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MAE 3241 Aerodynamics and Flight Mechanics Assigned: 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:

jiV

2

322

3xy

yxyyx

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.

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4. (10 pts.)

During a low-speed flight test, an airplane is equipped with some pressure gages in several

locations. The test altitude is 8 km. Gage 1 indicates the pressure of 1550 N/m2 above the

ambient pressure, while gage 2 shows the pressure of 3875 N/m2 below the ambient pressure.

a. If gage 1 is known to be at the stagnation point of the air flow, determine the airspeed and

Mach number of the flight.

b. Determine the speed of the air near gage 2 relative to the airplane and relative to the ground,

assuming no wind in the atmosphere.

c. What are the pressure coefficients at gage 1 and gage 2?

5. (10 pts.)

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. The freestream

velocity of the air in the test section is 60 m/s.

a. Determine the freestream static pressure inside the test section.

b. What is the value of the pressure coefficient at the stagnation point on a model tested in

the tunnel?

c. If the velocity of the air right at the inlet of the tunnel is 1 m/s, 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. (20 pts.)

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, as shown in the figure below. The upper part of the dividing/stagnation streamline

from this combination can be considered as the surface of the cliff.

a. Determine the expression for the height of the cliff (y) as a function of V∞, θ, and Λ. Hint:

siny r in polar coordinates.

b. Determine h, which represents the limit height of the cliff at the faraway distance.

c. Determine the vertical wind speed profile on the surface of the cliff as a function of V∞ and

θ. Hint: Vertical flow speed sin cosrv V V .

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7. (20 pts.)

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.

a. Determine the general forms of velocity potential and stream function for this tornado in

polar coordinates. 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]).

b. Radial lines from the tornado centerline will intersect with the streamlines. Determine the

expression for the angle between the streamline and the radial line at any intersection point.

What can you say about the dependency of this angle to r and θ coordinates?

c. At sea level, what is the local pressure and velocity at a radial distance of 50 m from the

centerline of the tornado?

Appendix Characteristics of the International Standard Atmosphere, SI Units

Altitude, h

km

Temperature, T

K

Pressure, P

N/m2

Density, ρ

kg/m3

Speed of

Sound, a

m/s

Viscosity, μ

kg/m s

0 288.16 101325 1.225 340.3 1.79E-05

0.5 284.91 95461 1.1673 338.4 1.77E-05

1 281.66 89876 1.1117 336.4 1.76E-05

1.5 278.41 84560 1.0581 334.5 1.74E-05

2 275.16 79501 1.0066 332.5 1.73E-05

2.5 271.92 74692 0.95696 330.6 1.71E-05

3 268.67 70121 0.90926 328.6 1.69E-05

3.5 265.42 65780 0.86341 326.6 1.68E-05

4 262.18 61660 0.81935 324.6 1.66E-05

4.5 258.93 57752 0.77704 322.6 1.65E-05

5 255.69 54048 0.73643 320.5 1.63E-05

5.5 252.44 50539 0.69747 318.5 1.61E-05

6 249.2 47217 0.66011 316.5 1.6E-05

6.5 245.95 44075 0.62431 314.4 1.58E-05

7 242.71 41105 0.59002 312.3 1.56E-05

7.5 239.47 38299 0.55719 310.2 1.54E-05

8 236.23 35651 0.52578 308.1 1.53E-05

8.5 232.98 33154 0.49575 306 1.51E-05

9 229.74 30800 0.46706 303.9 1.49E-05

9.5 226.5 28584 0.43966 301.7 1.48E-05

10 223.26 26500 0.41351 299.6 1.46E-05