71E-05 3 3.52578 0.81935 0. At sea level.0066 332.49E-05 1.