Drag of a cylinder using pendulum method

Shiv Kumar, Shivangi Sharma,

Shubham Maurya

Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala, 695547, India

The aim of the experiment is to find the drag force on the solid cylinder placed insubsonic wind tunnel using pendulum method. The equilibrium of the cylinder is used toobtain derive drag in terms of known parameters. The results indicated that as freestreamvelocity of air increased, drag also increased. Plot of CD against Re was nearly constant.

Nomenclature

CD Co-efficient of dragD Drag, Nq∝ Dynamic pressure, PaS Frontal area, m2

v Freestream velocity, m/sRe Reynolds numberρ Density of air, kg/m3

µ Dynamic viscosity, N s/m2

d Diameter of cylinder, ml Length of cylinder, mα Deflection of pendulum, degreeP2 Static pressure at test section inlet, PaP3 Static pressure at contraction inlet, Pamb mass of bob with pointer, kgmc mass of wooden cylinder, kgmp mass of pendulum rod , kgrb Length of pendulum rod, mrc Wooden cylinder rod length, m

I. Introduction

Force exerted in the flow direction by fluid on body is called drag force. It is due to combined effect ofpressure and shear force. So total drag is sum of skin friction and pressure drag (or form drag). Skin frictiondepends on body orientation, surface area and magnitude of wall shear stress while pressure drag dependson shape of body and frontal area. Drag force is strong function of angle of attack. It depends on density ρ,upstream velocity v, and the size, shape and orientation of body. The dimensionless number CD representdrag characteristics of a body:

Drag co-efficient, CD =D

12ρv

2S

where S = ldCD vary along a surface as a result of changes in velocity boundary layer in flow direction. So this is anaverage value. CD in some cases may also depend on Re and surface roughness. CD,friction is independentof surface roughness in laminar flow but strong function of it in turbulent flow due to surface roughness

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elements protruding into boundary layer. Re is inversely proportional to viscosity. For blunt bodies, frictiondrag less at higher Re. Skin friction dominates at low Re especially for streamlined bodies. Pressure drag islarger for blunt bodies and smaller for streamline bodies. Pressure drag becomes important when velocity istoo high for fluid to follow curvature of body and separates from body at some point creating low pressure inback because the total pressure in vicinity of front stagnation point is not recovered (Fig. 1). Streamliningdelays boundary layer separation by reducing pressure drag but skin friction increases as surface area isincreased.

Figure 1. Flow separation over cylinder2

The part of drag that is due directly to wall shear stress is called skin friction since it is caused byfrictional effects, and the part that is due directly to pressure P is called the pressure drag (also called theform drag because of its strong dependence on the form or shape of the body).

A. Experimental details

• It consists of a typical blower type wind tunnel.

• A circular wooden cylinder was hanged in the test section about a rod for free rotation about commonshaft due to the flow.

• There is multi-tube manometer which is inclined by 45 ◦ to measure pressure of wind tunnel which isconnected to contraction and suction point of blower.

• A pendulum with a deflection scale was attached to the cylinder using support and hinge mechanismin order to get the deflection of cylinder.

II. Background

The drag force exerted on wooden cylinder placed inside test section exerts a torque about common shaft.As a result, it deflects by an angle α. It comes to this equilibrium position due to torque exerted by weightsof cylinder, bob and connecting rod balancing the torque due to drag. At equilibrium, force balance andmoment balance can be used to derive the drag force (Fig. 2). However, moment balance directly gives theexpression for drag.

The equation for moment balance results in following expression for Drag

D =(mbrb +

mprb2

+mcrc)g.tan(α)

rc

CD vary along a surface as a result of changes in velocity boundary layer in flow direction. So this is anaverage value. CD in some cases may also depend on Re and surface roughness. For flow over cylinders (andspheres), we classify behaviour in different range of Re as follows (Fig. 3)

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Figure 2. Free body diagram of ball and pendulum bob

Figure 3. Variation of CD with Re for smooth cylinder1

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1. Re <= 1 or creeping flow approximation, fluid follows curvature and wraps completely around cylinderto meet on rear side of cylinder.

2. Separation starts at Re=10. Region of separation increases as Re increases upto Re=10,000. Cddecreases as Re increases.

3. For 1000¡Re¡100,000, Cd remains relatively constant. This is characteristic of blunt bodies. Flow inboundary layer is laminar.

4. CD drops suddenly for Re 105-106 because boundary layer becomes turbulent. So stronger adversepressure gradient required as inertial effects start dominating. Separation point moves further rear ofbody, wake size and pressure drag decreases.1

III. Procedure

• A relation was established between drag force and angle of deflection by balancing the torque of cylinderand pendulum about the common shaft.

• Power was switched on and RPM was gradually increased which caused to increased in deflection angle.

• The wind tunnel was operated for different Reynolds number by varying the RPM of the controller.

• For each RPM the pressure readings at necessary points and the angular deflections of the pendulumwere noted down.

• The same procedure was repeated for different deflection angle ranging from 1-12 degree with aninterval of 1 and corresponding reading ware noted down.

• From the pressure difference velocity and hence Reynolds number were calculated.

• The experimental observations of the drag force on the cylinder for each case was calculated and agraph is plotted between CD and Re.

IV. Results

1. The reduction in CD from 2.15 to 1.87 can be accounted by laminar to turbulent transition of flowoccurring around Re 30000-50000. The laminar flow separation creates adverse pressure gradient andhence the drag co-efficient is higher in the region of low Re. As Re increases, inertial effects dominateover adverse pressure gradient. The flow separation occurs further downstream because boundary layerremains attached to the surface. Thus, there is less pressure drag here.If cylinder is considered to be rough; surface roughness in general causes increase in CD in turbulentflow (especially for streamlined bodies). But for blunt body, increase in surface roughness may causeCd drop which is achieved by tripping boundary layer in turbulence at lower Re. Hence we wouldexpect CD to be lower than that at low Re which is evident through the result.

2. If we assume cylinder to be somewhat smooth, then by Fig.3, the Reynold numbers of our practicallie in the range 1000-100,000. So CD remains almost constant with slight variations (from (1.87-2.15)only. Since cylinder placed in transverse direction in flow, thus making it blunt body, this is justifiable.

V. Discussion and Conclusions

A. Conclusions

The co-efficient of drag (CD) of the wooden cylinder was determined for each Re using pendulum method.The cylinder surface could not be classified as being rough or smooth. Thus, the CD plot cannot be comparedto that of smooth cylinder. Furthermore, additional flow visualization techniques are required to explain theCD vs. Re graph and to say why it is varying like this.

Therefore, it is concluded that drag is not so simple phenomenon. It is the result of inertial, pressureand viscous forces around the cylinder. Thus, knowledge of flow transition becomes very important to studydrag on the cylinder.

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Figure 4. Variation of drag co-efficient(CD) with Reynolds number (Re) for the flow over cylinder

B. Limitations

1. The end effects are also present which leads to different flow behaviour at tips. The CD maybe higherthan the determined value owing to increased drag on the end surfaces.

2. The connecting rod also contributes to drag which we have not incorporated in the study.

3. Outside disturbances like wind blowing through the window affected the motion of pointer and it wasoscillating while reading of angular deflection was being taken.

4. Cylinder might have touched the wall of apparatus which might have caused erroneous readings.

5. The readings will also be affected by change in exit conditions of wind tunnel pressure (due to movementof students in front).

6. Large oscillations as angle increased. So mean value for angle was taken.

7. Roughness and small protrusions on cylinder surface would have caused turbulence earlier.

C. Error Analysis

The uncertainty analysis was done and plotted as shown in Fig.4 The maximum uncertainty in CD was 0.93.

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Appendix

1. Sample calculation

For reading-8Initial level of manometer = 18.5 cmAtmospheric pressure, patm = 750 mmHgmass of bob with pointer, mb = 225 gmass of wooden cylinder,mc = 58 gmass of pendulum rod ,mp = 111 gLength of pendulum rod, rb = 260 mmWooden cylinder rod length,rc = 130 mmDiameter of cylinder, d = 50 mmReference area, S = 0.00225 m2

Density of ethanol, ρe = 789 kg/m3

Density of air, ρ = 1.146 kg/m3

Viscosity of air, µ = 1.89 * 10−5 Ns/m2

Deflection of pendulum, α = 80

Drag,D =(mbrb +

mprb2

+mcrc)g.tan(α)

rc= 0.853N

P2 = Patm + ρeg(href − h2)sin(450)P3 = Patm + ρeg(href − h3)sin(450)

velocity, v =

√2 ∗ (P3 − P2)

ρ= 18.8 m/s

Re =ρvd

µ= 51296

CD =D

ρSv2

2

= 1.87

Error Calculationdh2 = dh3 = 0.001mdα = 0.50 ≈ 0.00873rad

Re =

√2gρρe(h2 − h3)d)

µ∂Re

∂h2=−∂Re∂h3

=

√2gρρed)

2µ(√h2 − h3)

= 823.929

∆Re =

√(∂Re

∂h2dh2)2 + (

∂Re

∂h3dh3)2 = 1.1625

CD =(mbrb +

mprb2

+mcrc)g.tan(α)

rcSgρe(h3 − h2)

∂CD

∂α=

(mbrb +mprb

2+mcrc)g.sec

2(α)

rcSgρe(h3 − h2)= 9.62

∂CD

∂h2= −∂CD

∂h3= −

(mbrb +mprb

2+mcrc)g.tan(α)

rcSgρe(h3 − h2)2= 35.83

∆CD =

√(∂CD

∂h2dh2)2 + (

∂CD

∂h3dh3)2 + (

∂CD

∂αdα)2 = 0.0980

Acknowledgments

We would like to acknowledge all the persons involved directly or indirectly in completion of this experi-ment. We would like to thank our lab supervisor Dr.B.R.Vinoth for their guidance and clearing our doubts.We would also like to thank our instructors Roshan Kumar and Prasanthi for providing us the needful forconducting the experiment.

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References

1Yunus A. Cengel and John M. Cimbala, Fluid Mechanics 2e., McGraw Hill Education, New York, 2010.2Ascher Shapiro, National Committee for Fluid Mechanics Films, MIT, 1961

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