ME 310: Fluid Mechanics Laboratory Cylinder in Cross Flow Investigation

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ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation






ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



1
I. Objective


A fundamental fluid mechanics problem of importance in many practical applications is that of a
circular cylinder in cross flow, i.e., with the free stream flow direction normal to the cylinder axis.
Examples include wind and water flow over offshore platform supports, flow across pipes or heat
exchanger tubes, and wind flow over power and phone lines. In this experiment, we will investigate
viscous flow around cylinders. The surface pressure distributions, wake velocity profiles, and drag
characteristics of smooth and rough cylinders will be studied.

II. Viscous Flow Over a Circular Cylinder


Depending on the Reynolds number, Re
D
, the flow pattern near a cylinder can vary significantly.


μ
DUρ
Re
D
⋅⋅
=
(1)

In Eq. 1 ρ is the density of the fluid, U is the velocity of the fluid, D is the diameter of the cylinder, and μ is
the dynamic viscosity of the fluid. For Re
D
< 5, the cylinder flow is unseparated, while for 5 < Re
D

< 40
two stationary eddies form immediately downstream of the cylinder. For Re
D
> 40 an unsteady wake flow
occurs, the width and nature of which depends on the Reynolds number.
In an actual viscous flow, the fluid velocity at the cylinder surface is zero by the ‘no-slip’ condition.
For Re
D
> 1000 this leads to formation of a boundary layer, a thin region adjacent to the surface where
viscous shear effects are important and the velocity increases from zero at the surface to the local free
stream value. Over the forward portion of the cylinder, the surface pressure decreases from the
stagnation point toward the shoulder. Thus, the boundary layer in this region develops under a favorable
pressure gradient
0ηP <∂∂
, where η is the streamwise coordinate measured along the surface. In this
region the net pressure force on a fluid element in the direction of flow is sufficient to overcome the
resisting shear force, and motion of the element in the flow direction is maintained.
However, as suggested by the inviscid theory, the surface pressure eventually reaches a
minimum and then begins increasing toward the rear of the cylinder. Thus, the boundary layer in this
downstream region develops under the influence of an adverse pressure gradient,
0ηP >∂∂
. Since the
pressure increases in the flow direction, a fluid element in the boundary layer experiences a net pressure
force opposite to its direction of motion. At some point the momentum of the fluid element will be
insufficient to carry it into the region of increasing pressure, the fluid adjacent to the solid surface is
brought to rest, and flow separation from the surface occurs. The resulting flow field is sketched in Figure
1.
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



2


Figure 1. Sketch of the flow field around a cylinder in cross flow.

Boundary layer separation results in the formation behind the cylinder of a relatively low pressure
region deficient in momentum. This region is called the wake. For separated flow over the cylinder, there
is a net imbalance of pressure forces in the direction of flow due to the relatively high pressure over the
forward portion and low pressure in the wake. This imbalance, in turn, results in a pressure drag on the
cylinder which dominates the total drag at large Reynolds numbers; the skin friction contribution to the
drag for Re
D
> 1000 is generally only a few percent of the total.
In order to quantify the net drag force F
D
on a cylinder, the drag coefficient C
D
is introduced.
Dimensional analysis for this flow situation shows that the drag coefficient is,














⋅⋅
=

=
⋅⋅
2DLUρ
F
2AUρ
F
C
2
1
D
proj
2
1
D
D
(2)

In this equation L is the cylinder length and A
projected
= LD is the projected area of the cylinder normal to
the approach flow.
A correlation of cylinder drag data is shown in Figure 2. As expected from dimensional analysis,
these data collapse to a single curve when plotted as C
D
vs. Re
D
. At low Reynolds numbers, Re
D
< 5,
there is no flow separation, the wake is laminar, and drag is predominantly due to skin friction. As the
Reynolds number is increased in the range 5 < Re
D
< 1000, the drag coefficient decreases continuously.
The total drag is made up of both skin friction and pressure drag. As Re
D
increases in the latter range,
the skin friction contribution decreases and pressure drag becomes dominant.
D
θ
Sta
g
nation Point
Se
p
aration Point
Boundar
y
La
y
er
Wake
U
1

P
1
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



3


Figure 2. Drag coefficient vs. Reynolds number for a cylinder in cross flow.

In the range 1000 < Re
D
< 2x10
5
the drag coefficient is approximately unity. At Re
D
> 2x10
5
, the
drag coefficient curve undergoes a relatively sharp drop. Experiments show that for Re
D
less than this
critical value, the boundary layer on the forward portion of the cylinder is laminar. Separation of the
boundary layer occurs just upstream of the cylinder midsection (θ
sep
~ 80°), and a relatively wide
turbulent wake is formed (Figure 3a). The pressure in the separated region behind the cylinder is
relatively constant and is lower than the surface pressure near the forward stagnation point, thus leading
to a large pressure drag component.
For Re
D
> 2x10
5
, transition to a turbulent boundary layer occurs on the forward portion of the
cylinder. Since a turbulent boundary layer has more momentum near the surface, (i.e., a ‘fuller’ velocity
profile) than does a laminar boundary layer, it can better resist separation under the action of an adverse
pressure gradient. As a result, the separation point is downstream of the cylinder shoulder, in this case

sep
~ 120°), and the wake is relatively narrow (Figure 3b). The net streamwise pressure force on the
cylinder is reduced as compared to the laminar boundary layer case and, as a result, the drag coefficient
is reduced substantially. Figure 4 compares the surface pressure coefficient, C
P
, to the angular position
on the cylinder for laminar, turbulent, and inviscid theory separation conditions.

10
-1

10
0

10
1

10
2

10
-1

10
0

10
1

10
2
10
3
10
4
10
5
10
6
Smooth cylinde
r
Increasing roughness or
free stream turbulence
Re
D
C
D
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



4

(a) (b)
Figure 3. Sketch of Laminar (a) and Turbulent (b) flow separation conditions and features.



Figure 4. Instantaneous surface pressure coefficient vs. angle showing the different separation
conditions.

Transition from laminar to turbulent flow in the boundary layer can be affected by surface
roughness and free stream turbulence effects. Therefore, the reduction in drag associated with transition
to a turbulent boundary layer does not occur at a unique Re
D
. Increasing surface roughness and/or free
stream turbulence shifts the critical Re
D
to smaller values as was shown in Figure 2.


Separation
Laminar B. L. Separation
Transition
Broad Wake
Laminar Se
p
aration Turbulent Se
p
aration
Narrow Wake
(Re
D
< 2 x 10
5
) (Re
D
> 2 x 10
5
)
θ
sep
~80°
θ
se
p
~120°
U
1

P
1
U
1

P
1
Turbulent B. L.

180
Turbulent
Laminar
Inviscid Theor
y

C
P
=1-4sin
2
θ
45 90 1350
-3
-2
-1
0
1
C
P
θ (°)
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



5
III. Drag Determination


A. Surface Pressure Integration Method
Consider the area element dA = LRdθ on the surface of the cylinder shown in Figure 5.



Figure 5. Pressure acting on a surface element of a cylinder in cross flow.

The component of the pressure-area force on this element projected into the streamwise direction is
given by Eq. 3,


( )
(
)
(
)
(
)
dθθcosRLθPθcosdAθPdF
SSD
⋅⋅⋅=⋅⋅=
(3)

where the dependence on θ is included explicitly to emphasize that P
s
is not constant. Integrating over
the entire surface of the cylinder to find the pressure drag, and recognizing the symmetry of the pressure
distribution about θ=0 we have,


( ) ( ) ( ) ( )
∫∫
⋅⋅⋅=⋅⋅⋅⋅=
π
0
S
π
0
SD
dθθcosθPDLdθθcosRLθP2F
(4)

where D=2R. Subtracting the constant incoming static pressure P
1
from the integrand does not change
the magnitude of the drag force since a constant pressure force normal to the surface gives the zero
vector when integrated over the cylinder surface. This is easily seen from Eq. 5.

R
P
U

P
dA=LRdθ

d
θ

ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



6

( )
0dθθcosP
π
0
1
=⋅

, for P
1
= constant. (5)

Combining Eq. 4 and Eq. 5 gives the following.


( )
[
]
( )

⋅−⋅⋅=
π
0
1S
D
dθθcosPθPDLF (6)

Substituting Eq. 6 into Eq. 2 yields the drag coefficient as a function of the pressure difference and the
angular position.


(
)
(
)
( )




=
⋅⋅
=












π
0
2
1
1S
proj
2
1
D
D
dθθcos
2Uρ
PθP
2AUρ
F
C
(7)

The term within the integral excluding the cos(θ) is known as the surface pressure coefficient,


(
)
(
)
( )
2Uρ
PθP
C
2
1
1S
P


=
(8)

Substituting Eq. 8 back into Eq. 7 gives the simplified version of the drag coefficient for surface pressure
integration.


( )
dθθcosCC
π
0
P
D

⋅=
(9)

This equation will be used in conjunction with surface static pressure measurements as one method of
determining the cylinder drag coefficient. This will require numerical integration to solve as shown in the
Appendix to this lab. Note that since surface shear forces have been neglected, this method ignores the
skin friction contribution to the total drag.

B. Control Volume Momentum Equation Method
A second method for determining the drag on the cylinder utilizes the control volume shown in
Figure 6. Because of symmetry about the x-axis, only half of the cylinder is considered; thus, the drag
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



7
force on the corresponding control volume (half-cylinder) is F
D
/2. In addition, the control volume extends
to y = H in the transverse direction, which is just outside the wake at downstream location 2. The usual
choices are made for the positive x- and y-directions.



Figure 6. Sketch showing the alteration in the velocity profile downstream of the cylinder.

Applying the integral continuity equation to this control volume,


0AdVρdρ
t
CSCV
=⋅⋅+∀⋅


∫∫
vv
(10)

The mass accumulation term (first term) vanishes assuming that the flow is steady, and the net mass flux
term (second term) can be evaluated as,


( )
∫ ∫
=+⋅⋅⋅+⋅⋅−

H
0
H
0
side21
0mdyLyUρdyLUρ
&
(11)

Notice that some mass,
side
m
&
must leave the top of the control volume since the mass leaving at 2 is
clearly less than that entering at 1 due to the mass defect in the wake. However, no mass leaves the
bottom of the control volume as this is a symmetry plane (streamline). Solving for
side
m
&
,

y
x
D
1
2

Wind Tunnel Wall
Control Volume
U
1

P
1
U
1
U
1
U
1
U
2
(y)

P
2
(y)
F
D
/2
H
side
m
&

ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



8

( )
[ ]

=
⋅⋅−⋅
H
0
21
side
dyLyUUρm
&
(12)

Now applying the x-component of the momentum equation to this control volume to find the drag
force, F
D
,


0AdVρVdρV
t
F
CS
x
CV
x
x
=⋅⋅⋅+∀⋅⋅


=
∫∫

vv
(13)

Again, the first term on the right-hand side (momentum accumulation term) vanishes for steady flow. The
force terms and the net momentum flux term are evaluated as follows,


( ) ( ) ( )
∫∫∫∫∫
⋅⋅⋅+⋅⋅⋅+⋅⋅⋅−=⋅⋅−⋅⋅+−
side
A
side
x
H
0
2
2
H
0
2
1
H
0
2
H
0
1
D
AdVρVdyLyUρdyLUρdyLyPdyLP
2
F
vv
(14)

Notice that shear forces are neglected in this momentum balance. This is a very good approximation for
the upper control surface, which is essentially in the free stream, and is certainly true along the lower
control surface since it is a symmetry plane. Assuming that the x-component of velocity everywhere
along the top control surface is (V
x
)
side
= U
1
= constant, this factor can be moved outside of the last
integral with the remaining integral factor being equal to
side
m
&
. Thus,


( )
[
]
( )
[
]
( )
[
]
∫∫∫
⋅−⋅⋅−⋅−⋅+⋅−=
H
0
21
1
H
0
2
2
2
1
H
0
21
D
dyyUUUρdyyUUρdyyPP
2L
F
(15)

Combining the last two integrals in Eq. 15,


( )
[ ]
( )
( )
[
]
∫∫
⋅−⋅⋅+⋅−=
H
0
21
2
H
0
21
D
dyyUUyUρdyyPP
2L
F
(16)

By introducing the definitions of the local stagnation pressures P
01
and P
02
(y),


2
1101

2
1
PP ⋅+=
(17)
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



9


( ) ( )
( )
yUρ
2
1
yPyP
2
2
2
02
⋅+=
(18)

and performing substantial algebra, the expression above for F
D
/2L can be recast as,


( )
[ ]
( ) ( )
dy
PP
yPyP
1
2

yPP
2L
F
H
0
2
21
101
202
2
1
0201
D





−−=

































(19)

Forming the drag coefficient C
D
, we then have,


( )
( ) ( )
D
dy
PP
yPyP
1
PP
yPP
2
DLPP
F
2DLUρ
F
C
H
0
2
21
101
202
101
0201
101
D
2
1
D
D



−−








⋅=
⋅⋅
=
⋅⋅⋅
=
















































(20)

Therefore, by measuring the three pressure differences P
01
– P
1
, P
01
– P
02
(y), and P
02
(y) – P
2
(y), and
performing the integral indicated above, the drag coefficient can be obtained. This integral will require
numerical integration to solve as shown in the Appendix to this lab. Note that both the pressure drag and
skin friction drag are included, in principle, in this control volume analysis.





IV. Experiments


A. Objectives
The objectives of this experiment are as follows:
1. To observe the flow patterns over a cylinder in cross-flow in a water tunnel using dye streak lines;
2. To determine the surface pressure distributions for smooth and rough cylinders;
3. To determine the wake velocity profiles for smooth and rough cylinders;
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



10
4. To determine the drag coefficients using both the surface pressure integration and control volume
momentum methods;
5. To compare the experimental findings to those expected based on theoretical considerations or
previous experimental results and to discuss the agreement and/or discrepancies.

B. Property Values
As discussed in Lab 3, Free Air Jet Investigation, gas flows satisfying Mach number less than 0.3
everywhere in the flow field can be treated as incompressible. Since the flow in the wind tunnel satisfies
this condition, the air density will be treated as constant. It can be computed from the ideal gas equation
of state using free stream pressure and temperature measurements,


constant
TR
P
ρρ
1
1
1
=

==
(21)

where R = 287 J/kg-K for air.
In addition, the viscosity of air at the wind tunnel operating conditions is required in order to
compute the Reynolds number. A power-law curve fit of the air dynamic viscosity data given in [5] over
the temperature range 250 K < T
1
< 800 K is given below,


0.7147
1
7
T103.15μ ⋅×=







(22)

where T
1
is in K, and μ has units N-s/m
2
.
Manometers are used extensively in this experiment to measure various pressure differences.
Typically, the manometer fluid is water. In order to convert the manometer height difference h to the
corresponding pressure difference the hydrostatic relation ΔP = ρ
W
gh is used with the value for the
density of water at atmospheric pressure and temperature equal to 998 kg/m
3
.

C. Equipment
1. Water Tunnel
The water tunnel is a closed-loop (e.g., recirculating) duct in which standard tap water flows. A
small electrically driven pump, controlled via a variable frequency drive, is used to set the flow rate. The
test section is a clear Plexiglas duct of 6” square cross-section with an open top. Various models (airfoils,
bluff bodies, cylinders, etc.) can be inserted into this section by removing the current piece and installing
the desired piece. For this experiment, a smooth brass cylinder will be used.
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



11

Figure 7. Closed-loop water tunnel apparatus.

A dye injection system is used to introduce a dye streak line over the cylinder. The probe is
adjustable, and must be positioned carefully to obtain the best flow visualization results. The best results
will be obtained for very low flow rates of water. At high flow rates, the dye disperses too rapidly, and it
becomes difficult to detect streak lines.

Flow Direction
Pump
Honeycomb Flow Conditione
r
6” x 6” Test
Cylinde
r
Dye Reservoirs
Dye Injection Tube
5
18
Plenum
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



12
2. Wind Tunnel
A schematic of the wind tunnel used in this experiment is shown in Figure 8. The wind tunnel is
of the in-draft type. Air is drawn by a centrifugal fan into the settling chamber through a faired inlet and
passes through a honeycomb screen flow conditioning section before being accelerated through the
contraction section into a Plexiglas test section. The flow then passes through the diffuser section into
the centrifugal fan and is discharged vertically through the silencer into the room. The test-section air
speed can be regulated by adjusting the remote speed control of the variable frequency controller.



Figure 8. The wind tunnel apparatus in the lab.

The cylinders used in this experiment are smooth-walled and rough-walled cylinders of nominal
3/4" (19.05 mm) diameter. The cylinder spans the entire test section width. The model has a static
pressure tap drilled in the surface so that the complete surface static pressure distribution (P
s
) can be
obtained by rotating the cylinder. The upstream static (P
1
) and stagnation (P
01
) pressures are measured
by means of a wall static tap and Pitot tube, while the downstream static (P
2
) and stagnation (P
02
)
pressure distributions are determined with a Pitot-static tube. In all cases, manometers are used to
measure the various pressure differences. In addition, a thermocouple is used to measure the approach
flow temperature (T
1
).






P
1
P
0
P
0
P
2
P
S
T
1
Contoured
Inlet Section
Settling
Chamber
Honeycomb screen
flow straightener
Contraction
section
C
y
linder
Remote S
p
eed Control
Plexiglass
Test Section
Acoustic
Silencer
Diffuser
Centrifu
g
al Fan
Electric
Motor
Variable Frequency
Speed Controller
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



13
3. Static and Stagnation Pressure Measurement Devices
A brief discussion of static and stagnation pressures was presented in Lab 3, Free Air Jet
Investigation. By definition, the static pressure is that pressure measured when moving along with the
flow at the local fluid velocity, i.e., with no deceleration involved in the measurements. This pressure can
be conveniently measured by means of a wall pressure ‘tap’, which is a small hole drilled carefully in the
wall, with its axis perpendicular to the surface (see Figure 9). If the hole is perpendicular to the wall, so
that no flow deceleration occurs, accurate measurements of static pressure can be made by connecting
the tap to a suitable measuring instrument. Also, if the wall tap is located in a region where the
streamlines are straight, the static pressure measured at the wall is equal to the value across the entire
test section.


Figure 9. Sketch of the devices used to measure the pressure in a moving fluid.

The stagnation pressure is defined as the pressure measured when a flowing fluid is decelerated
to zero velocity in a frictionless process. Using Bernoulli's equation, the stagnation pressure P
0
can be
related to the local static pressure P and speed U by


2

PP
2
0

=−
(23)

As shown in the schematic, a Pitot Tube, i.e., a hollow tube with a hole in its tip facing directly upstream,
is conveniently used to measure stagnation pressure. With this design the flow stagnates in the tube in a
very nearly frictionless manner.
Simultaneous measurements of the local static and stagnation pressures can be made with the
pitot-static tube. This probe consists of two concentric tubes; the inner tube measures the stagnation
pressure, while the static pressure is sensed by several small holes on the periphery of the outer tube.
Accurate use of this tube depends on the premise that the static pressure at the holes has returned to the
P
0
P
1

P
0
P
Static Wall Ta
p
Pitot Tube
Pitot Static Tube
Connections to pressure
measurement device
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



14
value at the tube tip. Generally, this requires that the static pressure orifices be placed at least three tube
diameters behind the nose. In other words, flow interference effects due to the nose extend about three
diameters along the tube stem.
If the static and stagnation ports of a Pitot-static tube are connected directly to a differential
pressure measurement device, the dynamic pressure, or difference between the static and stagnation
pressure, can be measured. In turn, this measurement can be used to determine the local fluid speed U.


1/2
0
ρ
P)(P2
U









=
(24)

D. Experimental Tasks
1. Water Tunnel Experiments
The model to be evaluated in the water tunnel is a smooth brass cylinder. Insert the model and
establish flow at a very slow rate. Position the dye injection probe just upstream of the cylinder, and
adjust it vertically to obtain as close to a stagnation point impingement as possible.
Observe the wake formation via dye streak lines. Record your observations with hand-drawn
sketches, or digital camera if available, noting the size of the wake, and the general pattern of the flow
region downstream of the cylinder. Some questions to consider are: how far downstream does it take for
the wake to dissipate? If you were to use a roughened cylinder, would you expect different behavior? Is
a either steady or unsteady pattern apparent in the wake region?

2. Wind Tunnel Experiments
Pressure distributions over smooth and rough brass cylinders in the wind tunnel will be
measured. The wind tunnel is to be run at its maximum speed (i.e., f = 60 Hz), so that the highest
possible cylinder Reynolds number is obtained. Once the flow is established, several initial
measurements will be taken to document the incoming flow conditions. These include the static pressure
P
1
, temperature T
1
, dynamic pressure P
01
-P
1
, and P
atm
-P
1
.
The objective of the first wind tunnel experiment is to obtain the static pressure distribution
around the periphery of the cylinder. This is accomplished by connecting the cylinder surface static
pressure tap P
S
and the free stream static pressure tap P
1
across a differential manometer, rotating the
cylinder to various angular positions θ, and recording the pressure difference P
S
– P
1
. With these
measurements the surface pressure coefficient, C
P
, can be found (see Eq. 8). The C
P
distribution should
also be integrated to obtain one value for the drag coefficient, C
D
(see Eq. 9). Note that the angle
θ should be measured in radians when performing the numerical integration (see Appendix)
The objective of the second wind tunnel experiment is to move a Pitot-static tube vertically across
the wake of the cylinder. Using the previous nomenclature, the measurements to be made and recorded
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



15
are the upstream-downstream stagnation pressure difference P
01
– P
02
(y) and downstream stagnation-
static pressure difference P
02
(y) – P
2
(y). With this latter quantity, the wake velocity profile U
2
(y) can be
determined from Eq. 24. In addition, these measurements can be used to obtain a second estimate for
the drag coefficient based on the control volume momentum analysis (see Eq. 20). Once again
numerical integration will be needed to solve the equation (see Appendix.)
A data sheet is supplied at the end of this lab section that details the various measurements to be
made, including suggested locations for the surface pressure distribution and wake velocity recordings.

V. Technical Report



A brief discussion of the material to include in your technical report for this experiment follows.
See the first section of this lab manual ‘Laboratory Technical Report Requirements’ for more information.
Your TA may also have their own requirements for format, style, etc.

Introduction
The introduction should contain a brief discussion of the objectives and motivation for the
experiment. A concise description of pertinent background information, such as cylinder flow patterns
and drag characteristics under various conditions, should be presented. In addition, the key
assumptions, equations, and variables used in the analyses should be described briefly, without in-depth
derivation or excessive detail.

Methods
A methods section must be included with an overview of the procedure and at least one
schematic of the experimental apparatus. An important part of engineering is being able to describe a
process with a picture or drawing. Do not scan the figures from the manual for this step. You must learn
to create your own schematics.

Experimental Results and Discussion
This is the most important section of the report. At a minimum, the results should include the
following plots, charts, and tables:
1. A plot of the surface pressure coefficient distribution as a function of angular position (C
P
vs. θ plot)
for the smooth and rough-walled cylinders, including the theoretical inviscid flow equation, C
P
=1–
4sin
2
θ;
2. A plot of the wake velocity profile, U
2
(y) vs. y, for the smooth and rough-walled cylinders, along with
the constant incoming velocity U
1
for comparison;
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



16
3. A table containing the drag coefficient values obtained from the surface pressure integration and
control volume momentum analyses for the two cylinders.
Comparison of the C
P
values and inviscid theory equation should be discussed. Likewise, the
wake velocity profile for the cylinder (both the one quantified in the wind tunnel, and observed
qualitatively in the water tunnel) should be discussed. In terms of the drag coefficient results, the
agreement or discrepancies between the two methods for obtaining C
D
should be described and
explained. In addition, the C
D
values obtained in these experiments should be compared to values
expected from experimental correlations.

Conclusions and Recommendations
This section should briefly and concisely restate the most significant results of the experiment. Any
recommendations for further study or improvements to be made in the experiment's design or procedure
should be included in this section.

VI. References


1. B. R. Munson, D. F. Young, and T. H. Okiishi, Fund. of Fluid Mech.
, Wiley 3
rd
ed., Ch.9.
2. R. W. Fox and A. T. McDonald, Introduction to Fluid Mechanics
, Wiley, 4th ed., Secs. 2-5.1, 6-6.5, 9-
7.3.
3. F. M. White, Fluid Mechanics
, McGraw-Hill, 2nd ed., Secs. 5.4, 7.6, 8.4.
4. F. M. White, Viscous Fluid Flow
, McGraw-Hill, Secs. 1-2, 3-11.3, 4-6.7.
5. F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer
, Wiley, 3rd ed., Sec. 7.4.

ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



17
VII. Appendix


Numerical Integration
Two of the simplest numerical integration techniques for equally spaced base points are the
trapezoidal rule and Simpson's rule. These are briefly described below.

1. Trapezoid Rule
The trapezoid rule is used to approximate the integral,



n
x
0
x
f(x)dx
(A1)

The interval x
0
≤ x ≤ x
n
is divided into n equal subintervals (n may be even or odd), each of width ∆x = x
i

x
i-1
, as shown in Figure A1. Approximating the integrand as a straight line over each interval, the area of
each section is given by,


Δx
2
)f(x)f(x
f(x)dx
i1-i
i
x
1-i
x







+


(A2)

Applying this equation repeatedly to each subinterval and summing the results, the integral over the
entire interval is approximated as,


[ ]
)f(x)f(x2)f(x2)f(x2)f(x
2
Δx
f(x)dx
n1-n210
n
x
0
x
+⋅++⋅+⋅+⋅≈

L
(A3)

where x
i
=x
0
+i*Δx. The absolute error in using the trapezoidal rule is bounded by
12)fmax(x)n(
3
′′
Δ
,
where
)fmax(
′′
is the largest value attained by the absolute value of the second derivative of f in the
interval x
0
≤ ξ ≤ x
n
. Thus, by reducing ∆x, the error is reduced.
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



18


Figure A1. Sketch of the trapezoid rule area integration. Notice that the hatched area is a trapezoid.

2. Simpson's Rule
To apply Simpson's rule to find the same integral as above, the entire interval x
0
≤ x ≤ x
n
is
divided into n equal subintervals, each of width ∆x = x
i
– x
i-1
, where n is even. However, the number of
subintervals n is now restricted to be even (odd number of base points). For Simpson's rule, a parabola
is fit to the integrand over each successive pair of intervals as shown in Figure A2. The area over one
such pair is then approximated as,


[ ]
)f(x)f(x4)f(x
Δx
f(x)dx
1ii1i
1i
x
1-i
x
+−
+
+⋅+⋅≈

3

(A4)


Applying this expression for each successive pair of intervals and summing gives the composite formula
for the integral over the entire interval as,


[ ]
)f(x)f(x4)f(x2)f(x4)f(x2)f(x4)f(x
3
Δx
f(x)dx
n1-n43210
n
x
0
x
+⋅++⋅+⋅+⋅+⋅+≈

K
(A5)

where x
i
=x
0
+i*Δx. The truncation error for Simpson's rule is bounded by
90)fmax(x)n(
(4)5
Δ
.
Therefore, for a given number of base points, an integral is more accurately approximated using
Simpson's rule than by the trapezoidal rule.

f

x










x
x
0
x
1
x
2
x
3
x
n-1
x
n
1 2 2 2 2 2 1





Line
Integrand
Weights
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



19


Figure A2. Sketch of the Simpson’s rule area integration. Notice that one side of the area is a parabola.








f

x







x
x
0
x
1
x
2
x
n-2
x
n-1
x
n
1 4 2 4 2 4 1
Parabola
Integrand
Weights
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation
Data Sheet


20

Constants and Initial Flow Conditions

Cylinder Texture: Smooth Rough


Diameter (mm): __________ ___________


Atmospheric Pressure, P
atm
(mm H
2
O): __________ From Book ___________


Pressure Difference, P
atm
-P
1
(mm H
2
O): __________ ___________


Static Pressure, P
1
(mm H
2
O): __________ Calculate ___________


Dynamic Pressure, P
01
-P
1
(mm H
2
O): __________ ___________


Temperature-initial, T
1
(°F): __________ ____________


Temperature-final, T
1
(°F): __________ ___________


Surface Pressure Measurements Wake Velocity Pressure Measurements


θ θ
P
S
- P
1
(mm H
2
O)
y y
P
01
-P
02
(mm H
2
O) P
02
-P
2
(mm H
2
O)
(°)
(radians
) Smooth Rough (in) (mm) Smooth Rough Smooth Rough
0 0
10 0.1
20 0.2
30 0.3
40 0.4
50 0.5
60 0.6
70 0.7
80 0.8
90 0.9
100 1.0
110 1.1
120 1.2
130 1.3
140 1.4
150 1.5
160 1.6
170 1.7
180 1.8

1.9

2.0
2.1
2.2
2.3


2.4
ME 310: Fluid Mechanics Laboratory
Cylinder in Cross Flow Investigation



21