Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
Page 1
Grading Sheet
~~~~~~~~~~~~~~
MIME 3470—Thermal Science Laboratory
~~~~~~~~~~~~~~
Laboratory №. 9
DRAG FORCE DETERMINATION
Students’ Names  Section №
APPEARANCE, ORGANIZATION, ENGLISH and GRAMMAR
MATHCAD
Data, Dimensions, Physical Properties
Calculated Drag Force Based on Fluids Text CD
Calculated Cp as Function of  from Wind Tunnel Data
Plot Calculated and Potential Flow Cp vs 
Computed Drag Force Using Trapezoidal Rule and Pressure Data
DISCUSSION OF RESULTS
Explain Differences Between Potential Theory & Actual Cp Profiles
Discuss Difference Between Drag Forces
 Calculated Using CD from a Fluids Text
 Calculated from Trapezoidal Integration of Data, &
 Measured Using the Wind Tunnel’s Drag Balance.
CONCLUSIONS
ORIGINAL DATASHEET
TOTAL
COMMENTS
d
GRADER—
POINTS
5
10
10
10
15
20
10
10
5
5
100
SCORE
TOTAL
Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
MIME 3470—Thermal Science Laboratory
~~~~~~~~~~~~~~
Laboratory №. 9
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one to determine the drag force from an experimentally measured
distribution. A typical pressure distribution is shown in Figure 2.
DRAG FORCE DETERMINATION
~~~~~~~~~~~~~~
LAB PARTNERS: NAME
NAME
NAME
SECTION
№
EXPERIMENT TIME/DATE:
NAME
NAME
NAME
TIME, DATE
~~~~~~~~~~~~~~
OBJECTIVE–To measure the pressure distribution around a circular
cylinder placed in a transverse air stream and calculate the drag force
based on the local pressure measurements. To compare the measured
pressure distribution and drag force with the values predicted by the
potential flow theory, and to identify the limitations of that theory.
INTRODUCTION–The theory of fluid flow around a smooth cylinder
dates back to 19th-century hydrodynamics. The potential flow theory
presented here predicts the velocity and pressure distributions about
the circumference of a circular cylinder immersed in a fluid stream.
The same theory predicts the lift and drag forces on the cylinder and
can be applied to airfoils in modern aircraft and turbomachinery
applications.
In the ideal potential theory for this experiment, the fluid is
assumed to be inviscid and incompressible. The flow at large distances from the immersed body is assumed to be uniform. In keeping
with the above assumptions, the flow is irrotational. When the fluid
encounters a smooth surface it is assumed to slip tangentially, flowing
without friction or separation along the surface; the fluid neither
penetrates the surface nor leaves a gap. This condition is sufficient to
yield a unique solution to the governing flow equations. This
boundary condition differs from the more realistic no-slip condition
applied to viscous flows.
Figure 1 shows a cross section of a circular cylinder of radius a in a
potential flow from left to right. The fluid speed at large distances
from the center is U and the corresponding static pressure is p. Note
that the angle  takes values from 0 to 2 radians and that the flow is
symmetrical about the vertical and horizontal axes of the cylinder.
Figure 2– Pressure Distribution Around a Circular Cylinder
Placed in a Real-Fluid Uniform Flow.
The length of the arrow at any point on the cylinder surface, in
Figure 2a, is proportional to the pressure at that point. Figure 2b is
a plot of the pressure data from Figure 2a. The direction of the
arrow in Figure 2a indicates that the pressure at the respective
point is greater than the free-stream static pressure (pointing
toward the center of the cylinder) or less than the free-stream static
pressure (pointing away). Note the existence of a separation point
and a separation region (or wake) (see Figure 3).
U
Figure 3 – Streamlines of Flow About a Circular Cylinder.
Figure 1 —Circular Cylinder in an Ideal (Potential Theory) Flow
The pressure in the back-flow (wake) region is nearly the same
as the pressure at the point of separation. The general result is a
net drag force equal to the sum of the forces due to pressure acting
on the front half (positive pressures) and on the rear half (negative
pressures) of the cylinder. To find the drag force, it is necessary to
sum the horizontal pressure components at each point along the
cylinder’s surface.
POTENTIAL FLOW PRESSURE DISTRIBUTION ABOUT A CYLINDER
For potential flow, the tangential velocity along the surface of
the cylinder is given by
(1)
U s  2U  sin 
For steady, inviscid, incompressible flow along a streamline,
Bernoulli’s equation is valid. Further, since the gravitational field is
conservative (i.e., does not change), it is justifiably ignored. By
applying Bernoulli’s equation to Equation 1, it can be established
without much difficulty that:
p   p
Cp  s
(2a)
1 U 2

2
For real, viscous fluids, the fluid exerts a pressure on the front half
of the cylinder in an amount that is greater than that exerted on the
rear half. The difference in pressure multiplied by the projected
frontal area of the cylinder gives the drag force due to pressure
(also known as form drag as opposed to skin-friction or surface
drag). Because this drag is due primarily to a pressure difference,
measurement of the pressure distribution about the cylinder allows
 1  4 sin2 
(2b)
where ρ and ps() denote the fluid density and the surface static
pressure at angle , respectively. Cp is called the pressure
coefficient or the Euler number. Observe that this coefficient is
dimensionless, taking the value of 1 at both the forward and
rearward stagnation points and a minimum value of –3 at  = /2.
U
r
2U 

Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
Also, note from Figure 4 that the pressure coefficient is symmetric
about both the x and y cylindrical axes.
U
y

x
p s  p0
forth his boundary layer hypothesis in 1904. Again, this paradox is
a result of the omission of the influence of viscosity.
In a practical sense, the approach necessary to calculate the drag
on the cylinder (or the drag coefficient) is to measure the pressure
along the entire cylinder surface and perform the integration of
Equation 3 using differential surface elements. If, for example, an
experimenter wishes to represent the pressure distribution by
making a large number of measurements (ps)i at discrete locations
i separated by intervals of  radians, the resulting calculation
for the drag force per unit length of cylinder would be:
N
1 U 2

2
FD 

i 1
The surface static pressure ps at  = 0, is equal to the stagnation (or
total) pressure, p0, of the flow. The stagnation pressure (the Bernoulli
constant in this case) is p0  p  12 U 2 . The quantity 12 U 2 , is
the dynamic pressure and is the kinetic energy per unit volume of the
distant stream, while p is the static pressure of the distant stream.
In this experiment, the student will measure the difference between
p and ps()pressure (where ps(0) = p0, such that this difference is
positive at the forward stagnation point) at various locations along
the surface of a cylinder placed in a wind tunnel stream. In order
to reduce the pressure data to the pressure coefficient Cp, the static
pressure and the dynamic pressure of the approaching free stream
are required. A manometer on the test bench will measure the
difference in pressure, ps() – p. The dynamic pressure can be
determined from the measured velocity of the free stream.
DRAG AND LIFT FORCES–Since there are no viscous shear stresses
(skin friction) in the potential model, the total force that the fluid
exerts on the cylinder is obtained by integrating the static pressure
force over the surface area A of the cylinder. The total force can be
resolved into components parallel and perpendicular to the flow
direction; they are called drag, FD, and lift, FL, respectively. In the
present case they are horizontal and vertical components:
  p  dA
(3)
  p  dA
(4)
A
s x
 p s  p s
i 1
 i
2


  i  i 1  
 cos
 rcyl Lcyl 




 
2



(5)
The cos term, in the Equation 5, generates the horizontal
projection of the pressure force. Since the tunnel static pressure is
assumed constant over the test section it is reasonable to replace (ps)i
in the above equation with (ps)i – p. This is the actual differential
pressure measured by the instrumentation in the lab and the p
term will simply cancel out of any calculation of the drag force.
The radius of the cylinder is rcyl and its length Lcyl.
deg 
vod
Figure 4 —Instantaneous Surface Pressure Coefficient vs. Angle
FD 
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APPARATUS–The apparatus in this experiment consists of an open (as
to a closed loop) wind tunnel with a 30cm diameter circular test
section (see Figure 5). The cylinder almost entirely spans this test
section and has a single pressure tap which moves with the cylinder as
it is rotated. The test section itself has a static pressure tap at its
underside for measuring the wind tunnel static pressure. A
differential pressure transducer (manometer) is connected between
the cylinder’s total pressure tap and the tunnel’s static pressure
tap. The pressure tap on the surface of the cylinder is a small
pinhole that is connected to a tubing connector that runs along the
axis of the cylinder.
Inlet
Inlet flow
flow
straighteners
straighteners
Nozzle
le
Test section
Diffuser
Diffuser
Fan
Fan
Cylinder
Cylinder
INLET FLOW
STRAIGHTENERS
NOZZLE
FAN
DIFFUSER
TEST
SECTION
and
FL 
A
s y
Due to the double symmetry discussed in connection with Figure 4,
both FD and FL obviously are zero for a cylinder in potential flow. It
is practical to conclude that there is no lift on a cylinder placed in a
transverse air stream; but, the prediction that there is no drag is not
easy to accept, and, in fact, it should not be. The inviscid flow assumption in the theory allows for complete pressure recovery along the
rear side of the cylinder. This does not happen in real flows as will
be seen from the measurements made in the laboratory.
Even less obvious is the fact that under the potential flow
assumption, FD = 0 independent of body shape! This outcome,
referred to as D’Alembert’s paradox, was a major point of
contention of 19th-century hydrodynamics until Ludwig Prandtl set
Figure 5 - The Experimental Setup.
CYLINDER–A hollow, plastic cylinder 25.4mm in diameter and
292mm long spans the breadth of the wind tunnel test section. The
cylinder can be rotated on its axis from one angular position to another
by hand. The angular position of the pressure tap relative to the
tunnel’s longitudinal axis is measured by means of a 360o protractor
located on the wind tunnel frame just below the cylinder. A pointer
is fastened on the threaded bar that supports the cylinder, indicating
the angle of rotation of the cylinder with respect to the stagnation
streamline that corresponds to zero degrees on the protractor.
MANOMETER–A differential manometer with a range of 0 to 2inH2O
is used to measure the pressure differences.
Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
PROCEDURE
1. The experiment is performed at a constant speed near 40mph.
2. Rotate the cylinder pointer to a protractor reading of  = 0o.
3. Record the pressure difference reading at  = 0o.
4. Rotate the cylinder pointer to  = 5o and record the pressure
difference again.
5. Continue taking readings in 5o increments for a total of 37
readings.
6. After all data have been taken, turn off the wind tunnel.
FOR THE REPORT
Calculations and Results
1. Calculate the drag force D  FD 
1
 C DU 2 A
2
where, CD – drag coefficient selected from a plot of CD vs. Re
(see data sheet),
U = U, the free-stream velocity,
A – is the area of the cylinder projected on a plane
normal to the flow direction,
ρ – is the air density.
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2. Calculate Cp as a function of angle (Equation 2a). Note, if the
value of Cp at the forward stagnation point is not equal to 1,
scale all values of ps i   p such that (Cp)0 = 1.
3. Plot calculated Cp vs.  for the wind tunnel speed. Also
calculate the potential theory value of Cp (Equation 2b) and
plot it on the same graph.
3. Use Trapezoidal integration (see Appendix A) to compute the
drag force on the cylinder based on the pressure difference
measurements (see Equation 5).
Discussion of Results
1. Explain the differences between the potential theory and actual
profiles of Cp.
2. Discuss the difference between the drag force calculated using
the CD from a fluids text (Item 1 above), calculated from
Trapezoidal integration of data (Item 3 above), and the drag
force measured using the wind tunnel’s drag balance.
Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
ORDERED DATA, CALCULATIONS, and RESULTS
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Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
DISCUSSION OF RESULTS
Explain differences between potential theory and actual Cp profiles
Answer
Discuss difference between drag forces 1) calculated using CD
from a fluids text, 2) Calculated from Trapezoidal integration of
data, and measured using the wind tunnel’s drag balance.
Answer
CONCLUSIONS
Page 6
Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
APPENDICES
APPENDIX A—TRAPEZOIDAL INTEGRATION
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Last Rev: 11 JUN 08
Drag Force Determination : MIME 3470
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APPENDIX B—DATA SHEET FOR DRAG FORCE EXPERIMENT
Time/Date
____________________________
DATA
Lab Partners
____________________________
Cylinder diameter
(______)
____________________________
Cylinder length
(______)
____________________________
Ambient temperature
(°____)
____________________________
Air Speed, U
(_____)
____________________________
Measured Wind Tunnel Drag.
(_____)
i
()
0
5
10
15
20
25
30
Surface Pressure Measurements
ps i   p
(inH2O)
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180