I. Introduction
In the field of fluid mechanics, it is often necessary to analyze or predict the characteristics and
effects of a form that is interrupting a stream flow. A simplified (yet still fairly complex) case can be seen
as a cylinder disrupting a cross flow. Examples of this could include a support beam that is encountering
high winds or a similar submerged system, perhaps supporting an oil platform in the ocean. Important
considerations in these cases include drag produced, the pressure distribution and the effect on the fluid
velocity profile.
For this experiment, both smooth and rough cylinders were examined while in a cross flow of air
and water. The air experiment was analyzed quantitatively, with a few more objectives than the water
experiment, in which a qualitative analysis of the streak lines produced by injected colored dye was
performed. In the air experiment, a variety of data sets were taken, which as intended allowed the
formation of a surface pressure distribution for analysis, as well as the generation of a wake velocity
profile for both the smooth and rough cylinders. The next objective is to determine drag coefficients
utilizing two methods: numerical surface pressure integration and control volume momentum analysis.
Finally, the experimental data is to be compared with theoretical data.
The theory behind this experiment takes heading from a few assumptions and the implications
of such when applied to fundamental fluid equations. Because the airflow in the wind tunnel was fairly
low, the assumption was made that the air could be considered incompressible. This allows a
manipulation of the ideal gas law to present the density of the air as:
𝑃
𝜌 = 𝑅∗𝑇
(1)
Where p is the free-stream pressure, T is the temperature in Kelvin and R is the ideal gas constant, 287
J/kg*k.
Equation 2 now presents a method of calculating the dynamic viscosity:
𝜇 = (3.15 ∗ 10−7 ) ∗ 𝑇 .7147
(2)
Together, these two quantities allow for the calculation of the Reynolds number, Re, as the
product of ρ, the fluid velocity and the cylinder diameter divided by μ. The Reynolds number is
important as it is responsible for the viscous flow pattern. For Re < 5, the cylinder flow is not separated,
for 5 < Re < 40 flow separation occurs and two eddies form downstream of the cylinder. For a Reynolds
number above 40, an unsteady wake flow occurs. In all occurrences, a boundary layer forms due to the
no-slip boundary condition. This layer thins significantly as the Reynolds number increases above 1000
and shear stresses become very important. With regards to pressure, the maximum pressure occurs as a
stagnation point at the front of the cylinder (facing oncoming flow). The pressure then decreases along
the cylinder's surface, until it reaches what is known as the separation point, where it begins to increase.
With separated flow, there is a resultant net force on the cylinder in the direction of the flow as a result
of a high pressure zone at the front, and a low pressure zone in the trailing wake. At high Reynold's
numbers, this pressure drag is the governing factor in the total drag, and the skin friction contribution
(shear stress) is less significant.
The dimensionless drag coefficient CD is introduced in equation (3) to quantify the drag force.
𝐶𝐷 =
𝐹𝐷
𝜌∗𝑈12 ∗𝐿∗𝐷⁄2
(3)
With FD as the drag force, U1 the free stream velocity, L the cylinder length and D the diameter. In this
experiment, two methods are used for the calculation of the drag force. First, surface pressure
integration assumes the pressure distribution is symmetric about a horizontal axis and is used to
calculate the drag as shown in equation (4).
𝜋
𝐹𝐷 = 2 ∗ 𝐿 ∗ 𝑅 ∗ ∫0 [(𝑃𝑠 (𝜃) − 𝑃1 )cos⁡(𝜃)]𝑑𝜃
(4)
Where P1 is the static pressure and PS is the streamwise pressure as a function of 𝜃, measured about the
center of the cylinder starting at the stagnation point. Combining eqn. (4) and (3) allows for the surface
pressure coefficient to be calculated at a specific point:
𝐶𝑃 =
𝑃𝑠 (𝜃)−𝑃1
𝜌∗𝑈12 /2
(5)
Which can be integrated for the drag coefficient, shown in (6).
𝜋
𝐶𝐷 = ∫0 𝐶𝑃 ∗ cos 𝜃𝑑𝜃
(6)
This method neglects the contributing viscous effects and is therefore more applicable in situations with
a high Re.
An alternate method observes a control volume and is governed by the momentum equation.
Again assuming symmetric flow about the horizontal axis, this idea is presented in Figure 1. The control
volume is set up such that U1=U2(H). The continuity equation can be solved for ṁ side assuming steady
flow and that no mass flows through the bottom surface of the control volume (due to the inherent
symmetry.
𝐻
ṁ𝑠𝑖𝑑𝑒 = ∫0 𝜌 ∗ [𝑈1 − 𝑈2 (𝑦)] ∗ 𝐿 ∗ 𝑑𝑦
(7)
Assuming the x component of velocity along the top surface of the control volume is constant, an
equation for the drag coefficient which accounts for both pressure and viscous drag can be derived.
𝐻 [𝑃01 −𝑃02 (𝑦)]
−
𝑃01 −𝑃1
𝐶𝐷 = 2 ∗ ∫0 {
2
𝑃02 (𝑦)−𝑃2 (𝑦) 1/2
)
]
} ∗ 𝑑𝑦/𝐷
𝑃01 −𝑃1
[1 − (
Also, the local fluid speed can be calculated using a Pitot tube, seen in equation (9).
(8)
𝑈 = √[
2(𝑃0 −𝑃)
]
𝜌
(9)
Where P0 is the static pressure and P is the stagnation pressure.
II. Methods
The wind tunnel draws air through a flow straightener, contraction section and clear test section
utilizing a centrifugal fan. Two cylinders with a small static pressure tap are inserted into the test
section independently and are free to rotate about an axis through the center of the circular cross
sections. One cylinder is smooth, and one is rough. The upstream and downstream static and stagnation
pressures are measured using a Pitot-static tube. The downstream Pitot-static tube is adjustable along
the vertical axis. Each of these is connected to a U-tube manometer. The temperature was recorded
before and after each experiment. For both cylinders, vertical pressure profiles where acquired by
adjusting the downstream Pitot tube and taking data at a constant height interval. An array of data
points were also recorded from the cylinder's pressure tap, with respect to angular position. A digital
photograph of the wind tunnel is included as Figure 2.
The water tunnel apparatus circulates standard tap water through a honeycomb flow
conditioner and a transparent test section. The test section has an opening which allows a selection of
objects to be inserted for comparison. A small adjustable tube allows for colored dye to be injected
upstream the object, such that streak lines can be easily observed. First, a wing shaped object is inserted
and observed, followed by a smooth cylinder. A digital photograph of the wind tunnel is included as
Figure 3.
III. Experimental Results and Discussion
The experimentally calculated surface pressure coefficient values are plotted with respect to
the angle, theta in Figure 4. The theoretical equation of an inviscid fluid is also shown, for comparison.
While the theoretical curve oscillates, with a maximum at both the front and rear of the cylinder, the
actual data does not follow this trend. Instead, the CP seems to level out as it approaches the rear side of
the cylinder. Starting from θ=0o and moving symmetrically outwards, the trend decrease, increases for a
short while and then decreases again. It is at this second local maxima that the separation point can be
seen, as discussed earlier. It is expected that with a more turbulent flow, the flow should remain
attached to the object for a longer distance than a less turbulent flow, and that seems to be the case.
Predictions were made that the smooth cylinder would produce a less turbulent flow than the rough
cylinder, and the smooth cylinder flow separates close to θ=100o while the rough cylinder flow separates
later on, within the range of 120o < θ < 135o.
The data points from Figure 4 are now represented as Figure 5. Here, surface pressure
coefficients are plotted with respect to the absolute value of their respective angles. This allows the
verification of the assumption that the flow was symmetric about a horizontal axis through the center of
the cylinder. While the data points do not line up perfectly, they are acceptably close. It would not be
expected that two sets of experimental data points taken for the same angle sets would line up exactly,
and as such exactly matching values were never expected. This is simply a qualification of the
assumption.
In Figure 6, the wake velocity data calculated from wake pressure measurements are shown for
both the smooth and rough cylinders, with respect to y, a measurement in inches above the cylinders'
vertical center. Included also is a straight line, U_1 which represents the incoming, constant upstream
velocity profile. It is very interesting to note the inflection point of each of these trends. The trends
transition from concave-up to concave-down just before the y = .4" = 10mm data point. This is a very
nice qualification of the data sets, as the cylinder radii were 9.5mm. Another point important to note is
the intersection between U_1 and the wake velocity profile trends at approximately y = 1". This is an
important value as it is interpreted as the height of the control volume used in the control-volume
momentum analysis.
To qualify the assumption of symmetry about a horizontal axis through the cylinder's center in
the downstream wake flow, Figure 7 shows the relationship between P01-P02 and the absolute value of
the vertical position, where P01 and P02 are the upstream and downstream stagnation pressures,
respectively and y=0 is taken as the vertical center of the cylinder. It may seem that the points do not
coincide, but the smooth and rough trends actually should not mimic each other. All that is important is
that the positive and negative data sets for each respective cylinder follow each other, which is
apparent.
As discussed previously, there are two methods by which the drag coefficient, Cd can be
calculated. The results of each method are summarized in Table 1. From these values, the scope of each
method of determining the drag coefficient is apparent. The Surface pressure integration method
produces two significantly lower values than the control volume momentum equation analysis, for both
the smooth and rough cylinder. This is due to the nature of each method, as each accounts for the
pressure difference and resultant net force, only the control volume method considers the drag due to
skin friction. As in Figure 4 each data set was compared to the theoretical, inviscid flow, it should be
noted that a revisitation and similar integration of this curve would result in an essentially zero value
drag coefficient. In this theoretical case, the flow would remain attached completely, there would be no
separation point and subsequently no pressure difference across the sphere to produce any unbalanced
force.
The Reynolds number, Re, was calculated for each the smooth and rough cylinder flows. These
values are presented as Table 2. Each value was approximately 55000, indicative of a very turbulent flow
and resulting in a thin boundary layer for each cylinder, as mentioned in the introduction.
In Figure 8, a digital photograph of a dye streakline in the water tunnel experiment is shown.
The dye was injected through a small hole in the cylinder facing the oncoming flow. The streakline is
attached to the cylinder for a short while, after which the separation point can be seen clearly. Because
the flow is fairly turbulent, the dye soon becomes fairly dissipated, although 1 vortex can be seen
clearly.
IV. Conclusions and Recommendations
The objectives of this experiment were successfully achieved, and assumptions made in doing so
where shown to be accurate. Key flow features, including separation point and wake were identified
both visually using dye in the water tunnel experiment, and analytically using various numerical
methods with the wind tunnel portion of the experiment. To do this, surface pressure distributions were
analyzed for both the smooth and rough cylinders, allowing the determination of the separation point
for each case. Also, wake velocity profiles were generated for each cylinder, allowing for a control
volume momentum analysis. Drag coefficients were calculated using both methods, and it was shown
that while the skin friction drag may only account for a small percent of the total drag, it is still
significant. These cases were also compared to the theoretical, inviscid case in which there is no
separation point, and therefore no drag. While the data varied slightly, the assumption of a symmetrical
flow was given merit. In the future it would be interesting to do a quantitative analysis of the water
tunnel system similar to what was performed on the wind tunnel experiment to greater tie the
connection between them rather than simply qualitatively viewing the flow.
V. Figures and Tables
ṁ side
control volume (dashed)
U 1, P 1
separation point
boundary layer
U2(y)
P2(y)
θ
wake
stagnation point
Figure 1: flow characteristics and control volume
intake
Pitot tubes
cylinder
U-tube manometers
Figure 2: Wind tunnel apparatus
viewing section
dye reservoir
cylinder
pump
U-tube manometers
Figure 3: Water tunnel apparatus
Figure 4: Surface pressure coefficient as a function of Theta
Figure 5: Surface pressure coefficient as a function of the absolute value of theta
Figure 6: Wake velocity as a function of height
Figure 7: stagnation pressure drop as a function of the absolute value of the vertical coordinate