PothiraajS Vortex

THE UNIVERSITY OF MANCHESTER SCHOOL OF MECHANICAL AEROSPACE and CIVIL ENGINEERING

LABORATORY REPORT VORTEX SHEDDING FROM A CYLINDER AND DATA ACQUISTION LECTURER : Mr. DENNIS COOPER

NAME: SRINATH POTHIRAAJ STUDENT ID: 8284494 COURSE: MSc THERMAL POWER AND FLUID ENGINEERING DATE: 3-11-2011

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Table of ContentsTitle...........................................................................................3

Appendix

TitleTitle of the experiment: Vortex Shedding from a Cylinder and Data Acquisition Author: Srinath Pothiraaj Date on which the experiment was conducted: 20-10-2011

Objectives To determine the vortex shedding frequency from a cylinder To determine the optimum sampling rate To find the effects of under and over sampling To plot graphs to interpret the variation of frequency spectra, velocity histogram, RMS, RTI and mean velocity profiles across the wake To analyse the errors which occurred during the experiment 2

IntroductionFluid flow past objects have been studied to a great extent because of their wide range of applications. One such case is the flow past a cylinder. [1] When airflow passes a cylinder it creates alternating lowpressure vortices on either side of the cylinder at a certain frequency that are governed by fluid properties, free stream velocity and dimensions of the object. This is called a Von Karman vortex street. If this frequency matches with the natural frequency of the object, the object begins to resonate and the movement can result in large displacement. Figure 1 shows the Von Karman vortex shedding behind a circular cylinder.

cylinder

[2]

Figure 1: Karman Vortex Street behind a

Equations used to determine Vortex Shedding Frequency [3] St = 0.198 1-19.7Re Eq. [1] St = f dUo Eq. [2] Re = Uo d Eq. [3] where, St is the Strouhal number 3

Re is the Reynolds number f is the vortex shedding frequency (Hz) d is the diameter of the cylinder (m) Uo is the free stream velocity (m/s) is the density of the fluid (kg/m3) is the dynamic viscosity of the fluid (kg/ms) The time interval between vortices from any given side is t, however, because the vortices alternate side to side the time interval between vortices from opposite sides of the cylinder is t/2. As there are two time intervals, two frequencies must be produced namely f and 2f.[3]

cylinder [4][3]

Figure 2: Vortex shedding from a

According to the Nyquist theory, maximum measured frequency in any given set of data is half the sampling frequency. Since the bandwidth of the signal needs to be considered, sampling frequency must be at least 2.5 times greater than the maximum frequency. [3] Set this as the sampling rate and choose a convenient number of samples at the nearest value of 2n. Hotwire Anemometer Hotwire anemometer is widely used for measurement of flow velocity and turbulence. The hotwire anemometer utilised in this experiment is[5]

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a constant temperature anemometer. It basically consists of an extremely fine wire of diameter 5 m and 2mm long, electrically heated up to a particular temperature. When this wire is introduced into airflow, the airflow cools down the wire. In order to maintain a constant temperature, a current is applied through the hotwire. Thus by measuring the electrical current required to maintain the wire at constant temperature, velocity can be determined. Hotwire anemometry provides an output voltage which, when converted to velocity is of the form[3]

U = U + u where, U is the mean velocity (m/s) u is the fluctuation component of velocity Equations used to calculate mean, RMS and RTI Mean Velocity U= i=1nUin Eq. [4] where, U is the mean velocity (m/s) Ui is the individual data value of velocity (m/s) n is the number of samples RMS Velocity RMS (u) = 1ni=1n(Ui- U)2 [5] where, RMS (u) is the RMS velocity (m/s) U is the mean velocity (m/s) Ui is the individual data value of velocity (m/s) 5 Eq.[4]

n is the number of samples Relative Turbulence Intensity (RTI) RTI = RMS(u)U 100% Eq.[6] where, RTI is the Relative Turbulence Intensity RMS (u) is the RMS velocity (m/s) U is the mean velocity (m/s) Eventually, the results obtained from this experiment are shown in graphs using Matlab, namely a) Frequency spectra graph. b) Velocity distribution graph. c) Mean, RMS and RTI velocity profiles.

Experimental Apparatus1. Armfield Wind tunnel[6] Dimensions: 300 mm wide x 300 mm high.

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Figure 3: Wind tunnel

2. Hotwire Probe[6] Dimensions: 5 m in diameter and 2mm in length

Figure 4: Hotwire probe

3. Data Acquisition board Make: National Instruments Model no: NI PCI 6071 E

[6]

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Figure 5: Data Acquisition board

4. Micromanometer [6]

Figure 6: Micromanometer

5. Traverse [6]

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Figure 7: Traverse

6. Circular cylinder [6] Dimension: 15mm in diameter

Figure 8: Circular cylinder

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7. Pitot tube probe [6]

Figure 9: Pitot tube probe

Experimental ProcedureThe first step was to calibrate a hotwire anemometer probe by exposing it to a set of velocities and then the coefficients of voltages that are obtained was recorded. These coefficients when entered into data acquisition software establish an output in terms of velocity. The next step was to calculate the vortex shedding frequency for the cylinder. In order to do this, an approximation of shedding frequency (f1) is calculated using Strouhal number, St=fdU0. The cylinder diameter, d, was 15mm and the free stream flow velocity U0 within the wind tunnel was 10m/s. By determining an approximate Reynolds number of 9833, a Strouhal number of roughly 0.2 was obtained. Therefore, vortex shedding frequency (f1) was found to be 132 Hz. According to the Nyquist sampling theorem, the maximum shedding frequency (f2) was twice the shedding frequency (f1), Maximum shedding frequency (f2)= 2 x 132 = 264 Hz Thus, knowing the shedding frequency, the next step was to determine the optimum sampling frequency. The optimum sampling frequency required was 2.5 times greater than the maximum frequency present. Sampling frequency = 2.5 x 264 = 660 Hz

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The sampling frequency was now approximated to 700 Hz. Once the optimum frequency of 700Hz was chosen, the next step was to analyse the effects of over and under sampling on the output. Now, the hot wire probe was brought to the cylinder centre line and the sampling frequency was set to the optimum sampling frequency of 700 Hz. The readings were recorded. The sampling frequency was then reduced and increased to a value of 350 Hz and 1400 Hz respectively. The readings were again recorded. The next step was to take readings at different points from the cylinder centre line at the optimum sampling frequency (700 Hz). So now the probe was lowered by 10mm using the traverse. The output of the experiment was saved again. The whole procedure was repeated until the probe was 100mm away from its initial position. All the data collected from the experiment were then analysed and visualized using a software called Matlab. This software enabled us to plot frequency graphs and velocity histograms that helped us to analyse the results.

Presentation of ResultsSampling Frequency: 350 Hz (Under Sampling) Position of Hotwire: Centre line of the cylinder.

equency spectra

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Graph 2: velocity histogram

Sampling Frequency: 1400 Hz (Over Sampling) Position of hotwire: Centre line of the cylinder

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Graph 3: frequency spectra


Sampling Frequency: Optimum frequency (700 Hz) Position of hotwire: Centre line of the cylinder


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Sampling Frequency: Optimum frequency (700 Hz) Position of hotwire: 10mm from the cylinder center line


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Sampling Frequency: Optimum frequency (700 Hz) Position of hotwire: 20mm from the cylinder centre line


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Sampling Frequency: Optimum frequency (700 Hz) Position of hotwire: 70mm from the cylinder center line


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Sampling Frequency: Optimum Sampling frequency (700 Hz) Position of hotwire: 100mm from the cylinder centre line


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Theory and CalculationsHere, the mean, RMS and RTI values was to be calculated in order to plot their velocity profiles across the wake. Below are the equations [7], [8] and [9] which were used to calculate these values, are restated from the introduction. Mean velocity U= Eq. [7] where, U is the mean velocity (m/s) Ui is the each individual data of velocity (m/s) n is the number of samples RMS velocity RMS(u) = 1ni=1n(Ui- U)2 [8] where, 24 Eq. i=1nUin

RMS(u) is the RMS velocity (m/s) U is the mean velocity (m/s) Ui is the individual data value of velocity (m/s) n is the number of samples Relative Turbulence Intensity (RTI) RTI = RMS(u)U 100% Eq.[9] where, RTI is the Relative Turbulence Intensity RMS(u) is the RMS velocity (m/s) U is the mean velocity (m/s) Example Calculation Since the calculations are repeated, one such calculation was done which can be seen below. Sampling Frequency: 700 Hz Position of Hotwire: 10mm from the centre line of the cylinder Using Equation [7] Mean Velocity, U= i=11024Uin=8.75 m/s Using Equation [9] RMS(u) = 11024i=11024(Ui- 8.75)2= 1.98 m/s Using Equation [8] RTI = 1.988.75 100%=0.23 Once all the calculations are performed, the results are then tabulated which can be seen below. 25

Distance of Hotwire from centre line of cylinder (mm) 0 0 0 10 20 30 40 50 60 70 80 90 100

Sampling Frequenc y (H z) 350 1400 700 700 700 700 700 700 700 700 700 700 700

Mean Velocity (m/ s) 8.63 8.57 8.67 8.75 9.53 10.09 10.30 10.28 10.37 10.39 10.37 10.40 10.42

RMS Velocity (m/ s) 2.14 1.93 1.94 1.98 1.81 1.14 0.70 0.28 0.11 0.11 0.08 0.07 0.08

RT I

0.25 0.23 0.22 0.23 0.19 0.11 0.06 0.03 0.01 0.01 0.007 0.006 0.0007

Table 1: Mean, RMS and RTI values

Now, using the values from the table graphs of mean, RMS and RTI are plotted across the wake.

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Graph 27: Velocity profile for mean and RMS values

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Graph 28: RTI values across the wake

Discussion of ResultsEffects of over sampling and under sampling The optimum sampling rate for this experiment was found to be 700 Hz. When the sampling frequency was reduced to a value of 350 Hz, it can be seen from Graph 1 that there are not enough samples to analyse the variation in the signal. This is phenomenon is known as under sampling and gives us wrong information about the signal, which can lead to aliasing. When the sampling frequency was increased to value of 1400 Hz, it can be seen from Graph 3 that there are more than enough samples to analyse the variation in the signal. This is known as over sampling which gives rise to two problems,

1. Noise. 2. Increased processing time to analyse excess data. Analysis of frequency spectra From Graph 5, the peaks clearly indicate the formation of upper and lower vortices when the probe is at cylinder centre at an optimum sampling frequency of 700Hz. When the probe is lowered by 10mm, frequency of lower vortex increases whereas upper vortex starts to decrease, which can be seen from Graph 7.This is because as we traverse down, the probe detects the lower vortex to a greater extent than the upper vortex. As the probe is lowered down to a great extent the magnitude of the graph keeps reducing constantly which can be seen from Graphs 9, 11, 13, 15, 17, 19, 21, 23 and 25. This is because the probe gradually moves out of the lower vortex too.

Analysis of velocity histogram From Graph 6, we can clearly see that the velocity is spread over a range of values, which shows that the flow is turbulent.

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As the probe is lowered down the range of values over which the velocity is spread out decreases which can be seen from the Graphs 10, 12, 14, 16, 18, 20, 22, 24 and 26. This shows that the turbulence of flow is reduced. Graph 27 shows the velocity profile for mean, RMS across the wake. From this it is evident that as the probe is lowered, mean velocity increases whereas the RMS velocity decreases.

Error Analysis The most common error is the human error caused by the experimenter when he misreads an instrument or when an error is made during calculations. The probe interfering with flow field causes an error of around 23 %.

Conclusion The maximum vortex shedding frequency was calculated to be 264 HZ. The optimum sampling frequency for the above vortex shedding frequency was approximately found to be 700 Hz. Under sampling was found to produce less number of data and over sampling produced too much data resulting in noise. The graphs of the frequency spectra, velocity histogram, RMS, RTI and mean velocity profiles across the wake were plotted. Errors that occurred during the experiment were analysed.

References[1] http://www.wikipedia.org/ [2] http://www.nonoscience.info/2009/03/12/avenue-de-henri-benard/ [3] Cooper D. (2011) Vortex Shedding from a Cylinder and Data Acquisition Laboratory Notes [4]http://www.grc.nasa.gov/WWW/Acoustics/code/adpac/sample/CYLIN DER_VORTEX_SHEDDING/index.html

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[5] Ardekani M.A. (2008) Hot-wire calibration using vortex shedding. http://www.sciencedirect.com/science/article/pii/S026322410800208X [6] Pictures taken from laboratory.

AppendixMatlab code for frequency spectra

optimum_samplingrate=700; under_sampling=350; over_sampling=1400; % no of samples optimum samples=1024; under_samples=512; over_samples=2048; i=0:1:optimum_samples-1; j=0:1:under_samples-1; k=0:1:over_samples-1; % Frequency calculation optimum_frequency=i*(optimum_samplingrate/optimum_samples); under_frequency=j*(under_sampling/under_samples); over_frequency=k*(over_sampling/over_samples); % Velocity data is read velocity_0mm_under= xlsread(filename,D1:D4); velocity_0mm_over= xlsread(filename,D1:D4); velocity_0mm_optimum = xlsread(filename,D1:D4); velocity_10mm = xlsread(filename,D1:D4); velocity_20mm = xlsread(filename,D1:D4); velocity_30mm = xlsread(filename,D1:D4); velocity_40mm = xlsread(filename,D1:D4); velocity_50mm = xlsread(filename,D1:D4); velocity_60mm = xlsread(filename,D1:D4); velocity_70mm = xlsread(filename,D1:D4); velocity_80mm = xlsread(filename,D1:D4); velocity_90mm = xlsread(filename,D1:D4); velocity_100mm = xlsread(filename,D1:D4); 30

% FFT fft_0mm_under=fft(velocity_0mm_under,512); fft_0mm_over=fft(velocity_0mm_over,2048); fft_0mm_optimum=fft(velocity_0mm_optimum,1024); fft_10mm=fft(velocity_10mm,1024); fft_20mm=fft(velocity_20mm,1024); fft_30mm=fft(velocity_30mm,1024); fft_40mm=fft(velocity_40mm,1024); fft_50mm=fft(velocity_50mm,1024); fft_60mm=fft(velocity_60mm,1024); fft_70mm=fft(velocity_70mm,1024); fft_80mm=fft(velocity_80mm,1024); fft_90mm=fft(velocity_90mm,1024); fft_100mm=fft(velocity_100mm,1024); % Since fft is symmetric, second half is thrown away m=1:1:513; r1=1:1:257; r2=1:1:1025; % take the magnitude of fft(x) magnitude_under=abs(fft_0mm_under(r1)); magnitude_over=abs(fft_0mm_over(r2)); magnitude_optimum=abs(fft_0mm_optimum); magnitude_10mm=abs(fft_10mm(m)); magnitude_20mm=abs(fft_20mm(m)); magnitude_30mm=abs(fft_30mm(m)); magnitude_40mm=abs(fft_40mm(m)); magnitude_50mm=abs(fft_50mm(m)); magnitude_60mm=abs(fft_60mm(m)); magnitude_70mm=abs(fft_70mm(m)); magnitude_80mm=abs(fft_80mm(m)); magnitude_90mm=abs(fft_90mm(m)); magnitude_100mm=abs(fft_100mm(m)); % Generate the plot,title,axis subplot(5,4,1) plot(under_frequency(m),magnitude_0mm_under); subplot(5,4,2) plot(over_frequency(m),magnitude_0mm_over); subplot(5,4,3) 31

plot(optimum_frequency(m),magnitude_0mm_optimum); subplot(5,4,4) plot(optimum_frequency(m),magnitude_10mm); subplot(5,4,5) plot(optimum_frequency(m),magnitude_20mm); subplot(5,4,6) plot(optimum_frequency(m),magnitude_30mm); subplot(5,4,7) plot(optimum_frequency(m),magnitude_40mm); subplot(5,4,8) plot(optimum_frequency(m),magnitude_50mm); subplot(5,4,9) plot(optimum_frequency(m),magnitude_60mm); subplot(5,4,10) plot(optimum_frequency(m),magnitude_70mm); subplot(5,4,11) plot(optimum_frequency(m),magnitude_80mm); subplot(5,4,12) plot(optimum_frequency(m),magnitude_90mm); subplot(5,4,13) plot(optimum_frequency(m),magnitude_100mm); axis([0,350,0,500]); ylabel('magnitude(Hz/w^2)'); xlabel('frequency(Hz)'); title('fft');

Matlab code for velocity histogram optimum_samplingrate = 700; no_of_samples=1024; bin=0:0.2:16; hist(velocity_0mm_optimum,bin); xlabel('velocity(m/s)'); ylabel('PDF');

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