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Materials and Methods

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Materials and Methods

    ME 309L Fluid Mechanics Laboratory

    Experiment 9 : Modeling and Dynamic Similarity

    Report Prepared by :

    Arun P Mohan

    Jeremy Schaeffer

    Brandon Wilson

     thOctober 16 , 2006

    Lab Division #4

Abstract

     The relationship between drag coefficients and Reynolds number was found

    experimentally using three different sized spheres inside a wind tunnel. The spheres were

    connected to a load cell that measured drag forces, which was turned into drag

    coefficients and Reynolds numbers for each sphere and wind speed. We have concluded

    that the Reynolds number is inversely related to drag coefficient in every case. The

    experimental values didn’t directly correspond to the theoretical or expected values, due

    to levels of uncertainty, and suggest digital readout on the ETD for a more precise

    experiment.

Introduction

     Drag is the component of force on a body acting parallel to the direction of

    relative motion. Drag force is very significant when it is calculated for flow around a

    sphere. This force is due mostly to the asymmetric pressure distribution created by the

    boundary layer separation, and thus the boundary layer is necessary to explain the drag

    on the sphere. The total drag force around a sphere is from a contribution of both friction

    drag and pressure drag. The overall objective of this procedure is to measure this drag

    force acting on different sized spheres as a function of air speed. This drag force

    measurement can then be used to compare the evolution of the drag coefficient as a

    , is related to the drag force by function of the Reynolds number. The drag coefficient, CD

    the following equation:

    FD (1) D?C1?VA?2

Where: F = drag force D

     ρ = density of the fluid

     V = velocity of the fluid ?

     A = frontal area of the object

    The Reynolds number; which is a dimensionless parameter relating typical inertial

    flows to typical viscous flows, determines if a flow is laminar or turbulent. The Reynolds

    number, Re, can be calculated using the following equation: d

    ?Vd?Re? (2) d?

    Where: ρ = density of fluid

     V = velocity of the fluid ?

     d = diameter of the sphere

     μ = dynamic viscosity

    If the Reynolds number is large then the flow is generally turbulent, while a small

    Reynolds number indicates a more laminar flow. If the number is relatively small i.e.

    ? 1then there is no flow separation from the sphere and the drag is predominantly Red

    friction drag.

    When comparing the Reynolds number to the drag coefficient a logarithmic scale

    should be used. A graph of these two numerical values has been plotted and our graph

    should resemble this as shown. This graph is from previous experiments and gives an

    idea of what our resulting graph should look like.

    Figure 1 : Drag coefficient of a smooth sphere as a function of

     Reynolds number (from Fox et al., 2004)

    Materials and Methods

     Drag forces were determined experimentally on spheres of three different sizes

    using a wind tunnel equipped with a load cell and instrumentation for measuring pressure

    and drag force. The wind tunnel utilized a velocity control knob along with a digital

    wind speed indicator for a higher level of usability. The experimental procedure was as

    follows:

    1. Determine the air density at room temperature using a barometer and

    thermometer in the lab where the experiment is to be done. Estimate dynamic

    viscosity of the air at room temperature (done using FMP Table A.10).

    2. Calibrate the measurement system.

    3. Place the smallest sphere (60 mm diameter) inside the wind tunnel, attaching it

    to the load cell.

    4. Turn on the Bridge Amplifier and Meter (BAM) and calibrate it:

    a. Turn on the POWER knob on the lower left corner clockwise to 4.

    Adjust the BALANCE knob in the upper right corner until the needle

    reads 0.

    b. The power switch is located beneath the BALANCE knob, switch power

    on. Using the calibration knob at the bottom of the BAM, adjust the

    bridge voltage until the needle reads 0.

    c. Set the calibration knob in the lower right corner to 20. Press and hold

    the calibration knob while adjusting the GAIN until the needle reads 84.

    5. Turn on the wind tunnel and set the motor frequency to 30 Hz. Position the

    stagnation tube several centimeters upstream from the sphere, so it measures

    undisturbed freestream velocity. Record the dynamic gage pressure (mm of H0) 2

    from the ETD instrument box.

    6. Record the reading from the top row of the BAM. Convert the reading (kg) to a

    drag force, D (in Newtons), using the following conversion:

    DNBAMreadingkgfNkgf[]([]/100*9.81[/]? (3)

    7. Repeat step 6 for air speed settings between 30 and 60 Hz, in 5 Hz increments.

    8. Set the wind tunnel speed to zero after collecting your data. Carefully replace

    the sphere with a new sphere of a different diameter. Repeat steps 5 through 8.

    9. Turn off the BAM. Be sure to turn off both power switches (knob in lower left

    corner and switch in upper right.

Results :

    The air density in the laboratory was calculated using the pressure from the barometer. The pressure was observed to be 731.5 mm of Hg after correction. Converting

    this to absolute pressure, the pressure was found to be 97.6 kPa. This gave us a value of 31.14 kg/m for the air density.

    Then using the acquired value for air density velocity for the fluid was found in each case. And with the flow velocity the drag coefficient and Reynolds number was also

    calculated in every case using equations (1) and (2). The spreadsheet showing all the

    calculations and data is attached in the appendix. The BAM reading was converted to

    force units using equation (3).

    The following two graphs show the drag force as a function of velocity, and the drag coefficient as a function of the Reynolds number with the uncertainties shown.

    Drag Force as a function of Wind Speed3.5

    3

    2.5

    2

    Sphere - 64mm1.5Sphere - 76mm1Sphere - 102mmDrag (N)

    0.5

    0

    0510152025303540-0.5

    -1

    -1.5

    Velocity (m/s)

    Figure 2 : Plot of Drag Force as a function of wind speeds for all spheres.

    Cd as a function of Reynolds number2.5

    2

    1.5

    Sphere - 64mm1Sphere - 76mm

    Sphere - 102mm0.5

    Drag Co-efficient

    0

    0.00E+005.00E+041.00E+051.50E+052.00E+052.50E+05

    -0.5

    -1

    Reynolds Number

    Figure 3 : Plot of drag coefficient as a function Reynolds number. Error bars have been

    inserted in each plot to highlight uncertainty for the data.

Table 1 : The following table shows the calculated uncertainties that were included

    during this experiment. The data is divided into three sets with sphere diameters of .064

    m, .076 m, and .102 m respectively. Each set has 6 values due to the varying frequencies

    from 30-55 Hz in 5 Hz increments.

    Sphere

    Uncertainty Pressure Drag Force Velocity Reynolds Drag CO

    Uncertainty Uncertainty Uncertainty Uncertainty Uncertainty

    0.0078125 0.002955015 1.27420999 0.002644277 0.00934152 1.274324309

    0.0078125 0.002020623 1.01936799 0.002414519 0.0092791 1.019508607

    0.0078125 0.001464807 0.72811999 0.002312053 0.00925297 0.728315512

    0.0078125 0.001101762 0.56631555 0.002261115 0.00924037 0.566566087

    0.0078125 0.000866471 0.46334909 0.002235367 0.00923411 0.463654765

    0.0078125 0.000703188 0.39206461 0.002220989 0.00923064 0.392425494

    0.006578947 0.002904646 1.27420999 0.002630288 0.00833293 1.274296331

    0.006578947 0.002004775 0.84947333 0.002411214 0.00826639 0.849600233

    0.006578947 0.001452323 0.56631555 0.002310083 0.00823746 0.566504207

    0.006578947 0.001099393 0.42473666 0.002260827 0.00822378 0.424987112

    0.006578947 0.000857748 0.36406 0.002234526 0.00821659 0.364351507

    0.006578947 0.000697432 0.26825473 0.002220535 0.0082128 0.268649758

    0.004901961 0.002808888 0.63710499 0.002604157 0.00707443 0.637216803

    0.004901961 0.001929123 0.42473666 0.002395736 0.00700039 0.424899453

    0.004901961 0.001408313 0.3185525 0.00230326 0.00696928 0.318766791

    0.004901961 0.001065037 0.254842 0.002256712 0.00695404 0.255108151

    0.004901961 0.000827213 0.21236833 0.002231646 0.00694594 0.212686584

    0.004901961 0.000668258 0.16989467 0.002218291 0.00694167 0.170291615

Discussions :

    From the two plots generated from the data we see that with an increase in the

    flow velocity of the air for each sphere, the drag experienced by it also increases. We also

    see that bigger the diameter of the sphere, the greater the increase of drag force with

    increase in air flow velocity. This could due to the increase of surface area to the flow of

    air or increase in frontal area of the sphere as the sphere size increases.

    From our results we confirmed that the value of Reynolds number is inversely

    related to the drag coefficient in every case. Thereby in general we observed that as the

    value of Reynolds number increased, the value of the drag coefficient decreases

    proportionately. This may be explained by looking at the mathematical formulas that

    describe these two constants [2].

     2C = D/ (0.5*ρ*V*A) Re = (ρ*V*d)/µ Dd

    Where C refers to drag coefficient and Re refers to Reynolds number. Dd

    Looking at the two formulas we see that density and velocity are the two common factors in each case, and we see that velocity is the main factor that both constants

    depend on. In the case of the drag coefficient, we see that flow velocity is in the

    denominator, hence with increase in velocity the drag coefficient would decrease. Now

    looking at Reynolds number, since velocity is in the numerator, with increase in flow

    velocity, the value of Reynolds number would increase proportionately. We did notice

    that the sphere with a diameter of 76mm showed an erratic behavior on the plot. This

    may have been due to some experimental errors that may have occurred at the time of the

    experiment. Other errors that may have occurred were, were that the surface of the sphere

    may not have been smooth, or perhaps the drag coefficient may have been altered due to

    the setup of the experiment and that may have been a source of error.

    From the data given in the laboratory handout we observed that drag coefficient of a smooth sphere as a function of Reynolds number had a linear region at lower

    Reynolds number values. Overall there was a negative linear slope on the given plot. In

    comparison to our data, our Reynolds number range was between the range of 50,000 to

    250,000. Comparing this range to the given plot in our handout (Figure 1), our graph

    (Figure 3) was similar as it showed the plot to be more or less having the same trend in

    data points. This same trend can be seen for the same range on the given plot from the

    handout.

    Uncertainty values were calculated and shown in each case and on each plot. The general trend in the uncertainty values were that its range decreased as the independent

    variable increased. This may be due to the presence of relative uncertainties, and we

    know that this would lead to lower uncertainty with higher measurement values.

    Assumptions that were made included steady flow for the fluid, viscid fluid flow and that the density of air remained constant in the wind tunnel.

    Conclusions :

     Within the limits of experimental uncertainty and human error reading the ETD,

    the results obtained for drag coefficient and its relationship to Reynolds number, the

    slopes and ranges of experimental data correspond to the theoretical, expected results in

    terms of the negative linear relationship. Although, the large level of uncertainties within

    the experimental data reduce the reliability of the data collected. The ETD used in the

    experiment should utilize a digital readout out for more reliable results.

    References :

     th1. Fox, R.W., McDonald, A.T., and Pritchard, P.J., Introduction to Fluid Mechanics, 6

    ed., Sections 7.2 through 7.5, Section 9.7, and Appendix F

2. Experiment 9: Modeling and Dynamic Similarity, 15 October 2006

     http://widget.ecn.purdue.edu/~me309

    Appendices

Relative Uncertainty Measurements

Relative Area Uncertainty

    DdA????ADAdD

    2D?A?

     4dAD??dD2AD?2??

Relative Velocity Uncertainty

    1???VPV?222 ????[()()]??VP??VPV?????

    2PV???

    dV12 ?dPP2?dVP12??3d2??

    111222???????[()()] VP??22

Relative Reynolds Number Uncertainty

    1????VRRRD?eee2222 ?????[()(??)()]??Re?VDRVRDRe?????ee

    DV??R?e?

    edRD???dV? ?edRDV?d???e

    dRV??dD?

    12222 ????????[()()()]Re?VD?

    Relative Drag Coefficient Uncertainty

    1FCCCVC?????A2222DDDDD?2 ??????????[()()()()]FAV?CD?DCFCCACV?????DDDDD?

    DdA????ADAdD

    2D?A?

     4dAD??dD2AD?2??

    122222 ?????????????[()(2)(2)]CFVV?DD??

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