This document summarizes an experiment analyzing potential flow theory for fluid flowing around a cylinder. Potential flow theory assumes an inviscid fluid and cannot account for drag. The experiment measured pressure coefficients around a cylinder in a wind tunnel and compared the results to potential flow theory. As expected, the experimental results showed drag due to viscosity that the theory could not capture. The boundary layer separation point varied with Reynolds number, supporting that viscosity affects the flow behavior.

1. 1
Abstract
During this experiment, we will analyze one of
the first applications of the potential flow
theory, the cylindrical test in a tunnel wind.
Throughout this report, we will determine the
differences between the theory and the practical
results getting to describe them by the
information obtained from the pressure
coefficients, because is by the pressure
coefficients that a difference will be noted due
to a fundamental assumption made during the
development of the potential theory, that is to
consider an non-viscous fluid. Also due to this
assumption, we will get to discover the
Alembert´s Paradox that states the non-
presence of drag in the theory but strongly
contradicted by the experimental results.
The realization of this experiment and
report will provide us the sufficient material to
study and comprehend the correlation between
the theoretical and practical results that any
person will get if they commit to study
aerodynamics.
Introduction
Potential Flow Theory
The experiment of a fluid flowing and
surrounding a cylinder can be mathematically
explained by the superposition of a doublet and
uniform flow, presented in the Potential Flow
Theory.
In the figure 1, it can be appreciated
how the superposition of those kinds of flow
create the flow around a cylinder.
Fig. 1. - Superposition of the types of flow uniform and
doublet to create flow around a cylinder.
According to the next chart, the equations that
describes the potential fluid are [1]
:
Chart 1. - Stream functions and potential functions for
elementary flow.
Ψ = Ψuniform flow + Ψdoublet =
U r sinθ – B/r sinθ (1)
Φ = Φuniform flow + Φdoublet =
U r cosθ + B/r cosθ (2)
PRESSION DISTRIBUTION AROUND A CILINDER AND
COMPARISON WITH POTENTIAL FLOW THEORY
Rodríguez Cárdenas Jesús Guillermo, Martínez Ruiz Pablo Elías, Cisneros Moreno Víctor
José.
Potential Flow Theory: Theory that says that the flow does not loose energy when passes
through the surface of a body.
Doublet: Type of flow that in conjunction with uniform flow creates de flow around a cylinder.

2. Jesús Guillermo Rodríguez Cárdenas, Pablo Elías Martínez Ruiz, Victor José Cisneros Moreno
2
The flow line that is on the stagnation point
always have the value of zero confirming the
condition of 𝑟 = √𝐵/𝑈 as a constant. Due to
the fact that the velocity is tangential to the
streamlines, the velocity Vr being
perpendicular to a circle of radius 𝑟 = 𝑅 =
√𝐵/𝑈 equals zero, it means that the circle can
be considerate as a streamline of the flow.
Replacing 𝑅2
𝑈 instead of B and deriving dΨ/dr
and dΦ/dr the functions of the streamline
velocity components are obtained:
According to the condition of R=r, Vr
will be zero, thus:
So with 𝜃 = 0 𝑜𝑟 𝜋 evaluation the velocity will
be null, corroborating that the stagnation points
can be evaluated and resembles with the
information of the experimental test.
Pressure distribution over circular cylinder
Being the velocity function of 𝜃 the local
pressures will be too. Applying Bernoulli’s
equation the local pressure distribution can be
found as shown in the next equation:
During wind tunnel experiments, the
pressure coefficient can be understood as a
dimensionless value that will be changing
among a body in concordance with static and
dynamic pressure in the point of analysis, that is
why Cp is:
While the experiment occurs, a small
hole will be changing its position, measuring
local static pressure at a given angle of the
cylinder, this data can be taken as Cp values if
they are obtained with the previous equation.
Starting again with Bernoulli’s equation
the next simplification can be made, taking any
point for its calculation:
Substituting and simplifying the
equation (3) in (4), we finally obtain:
Making a comparison between the
theoretical results of the potential theory and the
experimental test, we must suppose that they
will be very similar, but the consideration of an
inviscid fluid will affect in a great way the
results giving the property of drag due to a
viscous flow, property that in the theoretical
calculations will not appear. The non-
resemblance of this property is called the
Alembert´s paradox.
Variation by Reynolds
As we said in the last paragraph, the results will
vary between the theory and the experiment due
to the fundamental assumption of a non-viscous
fluid. Therefore, is good to make a prediction of
what we will be obtaining during this
experiment. For an angle between 0º and 180º
the results will change according to the value of
the Reynolds where the differentiation between
subcritical, critical and supercritical Reynolds
numbers is important for the values that will be
expected.
Therefore, since the air particles in the
boundary layer have been already slowed down
by the viscosity encounters an adverse gradient
of pressure, the boundary separation will occur,
nevertheless, not all the separations will occur
in the same place or time, notice that the
(3)
(4)
(5)
(6)
(7)
(8)
(9)

3. 3
PRESSION DISTRIBUTION AROUND A CILINDER AND
COMPARISON WITH POTENTIAL FLOW THEORY
separation will occur when the air particle
cannot overcome an adverse pressure gradient.
That is why the velocity and turbulence of the
particle have an important place in this
situation, and the only way to measure a value
of turbulence is by the Reynolds, therefore is by
this number that the boundary separation will
vary according to next statement: at higher
Reynolds, the detachment will occur in greater
values of θ [1]
.
To support this statements and results
we can see the chart above, which shows the
report made by Achenbach and Schlichting in
1968.
Fig. 2. - Theoretical pressure distribution around a
circular cylinder, compared with data for a subcritical
Reynolds and supercritical Reynolds numbers.
Fig. 3. - Location of the separation points on a circular
cylinder as a function of the Reynolds number.
Experiment Description
This experiment was realized in a medium
capacity wind tunnel with a test area of
approximately 50cmx50cm where the cylinder
tested had a diameter of 10 cm approximately.
The items required to proceed with the
experiment where:
 Cylinder with one static pressure intake
with the dimensions previously mentioned.
 Digital manometer
 Pitot tubes approximately 20 cm away from
the cylinder.
 Protactor from 0 to 360 degrees.
For a better perception in the next image we
will describe the components and procedure to
follow for a correct development of the
experiment.
Fig. 4. - Photo of the wind tunnel with the cylinder in
position of test.
Fig. 5. - Protactor and pitot´s tube location.
CYLINDER
PROTACTOR
PITOT´S TUBE

4. Jesús Guillermo Rodríguez Cárdenas, Pablo Elías Martínez Ruiz, Victor José Cisneros Moreno
4
Fig. 6. - Making a measurement and lecture with the
Digital manometer.
Procedure
1.-Estimate the wind tunnel velocity
2.-Turn the cylinder in order to accommodate
the pressure intake facing the wind
3.-Measure the pressure intake of θ 2 by 2
grades starting from zero. Where we have the
maximum pressure differential that corresponds
to the θ of stagnation.
4. - Find the Cp in this point, which must be 1.
5.-Measure the pressure differential and
calculate the Cp for θ from stagnation point to
180.
6.-For a Cp=0 the hoses must be inverted.
7.-Register every variation for all the θ solicited.
Results and discussions
The values of atmospheric pressure (P), air
temperature (T) and dynamic pressure (q) were
measured in the laboratory.
P=92400Pa
T=299.65K
q=205.8Pa
Based on values of P and T, density (𝜌) and
velocity (𝑉) were calculated:
𝜌 =
𝑃
𝑅𝑇
(10)
𝜌 = 1.0755
𝑘𝑔
𝑚3
𝑉 = √
2𝑞
𝜌
(11)
𝑉 = 19.5624
𝑚
𝑠
After that, kinematic viscosity (𝜇) and Reynolds
number (𝑅𝑒) were calculated:
𝜇 = (
𝑇
𝑇0
)
𝑛
𝜇0 (12)
𝜇 = 0.00003282
𝑘𝑔
𝑚𝑠
The reference distance used for the
calculation of 𝑅𝑒 usually is the diameter of the
cylinder (D), but because it was not measured,
an approximation was made, so the reference
distance used was 10 cm.
𝑅𝑒 =
𝜌𝑉𝐷
𝜇
(13)
𝑅𝑒 = 64105.4754
Below are two tables, one with the
results of pressure coefficient (Cp) using the
experimental data and another with the results
of Cp using the potential flow theory.
The experimental values were calculated
with the following equation:
𝐶𝑝 =
𝑃𝑠0−𝑃𝑠 𝜃
𝑃𝑑0
(14)
For the case of the potential flow theory,
another one was used:
𝐶𝑝 = 1 − 4𝑠𝑒𝑛2
(𝜃) (15)
MANOMETER

5. 5
PRESSION DISTRIBUTION AROUND A CILINDER AND
COMPARISON WITH POTENTIAL FLOW THEORY
Teta CP CP (potencial)
0 1 1
2 0,99659864 0,995128101
4 0,99076774 0,980536138
6 0,98396501 0,956295202
8 0,98104956 0,922523392
10 0,96355685 0,879385242
12 0,92031098 0,827090916
14 0,88532556 0,765895186
16 0,82701652 0,696096193
18 0,77113703 0,61803399
20 0,69727891 0,532088887
22 0,62876579 0,438679602
24 0,54421769 0,338261214
26 0,4494655 0,231322952
28 0,36103013 0,118385809
30 0,25510204 2,07257E-09
32 0,15889213 -0,123257704
34 0,0451895 -0,250786811
36 -0,09426628 -0,381966009
38 -0,20165209 -0,516156206
40 -0,32312925 -0,652703642
42 -0,43683188 -0,79094307
44 -0,54421769 -0,930201003
46 -0,66715258 -1,06979899
48 -0,77356657 -1,209056923
50 -0,8765792 -1,347296351
52 -0,97084548 -1,483843787
54 -1,0840622 -1,618033985
56 -1,16034985 -1,749213183
58 -1,25218659 -1,876742289
60 -1,31292517 -1,999999996
62 -1,38969874 -2,118385803
64 -1,4494655 -2,231322947
66 -1,47862002 -2,338261209
68 -1,49562682 -2,438679597
70 -1,50291545 -2,532088883
72 -1,49222546 -2,618033985
74 -1,45140914 -2,696096189
76 -1,41059281 -2,765895183
78 -1,35617104 -2,827090913
80 -1,30660836 -2,879385239
82 -1,28474247 -2,92252339
84 -1,2866861 -2,9562952
86 -1,2696793 -2,980536137
88 -1,24927114 -2,9951281
90 -1,24246842 -3
92 -1,2361516 -2,995128101
94 -1,24392614 -2,980536139
96 -1,25461613 -2,956295203
98 -1,25753158 -2,922523394
100 -1,24878523 -2,879385244
102 -1,24781341 -2,827090919
104 -1,25510204 -2,76589519
106 -1,2585034 -2,696096197
108 -1,26482021 -2,618033994
110 -1,26336249 -2,532088892
112 -1,26433431 -2,438679607
114 -1,26919339 -2,338261219
116 -1,27939747 -2,231322958
118 -1,28765792 -2,118385815
120 -1,28862974 -2,000000008
122 -1,30466472 -1,876742302
124 -1,29008746 -1,749213196
126 -1,29543246 -1,618033998
128 -1,30272109 -1,483843801
130 -1,27793975 -1,347296366
132 -1,29300292 -1,209056937
134 -1,28474247 -1,069799004
136 -1,28862974 -0,930201017
138 -1,28425656 -0,790943084
140 -1,27793975 -0,652703656
142 -1,27842566 -0,51615622
144 -1,2755102 -0,381966022
146 -1,27162293 -0,250786824
148 -1,25364431 -0,123257717
150 -1,27842566 -1,03628E-08
152 -1,27016521 0,118385797
154 -1,26724976 0,231322941
156 -1,27308066 0,338261203
158 -1,2585034 0,438679592
160 -1,27210884 0,532088878
162 -1,26093294 0,618033981
164 -1,27065112 0,696096185
166 -1,25947522 0,76589518
168 -1,26093294 0,82709091
170 -1,26579203 0,879385237
172 -1,27696793 0,922523388

6. Jesús Guillermo Rodríguez Cárdenas, Pablo Elías Martínez Ruiz, Victor José Cisneros Moreno
6
174 -1,26530612 0,956295199
176 -1,26579203 0,980536136
178 -1,26870748 0,9951281
180 -1,2585034 1
Graphic 1.-Comparison between values of CP from
experimental data and potential flow theory.
To finalize, the solution of the integral to
calculate de drag coefficient (Cd) was solved,
proving that in an inviscid there is no drag.
𝐶𝑑 = 2 [∫ 𝐶𝑝𝑑(𝑠𝑒𝑛𝜃) −
𝜋
2
0
∫ 𝐶𝑝𝑑(𝑠𝑒𝑛𝜃)
𝜋
𝜋
2
]
𝐶𝑑 = 2 {[−𝐶𝑝(𝑐𝑜𝑠𝜃)]0
𝜋
2
+ [𝐶𝑝(𝑐𝑜𝑠𝜃)] 𝜋
2
𝜋
}
𝐶𝑑 = 2 {𝐶𝑝 [− cos (
𝜋
2
) + 𝑐𝑜𝑠(0) + 𝑐𝑜𝑠(𝜋) − cos (
𝜋
2
)]}
𝐶𝑑 = 2𝐶𝑝[cos(0) + cos(𝜋)]
𝐶𝑑 = 2𝐶𝑝(1 − 1)
𝐶𝑑 = 0
Therefore, the drag obtained by the
integral equals zero, which can be explained by
the potential flow theory, assuming that an
inviscid flow is used. However, analyzing the
graphic obtained by using the experimental data
and the graphic obtained from potential flow
theory, can be observed that exists a significant
difference between both. The experimental
results don`t reach values of Cp below -1.3 and
once they pass from positive to negative they
keep being negative and the results from
potential flow theory reach a value of -3, they
become negative but return to the positive side
and are symmetrical. These differences are due
to viscosity, because an inviscid flow does not
experiments separation but a viscous one
separates when flowing through the surface of
an object, so, when it passes through the surface
of the cylinder, the Cp does not reach lower
values because once the flow separates, the
pressure stops decreasing. In addition, the Cp
does not return to being positive because the
flow already separated keeps approximately the
pressure that it had when the separation
occurred, that’s why in the graphic we can see
that after certain point the Cp becomes cuasi-
stable.
Conclusion
As told before, the difference between an
inviscid flow (potential flow theory) and a
viscous flow is something worth of attention.
The study of an inviscid flow helps us realize
how a flow under certain circumstances will
behave, which is easier than analyzing the
problem counting with viscosity, and after
understanding his behavior that way, viscosity
can be added to the equation. Once we start
approaching the problem without ignoring
viscosity, drag appears, and everything gets a
little more difficult to explain. Making
comparisons between viscous and inviscid flow
helps to identify which things can make the
flow act different. For example, from the
comparison made between the graphics showed
before, can be deduced that when the flow never
separates the Cp changes in a symmetrical way
and goes down and up again, but in the case
where drag is not cero after the flow separates
the Cp stabilizes. Also, when comparing the
graphic from experimental data showed above
with graphics at higher Re becomes almost
obvious that at higher Re the separation of the
flow delays, causing the Cp to have a different
behavior along the surface of the cylinder.
References
[1] John J. Bertin and Russell M. Cummings. Aerodynamics
for Engineers. 5th edition, Pearson, 2009.
-3,5
-3
-2,5
-2
-1,5
-1
-0,5
0
0,5
1
1,5
0 50 100 150 200
CP
Teta (°)
CP Vs Teta
Potential flow
theory
Experimental
Data
Table 1.-Data from the calculation of Cp X θ

7. 7
PRESSION DISTRIBUTION AROUND A CILINDER AND
COMPARISON WITH POTENTIAL FLOW THEORY
[2] Barnes W. Mccormick. Aerodynamics, Aeronautics and
Flight Mechanics. 1st edition, John Wiley & Sons, Inc.
1995.
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