Lab 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Brian Park, David Gentry, Aron Dobos

 

October 4, 2004

 

Second Monday

 

Engineering 41

                                                                                       


Abstract

 

This lab consisted of two parts: the first involving pipe flow, the second involving air around a cylinder.  In the first section, air is moved through a pipe using fan.  How the velocity profile is acting within the pipe is determined using different methods, both graphical and numerical.  Also, the two different types of flow, laminar and turbulent, were compared.  The second section consisted of blowing air through a wind tunnel around a cylindrical rod.  The velocity of the air flow is then related to the vortex shedding frequendy of the rod through the Strouhal number.

 

 

Introduction

 

Part I: Pipe flow

            When fluid moves through a pipe, there are three different states of flow: laminar, transitional, and turbulent.  Which state the flow is in depends on many variables, including the velocity of the fluid, diameter of the pipe, and properties of the fluid.  Laminar flow occurs when the fluid moves slowly through the pipe, the velocity profile taking on a parabolic shape where the center of the fluid travels faster than at the pipe walls due to viscous and pressure forces.  Turbulence occurs when the velocity of the fluid overcomes the viscous forces acting on the fluid.  Therefore, the velocity profile of the liquid no longer takes on a perfectly parabolic shape, but becomes flatter towards the center, and steeper near the walls.  Also, the liquid distorts itself, mixing together, and creating error.  It is not completely understood how the flow shapes during turbulence, and much research has been done on the subject.

There are many ways to determine exactly which state the flow is in.  One way is to fit the profile of the flow to different curves, and determine what type it is based on the coefficients of the fitted curve.  The most popular way, however, is by using the Reynolds number.  The Reynolds number takes into account the variables stated in the previous paragraph, and ultimately determines pipe flow characteristics.  A greater Reynolds number means more turbulence in the flow.  This number can then be used to find things such as frictional force caused by the pipe.

 

Part II: Vortex Shedding

Vortex shedding occurs because a large shear force develops close to the cylinder.  At the surface of the cylinder, the velocity of the fluid is zero, but due to the decreased pressure of the fluid moving around the cylinder, the outer velocity is much larger than the initial velocity.  This velocity differential creates a shear on the outside of the streamline causing the fluid to turn inwards after passing the cylinder. 

 

For generally small initial velocities, the flow does not have enough energy to break the vortices off into little independently spinning cyclones, and the flow increases in pressure around the back side of the cylinder.  Eventually the turbulence diminishes and a smooth flow pattern is achieved again behind the cylinder. 

For greater initial velocities, the energy in the flow is great enough to break the spinning vortices off the mainline flow.  They continue to break away from the cylinder at a specific frequency that is directly proportional to the Reynolds number.  As the vortices move away from the cylinder, their energies slowly decrease, and the result is that the vortex eddies move away from each other and spread out.  It should be noted that the Strouhal number does not depend whatsoever on the diameter of the cylinder: it is simply a function of the energy: i.e: the velocity (Re).

The staggered vortex pattern happens because the vortex “packets” fly off the cylinder once on one side and then on the other.  The reason why the vortices are shed at a certain frequency is not well understood.  It could be partially explained by considering very rapid and small pressure fluctuations that travel along the circumference of the cylinder and excite the vortex shedding to happen alternately from both sides of the cylinder.  The staggered vortices create pressure differentials in the water perpendicular to the original streamlines that result in the side to side motion observable in the wake of the cylinder. 

Consider a flag on a pole.  With a strong enough wind, the flag flaps in the air because of the same side to side forces on the flag as the vortices are shed off the cylindrical pole.  It is commonly observable that higher wind speeds result in stronger thrashing of a flag.

The frequency of vortex shedding explains exactly the audible whine of a fluid (or air) moving across a cylinder.  The tones are not pure because of the various frequencies within the vortexes themselves, unlike a flute tone.  In fact, if the wind tunnel were entirely silent, the lab could have been performed by holding a sensitive microphone close to the cylinder and measuring the audio frequencies picked up.

 

 

Procedure

 

            See Appendix A for instruction and procedure of the experiment.

 

Theory

 

In order to determine whether the pipe flow is laminar, transitional, or turbulent can be found by using a non-dimensional parameter, known as the Reynolds number.  It is a ratio of the inertial effects to the viscous effects.  It is denoted by

,

where ρ is the density of the fluid, V is the average velocity, μ is the fluid shear viscosity, and D is the pipe diameter.  For air at 20 degrees Celsius, , and .  Generally, a Reynolds number less than 2100 is considered laminar, while a Reynolds number greater than 4000 is turbulent.  These numbers are only a rough estimate, however, and do not account for inconsistencies in the pipe and/or outside effects.

The velocity of the air was calibrated using the formula:

 

,

 

where Volt refers to the voltage output of the hotwire anemometer and the coefficients listed in the Table 1.

 

Table 1.  Coefficients for Voltage to Velocity Conversion

k

 2.12

a

 2.67e-1

b

-1.71

c

-4.43e-2

d

 3.27e-1

   

The velocity profile for fully developed laminar flow in a pipe is given by:

                                                   ,

where Ro is the radius of the pipe, μ is the dynamic viscosity, and dP/dx is the pressure drop due to friction.  Therefore the velocity profile for smooth laminar flow is parabolic and dependent on the radial distance from the centerline; the velocity decreases as the radial distance increases.  The equation used to curve fit data in KaleidaGraph for laminar flow was: m1*((M0-m2)/13.2)^ 2-m3.

            Turbulent flow profiles are commonly modeled using the empirical power-law velocity profile:   

                                                           ,

where Vo is the centerline velocity and n is a function of the Reynolds number.  As the Reynolds number increases, the power n increases.  Therefore as n increases the velocity profile becomes flatter and takes on a more uniform velocity profile.  Turbulence begins when n ≈ 7.  The equation used to curve fit data in KaleidaGraph for turbulent flow was: abs(m2-M0)^(1/m1).  

 

Part II

 

The Strouhal number relates the vortex shedding frequency to the diameter of the rod, and the velocity (U0).  For large velocities, the Strouhal number is constant.  The formula is shown below

 

Results

 

            The results for pipe flow are summarized in Table 2.

 

Table 2.  Reynolds Number at Various Air Velocities

Speed Input (V)

Avg Velocity (m/s)

Reynolds Number

n-value

70

0.60

1007

1.37

78

1.20

2011

1.37

83 (Laminar)

1.81

3043

4.03

83

2.14

3596

5.14

90

2.58

4369

7.20

 

The speed input is the speed of the fan at a particular voltage, which was varied by professor Orthlieb during the experiment; the higher the voltage, the faster the velocity.  The voltage output of the MATLAB program was converted into velocity, averaged, and then plugged into the Reynolds number formula as stated in the theory section. 

According to the theory, in order for the flow to be laminar, the Reynolds number should be less than 2100, and for the flow to be turbulent, the Reynolds number should by greater than 4000.  For the speed inputs of 70V and 78V, the fluid flow in the pipe is laminar.  For both speed inputs at 83V, the flow is considered to be transitional.  In the transitional state, or the fluid is changing from laminar flow to turbulent flow or vice versa.  The (Laminar) indication at 83V means that the flow is more laminar than turbulent in the transitional state, as can be seen in the Reynolds numbers.  At 90V speed input, the air flow through the pipe is turbulent.  These results correspond accordingly with our figures.

Figure 1 through Figure 4 display the velocity profiles at each speed input.  The velocity profiles correlate the velocity of a fluid mass particle to the radial distance away from the centerline.  The error bars shown indicate the standard deviation of each data point calculated using a MATLAB script (Appendix B). 

Figure 1.  Velocity profile at 70V speed input.

 

            Figure 1 shows almost a perfect parabolic velocity profile, as shown with an extremely high correlation coefficient.  Likewise, the standard deviation at each point is very small.  Therefore, the air flow at this speed was considered to be laminar.  The same argument holds for Figure 2.  The fluid velocity at 78V speed input continues to have a strong parabolic relationship to the radial distance.  However, the standard deviation of each data point increases but remains to be small enough to conclude that the flow is laminar.     

           

Figure 2.  Velocity profile at 78V speed input.

 

For the cases where the speed inputs were 83V, the flow in the tube was determined to be transitional.  The top of the profiles are beginning to flatten out and velocities drop off steeper towards the walls of the pipe.  The speed input of 83V was subjected to disturbances from the outside environment while the speed input of 83V (Laminar) was controlled from disturbances.  There exists a stronger parabolic fit for the 83V (Laminar) input than the normal 83V input.  In addition, the standard deviation of each data point was overall lower in the 83V (Laminar) input than in the 83V input.  This suggests that the air flow is closer to laminar flow at 83V (Laminar) input and more turbulent at 83V input.  Figure 5 exhibits a turbulent velocity profile, where the top is flat and the sides drop off very sharply.  The standard deviation at each data point was generally the greatest at 90V.         

Figure 3.  Velocity profile at 83V speed input with no disturbances.

 

 

Figure 4.  Velocity profile at 83V speed input subject to disturbances.

 

Figure 5.  Velocity profile at 90V speed input.

 

 

 

The power-law velocity profiles are shown in Figures 6 – 10.  The n value increases as the Reynolds number increases, as was shown in Table 2.  Turbulent cases begin when n ≈ 7.

 

Figure 6.  Power-law velocity profile for 70V speed input.

  

Figure 7.  Power-law velocity profile for 78V speed input.

The relationship between the velocity and the radial distance is almost a linear relationship in Figures 6 and 7.  As a result, flow is laminar for speed inputs of 70V and 78V.

Figure 8.  Power-law velocity profile for 83V (Laminar) speed input.

 

 

            Evidence that the 83V (Laminar) is in fact more laminar than at the 83V input speed can be seen according to the n values.  The n value at 83V speed input is close to reaching turbulence, but it is not obvious that the flow is completely turbulent. 

 

Figure 9.  Power-law velocity profile for 83V speed input.

Figure 10.  Power-law velocity profile for 90V speed input.

Flow at a fan speed of 90V is turbulent.  Fluid flow is defined to be turbulent when n ≥ 7.  In conclusion, all data and analysis are consistent with each other.  Flow is laminar at 70V and 78V; transitional at 83V(s); and turbulent at 90V. 

There could be error in the data for a few different reasons.  One is that data points were not taken from wall to wall, as the hotwire anemometer might have been damaged had it touched the wall of the pipe. Therefore, the velocities at the two extreme ends of the pipe were not found.  Also, the hotwire anemometer may not have been positioned exactly at the horizontal centroidal axis.  This potential error may have caused the data to show a smaller maximum velocity than what was actually in the pipe.  Further error may have been caused by air movement in the room, as people were constantly moving and the window was open.

There exists many important applications of laminar and turbulent flow.  If water velocities are too great in fire hoses, turbulence occurs and the hose starts squirming and swinging uncontrollably.  Laminar flow is important in electrical applications.  DC circuits need a constant laminar flow of electrons.  If a turbulent flow of electrons exists, such as after a lightning strike, it could cause a short circuit and damage the circuit.  However, in situations in dealing with cigarette smoke and car exhaust, turbulence is needed to quickly mix and diffuse the gases into the atmosphere.         

 

 

Part II

 

After a certain Reynold’s number, the Strouhal number becomes constant.  Since the diameter of the cylinder is constant, the frequency must increase linearly with the velocity.  The data obtained from the wind tunnel make this relationship clear.

 

 

Measured Wind Tunnel Velocity and Frequencies

Tunnel

V (m/s)

f (0.056)

f (0.092)

f (0.373)

10

3.0192

0.43

0.27

0.06

20

6.8107

0.96

0.61

0.14

30

10.602

1.53

0.95

0.21

40

14.394

2.13

1.32

0.31

50

18.185

2.71

1.67

0.38

60

21.977

3.3

2.03

0.46

70

25.768

3.9

2.41

0.55

80

29.56

4.47

2.76

0.64

 

Calculated Reynolds Numbers and Strouhal Numbers

Re(0.056)

Re(0.092)

Re(0.373)

Strhl.(0.056)

Strhl. (0.092)

Strhl. (0.373)

27.354

44.9386

182.1967

0.20258

0.20897

0.18828

61.704

101.3716

410.9956

0.20049

0.20929

0.19475

96.055

157.8046

639.7945

0.20527

0.20939

0.18766

130.41

214.2375

868.5934

0.21049

0.2143

0.20405

164.76

270.6705

1097.392

0.21197

0.21459

0.19797

199.11

327.1035

1326.191

0.21359

0.21585

0.19831

233.46

383.5365

1554.99

0.21528

0.21855

0.20222

267.81

439.9694

1783.789

0.21509

0.21819

0.20513

 

In the graphs below, the Strouhal number is plotted as a function of the Reynolds number.  On the first graph, only the two smaller cylinders are shown for clarity.  The second graph includes the large cylinder also whose Reynolds numbers are much greater.

 

 

 

 

 

The calculated Strouhal numbers for all the cylinders are approximately 0.21, with the exception of the largest cylinder whose Strouhal numbers turn out to be slightly lower.  Looking at the graph of wind tunnel tests from Essentials of Fluid Dynamics by Ludwig Prandtl, the expected Strouhal number is 0.21, which compares very favorably with the acquired data.  The data obtained confirms that the frequency of vortex shedding is proportional to only the Reynolds number, provided that the flow has enough initial energy to initiate the vortices to spin off the cylinder.

 

 

Implications of Vortex Shedding

 

Vortex shedding possesses many implications for bridges and power lines, or any cylindrical structure exposed to a constant wind.  The cables in suspension bridges probably are not terribly affected by vortex shedding.  The heavy duty cables are made of many finer strands, resulting in a cylindrical cable that does not have a smooth surface.  Rather, each small strand may shed small vortices, but the small eddies between the individual strands provide a zone of turbulent fluid motion with non-zero velocity.  As a result, the pressure differential between the overall cable surface and an oncoming streamline is not as great, meaning that the wind velocity must be much greater to cause the vortices to spin off.  Due to the rough surface of the cable, the vortices probably would not spin off in straight lines, but would have a more random distribution, shielding the structure from a strong onslaught of staggered vortices that could cause large vibrations.  The larger the cylindrical surface area, the more susceptible the structure is to destructive vibrations due to vortex shedding.  To prevent vortex shedding from flapping a bridge around like a flag on a pole, the bridge must be reinforced to resist torsion along its axis.  It should be designed aerodynamically to allow the wind to pass over its surface without creating torque differentials across its surface that arise from the alternating pressures due to vortex shedding.  In this way, the bridge can potentially be shielded from resonating at the vortex shedding frequency.

 

 

 

 

Acknowledgments

 

Fred Orthlieb.  Thermofluid Mechanics Lab.  Engineering 41.  Swarthmore College

Department of Engineering.  Sept. 20, 2004.

 

Munson, Okiishi, and Young.  Fundamentals of Fluid Mechanics.  Hoboken: John Wiley

and Sons, Inc., 2002.  444-478.

 

Prandtl, Ludwig.  Essentials of Fluid Dynamics.  New York: Hafner Publishing

Company, 1989.  183,187.