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

(θ

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

Uρ

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

Uρ

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

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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

Uρ

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

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