Lab 4:  Fluid Dynamics Week 1

Introduction:

Basic Concepts

Mass is conserved throughout the wind tunnel.  The continuity equation is:

Momentum is also conversed within the wind tunnel.  The   momentum equation is:

By combining the continuity equation and momentum equation with the viscous forces of the fluid the Navier-Stokes Equations are derived.

The Reynolds number is a dimensionless parameter which is a measure of the flow’s inertial forces over its viscous forces. 

A large Reynolds number indicates higher inertial forces, which infers a high fluid velocity and a greater likelihood that the flow is turbulent. Reynolds number is an excellent way to tell if the flow is laminar or turbulent. High Reynolds number usually indicates that the flow is turbulent and low Reynolds number indicates laminar flow. A good rule of thumb is that turbulence occurs approximately at and above a Re = 3*10^5 for circular cylinder.  

Bernoulli’s Equation

Flow is assumed to be along a streamline and cannot cross streamlines.  At wind speeds where Mach<<1 the flow is assumed to be incompressible.  The wind tunnel is designed to minimize the amount of turbulence in the test section and therefore the flow is assumed to be steady and time independent.  Friction is negligible. Flow assumed irrotational.  From these assumptions the Navier-Stokes equation reduces to Bernoulli’s Equation.

Further simplification can be made when the gravity is neglected since any height difference would be small.  Then, each term in the Bernoulli’s Equation represents a particular pressure. 

Wind-Tunnel Fan Speed to Airflow Velocity Conversion

Note that the wind tunnel program input is in Hertz (fan speed). However, your calculations require airflow velocity. You will create an equation to do this conversion. When performing calculations use only SI units. No Mixing.

Pitot-Static Tube

The instrument that is used to measure the total and static pressure is called a Pitot-static tube, which is pictured below.

Figure1.  Pitot-Static Tube

A Pitot-static tube is a pressure measuring device which is essentially a thin hollow tube that has two hole taps in it.  The hole at the end of the tube (parallel with flow) brings the flow to zero velocity (i.e. ) which yields the total pressure (or stagnation pressure) of the flow.  The holes on the side of the tube (perpendicular to flow) measure the static (ambient) pressure of the flow.  Using Bernoulli’s equation, subtracting the static pressure from the total pressure yields the dynamic pressure.  Velocity now can be obtained by the dynamic pressure that was measured with the following equation.

Wake

As flow progresses over a bluff body the boundary layer separates from the object and forms a wake behind the body.  The separation of the flow from the bluff body causes a large amount of drag on the object (high pressure stagnation point in the front of the object and a low pressure wake behind the object).  The bluff body in the flow produces a certain amount of momentum deficiency behind the bluff body.  This momentum deficiency can be seen as a loss in velocity within the wake of the bluff body.  Directly behind the bluff body the static pressure drops but with distance (roughly 10 diameters) behind the bluff body the static pressure in the wake recovers to the static pressure in the freestream.  The momentum deficiency is determined by finding the mass flux from in front to behind the bluff body.

Figure2.  Wake behind a Bluff Body

The formula to calculate the drag is:

Where ρ is the density of the air, L is the total length of the cylinder, u₁ is the freestream velocity, u₂ is the velocity behind the cylinder, w is the width of the control volume and y is the direction along the pressure rake.

The coefficient of drag is defined as:

Where S is the frontal surface area. (diameter*Length)

Figure3.  Control Volume for the Bluff Body

Von Karman Vortex Shedding – Unsteady Aerodynamics

Up until now most of your classes have had you make a number of over-riding assumptions when solving problems. For instance, when dealing with continuity and momentum equations you have mostly assumed steady, inviscid, and irrotational flow (thus yielding Bernoulli’s eqn.). It is important for you to realize, however, that even simple everyday phenomena are far from that easy. We simply make those assumptions to get a grip on at least some of the parameters at work. In this lab you will get your first real bit of exposure to unsteady fluid behavior.

A Von Kármán vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of flow over bluff bodies. Over a large Re range (47<Re<10⁷ for circular cylinders), eddies are shed continuously from each side of the body, forming rows of vortices in its wake. Ultimately, the energy of the vortices is consumed by viscosity as they move further down stream and the regular pattern disappears.

When a vortex is shed, an unsymmetrical flow pattern forms around the body, which therefore changes the pressure distribution. This means that the alternate shedding of vortices can create periodic lateral forces on the body in question, causing it to vibrate. If the vortex shedding frequency is similar to the natural frequency of a body or structure, it causes resonance. It is this forced vibration which, when at the correct frequency, causes suspended telephone or power lines to 'sing', the antennae on your car to vibrate more strongly at certain speeds and it is also responsible for the fluttering of Venetian blinds as the wind passes through them.

When considering a long circular cylinder, the frequency of vortex shedding is given by the empirical formula

where f is the vortex shedding frequency, d is the cylinder diameter, and Re is the Reynolds number. This formula will generally hold true for the range 250 < Re < 2 × 10⁵. The dimensionless parameter fd/V is known as the Strouhal number.

Figure 4: Von Karman Vortex Street shed from cylinder (top view of test section). Adapted from Van Dyke’s book of Fluid Motion

Equipment

• Closed loop wind tunnel
• Pitot-static tube
• Aerolab 6 Degree of Freedom Sting Balance
• Daytronic System 10 DataPac
• Pressure Systems 9010 Optomux
• 3x 2” O.D. Cylinder
• Pitot tube rake
• Thermometer
• Labview

Procedure

Part 1: Wind Tunnel Calibration

1. Place the Pitot-static tube in the 1^st hole in the middle of the test section (~16” from wall). Align the Pitot-static tube parallel to the freestream flow.
2. Record 3 readings from the pressure transducer and the temperature readout in your notebook. The tunnel should be off at this point. This will serve as a value to zero the rest of your calibration from.
3. Increase the tunnel speed to 5 Hz and repeat step 2.
4. Repeat this process until the tunnel is at 50 Hz.
5. This data set will be used to correlate velocity and frequency.

Part 2: Vortex shedding

LabVIEW Program: Accelerometer-Cylinder.vi/Transducer-Cylinder.vi

1. Measure the length and diameter of the cylinder.
2. Secure cylinder with accelerometer into first hole in bottom of test section.
3. Turn on the wind tunnel to 30 Hz.
4. Set sampling frequency to 6000 Hz and sample 16384 times total
5. Collect accelerometer data
6. Turn the wind tunnel up to 45Hz and repeat steps 3-6
7. Place cylinder with Pressure Transducers in tunnel
8. Turn tunnel to 30Hz, then 45Hz
9. Record pressure data for several seconds

Part 3: Wake

1. The wind tunnel will be run at 25Hz and 45Hz.
2. Measure distance between each tap on rake.
3. Place rake inside the wind tunnel (without the presence of the cylinder).
4. Run Labview.  Read and Record Pressure measurements (20 times for average).
5. Mount the cylinder 8 diameters ahead of the rake in the wind tunnel.

6. Turn wind tunnel on.  Record the pressure (20 times for average) with Labview.

VERY IMPORTANT TIPS and Questions!!!: 

Tips-

• Be consistent with units (***Convert everything to SI units***). All calculations and results should be presented in SI units.
• Clearly label all graphs and tables (so that it is immediately obvious what trend is trying to be conveyed).
• If trying to compare results from two similar experiments combine plots onto one graph (consolidation and/or direct comparison of results is very effective method of presentation).
• Pay attention to neatness of graphs and tables (i.e. white background, thick dark axes, thick colored trendlines, appropriately scaled axes, # of digits, etc).
• Address all questions posed in each lab concisely including all directly relevant information. You may address additional concerns for completeness.
• Do NOT attach any raw data anywhere in lab (only averaged data and short samples may be included when necessary)---No treestumps!
• Read outside sources for additional context (Wikepedia (online) is a great source for basic information when lost but use with caution, it can be misleading). Reference websites when necessary, it is easy enough.
• Reference any FIGURES, PICTURES, ETC used in your report directly under the figure. We read the same websites, books, and reports that you do. Save yourself from trouble and write a reference if you didn’t create it yourself!
• Get started on calculations immediately after lab, this is a long one!

Some Questions-

Part 1:  (Wind Tunnel Calibration)

• Plot Frequency vs. Velocity of the wind tunnel.  CLEARLY LABEL EVERYTHING!
• Create an equation for velocity as a function of frequency.
• How well does your data set match your equation? Use the “R Squared” value to determine this.

Part 2: (Vortex Shedding)

• Provide the Strouhal number, Reynolds number, and tunnel velocity for every run.
• What is the vortex shedding frequency as determined by the empirical formula?
• Talk about flow unsteadiness. Relate it to the pressure transducer data. What does the pressure signature look like as time evolves? Are there any notable trends in the data?
• Plot fluctuating pressure (Voltage) vs. time. Plot also in Fourier space (e.g. frequency).
• What is the vortex shedding frequency as determined by the pressure transducer data? Does this value agree with the empirical result?
• From accelerometer data: plot amplitude vs. time, plot also in Fourier domain (e.g. voltage vs. frequency).
• What is the vortex shedding frequency as determined by the accelerometer?
• Does this value agree with the two previous results?
• At what tunnel speed did the cylinder vibrate the most (based on noise and amplitude)? Why does the cylinder vibrate more violently at certain tunnel speeds (not necessarily at the fastest speeds either)?
• Discuss why we can deduce vortex shedding frequency using all three methods. How come we can use the same accelerometer in the vibrations lab to find the shedding frequency in a fluids lab? How does the pressure transducer allow us to capture the shedding frequency as well?

Part 3: (Cylinder Wake)

• Plot freestream velocity (m/s) vs. rake tube location (cm) with and without the presence of the bluff body (refer to Figure 3).

• Calculate the momentum deficit caused by the cylinder.
• How is momentum deficit related to the drag created by the object?
• What is the coefficient of drag for the entire cylinder as deduced from pressure data? (Note: In Part II of this lab we will calculate the cylinder drag coefficient by direct measurement with the force balance instead)
• What is the Reynolds number?
• Compare your calculated coefficient of drag to published values at similar Reynolds numbers.

Useful Conversions can be found at:

http://www.onlineconversion.com/

Also:

R=287 J/(kg-K)

1 psi = 6894.757 Pa

1 m = 3.280839895 ft

Viscosity = 0.00001789 kg/(m-s)