DEPARTMENT OFCIVILENVIRONMENTAL& GEOMATICENGINEERING

FLUID MECHANICS II

PRESSURE AROUND A CYLINDER AND CYLINDER DRAG

OBJECTIVES

1. To measure the pressure around the circumference of a circular cylinder in a uniformsteady air flow for different flow velocities and to plot diagrams for distribution of thecorresponding pressure coefficients.

2. To compare the measured pressure coefficient distribution with standard measurements inthe literature (Figure 3).

3. To integrate the distribution of measured pressure coefficient around the cylinder andobtain the form drag coefficient. To plot the relation between the drag coefficient and theReynolds number and to compare this with classical results (Figure 2).

4. To measure the velocity profile in the wake downstream of the cylinder. To understandthe qualitative relation between the form of this profile and the drag.

5. To use spreadsheets for collecting, processing and analysing experimental data.

Lab reports must be prepared individually and submitted before the specified deadline.

THEORY AND DEFINITIONS

1. Drag force

The force exerted on a body by a flowing fluid and acting in the direction of flow is called drag.The drag is due to pressure and shear stresses applied by the fluid to the surface of the body.The part of the drag associated with pressure very much depends on the shape of the body andis often called form drag. For bluff bodies, form drag can constitute a considerable part oftotal drag. The form drag can be obtained by integration of pressure acting on a projection ofthe body perpendicular to the flow direction over all the surface of the body. For a cylinderperpendicular to the flow (Figure 1) the elementary drag force per unit width at angle α is

dD = P cos α r dα

A

Uαd

αddl = r

r

α

P

D O

Figure 1

1

and the total drag per unit width can be obtained by integration over the cylinder circumference.The flow is symmetric and pressure distributions on the upper and lower surfaces are the same.Thus, we can integrate from 0 to π and multiply the result by two. We have:

D = 2 r

π∫0

(P − P∞) cos α dα . (1)

Here we subtracted the pressure in the flow far from the body, which is often used as the referencepressure.

2. Drag coefficient and pressure coefficient

It is convenient for both practical applications and theoretical analysis to describe physicalphenomena in non-dimensional form. This allows us to find universal relations which can beapplied to different physical conditions, for example to experimental models and real life proto-types. Such a non-dimensional parameter describing drag of bodies in a fluid flow is the dragcoefficient. Using your common sense and everyday experience you could deduce that dragshould depend on flow velocity (higher velocity → higher drag) and shape and size of a body(larger body→ higher drag). The drag force can be represented as the product of some averagedpressure by the area of body projection perpendicular to the flow. For example, for a cylinderperpendicular to flow, this area is A = d b, where d is cylinder diameter and b is span. Whenthe cylinder is very long the flow can be considered two-dimensional. In this case we can usedrag per unit width and take b = 1 unit. For identical bodies the pressure distribution dependson fluid velocity. The value of ρ U2

∞ has the swame dimension as the pressure, and one cansuggest that P ∼ ρ U2

∞. We can see now that the drag can be represented as D ∼ ρ U2∞A,

and the corresponding non-dimensional coefficient of proportionality is the drag coefficient. Theconventional definition for the drag coefficient is

CD =D

A ρU2∞/2

, (2)

where U∞ is the velocity of the uniform flow far from the body, ρ is the fluid density and A isthe area of a body projection perpendicular to velocity.

For geometrically similar bodies in the flow of an incompressible viscous fluid the dragcoefficient depends only on the Reynolds number, which describes the importance of viscosity.For our case the Reynolds number can be specified as

Re =ρ d U∞

µ,

where ρ and µ are the density and dynamic viscosity of air, d is cylinder diameter and U∞ is thefree stream velocity. Therefore, specifying the particular form of the relation of drag coefficientwith Reynolds number CD = CD(Re) is a problem of great practical importance. By using asingle plot of CD as a function of Re one can calculate drag for similar bodies of various sizesand for various flow velocities from the formula

D = CDρ U2

∞2

A . (3)

Wind tunnel experiments is one of the conventional methods of obtaining the relation of CD

from Re. Examples of experimental results for the relation of pressure coefficients to Reynoldsnumber for different body shapes are represented on Figure 2.

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Similarly, pressure distribution over a body surface can be specified via the non-dimensionalpressure coefficient

CP =P − P∞ρ U2

∞/2. (4)

Pressure coefficients around similar bodies of different sizes in flows of different velocities willbe the same if the Reynolds numbers of these flows are the same, although the correspondingpressures can be different. An example of pressure coefficients around a cylinder for differentReynolds numbers is shown in Figure 3

To measure pressure coefficients in experiments one should measure pressure difference be-tween two static pressure tapings, one being on the surface of the cylinder and the other in thearea of undisturbed flow, ideally far upstream of the cylinder.

It is convenient to use the measurement at the front stagnation point (point O, Figure 1),where velocity is zero, to calculate the velocity of the undisturbed flow. The Bernoulli equationalong the streamline AO reads:

P∞ +ρ U2

∞2

= P0 .

This givesρ U2

∞2

= P0 − P∞ ,

which is simply the manometer reading at α = 0. Then to obtain the pressure coefficient at anyα it is sufficient to divide the corresponding manometer reading by the stagnation point reading

CP (α) =P (α)− P∞P0 − P∞

.

The stagnation point reading can also be used to estimate the undisturbed flow velocity:

U∞ =√

2 (P0 − P∞)/ρ .

Combining equation (1) with (4) and (2) you can find that the drag coefficient can be obtainedby integrating the pressure coefficient over the cylinder surface:

CD =

π∫0

CP (α) cos α dα . (5)

3. Using a Pitot-static tube for measuring flow velocity

Static pressure

P

0P

PU

P

To manometer

Total pressure0

Figure 4

A Pitot-static tube is the combination of astatic pressure tapping with a total pres-sure tapping (Figure 4). According to theBernoulli equation the pressure at the stag-nation point (the total pressure) is P0 =P + ρ U2/2, and measuring the differencebetween P and P0 one can calculate thevelocity as

U =√

2 (P0 − P )/ρ

4

Figure 5

A

C

B

Figure 6

U

C

A

B

Figure 7

U

B

C

D

Figure 8

A

B

E G

C D

F

H I

J

Figure 9 Figure 10

5

Figure 11

A

B

Figure 12

EQUIPMENT

Experiments are performed in the wind tunnel of the Civil Engineering Fluids Laboratory (Fig-ure 5). A cylinder, 114mm or 60mm in diameter, is placed across the working section (Fig-ure 7,8). A fan (A) powered by an electric motor creates an air flow along the tunnel. Flowvelocity U∞ can be regulated by changing fan shaft power from 20% to 100%. Static pressuretappings on the cylinder surface (C) and on the floor of the working section (D) are used tomeasure the difference between pressures on the cylinder’s surface and in the uniform flow. APitot tube (B) is used to measure the flow velocity in the wake downstream of the cylinder.

Pressure differences are measured by a micro manometer (Figure 6 A). The analogue voltagesignal proportional to pressure difference is transmitted from the micro manometer (A) to thedigital multimeter (B) and then to a computer (C) to be acquired by acquisition software. Thedata then can be copied to a spreadsheet for processing and analysis.

EXPERIMENTAL PROCEDURE

1. Before you start

1. Read the Fluids Laboratory safety rules. Read and sign the Risk Assessment Form. Makesure that you understand possible risks and risk control measures.

2. - Set the pneumatic valve switch of the micromanometer (Figure 9B) to ”=” (upposition).

- Set the operation mode switch of the micromanometer (C) to ”±∆P ” (down posi-tion).

- Set the pressure range switch (D) to the desired position. Use ”1%” for fan powerless than 60% and ”10%” othervise.

- Set the automatic zero switch to ”AUTO ” (up position).

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- Check that plastic pressure tubes are connected to the positive and negative airpressure ports (J). Make sure that the high pressure tube (red) is connected to thepositive (right) port and the low pressure tube (yellow) is connected to the negative(left) port.

- Turn on the mains switch (F), the green indicator light should glow.

- Use the ”ZERO ” control (G) to set zero on the micromanometer indicator (A). Ithelps to have the ”TIME CONSTANT” control (H) in the fully anti-clockwise positionat this stage.

- Set the ”TIME CONSTANT” control (H) to the left horizontal position.

WARNING! THE PNEUMATIC VALVE SWITCH (B) MUST BE SET TO ”=” (UPPOSITION) ALL THE TIME WHEN READINGS ARE NOT TAKEN. DO NOTSQUEEZE OR APPLY PRESSURE TO THE PLASTIC PRESSURE TUBES.

3. Switch on the multimeter, check that DC and V options are chosen. Use ”∨ ” and ”∧ ”buttons to set 3 digits after the decimal point on the multimeter digital indicator. Notethat the voltage output of the micromanometer is ±5V for the full pressure scale, whichis 100, 10 or 1 mm of water depending on the position of the pressure range switch of themicromanometer. Check that the error in the multimeter reading is within 2% of the fullvoltage range.

4. Switch on the computer. Start the “PC1604” data acquisition program (Figure 10). Set“Log interval” to 2 seconds and “Log Count” to 9. Press “START” and then “Cancel”on the file opening prompt. Nine subsequent readings of the multimeter with 2 secondsinterval will be acquired and appear in the ”LOGGER ” window of the acquisition program.

5. Open the “Cylinder Data” spreadsheet in the folder C:\Cylinder Experiment and choosethe worksheet according to your fan power. The spreadsheet is developed for collectingand primary processing of your experimental data. Study it carefully. Average manometerreadings and the corresponding pressures will be calculated automatically after entering thedata from the acquisition program into the spreadsheet. Check that formulas calculatingthe values of average readings, pressures and the Reynolds number are correct. To obtainthe values for velocities and pressure coefficients, and to get the points on the graphs youshould insert correct formulas into the corresponding rows of the table.

Note: This spreadsheet is an example how Excel can be used for processing and analysingexperimental data. You should be able to develop similar spreadsheets.

Try copying data from the window of the acquisition program to the spreadsheet. SeeAppendix 1 for details.

6. Use the optical level (Figure 11) to set the position of the Pitot-static tube at the levelwith the cylinder axis. Fix the slider of the Pitot tube ruler (Figure 12 A) at 40cm.

7. Switch on the wind tunnel motor and gradually increase the power to the required level.The flow velocity at the position of the cylinder for each power can be estimated by thefollowing calibration relation

U = 0.079 P− 0.9 , (6)

where U is the velocity of the flow in m/s and P is the fan power in %.

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2. Pressure around the cylinder

1. Check that the pressure tubes from the cylinder tapping and from the steady flow tappingare connected to the micromanometer.

WARNING! THE PNEUMATIC VALVE SWITCH (Figure 9B) MUST BE SET TO”=” (UP POSITION) WHEN PERFORMING ANY OPERATIONS WITH PRES-SURE TUBES.

2. Set the cylinder pressure tapping at the upstream stagnation point (α = 0◦).

3. Set the pneumatic valve switch on the micromanometer panel (Figure 9B) to ”READ ”(down position).

Note: Wait at least 20 seconds before starting data acquisition.

4. Click the “START” button on the acquisition program window (Figure 10) and then“Cancel” on the file opening prompt. Wait until readings are acquired.

5. Close the pneumatic valve switch (up position).

6. Transfer data to the spreadsheet (see Appendix 1).

7. Rotate the cylinder anti-clockwise with 15 degrees interval and repeat measurements forpoints over the upper cylinder surface including the rear stagnation point (α = 180◦)

3. Velocity profile in the wake

1. Connect the pressure tubes from the Pitot-static tube to the micromanometer. Make surethe the pneumatic valve switch (Figure 9B) is closed (up position).

2. Check that the Pitot-static tube is on the level with the cylinder axis and the reading ofthe ruler is 40cm (Figure 12).

3. Open the pneumatic valve switch on the micromanometer panel (Figure 9B).

Note: Wait at least 20 seconds before starting data acquisition.

4. Click the “START” button on the acquisition program window (Figure 10) and then“Cancel” on the file opening prompt. Wait until readings are acquired.

5. Close the pneumatic valve on the micromanometer panel.

6. Transfer data to the spreadsheet (see Appendix 1).

7. Release the screw of the Pitot-tube holder (Figure 12B). Carefully move the Pitot-statictube and fix it at new position according to entries of the spreadsheet table. Repeatmeasurements for all coordinates shown on the table.

4. After you finish

1. Switch off the wind tunnel fan.

2. Assess your experimental data.

- Check that all data in the header of your spreadsheet are correct.

- Check that the measured uniform flow velocity is reasonably close to a value obtainedby the calibration relation (6).

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- Check that the Reynolds number is calculated correctly.- Insert appropriate formulas for pressure coefficients and velocity into the correspond-

ing rows of the spreadsheet. Check if your results appear in the ”Results” worksheet.- Compare the plot for pressure coefficient with the standard result on Figure 3.- Discuss with your colleagues the shape of the velocity profile in the wake. Does it

look reasonable?

3. Switch off the equipment. Make sure that the pneumatic valve switch of the micromanome-ter is closed (up position).

Note: Your results and results of other groups will be available on the course web page forfurther analysis and preparation of lab reports.

REPORT WRITING AND DISCUSSION

1. Use the spreadsheet with your own experimental results and the two spreadsheets sum-marising all groups results for preparing your report. Study the summary spreadsheet andfind out how it was created. If you feel confident, you can develop a similar spreadsheetof you own.

2. Include the graph of Cp versus α to your report. Compare this graph with the standardresults on Figure 3. Comment on the shape of the graph and on differences between youresults and standard results.

3. Using the data from the summary spreadsheets calculate the drag coefficient using equation(5) and the drag per 1m span of the cylinder by equation (3) for two cylinder diameters.A method of numerical integration using spreadsheets is discussed in Appendix 2. Youcan use the template provided in the “Integration template” worksheet.

4. Plot (i) drag D as a function of free-stream velocity U∞ for two cylinder diameters onone plot and (ii) drag coefficient CD as a function of Reynolds number for two cylinderdiameters on another plot. Compare the graphs and discuss which of them is more usefulfor practical applications.

5. Compare the observed values of the drag coefficient with the standard results presentedon Figure 2. Comment on differences between the measured and standard results. Is thedrag due to shear stresses important for the studied range of Reynolds numbers? Discussthe variations of measured drag coefficients with Reynolds number.

6. Include the graph of the wake velocity profile in your report. Comment on the shape ofthe graph. Why is the velocity in the wake lower then the free-stream velocity? Why isthe uniform velocity outside the wake higher than the free-stream velocity?

7. Give a general discussion of your results. Discuss the physics of drag generation. Whyis the representation of drag in the non-dimensional form of drag coefficient important?What is the role of experiments in obtaining universal relations of drag coefficients toReynolds number? What is the practical importance of such relations?

8. Apply the results of your experiment to the following practical problem:

A radio mast has a 5m long cylindrical section 0.2m in diameter. Find the wind speedwhen the results of your experiment can be used for calculating wind load on the mast.Calculate the total load on the section at this speed assuming a uniform wind profile.

9. Make general conclusions to your report.

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APPENDIX 1

Copying data from the acquisition program to the spreadsheet

1. Select logger output on the panel of the “PC1604” acquisition program and copy it to thebuffer (Ctrl+c).

2. Select appropriate cells of the spreadsheet and paste the data from the buffer (Ctrl+p).Select “Use text import wizard” in “Paste Options”.

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3. Select “Fixed width”. Press “Next”

4. Set column breaks to include numerical data (including minus signs) into one column.Press “Next”.

11

5. Select “Do not import column (skip)” for all columns not containing numerical data. Press“Finish”.

6. Numerical data are now imported to the spreadsheet. Average reading and the pressuredifference will be calculated in the corresponding cells.

7. Save the changes (Ctrl+s).

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x2 x3 x4 x5 x6 x7

4F(x )

1F(x )2F(x )

3F(x )

5F(x )

6F(x )

7F(x )A5

A2A1

A3

A4

A6

F(x)

x

bax

A

1

Figure 13

APPENDIX 2

Using spreadsheets for numerical evaluation of integrals

A value of a definite integral I =∫ ba F (x) dx of some function F (x) from x = a to x = b

can be defined as the area between the graph of this function and an x-axis on the intervala ≤ x ≤ b. If the value of F (x) is known at n points x1, x2, x3, . . . xn on the interval a ≤ x ≤ bincluding points a and b, the value of the integral can be approximately found as sum of areas oftrapezoids A1, A2, . . . An−1, as shown on Figure 13 for the case of n = 7. Representation by asum is convenient for numerical evaluation of integrals by computer, and the method describedhere is called the trapezoidal rule. The evaluation formula for the trapezoidal rule is:

I ≈n−1∑i=1

12

( F (xi+1) + F (xi) ) ( xi+1 − xi ) . (7)

The accuracy of approximation increases for larger number of points n.

An example below shows how Excel spreadsheets can be used for evaluation of integrals. We willevaluate integrals of functions F (x) =

√x and G(x) = x2 from 0 to 10 using 11 points evenly

distributed over the interval. First table on the spreadsheet includes values of x in cells B2-L2,and values of functions F (x) and G(x) at these points in cells B3-L3 and B4-L4 respectively:

Cells C6-L6 and C7-L7 include values of individual additives in the sum (7) and zeros are insertedto cells B6 and B7 to preserve the length of the rows. Formulas C6=(C2-B2)*(B3+C3)/2 andC7=(C2-B2)*(B4+C4)/2 are inserted manually and then copied to the rest of the row:

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After this sums of values in B6-L6 and B7-L7 are calculated and written to the cells M6 and M7respectively, which are the required numerical estimates of integrals:

The number of points in the example is small, and the approximation is rather rude. Howeverthe obtained estimates 20.89 and 335.00 are fairly close to the exact values 21.08 and 333.33.Better approximations can be obtained by taking larger number of points. A different exampleand an integration template are provided in the lab spreadsheet.

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READING

Massey,B.S. Mechanics of Fluids, 8th edition, Taylor & Francis, 2006.

8.8.3 Components of drag

8.8.4 Profile drag of two-dimensional bodies

3.7.1 The Pitot tube and the Pitot-static tube

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