Themofluids 2 Fluids Laboratory B: Flowfieldsand forces on bodies in an airstream

M.A.Gilbertson with material from the TQ training manuals

1 Objectives of the laboratory

1. Observation of the flow field generated about different shapes in a highReynolds number flow.

2. Relation of the flow field around objects to the drag experienced bythem.

3. Understanding of the relation between drag and speed and the conceptof the drag coefficient.

4. Relation of the pressure distribution around a sphere to the drag itexperiences.

5. The use of velocity defect in the wake to calculate drag.

2 Background and theory

2.1 Flow and drag

When there is relative motion between a body and a fluid, as when a bodymoves through a fluid at rest or when a fluid flows over a body, there is aforce on the body. The component of force in the direction of the relativemotion is called drag, and the component of force normal to the direction ofrelative motion is called lift.

The force on a body depends on the shape, attitude and size of the body,the density and viscosity of the fluid, and the velocity of relative motion. Itmay also in some circumstances depend on things such as the surface rough-ness of the body and the unsteadiness or turbulence in the fluid stream, butthese will not be considered in this experiment. It will also depend on otherparameters when the relative velocity is sufficiently high to be comparable

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with the speed of sound in the fluid, but in this experiment, velocities arerelatively low.

In these experiments you will be examining the flow around various ob-jects, and the forces generated on them at fairly high Reynolds numbers: theyare large enough for viscous forces generated in the boundary layers aroundthe shapes (skin friction) not to be significant, but not so large that theboundary layers are turbulent. At these Reynolds numbers, the flow aroundthe back of the shapes can separate and cause a wake to form. This flowpattern generates an asymmetrical pressure distribution around the objectthat causes it to feel a force.

2.2 Force coefficients

Under the conditions being investigated, the force on a particular shape bodyat a particular attitude will be a function of the size of the body, the densityand viscosity of the air, and the velocity i.e. we can write

F = f (l, ρ, µ, u) .

By dimensional analysis, because any relationship between the above fivevariables must be dimensionally homogeneous in three independent dimen-sions (mass, length and time), we can deduce that there will be a relation-ship between two (from Buckingham’s Pi theorem, five minus three) non-dimensional parameters i.e. we can write

F12ρu2l2

= f

(ρul

µ

).

where a factor of one half is introduced in the parameter involving the force torelate it directly to the dynamic pressure in the flow, and l2 is re-interpretedas an area, A. The left hand side of the equation is an Euler number, theforce coefficient, CF , and the right hand side is the Reynolds number, Re sothat

CF = f (Re) .

In terms of the components of force, we have the drag coefficient and liftcoefficient defined as the Euler numbers

CD =Drag12ρu2A

CL =Lift

12ρu2A

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For bluff bodies such as spheres, the area A is the frontal area or maximumcross sectional area, and the characteristic length taken in the evaluation ofthe Reynolds number is the diameter. For lifting bodies, the area taken isusually the frontal projected area at zero incidence (i.e. span × thickness),and the characteristic length for Reynolds number is the thickness.

Thus we have that in general any force coefficient is a function of Reynoldsnumber, though sometimes it is a function of other parameters as well, asmentioned above. However, for certain ranges of Reynolds number, theforce coefficients are independent or almost independent of Reynolds num-ber. Where this is so, then the force is directly proportional to the square ofthe velocity, and to the density of the fluid and the representative area of thebody. Of course, the force coefficients are different for different shape bodiesand for different attitudes or angles of incidence.

2.3 Drag on a body

There are generally two major contributions to drag: skin friction drag andpressure drag. Skin friction drag is caused by the generation of boundarylayers on the surface of the object. It is therefore generated by viscosityand so is dominant at low Reynolds numbers. Pressure drag is caused bythe inertia of the fluid which means that at high speeds, a sufficiently largepressure gradient can often not be generated on a curving body that will allowthe fluid to follow it, so it separates, causing the formation of a turbulent areof fluid called a wake. There is low pressure in the wake compared with thestagnation pressure at the front of the object, and so generates a net dragforce. This contribution is controlled by the fluid’s density and is dominantat high Reynolds numbers.

For high Reynolds numbers when the contibution to drag from the pres-sure distribution is much larger than the contribution from shear stress inthe boundary layers (this is experimentally a convenient assumption to makebecause pressure is easy to measure, shear force is not), then for the cylindershown in figure 1, the drag per unit length on an element of length δs is

δD = p cos θδs,

or for the entire cylinder,

D =

∮p cos θds

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Figure 1: Schematic diagram of flow around a cylinder (from TQ Educationand Training manual).

or in a dimensionless form

D12ρU2d

=1

d

∮p

12ρU2

cos θds,

CD =1

d

∮Cp cos θds,

where the pressure coefficient,

Cp =p

12ρU2

.

For a cylinder

δs =d

2δθ

where d is the diameter of a cylinder, so that

CD =1

2

∫ 2π

0

Cp cos θdθ,

or, by symmetry,

CD =

∫ π

0

Cp cos θdθ. (1)

For an inviscid flow, it can be shown that the pressure coefficient distributionaround a cylinder will be given by

Cp = 1− 4 sin2 θ.

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Figure 2: Schematic diagram for estimating the drag on a cylinder from thewake defect (from TQ Education and Training manual).

2.4 Drag estimation from a wake traverse

Following Newton’s second law of motion, the drag on an object will causethe momentum of a gas flow to drop. It is then possible to estimate dragby measuring and comparing the momentum in a flow before and after anobject; so, from Bernoulli’s equation, for the object shown in figure 2, thedrag force

D = 2hp0 +

∫ h

−h

ρU2dy −(

2hpe +

∫ h

−h

ρu2dy

).

Putting this in terms of a drag coefficient, per unit length of cylinder,

CD =D

12ρU2d

=2h

d

p0 − pe

12ρU2

+2

d

∫ h

−h

1− u2

U2dy.

Now assuming that the static pressure drop along the duct is much smallerthan that generated by the action of drag so the first term is negligiblecompared with the second, and non-dimensionalising distance using η = y/h,

CD =2h

d

∫ 1

−1

1− u2

U2dη.

Now defining the velocity defect u′ = U − u, and substituting it in, for theexperimental situation,

CD = 2

∫ yd

− yd

(u′

U

(1− u′

U

))dy

d. (2)

This form of the equation only applies to objects in straight-sided ducts.

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3 The experiment

There are four parts to the experiment:

1. Observing the flow patterns of obstacles in a flow field.

2. Measuring the drag force on a cylinder and an aerofoil directly.

3. Estimating the drag on an object from measurements made in its wake.

4. Calculating drag from the pressure distribution about the cylinder.

Each section of the laboratory requires some practical work and for youto answer questions for the write-up, which you must hand in.

3.1 Flow visualisation

In this experiment, you will visualise the flow around objects using smoke.The purpose of the experiment is for you to see the flow field and connect itwith the forces that are exerted in the flow.

YOU WILL RECEIVE INSTRUCTION ON HOW TO USE THE APPA-RATUS FROM THE DEMONSTRATOR. DO NOT ATTEMPT TO USETHE EQUIPMENT UNTIL THIS HAS TAKEN PLACE. IF AT ANY TIMEYOU ARE UNSURE OF WHAT YOU ARE DOING, ASK THE DEMON-STRATOR.

There are three shapes for you to experiment upon: a cylinder, a flat plate(with angled ends), and an aerofoil. At each test either sketch or photographthe flow patterns observed (neither form of recording is preferred over theother). It is also important that you take careful note of the extent andform of separated regions: where are there stagnation and separation points?What do any vortices look like? Where is the flow unsteady?

3.1.1 Cylinder

Put the cylinder within the working section and switch on the smoke. Ob-serve and sketch the flow pattern seen for three or four different flow rates.Note carefully any changes that you can see in the size and form of the wake.

3.1.2 Flat plate

Put the flat plate into the apparatus with the widest side facing upstream.Sketch or photograph the wake at a low and a high speed. Note againcarefully any changes you can see in the size and form of the wake. What ishappening in front of the flat plate?

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Figure 3: Diagram of the different apparatus configurations for directly mea-suring drag (from TQ Education and Training manual).

3.1.3 Aerofoil

The aerofoil differs from the previous shapes because it now can be asymmet-rical with respect to the air stream. Fix the air speed and then record theflowfield around the aerofoil when the angle of incidence is 0, 5, 10, 15, and30 degrees to the vertical. Again, note carefully any changes in the extentand structure of the wake region.

3.2 The drag on bodies in an airstream

3.2.1 Drag and pressure distribution

In this experiment you will measure the drag on a cylinder and an aerofoilin an airstream and then calculate drag for the cylinder directly from thepressure distribution around it.

YOU WILL RECEIVE INSTRUCTION ON HOW TO USE THE APPA-RATUS FROM THE DEMONSTRATOR. DO NOT ATTEMPT TO USETHE EQUIPMENT UNTIL THIS HAS TAKEN PLACE. IF AT ANY TIMEYOU ARE UNSURE OF WHAT YOU ARE DOING, ASK THE DEMON-STRATOR.

Configure the apparatus as shown in figure 3. Fit the cylinder in theapparatus and adjust the weights so that equilbrium is achieved. Note thevalue of the weights; this will need to be subtracted from subsequent readingsto give the force on the cylinder owing to the air flow past it. Switch onthe apparatus on a low gas speed and note down the upstream stagnationpressure (tapping from the top of the apparatus) and static pressure (from

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Figure 4: Diagram of the different apparatus configurations for indirectlymeasuring drag from a wake traverse (from TQ Education and Training man-ual).

which you can calculate air speed from Bernoulli’s equation) and the weightnecessary for equilibrium to be achieved. Repeat when incrementing the airflow to give around eight readings. Repeat for the aerfoil at an angle ofincidence of zero degrees.

Measure the diameter and length of the cylinder and the maximum thick-ness and span of the aerfoil.

3.2.2 Wake traverse

Now reconfigure the apparatus again as shown in figure 4. Place the cylinderin the model and set a fairly high constant wind speed. Connect up themanometer so that it is comparing the total pressure (as measured in thePitot tube) with the static presure at the exit. Traverse the Pitot tube acrossthe wake of the cylinder, noting the pressure difference at each increment.Away from the centreline of the cylinder, the total pressure will be nearlyconstant, and the increments can be about 5 or 10mm; however, close tothe centreline, the pressure will be changing significantly, and the incrementsshould be only about 2mm. Because of turbulence, the readings will fluctuatequite alot. Try to record the average. The fluctuations can be damped out alittle by gently pinching the tube to the manometer, but if this is overdone,the results will be distorted.

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Figure 5: Diagram of the different apparatus configurations for measuringthe pressure distribution around a cylinder (from TQ Education and Trainingmanual).

3.2.3 Pressure distribution around a cylinder

Reconfigure the apparatus as shown in figure 5. Connect the model tothe manometer, set the wind speed to maximum, and the protractor to 0o.Record the static pressure, and make sure that this and the stagnation pres-sure remain constant throughout the experiment (so that the air speed isconstant). Record the surface pressure on the cylinder and then between 0and 180◦ rotate the model by 5◦ and note the surface pressure each time.

4 Write-up

The following piece of work is compulsory for all students and in additionto any write-up required by your tutor. This is a group report and onlyone report is required from each laboratory group of typically four students.You must deposit your work in the box on the second floor landing in theQueen’s Building within a week of you performing the lab, i.e. by midday thefollowing Monday or Thursday depending on which day it was performed.Late reports will not be assessed. We will discuss the laboratory and write-up

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at a compulsory seminar a fortnight after the laboratory in Design Studio 1.There will also be an opportunity for peer assessment. If you do not attendthe seminar, your individual mark for the lab will be halved.

Please address the points below (and only the points below: a full labo-ratory write-up is not required). No credit will be given for verbosity. Pleasemake sure that your report is clearly labelled with your names, tutors, andlaboratory group number. You will almost certainly not be able to answerthese questions on the basis of your current knowledge and the theory sectionabove, and will need to refer the recommended textbooks for the unit.

For air, use the standard sea level values for density (1.225 kg/m3 ) andviscosity (1.79× 10−5 kg/ms).

4.1 Flow visualisation

1. Display the sketches or photographs of all three shapes. Annotate anyinteresting features.

2. For the sphere, how does the wake region vary with increasing flowspeed? Describe this in terms of factors such as the size, nature andextent of unsteady regions, and the position of stagnation and separa-tion points.

3. How does the wake region around the flat plate compare with that forthe sphere? Does changing the flow speed make any difference?

4. What happens to the wake around the aerofoil as the angle of incidencechanges? How will this affect lift? What is stall and when is this pointreached?

4.2 Drag measurement

1. For the cylinder, from your readings calculate the gas flow speeds andthe corresponding drag forces on it. Hence calculate the Reynolds num-bers and corresponding drag coefficients. Present all this informationin a table.

2. Repeat for the aerofoil .

3. On a single graph, plot the variation of drag coefficient with Reynoldsnumber for the two shapes.

4. How does the drag coefficient vary with Reynolds number? For a cylin-der, over what range of Reynolds numbers would this be true?

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5. Why is the drag coefficient different for the aerofoil from that for thecylinder? How can this difference be related to the flow field aroundthe bodies? To minimise drag, what can you say about the shape ofthe forebody and afterbody of an object in an airstream?

6. If you were to do this experiment in a viscous liquid so that theReynolds numbers were small, how would drag coefficient vary withReynolds number?

4.3 Pressure distribution around a cylinder

1. Set out a table where for each angle θ, you calculate the pressure coef-ficient, Cp and the quantity Cp cos θ.

2. Plot out the values you have measured of Cp against θ. On the samegraph, plot out the expected variation of Cp against θ for an inviscidfluid. Account for the difference between the two curves and relate thisto the visualised flow field you observed around a cylinder.

3. Plot out Cp cos θ against θ for your measurements. Using a suitableapproximate method (counting squares, Simpson’s rule, trapezium rule&tc), calculate the area under this graph, whence calculate the dragcoefficient, as from equation 1.

4.4 Wake traverse

1. Set out your results for position y, normalised with respect to the cylin-der diameter d, and the corresponding difference between the upstreamstagnation pressure and the static pressure recorded at the surface ofthe cylinder.

2. For each position, from the pressure differences, calculate the gas ve-locity in the wake u′ using Bernoulli’s equation so that

u′ =

√2(Pe − pe)

ρ.

3. From the constant pressure difference away from the centreline of thecylinder, calculate the freestream gas speed U using Bernoulli’s equa-tion.

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4. Plot outu′

U

(1− u′

U

)against y

d, and approximately integrate the area under this curve to

calculate the drag coefficient using equation 2.

4.5 Comparison of drag coefficients

1. Compare the three measured values of drag coefficient.

2. For the Reynolds numbers you have been using, what would you haveexpected the value of drag coefficient to be?

3. Account for any difference between the theoretical and measured dragcoefficients.

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