Cover Page for Precalculations – Individual Portion
Drag on Spheres
Prepared by Professor J. M. Cimbala, Penn State University
Latest revision: 11 January 2012
Name:
________________________________________
Date:
________________________________________
Section number:
ME 325._____
Group letter: (A, B, ...) _____
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Precalculations
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Precalculations
A.
Dimensional Analysis
We start this lab with a quick review of dimensional analysis. The drag force, FD, on a sphere is a function of sphere
diameter, d, freestream velocity, V, fluid density, , the coefficient of viscosity, , of the fluid, the speed of sound, a, in the
fluid, and the average roughness height, , on the sphere surface in the fluid. This parametric relation is written as
FD  FD  d ,V ,  ,  , a,  
(1)
The dimensions of each variable are given below in M-L-t (mass-length-time) primary dimension system:
Variable
Dimension
{ML/t2}
{L}
{L/t}
{M/L3}
{M/(Lt)}
{L/t}
{L}
FD
d
V


a

(8)
1.
Since there are seven variables and three primary dimensions (M, L, and t), we expect 7 – 3 = 4 dimensionless
parameters, or s, based on the Buckingham pi technique.
2.
Choosing d, V, and  as the “repeating variables”, work through the method of repeating variables, showing all
your work in the space below. Summarize your results (list your s) in the spaces provided.
1 =
2 =
3 =
4 =
Now, recall that these Pi’s can be multiplied by any constant or raised to any exponent (including -1). Be sure that
your Pi’s reduce to the following relationship between dimensionless parameters:


CD  CD  Re d , Ma, 
(2)
d

where CD is the drag coefficient, A is the sphere’s projected frontal area, Red is the Reynolds number based on sphere
diameter, and Ma is the Mach number. All of these dimensionless parameters are defined in the Nomenclature section of this
lab.
(2)
3.
Flows can be considered incompressible when the Mach number is less than about 0.3. Our wind tunnel has a
maximum speed of about 55 m/s. The speed of sound in room temperature air is approximately 340 m/s. Are
compressibility effects important in this lab? Explain your answer.
For the remainder of this lab, Mach number will be neglected. Eq. (2) then reduces to


CD  CD  Re d , 
d

(5)
4.
(3)
For a smooth sphere, /d = 0, and the drag coefficient reduces to a function of Reynolds number only, regardless
of sphere diameter. Thus, one plot of CD as a function of Red is sufficient to fully describe the drag force on a
sphere, as illustrated in Figure 1 of the Background section of this lab. Estimate the drag force in newtons on the
smooth yellow-orange ball used in this lab, for conditions when the wind tunnel operates at a freestream speed of
50.0 m/s, with air at 25oC. Show all your work in the space below. Note: As always, be careful with units.
FD = __________________ N.
(5)
5.
In your own words, explain the “drag crisis”. In other words, discuss why the drag coefficient suddenly drops at a
Reynolds number greater than about 2  105.
(5)
6.
The maximum wind tunnel speed in our lab is about 55 m/s when the air is about 25 oC. Look at the list of spheres
used in this lab. With which of the smooth spheres (if any), do you expect to be able to achieve high enough
Reynolds numbers to measure the drag crisis? Show all your work and explain your answer in the space below.
B.
Pitot-static probe
A Pitot-static probe is used in this experiment to measure the freestream velocity of the wind tunnel. A sketch of the
probe is shown in Figure 3:
Pitot-static probe
V

Stagnation
pressure, PT
Static
pressure, P
To low port (static pressure port) of transducer
To high port (stagnation pressure port) of transducer
Figure 3. Schematic diagram of a Pitot-static probe.
The velocity at the static pressure holes is approximately equal to freestream velocity V, while the velocity at the total
pressure hole is nearly zero, since the hole is located at the stagnation point of the probe.
(5)
1.
Neglecting friction, use Bernoulli's equation to find an equation for velocity V as a function of pressure difference
(PT – P) and fluid density . Show all your calculations in the space below:
Result: V =
(4)
Write down Eq. (4) on a separate sheet of paper, and bring it with you to the lab. (This is necessary since you need the
equation to perform the lab, but you will hand in these Precalculations before starting the lab.)
In our setup, the pressure difference (PT – P) is measured in units of “inches of water”. This is actually the column
height of water which is required to support this pressure difference. The conversion from inches to water to standard SI units
is:
1 inch of water column height = 249.09
N
m2
Equation (5), along with your Equation (4), will be used to measure the wind tunnel velocity in this lab experiment.
Experimental Objectives
a.
b.
c.
d.
e.
Calibrate the electronic pressure transducer.
Calibrate the drag balance.
Become familiar with operation of the wind tunnel.
Measure drag on various spheres.
Display data in non-dimensional form, and compare results with published data.
(5)