I. Introduction In the field of fluid mechanics, it is often necessary to ...

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24 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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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 distribut
ion 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
experim
ent, 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 w
ell 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 c
ontrol 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
l
ow, 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:






















(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:

























(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 patter
n.

F
or 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
n
o
-
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 begi
ns 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 i
s the governing factor in the total drag, and the skin friction contribution
(shear stress) is
less significant
.


The dimensionless drag coefficient C
D

is introduced in equation (3) to quantify the drag force.

























(3)

With F
D

as the drag force, U
1

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).
















[















]







(4)

Where P
1

is the static pressure and P
S

is the streamwise pressure as a function of

, measured
about the
center of the cylinder s
tarting at the stagnation point. Combining eqn. (4) and (3) allows for the surface
pressure coefficient to be calculated at a specific point:



























(5)

Which can be integrated for the drag coefficient, shown in (6).



















(6)

This method neglects the con
tributing 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 present
ed in Figure 1. The control
volume is set up such that U
1
=U
2
(
H
)
. The continuity equation can be solved for


side assuming steady
flow and that no mass flows through the bottom surface of the control volume (due to the inherent
symmetry.












[








]







(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.










{
[








]






[


(
















)



]

}







(8)

Also, the local fluid speed can be calculated using a Pitot tube, seen in equation (9).










[








]

(9)

Where P
0

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
bef
ore 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 wat
er tunnel apparatus circulates standard tap
water through a honeycomb flow
conditioner and a transparent test section. The test section has an openin
g 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, fol
lowed 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 Figur
e
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 C
P

seems to level
out as it approaches the rear side of
the cylinder.
Starting from θ=0
o

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 c
ylinder would produce a less turbulent flow than the rough
cylinder, and t
he smo
oth cylinder flow separates close to

θ=100
o

while the rough cylinder flow separates
later on, within the range of 120
o
< θ < 135
o
.



The

data points from Figure
4

are now repr
esented 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 exp
ected. 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 cylinder
s
, with respect to y, a measurement in inches above the cylinders
'

vert
ical 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 be
fore the y = .4" =
10m
m data point.

This is a very
nice qualification of the data sets, as the cylinder radii w
ere 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

sho
ws the relationship between

P
01
-
P
02

and the absolute value of
the vertical position, where P
01

and P
02

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, C
d

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
prod
uces two significantly lower values than the control vo
lume 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
dra
g 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 5
5000
, indicative of a
very
turbulent flow
and resulting in a thin boundary layer for each cylinder, as mentioned in the introduction.


In F
igure
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 separatio
n 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 assumpt
ions 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 o
f 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 fo
r 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 t
o 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












Figure 1: flow characteristics and control volume


Figure 2: Wind tunnel apparatus



side


boundary layer


separation point


wake



stagnation point



θ

control volume (dashed)


U
1
, P
1


U
2
(y)


P
2
(y)


Pitot tubes



U
-
tube manometers



cylinder



intake





Figure 3: Water tunnel apparatus


viewing section



U
-
tube manometers



pump



dye reservoir




cylinder




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