DRAG FORCE AND VELOCITY
PROFILES FOR A CYLINDER IN
CROSSFLOW
Tuesday May 6th, 2014
Xie Zheng – First Author, Theory, List of Equipment and Experimental Procedure
Kanchan Bhattacharyya – Discussion and Conclusion
Matthew Stevens–Abstract, Introduction
Ting Zhang – Results
Abstract
In this experiment we determine the drag force on a cylinder in an airflow, as well as velocity
profile of the incoming and exit streams of the airflow. The drag force acting on the cylinder is
determined via measurement of the pressure distribution around the cylinder, which is also used to
calculate the velocities of the respective inlet and exit streams. It is with these velocities that we are
then able to quantify the time rate of change of mass within the control volume and verify that steady
state conditions exist. Through the measured pressure and velocity profiles we are then able to verify
the respective profiles, the existence of large Reynolds Numbers that arise in turbulent flow regimes,
the respective contributions of pressure and friction to the overall drag force acting on the cylinder, and
relatively constant mass flow rates throughout the system. We find that in such turbulent situations that
pressure is the primary contributor to total drag experienced by the sphere, and that considerable mass
flow defects exist in regions near the cylinder as opposed to regions further away from the surface of
the cylinder. Finally, we are able to quantify the effects of boundary layers on stream velocity as we find
slight variances in the steady state inlet and exit stream velocities.
Introduction
The ability to quantify the forces and velocities associated with fluid flow around objects is of
great importance in the Engineering Sciences, as the measurement of such quantities enables us to
fundamentally understand many of the physical phenomena we are exposed to on a regular basis and
their mechanisms. For example, we can understand the variance in drag effects experienced by
airplanes in flight and those experienced by cars, as well as the kinematic variances in the flight path of a
baseball when a pitcher throws pitches with different grips and arm angles. [1] In such cases, the
measurement of the pressure distribution acting on the surface of an object allows us to quantify the
effects of forces such as drag, and give us the ability to identify the contributors to these effects. For
1
example, we can definitively attribute the presence of the drag acting on buildings to a dominating
pressure drag in contrast to drag to friction. [1]
Through these concepts and experimental evidence, we can identify many established trends
while simultaneously finding validity in the laws that govern the world around us. For example, in
measuring the pressure distribution around an object in turbulent flow we can determine the velocity of
the fluid at various regions near the object with the help of Bernoulli’s equation describing the flow
energy of a fluid in motion. Such velocities can then be used to calculate the Reynolds Number
describing the flow scenario, and further quantify the existence of dominant inertial forces giving rise to
the large Reynolds Numbers associated with Turbulent flow regimes.
Through direct measurement of the pressure acting on a cylinder subjected to high velocity air
in turbulent cross-flow, we will be able to determine the velocity of the air flow at various points. It is
with these velocities that we can quantify the dominant presence of pressure as the primary contributor
to the drag force experienced by the cylinder, as well as the existence of boundary layers that disrupt
the flow in the regions neighboring the cylinder and large Reynolds numbers that describe such
scenarios.
Theory
1. Pressure Drag Measurement
To determine the drag force on a cylinder in an airflow, we measure the pressure distribution
around the cylinder. Referring to the Figure 1 in lab manual [1], the sum of two forces, pressure and a
viscous shear, represent the total drag force experienced by the cylinder in the air stream. And
mathematically the total drag force per unit length is:
𝐷𝑥
𝑙
= ∫2𝜋[𝑃(𝜃)𝑐𝑜𝑠𝜃 + 𝜏(𝜃)𝑠𝑖𝑛𝜃]𝑅𝑑𝜃
2
Eq.1
where l is the length of cylinder, Dx is total drag force in x-direction, P is pressure, R is cylinder
radius, τ is shear stress and 𝜃 is angle along cylinder.
And the drag coefficient, CD , is defined as:
𝐶𝐷= 𝐷𝑥
Eq.2
1
2𝑅𝑙1/2𝜌𝑈2
0
where ρ is the air density and U0 is the upstream velocity.
Inserting Eq.1 for Dx/l and canceling common terms, Eq.2 becomes
𝐶𝐷 =
2𝜋
1
∫
[𝑃(𝜃) − 𝑃0 ]𝑐𝑜𝑠𝜃𝑑𝜃
𝜌𝑈02 0
Eq. 3
2. Velocity Profiles and Mass
The cylinder retards the air flwoing over it, resulting in a decreased velocity in the wake behind the
cylinder. Flowing fluids have two kinds of energy which is kinetic energy and potential energy. By
inserting a thin tube, pitot tube, with the opening facing into the flow, the velocity at the opening to the
tube can be measured as follows. The tube is connected to a manometer to measure the pressure inside
the tube. Just outside the tube, the flow has a velocity and static pressure. The relation between
stagnation pressure and static pressure could be written as:
1
𝑃𝑠𝑡𝑎𝑔 = 𝑃𝑠 + 2 𝜌𝑢2
And 𝑢 = √2
Eq. 4
𝑃𝑠𝑡𝑎𝑔 −𝑃𝑠
Eq. 5
𝜌
where Ps=101 Pa
the inlet velocity distribution into the test cell is uniform. To measure the inlet velocity, only a
single velocity measurement is required. The cylinder is rotated so that the pressure tap hole is oriented
into the oncoming stream. Bernoulli’s equation can be written as
3
1
𝑃𝑠𝑡𝑎𝑔,𝑐𝑦𝑙 = 𝑃𝑜 + 2 𝜌𝑈02
Eq.6
Procedure
Since the experiment was already setup in the lab, we check the connections of the equipment and
make sure we have all the instruments we need. First of all, we opened up the blower to make it keeps
flowing the air. After around 10 minutes, the reading of temperature shown on the thermometer first
raised several degrees and then became stable, and we recorded the reading when the reading was
stabilized. As for the height of the manometer, we also recorded the initial place of the top of the fluid.
1. Pressure Drag Measurement
We inserted the circular cylinder with attached protractor through the back of the section, and then
connected the cylinder pressure tap to the left port on the manometer. Also, for the upstream pressure
port tap, we connected it to the right port of the manometer. So, the reading on manometer became
different. We rotated the cylinder from 0 degree to 360 degree with increment of 5 degrees and also
recorded the position of the top of the fluid.
2. Velocity Profiles and Mass conservation
We connected the Pitot tube to the left port on the manometer and left the right manometer port
open to the air. Then, we moved the pitot tube from y=0,, to y=100mm and recorded the position of the
top of fluid at manometer.
List of equipment
Manometer (manufacturer: Dwyer): we used the manometer to measure the difference in
pressure by comparing the heights of the fluid.
Blower (AIR FLOW BENCH AF10): we use the blower to generate the air flow for experiment.
4
Results
b. Plot P(θ)-P0 and (P-P0)COSθvs. θ
Figure 1.This figure shows the relationship between P-P0 vs. θ.
P-P0 vs. θ
1.5
1
P-P0
0.5
0
0
100
200
300
400
-0.5
-1
-1.5
-2
θ
Figure2. This figure shows the relationship between (P-P0 )cosθ vs. θ.
(P-P0)COSθ vs. θ
1.2000
1.0000
0.8000
(P-P0)COSθ
0.6000
0.4000
0.2000
0.0000
-0.2000 0
100
200
300
-0.4000
-0.6000
-0.8000
θ
5
400
c. Calculate the pressure drag coefficient
governing equation:
𝐶𝐷 =
2𝜋
1
∫
[𝑃(𝜃) − 𝑃0 ]𝑐𝑜𝑠𝜃𝑑𝜃
𝜌𝑈02 0
Eq. 3
1
where 𝑃𝑠𝑡𝑎𝑔 − 𝑃0 = 1.05 𝑖𝑛 𝐻2 𝑂, according to Eq.2, 𝜌𝑈2 = 0.001913641.
0
2𝜋
In order to calculate ∫0 [𝑃(𝜃) − 𝑃0 ]𝑐𝑜𝑠𝜃𝑑𝜃, we use Simpson’s Rule, we found 𝐶𝐷 = 1.13
d. Calculate the Reynolds number, 𝑹𝒆 = 𝝆𝑼𝟎 𝒅/𝝁
Table 1. This figure shows the values of μ, ρ, diameter d, velocity U0 and calculated value for Reynolds
number Re
μ
0.00001511
density ρ (kg/m3)
1.1225
diameter (m)
0.01262
Velocity (U0)
21.5762
Re
20228.15158
According to the graph below, we can find that the CD=1.15
Figure 3 . This figure shows the relationship between CD and Reynolds number
6
Part2.Velocity Profile and Mass Conservation Measurement
Calculate the exit velocity of the airflow by using the Eq.4 and Eq.5.
Figure 4. . The Exit Velocity Profile Derived from Pitot Tube Measurement
35
30
Velocity m/s
25
20
15
10
5
0
0
20
40
60
80
100
120
Distance [mm]
Calculate the mass flow defect by using the following equation:
𝐷𝑒𝑓𝑒𝑐𝑡 = 1 −
𝑢(𝑦)
𝑈0
Eq. 7
Figure 5. Mass Flow Defect vs. distance
0.25
Flow Defect
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0
20
40
60
Distance [mm]
7
80
100
120
Use Simpson’s rule again to integrate the exit velocity and calculate the error between inflow and
outflow,
𝐻
∫0 𝜌𝑢 𝑑𝑦 = 𝜌𝑈0 𝐻 (Simplified intergration)
𝜀𝑚𝑎𝑠𝑠 =
𝑚̇𝑖𝑛 −𝑚̇𝑜𝑢𝑡
𝑚̇𝑖𝑛
× 100%
𝜀𝑚𝑎𝑠𝑠 = −11.34%
Eq. 8
Eq.9
Eq.10
DISCUSSION
For a bluff body like a cylinder in a cross-flow, a boundary layer forms from the stagnation point
and over both sides. Since the surface normal pressure is the greatest at the stagnation point and drops
as a function of θ until a point near the top and bottom of the cylinder, there is a negative pressure
gradient. In this region, where P-P0 decreases as a function of θ, the boundary layer “adheres” to the
cylinder. However, the surface normal pressure reaches a minimum at the top and the bottom and
gradually increases towards the back of the cylinder. In this region, the boundary layer experiences a
positive pressure gradient opposite the flow direction, which will cause the fluid near the surface to
eventually come to rest and cause the fluid to separate from the surface, leaving behind a wake region.
In Figure 1, P-P0 vs. θ is plotted. In the region where 𝜃 ≤ 70°, P-P0 continues to drop until
reaching the minimum described earlier. After this point, there is a small increase until P-P0 remains
constant, which is the wake region. The small increase in pressure is where the fluid will start to
separate from the surface of the cylinder. At the end of the wake region, around 𝜃 ≤ 300°, the pressure
difference drops slightly and then rises steadily, and here we return to the boundary layer forming on
the bottom face. In Figure 2, (P-P0)cosθ vs. θ is plotted, producing a very different graph. The area under
this curve adds up to the pressure drag coefficient, 𝐶𝐷 where the body is assumed to be bluff and shear
drag is negligible. The graph is cyclic, and has peaks in the pressure differences, appear at 𝜃 =
8
0°, 170°, 360° where the surface is normal to the flow and therefore the pressure normal is the highest,
and has troughs at 𝜃 = 60° 𝑎𝑛𝑑 305° which are near the bounds of the wake region, where the fluid
begins to separate from the surface, and have the lowest pressure normals.
According to Table 1, the experimental Reynolds number Re ≈ 20228, and from Figure 3, this
corresponds to 𝐶𝐷 = 1.15. In contrast, the 𝐶𝐷 obtained from this experiment’s results and using
Simpson’s Rule to integrate numerically, was 1.13. These values are virtually identical and the
discrepancy is likely due to the limits of accuracy of numerical integration. This strong agreement attests
to the power of this experimental setup and the acceptability of neglecting the shear drag.
Now examining the velocity profile in Figure 4, there is a clear dip in the constant free-stream
𝑚
velocity 𝑈0 ≈ 32 𝑠 , in the middle of the pitot tube range. This is due to the fact that the air velocity in
this region is what has merged together after passing over both sides of the cylinder, with some of the
air trapped in the wake region, resulting in a lower velocity reading by the pitot tube. This effect is
obviously less apparent the farther away from the cylinder, for streamlines that weren’t part of the
viscous boundary layer. Similarly, the mass flow defect is also greatest in the pitot tube region blocked
by the cylinder as shown in Fig. 5, as much of the air is trapped in the wake region. Using Simpson’s rule,
the error in the inflow and the outflow of mass is 𝜀𝑚𝑎𝑠𝑠 = −11.34% from Eq. 10. This negative result
suggests that the outflow is higher than the inflow, which can only be due to overestimating the exit
cross-sectional area, which is reduced because of boundary layers forming on the inner sides of the
apparatus.
Conclusion
The apparatus used in this experiment involving systematically rotating about a cylinder and
measuring the pressures normal to the surface as well as those parallel to the flow-lines, proved to be
an extremely powerful setup. The pressure drag coefficient, 𝐶𝐷 calculated through the numerical
9
integration of (P-P0)cosθ over the domain of θ via Simpson’s Rule, was calculated to be 1.13, which
compared to the value from the literature that is expected at the experimental Reynolds number Re ≈
20228 of 1.15, is practically identical. Such agreement is uncanny and illustrates how accurate this direct
measurement of pressure as a function of θ in calculating parameters such as 𝐶𝐷 , provided the
increment in θ is sufficiently small and efforts are made to achieve a reasonable step size for the
numerical integration. The separation of boundary layer flow from the cylinder surface when the
pressure normal reaches a minimum and begins to rise is also well represented from the graphs
obtained in Fig. 1 and Fig. 2. If there is an issue with this experimental setup, is that there is a significant
−11.34% mass flow defect coming from a reduced effective area from boundary layers growing on the
inner surfaces of the chamber, causing the actual exit velocity to be higher than assumed. To reduce this
effect, introducing a correction factor for the area may prove helpful, which can be done by
approximating the size of the boundary layer where the pitot tube measurements are made – that is at
the end of the chamber. This can be done after simply after determining whether the flow on the inside
of the chamber (different surface) is laminar or turbulent, and applying the correct semi-empirical
relation between 𝛿 and 𝑥. In general, ~ 𝑅𝑒 1/2, and since 𝑅𝑒 is a function of 𝑥, it is simple to
approximate what the true effective area is at the end of the chamber.
REFERENCES
1. F. P. Chiang and T. Y. Hsu, Manual for experiments in solid Mechanics, 2014
10
Appendix
Table 2. Data measured and calculated for Pressure Drag Measurement
θ (deg.)
P(θ)
P(θ)-P0
cosθ
(P-P0)COSθ
0
2.89
1.05
1.0000
1.0500
5
2.91
1.07
0.9962
1.0659
10
2.91
1.07
0.9848
1.0537
15
2.86
1.02
0.9659
0.9852
20
2.74
0.9
0.9397
0.8457
25
2.63
0.79
0.9063
0.7160
30
2.4
0.56
0.8660
0.4850
35
2.14
0.3
0.8192
0.2457
40
1.88
0.04
0.7660
0.0306
45
1.73
-0.11
0.7071
-0.0778
50
1.30
-0.54
0.6428
-0.3471
55
1.06
-0.78
0.5736
-0.4474
60
0.84
-1
0.5000
-0.5000
65
0.65
-1.19
0.4226
-0.5029
70
0.46
-1.38
0.3420
-0.4720
75
0.52
-1.32
0.2588
-0.3416
80
0.58
-1.26
0.1736
-0.2188
85
0.68
-1.16
0.0872
-0.1011
90
0.74
-1.1
0.0000
0.0000
95
0.78
-1.06
-0.0872
0.0924
100
0.79
-1.05
-0.1736
0.1823
105
0.8
-1.04
-0.2588
0.2692
110
0.8
-1.04
-0.3420
0.3557
115
0.79
-1.05
-0.4226
0.4437
11
120
0.77
-1.07
-0.5000
0.5350
125
0.77
-1.07
-0.5736
0.6137
130
0.76
-1.08
-0.6428
0.6942
135
0.76
-1.08
-0.7071
0.7637
140
0.76
-1.08
-0.7660
0.8273
145
0.75
-1.09
-0.8192
0.8929
150
0.75
-1.09
-0.8660
0.9440
155
0.74
-1.1
-0.9063
0.9969
160
0.75
-1.09
-0.9397
1.0243
165
0.75
-1.09
-0.9659
1.0529
170
0.75
-1.09
-0.9848
1.0734
175
0.76
-1.08
-0.9962
1.0759
180
0.76
-1.08
-1.0000
1.0800
185
0.76
-1.08
-0.9962
1.0759
190
0.76
-1.08
-0.9848
1.0636
195
0.76
-1.08
-0.9659
1.0432
200
0.76
-1.08
-0.9397
1.0149
205
0.76
-1.08
-0.9063
0.9788
210
0.75
-1.09
-0.8660
0.9440
215
0.74
-1.1
-0.8192
0.9011
220
0.75
-1.09
-0.7660
0.8350
225
0.74
-1.1
-0.7071
0.7778
230
0.75
-1.09
-0.6428
0.7006
235
0.75
-1.09
-0.5736
0.6252
240
0.76
-1.08
-0.5000
0.5400
245
0.76
-1.08
-0.4226
0.4564
250
0.77
-1.07
-0.3420
0.3660
255
0.78
-1.06
-0.2588
0.2743
12
260
0.78
-1.06
-0.1736
0.1841
265
0.79
-1.05
-0.0872
0.0915
270
0.78
-1.06
0.0000
0.0000
275
0.78
-1.06
0.0872
-0.0924
280
0.77
-1.07
0.1736
-0.1858
285
0.72
-1.12
0.2588
-0.2899
290
0.62
-1.22
0.3420
-0.4173
295
0.55
-1.29
0.4226
-0.5452
300
0.55
-1.29
0.5000
-0.6450
305
0.62
-1.22
0.5736
-0.6998
310
0.76
-1.08
0.6428
-0.6942
315
0.98
-0.86
0.7071
-0.6081
320
1.25
-0.59
0.7660
-0.4520
325
1.55
-0.29
0.8192
-0.2376
330
1.82
-0.02
0.8660
-0.0173
335
2.05
0.21
0.9063
0.1903
340
2.34
0.5
0.9397
0.4698
345
2.54
0.7
0.9659
0.6761
350
2.73
0.89
0.9848
0.8765
355
2.84
1
0.9962
0.9962
360
2.9
1.06
1.0000
1.0600
Table3 Data calculated for Profile and Mass Conservation Measurement
Distance [mm] Pstage [Pa] Pstage-Ps [Pa] Density kg/m3 u [m/s]
0
642.0072
541.0072
1.1225
30.28137
2
637.0304
536.0304
1.1225
30.14177
4
651.9608
550.9608
1.1225
30.55866
6
651.9608
550.9608
1.1225
30.55866
13
8
651.9608
550.9608
1.1225
30.55866
10
651.9608
550.9608
1.1225
30.55866
12
656.9376
555.9376
1.1225
30.69637
14
681.8216
580.8216
1.1225
31.37584
16
691.7752
590.7752
1.1225
31.64354
18
691.7752
590.7752
1.1225
31.64354
20
691.7752
590.7752
1.1225
31.64354
22
681.8216
580.8216
1.1225
31.37584
24
681.8216
580.8216
1.1225
31.37584
26
681.8216
580.8216
1.1225
31.37584
28
681.8216
580.8216
1.1225
31.37584
30
681.8216
580.8216
1.1225
31.37584
32
676.8448
575.8448
1.1225
31.24113
34
661.9144
560.9144
1.1225
30.83346
36
637.0304
536.0304
1.1225
30.14177
38
592.2392
491.2392
1.1225
28.85496
40
547.448
446.448
1.1225
27.50803
42
502.6568
401.6568
1.1225
26.09165
44
487.7264
386.7264
1.1225
25.60212
46
457.8656
356.8656
1.1225
24.59384
48
447.912
346.912
1.1225
24.24843
50
447.912
346.912
1.1225
24.24843
52
452.8888
351.8888
1.1225
24.42175
54
472.796
371.796
1.1225
25.10304
56
497.68
396.68
1.1225
25.9295
58
522.564
421.564
1.1225
26.73042
60
572.332
471.332
1.1225
28.26425
62
612.1464
511.1464
1.1225
29.43382
14
64
646.984
545.984
1.1225
30.42033
66
666.8912
565.8912
1.1225
30.96995
68
691.7752
590.7752
1.1225
31.64354
70
696.752
595.752
1.1225
31.77655
72
696.752
595.752
1.1225
31.77655
74
696.752
595.752
1.1225
31.77655
76
696.752
595.752
1.1225
31.77655
78
696.752
595.752
1.1225
31.77655
80
696.752
595.752
1.1225
31.77655
82
696.752
595.752
1.1225
31.77655
84
691.7752
590.7752
1.1225
31.64354
86
686.7984
585.7984
1.1225
31.50998
88
676.8448
575.8448
1.1225
31.24113
90
671.868
570.868
1.1225
31.10583
92
671.868
570.868
1.1225
31.10583
94
666.8912
565.8912
1.1225
30.96995
96
666.8912
565.8912
1.1225
30.96995
98
646.984
545.984
1.1225
30.42033
100
537.4944
436.4944
1.1225
27.19965
15