1 Drag of a Cylinder Using Pendulum Method

Drag of a cylinder using pendulum method Joshi Yash∗, Kiran Kalbhor†, Kalicharan Hansda‡, Kartikey Sharma§ Aerodynamics Lab, Department of Aerospace Engineering, Indian Institute of Space Science and Technology

Experiment was performed to calculate the drag force on the solid cylinder. Drag force variation for different Reynolds numbers was also observed. Experiment was conducted at low subsonic free stream velocities. Drag was observed to have increased with the increase in free stream velocity of air.

Nomenclature D mb mr mc rb rc g θ θm CD Dwc S Re v ν ρ P Patm Tatm Mair R

Drag force, N Mass of bob with pointer, Kg Mass of pendulum rod, Kg Mass of wooden cylinder, Kg Length of pendulum rod, m Wooden cylinder rod length, m Acceleration due to gravity, m2 /s Deflection angle Inclination angle of manometer tubes Coefficient of drag Diameter of Wooden Cylinder, m Projected area of cylinder in flow direction, m2 Reynolds Number Velocity of air, m/s Kinematic viscosity, m2 /s Density of air, Kg/m3 Static Pressure of air, mm of ethanol Atmospheric pressure, mm of Hg Atmospheric temperature, ◦ C Molecular mass of air, g/mole Universal Gas Constant, J/kg.K

Subscripts ∞ Free stream properties

I.

Introduction

When a body is placed inside the flow domain of the fluid, fluid exerts force on the body and vice versa. Generally this force is classified into lift and drag. Drag force is the force exerted by the fluid on the body in the flow direction. Study of drag force is very crucial for aircrafts, rockets as well as vehicles. Most of the times, main aim of design is to reduce drag. ∗ Student,B.

Tech. Tech. ‡ Student,B. Tech. § Student,B. Tech. † Student,B.

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SC13B022 SC13B023 SC13B024 SC13B025

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Here drag force on a cylinder was measured. A wooden cylinder was attached with a pendulum. And the cylinder was placed inside the wind tunnel. The experimental set up has been covered in the later portion of the report. Cd , the drag co-efficient is given by the formula stated below: CD =

D 1 2 2 ρ∞ v∞ S

(1)

CD consists of two kinds of drag forces. Skin friction and pressure drag. Pressure drag is produced because of differential pressure generated between upstream of the body and downstream of the body. Skin friction, as name suggests, is caused by the friction (wall sheer stress) between body and the flow. Skin friction dominates for low Re values (especially for stream lined bodies). Pressure drag becomes dominating at high velocities and for blunt bodies. Flow fails to keep itself aligned with the surface and separates. Separation in turn creates low pressure in the downstream of the body which contributes to pressure drag.

II.

Background

The air flow over the cylinder exerts drag force on the cylinder causing it to rotate about the fixed shaft. Depending on the amount of drag force the cylinder deflects to a particular angle till it comes to equilibrium due to drag and weights of cylinder, rod and bob as shown in fig. 1. Thus by using force balance or moment balance the drag force on the cylinder can be calculated if angle of deflection is known.
mb rb + mr2rb + mc rc g tan θ (2) D= rc

Figure 1. Free body Diagram at equilibrium (Only required forces are shown)

The characteristic length for a circular cylinder or sphere is taken to be the external diameter D. Thus, the Reynolds number is defined as Re = VD/ν where V is the uniform velocity of the fluid as it approaches the cylinder or sphere. The critical Reynolds number for flow across a circular cylinder or sphere is about Recr ≈ 2 × 105 . That is, the boundary layer remains laminar for about Re . 2 × 105 and becomes turbulent for Re & 2 × 105 .

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Figure 2. Average drag coefficient for crossflow over a smooth circular cylinder and a smooth sphere.1

The nature of the flow across a cylinder or sphere strongly affects the total drag coefficient CD . Both the friction drag and the pressure drag can be significant. The high pressure in the vicinity of the stagnation point and the low pressure on the opposite side in the wake produce a net force on the body in the direction of flow. The drag force is primarily due to friction drag at low Reynolds numbers (Re < 10) and to pressure drag at high Reynolds numbers (Re > 5000). Both effects are significant at intermediate Reynolds numbers. Variation of CD with Reynolds number is shown in fig. 2 and visualisation of different regimes of flow are shown in fig. 3

Figure 3.

Visualisation of flow over cylinder at different ranges of Reynolds number2

• For Re 6 1 and the drag coefficient decreases with increasing Reynolds number. There is no flow separation in this regime.

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• At about Re = 10, separation starts occurring on the rear of the body with vortex shedding starting at about Re ∼ = 90. The region of separation increases with increasing Reynolds number up to about Re=103 . At this point, the drag is mostly (about 95 percent) due to pressure drag. The drag coefficient continues to decrease with increasing Reynolds number in this range of 10 < Re < 103 . • In the moderate range of 103 < Re < 105 , the drag coefficient remains relatively constant. This behavior is characteristic of blunt bodies. The flow in the boundary layer is laminar in this range, but the flow in the separated region past the cylinder or sphere is highly turbulent with a wide turbulent wake. • There is a sudden drop in the drag coefficient somewhere in the range of 105 < Re < 106 (usually, at about 2 × 105 ). This large reduction in CD is due to the flow in the boundary layer becoming turbulent, which moves the separation point further on the rear of the body, reducing the size of the wake and thus the magnitude of the pressure drag. This is in contrast to streamlined bodies, which experience an increase in the drag coefficient (mostly due to friction drag) when the boundary layer becomes turbulent.1

III.

Experimental Set-up and Procedure

The experimental set-up consists of a blower type wind tunnel. Wind tunnel had radially blowing fan which was driven by a motor whose RPM could be changed vary the air speed. There were tapping at entry of contraction cone and test section for measurement of static pressure. A multi-tube manometer kept at 45 deg inclination was used to measure these pressures. A spirit level was also provided to make sure that ethanol in the manometer tubes was at same level initially. A wooden cylinder was suspended in test section; Pointer attached to the pendulum was used to find deflection angle of cylinder from the scale as shown in fig.4.

Figure 4. Test-section of experimental set-up for drag on a cylinder using pendulum method experiment

• Before starting the experiment, the formula to find the drag on the cylinder using the angle of deflection was found out. • It was checked that cylinder was hanging vertically and the levels of ethanol in manometer tubes was same. • Motor was switched on and rpm of the motor was changed gradually and for some deflection of cylinder its corresponding angle, pressure were measured using the pointer attached to the pendulum and manometer tubes respectively. • This was repeated upto 12◦ of deflection. 4 of 10 Aerodynamics Lab Report, IIST

• While noting down the deflection values, the oscillations of the cylinder were observed and correspondingly angle of oscillation was also noted. • From the observed values, Reynolds number and coefficient of drag were calculated and were plotted.

IV.

Results

At steady state a flow field forms around the object, force and coefficients changes with different flow parameters. For incompressible flows Reynolds number is an important parameter that decides pressure distribution over an object, hence aero dynamic force on a body.

C o e f f ic ie n t o f D r a g

2 .4 2 .2 2 .0

C o e ffic ie n t o f D r a g

1 .8 1 .6 1 .4 1 .2 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 2 0 0 0 0

3 0 0 0 0

4 0 0 0 0

5 0 0 0 0

6 0 0 0 0

7 0 0 0 0

R e y n o ld s N u m b e r

Figure 5. Plot of CD Vs. Re along with uncertainties

1. When compared to reference ideal graph (refer plot 2), the plot of CD Vs Re shows slight variation. Value of CD is almost constant for 2/3rd of the readings.

1 .2

1 .0

D ra g (N )

0 .8

0 .6

D ra g 0 .4

0 .2

0 .0 2 0 0 0 0

3 0 0 0 0

4 0 0 0 0

5 0 0 0 0

6 0 0 0 0

R e y n o ld s N u m b e r

Figure 6. Plot of Drag vs Re

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7 0 0 0 0

2. Initially at lower values of Re, oscillations of pendulum were negligible. At higher Re values, both frequency and aptitude of oscillations increased, causing significant error. D ra g 1 .2

1 .0

D ra g (N )

0 .8

0 .6

0 .4

0 .2

0 .0 0

1 0 0

2 0 0

3 0 0

V e lo c ity ^ 2 ( m

Figure 7.

4 0 0 2

5 0 0

/s 2)

Plot of Drag vs v2

3. Plot 6 and plot 7 shows that Drag is proportional to Re and square of velocity for the experimantal range of data.

1 .2

1 .0

D ra g (N )

0 .8

0 .6

D ra g

0 .4

0 .2

0 .0 0

2

4

6

8

1 0

1 2

A n g u la r D e fle c tio n ( D e g r e e )

Figure 8. plot of Drag vs Angular Deflection

4. Drag force varies linearly with the angular deflection of the cylinder as shown in fig.8.

V.

Conclusion

With increasing Reynold’s number variation of CD of the cylinder is not exactly monotonic. The fluctuations are due to different degree of turbulence and fluctuating form drag over the cylinder. 1. Reynolds no. varies from 21000 to 69000 in this regime the flow is a region where the vortex sheet is fully turbulent refer fig 3. Its a turbulent region some fluctuations are expected and inherent. 2. For this kind of fully turbulent flow, the value for Cd is relatively constant. 6 of 10 Aerodynamics Lab Report, IIST

3. As CD is nearly constant in the experimental range of Re values, hence drag force is found to be proportional to square of the velocity as expected from equation 7. 4. The drag force on the cylinder is proportional to tan which implies that drag force increases with the increase in angular deflection. For small of deflection tanθ is linear this explains linear nature of plot 8. 5. Cd not only depends on the flow velocity but will also depend upon flow direction, object position, object size, fluid density and fluid viscosity. 6. The oscillation of the pendulum at higher velocities of air may be because of the change of flow from laminar to turbulent regime. Oscillatory behaviour of the cylinder is combine effect of shedding vortexes in the wake region and mechanical vibration of motor. Causes of variation and uncertainty • The wooden cylinder is misaligned with the flow . • Manometer’s base is not aligned to the ground. • The ideal Cd plot (Refer fig. 2) was a 2D case (or infinite cylinder) but in experiment its a 3D object (Finite cylinder) will have some finite shear forces(boundary layer effects) acting on the side walls which will contribute to drag. • Mechanical Vibration of motor is causes the cylinder to oscillate causing error in reading. Oscillations started around 800 rpm. At higher rotational speeds, the amplitude of oscillations were fluctuating, possible reason being the fully turbulent nature of vortex sheet (as shown in fig 3). • It was a subsonic experiment so any disturbance in aft field can also causes variation in result. Also the rod used for suspending the cylinder will have some contribution in variation in readings. • Steady state may not have be achieved due to variable velocity and pressure distribution. • The leakage in wind tunnel causes the internal pressure to drop and induces disturbances in the flow which consequently reduces the drag force experienced by the cylinder. • Due to prolonged uses the tube of manometer is having some deposited residue around 5 cm from the base so that might contribute to error due variation in density of fluid . The blue coloured substance can be ”CuSO4 ” usually added in industrial Ethanol. Uncertainty Analysis Error analysis is done for θ = 5o . Drag on the cylinder is given by the following moment balance equation:
mb rb + mr2rb + mc rc g tan θ D= (3) rc The above equation has only one variable in the form of deflection. Therefore uncertainty in drag is given by: ∂D LMr 2LM = sec2 θ × g × (2m + + ) ∂θ l l = 12.22 Net uncertainty is given by: ∂D δθ × = 1.236477 ∂θ D

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Reynolds number is given by: Re =

ρ×V ×D µ

Uncertainty in Reynolds number depends on velocity of flow which is dependent on rise of ethanol on the multi-tube manometer. Uncertainty in Reynolds number due to h1 is given by: 289694.117 × −1 ∂Re √ = ∂h1 2 × h2 − h1 Therefore total error due to h1 is given by: ∂Re ∂h1

× δh1 Re

= −0.02132

Similarly total error due to h2 is given by: ∂Re ∂h2

× δh2 Re

Net error due to h1 and h2 is given by: s e 2 ( ∂R ∂h1 × δh1 ) Re

×

= 0.02132

e 2 ( ∂R ∂h2 × δh2 )

Re

=

p

−0.021322 × 0.021322

= 0.03015 Similarly error in Cd can be found out. The parameters effecting the uncertainty in Coefficient of drag are deflection and the height of the rise in ethanol level in the multi-tube manometer. Maximum uncertainty value for CD was obtained as ±0.572 while the minimum value was ±0.061. Similarly maximum uncertainty in calculation of Reynolds number was ±0.145 and minimum value was ±0.014.

Appendix Sample Calculation Experiment is well within the incompressible regime of fluid flow (i.e. Mach no. less than 0.3), So Bernoulli’s equation is valid. Patm Tatm Mair R

750 mm of Hg 314.15 K 29.1 gm/mole 8.314 J/(kg.K)

Table 1. Atmospheric Properties

By using Bernoulli’s equation for two points at the same height, we have: 1 1 P1 + (ρ(v1 )2 ) = P2 + (ρ(v2 )2 ) 2 2

(4)

Density of air is calculated by: ρ=

PM RT

Using properties given in table V. ρ=

13600 × 0.75 × 9.81 × 29.1 8.314 × 304 8 of 10

Aerodynamics Lab Report, IIST

(5)

Measurement rb rc Dwc Wwc mr mc mb θm

Values 260 mm 130 mm 50 mm 45 mm 111 gm 58 gm 225 gm 45 Degrees

Table 2. Apparatus Measurements

ρ = 1.152kgm−3 Drag force on the given cylinder is calculated using moment balance
mb rb + mr2rb + mc rc g tan θ D= rc

(6)

Sample calculation for reading 5: θ = 5o ; D = 9.81 × tan(5)(2 × 0.058 +

0.260 × 0.111 2 × 0.260 × 0.225 + ) 0.130 0.130

D = 1.0625kgms−2 ; (0.166 − 0.142) × sin45 × 9.81 × 789 =

1 × 1.152 × v 2 2

v = 15.454m/s CD =

D 1 2 2 ρv S

(7)

CD = 1.2697

Acknowledgments We would like to acknowledge all the people involved directly or indirectly in completion of this experiment and report. We would like to thank our lab supervisor Dr. B. R. Vinoth for providing us an opportunity to experimentally understand various aspects of aerodynamics. We would also like to thank our instructors Mr. Roshan Kumar and Ms. Prasanthi for providing us the needful guidance for conducting the experiment.

References 1 Yunus A. Cengel, John M. Cimbala, FLUID MECHANICS: FUNDAMENTALS AND APPLICATIONS, SI Units, 1st Edition, Mc Graw Hill Education, chapter 11, pp 583-586 2 Sunden, Bengt Tubes, Crossflow Over, Article, Thermopedia DOI: http://www.thermopedia.com/content/1216/

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