lab report

Wind Tunnel Background

Important Dimensions Diameter of the cylinder � = 0.75 �� Length of the cylinder � = 12 �� = 1 �� Height of the test section " = 12 �� = 1 ��

IMPORTANT NOTE!!!

I HIGHLY recommend that you do all of the lab calculations in SI units, NOT British units!

Velocity Profile

• How to get position into units of in

The data acquisition system for the wind tunnel gives the position of the pitot tube in units of

volts. You can see this in your excel files for the velocity profiles. To get the position in terms of length

and not voltage, we will use a linear relation. Taking your minimum value in volts and maximum value in

volts, match them with the minimum value in height, 0 in, and the maximum value of height, 12 in,

respectively. From there, you should get a linear equation relating the two quantities. Here is an

example of how you would get the equation:

Note: This is just an example! Your equation will be different!

• Calculating the velocity from the pressures

You will see in the excel files for the velocity profiles that there are two different pressures: Diff

P and Total P. You will need to use Diff P to find the velocity. The total pressure could be used to find

the changes in density throughout the velocity profile, but we are interested in finding the velocity

profile and the density changes will be negligible. By understanding how a pitot tube works, the

differential pressure is related to velocity through Bernoulli’s equation.

y = 0.745x + 2.5

0

2

4

6

8

10

12

14

0 5 10 15

P o

si ti

o n

i n

u n

it s

o f

in

Position in units of V

Example of Relation between V and in

Series1

Linear (Series1)

#$ + $& '($& + ')*$ = #& + $& '(&& + ')*&

It is a stagnation point at the tip of the pitot

tube, so the velocity is equal to zero. Assuming

the difference is height is negligble, Bernoulli’s

equation reduces down to

∆# = 12 '( &

( = ,2∆#'

Now use this equation to find the velocity for every pressure under the “differential

pressure” column in the velocity profile excel files.

Now plot the four velocity profiles and explain their behavior.

Calculating the Drag (-./01) and Coefficient of Drag (23) • Converting the manometer pressure readings

As noted in the lab, the manometer pressure readings are opposite the real pressure. When the

pinpoint hole on the cylinder is facing the flow (i.e. the stagnation point), the pressure should be

a maximum, however on the manometer, it was a minimum. To convert the manometer data

into proper pressure data, you must do the following:

1. Take the absolute value of all of the data

2. Find the maximum manometer value.

3. Subtract all of the values from the maximum pressure value.

As an equation, you want to do the following:

For a pressure reading “S”: "converted pressure reading S" = |maximum manometer value| − |pressure reading S|

By doing this, you will get the proper pressure behavior around the cylinder.

Now you will plot the pressures around the cylinder and explain the behavior.

#<=>?@>=AB@

(CDEE <=DE>F

To pressure

instrument

• Calculating the Drag and Drag Coefficient

By integrating the x-component of the surface pressure on the cylinder over the surface area,

the resultant x-component of the force can be obtained:

GH = I #< JKLMNO<PQ Note that #<JKLM is the component of the surface pressure, #<, in the x-direction. As a result, you are multiplying a differential surface area by the x component of surface pressure and integrating over the

entire surface area.

Be sure to convert the #< value from inches of water, which was the unit of measure of the manometer, to Pa or Psi. Here is how to convert inches of water to Pascals:

#< R#ST = ')ℎ = ')(#< R�� "&WT) Note that ℎ, or as I wrote above #< R�� "&WT, is the inverted manometer water height.

' ≈ 1000 Z?F[ and ) = 9.8 F<^

Now, since our experiment was taken in discrete measurements, we have to use approximation

methods to evaluate the above integral. G_D>? = #H NO< Now we need to find NO< . The surface area of a cylinder is defined as: O< = 2`a� Since there are 72 pressure taps (each 5° apart) on the sphere, that means that there are 72 segments (a

segment is the arc length between two pressure taps). However, since we took pressure measurements

in 10° increments, our differential surface area can be described in the following way:

NO< = �K�Sb Lca�SJd SadS K� S Jeb��Nda# gadLLcad �SgL hdSLcadN = 2`a�36 Then the force equation becomes the following. At each pressure tap, the force acting on the cylinder is G_D>? = #H NO<

G_D>? = (#< JKLM) k2`a�36 l Thus the total drag on the cylinder is the following:

G_D>?,=B=>n = o #<,A JKLMA k2`a�36 l pq

Ar$

Now using the definition of drag:

G_D>?,=B=>n = Gs = 12 t_D>?'(CDEE <=DE>F& OCDB@=>n

Rearranging the above equation to solve for the coefficient of drag, you can calculate t_D>?: t_D>? = ts = G_D>?,=B=>n12 '(CDEE <=DE>F& OCDB@=>n

Where U is the velocity and A is the projected area NOT the cross-sectional area of the cylinder. When

you look at the cylinder from the flow’s perspective, the cylinder appears to be a rectangle. As a result,

the projected area is: OCDB@=>n = �� Where D is the diameter of the cylinder and L is the length.

In the calculation of the drag and drag coefficient, you must use the free stream velocity, which was

found by using the pitot tube upstream of the cylinder.

• Comparing your ts value to theoretical values In order to properly compare your data to theoretical data, you must calculate the Reynolds

number, a dimensionless fluids parameter.

uds = '(�v Where V is the free stream velocity and D is the diameter of the cylinder. Then after calculating this

quantity, you can use the chart below to see the ts value that you should have gotten for that Reynolds number.

Figure 1: Drag coefficient as a function of Reynolds number for a smooth circular cylinder and a smooth sphere. [1]

Make sure you use the curve for the smooth cylinder, NOT the smooth sphere.

Compare your calculated ts to the theoretical ts and explain the results.

Some other helpful illustrations from a fluid dynamics book:

Figure 2: Typical flow patterns for flow past a circular cylinder at various Reynolds numbers as indicated in Figure 1. [1]

Figure 3: Character of the steady, viscous flow past a circular cylinder: (a) low Reynolds number flow, (b) moderate Reynolds

number flow, (c) large Reynolds number flow. [1]

Figure 4: A comparison of theoretical (inviscid) pressure distribution on the surface of a circular cylinder with typical

experimental distribution. [1]

Figure 5: Inviscid flow past a circular cylinder: (a) streamlines for the flow if there were no viscous effects, (b) pressure

distribution on the cylinder’s surface, (c) free-stream velocity on the cylinder’s surface. [1]

Figure 6: Explanation of boundary layer behavior around a cylinder. [1]

References

1. Munson, Bruce R. et al. Fundamentals of Fluid Mechanics, Fifth Edition. Ames, Iowa: John Wiley

& Sons, 2006.