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MAE 3241 Aerodynamics and Flight Mechanics Assigned: Jan 30, 2015

Homework #2 Due: Feb 6, 2015

Submit your answers to all questions below. Show your working steps in sufficient details.

1. (20 pts.)The velocity components of some flows are given below. Determine for each flow below

whether it is rotational or irrotational and calculate its vorticity.

a. Two-dimensional Couette flow with u=Kyand v= 0.

b. Circulating flow with Vr= 0 and V= r, where is a constant.

c. Hyperbolic flow with u=Axand v= Ay, whereAis a constant.

d. Two-dimensional vortex flow with Vr= 0 and V= k/r, where kis a constant.

2. (5 pts.)

Two of three velocity components for an incompressible flow are:

yzyvxzxu 22 22

Determine the velocity component w(x,y,z) to satisfy the continuity equation.

3. (5 pts.)

The velocity components for a two-dimensional incompressible flow are

222222

22

)(

2

)(

)(

yx

Cxyv

yx

xyCu

where Cis a constant. Does this flow satisfy the continuity equation? Provide justification to

your answer.

4. (25 pts.)

A 90opipe bend in the figure has a cross-sectional area of 0.01 m2at the inlet and 0.0025 m2at the outlet. Air enters from the inlet of the pipe at the speed of 100 m/s in the xdirection

(assumed uniform across the cross-sectional area). The pressure of the air flow at the inlet and

outlet is the same, which is the ambient pressure of the surrounding. Assume the air isincompressible and has the density of 1.225 kg/m3.

a. Determine the speed of the flow at the outlet.

b. Calculate the force exerted on the pipe by the airflow in thexandydirections.

Hint: define the control volume used in your analysis appropriately and apply the relevantfundamental fluid flow principles.

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5. (20 pts.)

In the lecture notes, drag calculation of a two-dimensional object by measuring velocityprofiles upstream and downstream of the control volume surfaces has been discussed. Consider

now an airfoil in a wind tunnel (i.e., a wing that spans the entire test section). Show that the

lift per unit span can be obtained from the pressure distributions on the top and bottom parallelwalls of the wind-tunnel test section (i.e., from the pressure distributions on the walls aboveand below the airfoil). Define the control volume used in your analysis in detail in your answer.

6. (25 pts.)

The figure below shows a control volume used in analyzing a two dimensional flow around

the airfoil.HUandHDare half of the vertical dimensions of the upstream end (surface 1) and

downstream end (surface 2) of the control volume. Velocity profiles at surfaces 1 and 2 of thecontrol volume are indicated in terms of U(= constant) in the figure. Surfaces 3 and 4 coincide

with the streamlines of the flow. Assume the flow to be steady and incompressible. The

pressure over the entire surface of the control volume is constant.a. What is the relationship betweenHUandHD?

b. IfHD= 0.025c, calculate the drag coefficient of the airfoil.