Vortex Saad Zafar

School of Mechanical, Aerospace and Civil Engineering

Lab Report

Vortex Shedding from a Cylinder & Data Acquisition

Name : Saad Zafar

Student ID : 7564644

Course : M.Sc. Thermal Power and Fluid Engineering

Lab Date : 27/10/2011

Due Date : 10/11/2011

Objectives:

To calculate the optimum sampling rate and see the effect of over and under

sampling.

To plot the frequency spectra and the velocity histogram by using the Matlab software

and also see the effect of vortices on the fluid velocity.

To study the variation of Mean Velocity, RMS Velocity and RTI across the different

position of the wake.

Introduction:

When flow is passed over a cylinder it also generates thin boundary layer, in characteristic

this boundary layer is unstable and breaks away as the flow is passed over the cylinder as a

result it generates a pair of travelling vortices. The vortices are generated from the both sides

of the cylinder at regular intervals. The generation frequency of these vortices or vortex

shedding frequency depends upon the Reynolds number.

Figure No. 1: Heart-shaped clouds float over a Mexican island in a photo taken 200 miles

above Earth [1].

The relationship between the vortex shedding frequency and the Reynolds number is

explained below

(1)

(2)

(3)

Where

ρ = Density of the fluid (kg/m3)

= Free stream velocity (m/s)

d = Diameter of the cylinder (mm)

= Dynamic viscosity of the fluid (pa/s)

= Strouhal number

= Reynolds number

= Vortex shedding frequency (Hz)

When a fluid flows over a cylinder the vortices are generated from the both sides of the

cylinder at regular intervals. The time interval between the generation of vortices from the

each side the cylinder is t. But we know as a matter of fact that these vortices are generated

alternatively from both sides, the time interval between two vortices generated from the

opposite sides is t/2. So in the light of this discussion we can say that two frequencies are

generated f1 and f2.

f1

(4)

f2 (5)

fmax = 2f1 (6)

Where fmax (Hz) is the maximum frequency in the data.

This experiment is designed to see the effect of vortex shedding when a fluid passes over a

cylinder. In order to get reasonable results it is very important that we chose the right

sampling rate. The optimum sampling rate can be found by using Nyquist theory which states

“The sampling frequency should be at least twice the highest frequency contained in the

signal”. But keeping in mind the band width of the signal as rule sampling frequency fs (Hz)

must be 2.5 times greater than the maximum frequency in the signal. So

fs = 2.5× fmax (7)

Hot Wire Anemometry Principal

Anemometer is device that is generally used to determine the velocity and pressure of the

fluid. The hot wire anemometer is used to measure the velocity of fluid flow and its working

principal states “Hot-wire anemometers are devices used to measure the variables occurring

in turbulent flows, such as mean- and fluctuating velocity components, mean and fluctuating

temperature, etc. The sensors are thin metallic elements heated by an electric current (Joule

effect) and cooled by the incident flow, which acts by virtue of its mass flux and its

temperature (through various effects, but with forced convection usually predominant). From

the temperature (or resistance) attained by the sensor, it is then possible to deduce

information on the flow” [2]. The hot wire anemometer’s resistances is connected in a

Wheatstone bridge as it changes the output voltage from the Wheatstone bridge also changes

so we get the velocity of the flow in terms of voltage.

U = U + u’ (8)

1

ni

i

UU

n

(9)

Where

U = Total instantaneous velocity (m/s)

u’ = Fluctuating component of the velocity

U = Mean velocity (m/s)

iU = Instantaneous velocity at each sample point (m/s)

n = Number of samples

The RMS value of the fluctuating component of the velocity can be found by using following

equation.

2

1

1( ') ( )

n

i

i

RMS u U Un

(10)

( ')100%

RMS uRTI

U

(11)

Where RTI is relative turbulence intensity.

In this experiment we will calculate the maximum frequency in the flow, optimum sampling

rate and the effect of over and under sampling.

We will also plot the variations of mean velocity, RMS velocity and RTI across the different

positions of wake [3].

Experimental Apparatus

The apparatus used in this experiment is listed below.

Arm Field Wind Tunnel

This wind tunnel is used to generate air flows of various speeds and its cross sectional

area is nearly 300 mm * 300 mm.

Traverse Gear Mechanism

With the help of this mechanism the hot wire probe is placed at different positions

across the wake.

Pitot Tube

It is used to determine the pressure of the flow.

Hot Wire Anemometer Probe

The diameter of the hot wire is 0.005mm. This is placed inside the wind tunnel.

Hot Wire Anemometer Control Box

This instrument is used to convert the changes in the resistance of the hot wire probe,

due to the change in the velocity of wind, in the form of voltage.

Micro Manometer

This instrument is used for the measurement of the pressure.

Computer

The computer is equipped with the software that are used receive signal from the

Analogue to Digital Converter. It also stores the date of the experiment that is

performed.

Circular Cylinder

A circular cylinder of 15 mm is placed inside the wind tunnel to generate the vortex

shedding effect.

Spectrum Analyzer

Experimental Procedure

In the first step the velocity of the flow was calibrated in terms of voltage. Each time when

we increase the pressure of the fluid flow in the arm field wind tunnel, the resistance of the

hot wire probe changes and the output voltage also changes. So for different values of

pressure we noted the output voltage. For every value of pressure the value of the velocity is

calculated by using Bernoulli equation. So in this way we calibrated fluid velocity.

In the next phase we determine the maximum frequency in the signal by using the equation

number (1) and (7) its value was 266.7 Hz. Then the optimum sampling rate was determined

by using the equation number (8) and its value was 667 Hz. We rounded this frequency to

700 Hz. The number of samples collected per second was set to the 210

= 1024.

Now we place the hot wire probe at the centre line of the circular cylinder and take three set

of data for the three set of sampling rates (optimum sampling rate, under sampling and over

sampling rate).

At the final stage the hot wire probe was placed at different locations across the wake, each

position was 10 mm apart. In order to perform the experiment the number of samples

collected per second was set to 1024 and sampling rate was set to 700 Hz.

In that way we had enough data to discuss the effects of over and under sampling; to discuss

the variations of mean velocity, RMS velocity and RTI across the different positions of wake.

Results

Graph No. 1: Frequency Spectra for Optimum Sampling Frequency of 700 Hz.

Graph No. 2: Frequency Spectra for Under Sampling Frequency of 350 Hz.

Graph No. 3: Frequency Spectra for Over Sampling Frequency of 1400 Hz.

Graph No. 4: Histogram for Optimum Sampling Frequency of 700 Hz.

Graph No. 5: Histogram for Under Sampling Frequency of 350 Hz.

Graph No. 6: Histogram for Over Sampling Frequency of 1400 Hz.

The Following Set of Graphs Will Represent the Frequency Spectra across

the Wake

Graph No. 7: Frequency Spectra for Sampling Frequency of 700 Hz at 0mm.

Graph No. 8: Frequency Spectra for Sampling Frequency of 700 Hz at 40mm along the

wake.


wake.


wake.

The Following Set of Graphs Will Represent the Velocity Histogram across

the Wake

Graph No. 11: Velocity Histogram for Sampling Frequency of 700 Hz at 0mm.




The Following Set of Graphs Will Represent the Variation of Mean

Velocity, RMS Velocity and RTI across the Wake

Graph No. 15: Variation of the Mean Velocity along the Wake.

Graph No. 16: Variation of the RMS Velocity along the Wake.

Graph No. 17: Variation of the RTI % along the Wake.

Wake Position

mm

Mean Velocity

m/s

RMS Velocity

m/s

RTI %

0 8.61 2.10 24.44

10 8.86 2.07 23.44

20 9.61 1.78 18.55

30 10.24 1.18 11.57

40 10.34 0.84 8.15

50 10.39 0.28 2.68

60 10.48 0.13 1.29

70 10.43 0.08 0.80

80 10.50 0.08 0.80

90 10.51 0.09 0.87

100 10.38 0.07 0.69

Table No. 1: Experimental values of Mean Velocity, RMS Velocity and RTI %

across the wake.

Theory and Calculations

In order to calibrate the velocity in terms of voltage following relationship is used

2 + 7.6416x + 0.1198 (12)

x = (E-Eo)2

(13)

Where

E = Value of Voltage at a respective point

E0 = Reference Voltage

The sampling frequency can be calculated by the following method.

U0 = 10 m/s

d = 0.015 m

St = 0.2

Hz

The maximum frequency in the flow is given by the following equation.

fmax = 2f

fmax = 2×133.33

fmax = 266.67 Hz

In order to calculate the sampling frequency fs we used the Nyquist theory according to which

fs = 2.5× 266.67

fs = 666.67 Hz

fs ≈ 700 Hz

So the optimum sampling frequency is nearly 700 Hz.

Sr. No E

V

(E-Eo)

V

(E-Eo)2

V2

U

m/s 1 3.513184 0.871797 0.760029 6.946396

2 3.491211 0.849824 0.722201 6.573431

3 3.605957 0.96457 0.930395 8.675006

4 3.364258 0.722871 0.522542 4.670251

5 3.662109 1.020722 1.041874 9.84943

6 3.435059 0.793672 0.629915 5.680089

7 3.417969 0.776582 0.603079 5.424719

8 3.569336 0.927949 0.861089 7.962143

9 3.515625 0.874238 0.764292 6.988675

10 3.554687 0.913301 0.834118 7.688305

Table No. 2: Sample date set obtained at the 700 Hz and 1024 samples per

second.

2 + 7.6416x + 0.1198

Consider the data of first row of table number 2

U = 1.0014×(0.760029)2 + 7.64×0.760029 + 0.1198

U = 6.51 m/s

So the value of U measured through the equation is 6.51 m/s and the experimental value is

6.94 m/s.

The values of the Mean Velocities, RMS Velocities and RTI across the wake are calculated

by using Matlab software and they are represented in the table below.

Wake Position

mm

Mean Velocity

m/s

RMS Velocity

m/s

RTI %

0 8.61 2.10 24.44

10 8.86 2.07 23.44

20 9.61 1.78 18.55

30 10.24 1.18 11.57

40 10.34 0.84 8.15

50 10.39 0.28 2.68

60 10.48 0.13 1.29

70 10.43 0.08 0.80

80 10.50 0.08 0.80

90 10.51 0.09 0.87

100 10.38 0.07 0.69

Table No. 1: Experimental values of Mean Velocity, RMS Velocity and RTI %

across the wake.

Result Discussion

Graph No. 1 shows the results of frequency spectra at the optimum sampling rate. Frequency

values are plotted along the x-axis while there magnitudes are plotted along the y-axis. By

analyzing the signal we say that there are two distinct peeks that can be seen. The 1st peak

appears in between 130 Hz to 135 Hz. While the 2nd

peaks appears in between 255 Hz to 265

Hz. These two frequency peaks represents the presence of two vortices that are generated as

an effect of vortex shedding phenomena due to the fluid flow over the circular cylinder.

Graph No. 2 shows the result obtained from the under sampling frequency data. Here we can

see lots of peaks at nearly the same height. We are unable to find a distinct peak. The reason

of this behaviour is the un-availability of sufficient data points so we cannot predict about the

original signal and we lose some important information about the original sampling. The 3rd

graph shows the results of over sampling. Here due to the excessive data points the signal is

too densely packed that we cannot find the distinct peaks and this signal also contains lots of

noise. It also makes it difficult to analyze.

In the graph 7 one can see two peaks which show the presence of two vortices on the both

sides of cylinder. In the next graph when probe is moved downwards we see only one distinct

peak. As the probe is moved further down across the wake, the first peak also started to

diminish. In the graph 9 at a distance of 70 mm one can only see a hint of a peek and when

we finally reach the distance of 100 mm away from the centre line the both peaks are

completely diminished.

The graphs from 11 to 14 show the velocity histogram at different positions across the wake.

In the graph 11 the velocity is distributed from 5 m/s to 13 m/s. When the probe is placed at a

distance of 40 mm from the centreline the velocity is mainly distributed to the higher values.

If we keep moving away from the centreline of the cylinder the velocity distributed towards

the higher value.

The graph 15 shows the variation of the mean velocity across the wake. As we move away

from the wake the mean velocity keeps on increasing. At the later stages away from the wake

its value starts stabilizing.

The graph 16 shows the variation of RMS velocity across the wake and graph 17 shows the

variation of RTI % across the wake. As we move away from the wake the RMS and RTI %

values decreases showing that the turbulence effects become less and less, away from the

wake.

Conclusion

Under sampling will result in the loss of important information about the data and in

the case of over sampling there is too much noise to predict the original signal. So,

under and over sampling both are not suitable to analyze the original signal. For

analyzing a set of data always use the optimum sapling rate as explained in the report.

The mean velocity is increased as we moves away from the wakes and after a suitable

distance it stabilizes.

RMS value and RTI % is decreased as we move away from the wake means that the

effect of turbulence go on decreasing.

References

[1] WEATHERVORTEX (2011) Heart-shaped clouds float over a Mexican island in a photo

taken 200 miles above Earth [Online Image] Available from:

http://weathervortex.com/wakes.htm [Accessed 09/11/2011].

[2] Comte-Bellot,G. (1976) Hot-Wire Anemometry. 8th

ed. Annual Review of Fluid

Mechanics.

[3] Cooper, D. (2010). Vortex Shedding Lab manual

http://weathervortex.com/wakes.htm

Index

Fast Fourier Transform

%FFT at 700 Hz at 0 mm

Data_Velocities=importdata('Velocity Results.xlsx') % Data input from Axel

to Matlab Velocity_0mm=Data_Velocities(:,1); % velocity data at 0 mm Fs=700; % sampling rate p=1024;

t=Fs/p; FFT_0mm=fft(Velocity_0mm); Frequency_0mm=[0:1:512].*t; Magnitude_0mm=abs(FFT_0mm(1:513,:)); plot(Frequency_0mm,Magnitude_0mm) %plotting results axis([0 350 -100 300]) xlabel('Frequency (Hz)') ylabel(' Magnitude (Hz/ W^2)') title( ' Fast Fourier Transform Analysis at the Centre Line of the

Cylinder')




t=Fs/p; FFT_0mm=fft(Velocity_0mm); Frequency_0mm=[0:1:512].*t; Magnitude_0mm=abs(FFT_0mm(1:513,:)); plot(Frequency_0mm,Magnitude_0mm) %plotting results axis([0 350 -100 300]) xlabel('Frequency (Hz)') ylabel(' Magnitude (Hz/ W^2)') title( ' Fast Fourier Transform Analysis at 40 mm along the Wake')









Histograms % histogram for optimum frequency at 0 mm

Velocity_0mm= Data_Velocities(:,1); %velocity data at 0 mm hist(Velocity_0mm,0:0.1:20) axis([0 20 0 50]) xlabel('Velocity (m/s)') ylabel('Velocity Probability Distribution (PDF)') title('Histogram for Sampling Frequency of 700 Hz at 0mm along the Wake')

% histogram for optimum frequency at 40 mm






Mean Velocity Data_Velocities=importdata('Velocity Results.xlsx'); % Data input from Axel

to Matlab

Velocity_0mm=Data_Velocities(:,1); % velocity data at 0 mm Velocity_10mm=Data_Velocities(:,2); % velocity data at 10 mm Velocity_20mm=Data_Velocities(:,3); % velocity data at 20 mm Velocity_30mm=Data_Velocities(:,4); % velocity data at 30 mm Velocity_40mm=Data_Velocities(:,5); % velocity data at 40 mm Velocity_50mm=Data_Velocities(:,6); % velocity data at 50 mm Velocity_60mm=Data_Velocities(:,7); % velocity data at 60 mm Velocity_70mm=Data_Velocities(:,8); % velocity data at 70 mm Velocity_80mm=Data_Velocities(:,9); % velocity data at 80 mm Velocity_90mm=Data_Velocities(:,10); % velocity data at 90 mm Velocity_100mm=Data_Velocities(:,11); % velocity data at 100 mm

Mean_Velocity_0mm=mean(Velocity_0mm); % Mean velocity at 0 mm Mean_Velocity_10mm=mean(Velocity_10mm); % Mean velocity at 10 mm Mean_Velocity_20mm=mean(Velocity_20mm); % Mean velocity at 20 mm Mean_Velocity_30mm=mean(Velocity_30mm); % Mean velocity at 30 mm Mean_Velocity_40mm=mean(Velocity_40mm); % Mean velocity at 40 mm Mean_Velocity_50mm=mean(Velocity_50mm); % Mean velocity at 50 mm Mean_Velocity_60mm=mean(Velocity_60mm); % Mean velocity at 60 mm Mean_Velocity_70mm=mean(Velocity_70mm); % Mean velocity at 70 mm Mean_Velocity_80mm=mean(Velocity_80mm); % Mean velocity at 80 mm Mean_Velocity_90mm=mean(Velocity_90mm); % Mean velocity at 90 mm Mean_Velocity_100mm=mean(Velocity_100mm); % Mean velocity at 100 mm

Mean_Velocity=[Mean_Velocity_0mm,Mean_Velocity_10mm,Mean_Velocity_20mm,Mean_

Velocity_30mm,Mean_Velocity_40mm,Mean_Velocity_50mm,Mean_Velocity_60mm,Mean_V

elocity_70mm,Mean_Velocity_80mm,Mean_Velocity_90mm,Mean_Velocity_100mm]

wake= (0:10:100)

plot(wake, Mean_Velocity) % ploting the mean velocity along the wake

xlabel('Position of Hot Wire Probe along the wake (mm)') ylabel('Mean Velocity (m/s)') title('Variation of Mean Velocity along the wake')

RMS velocity

Data_Velocities=importdata('Velocity Results.xlsx'); % Data input from Axel

to Matlab


Rms_Velocity_0mm=std(Velocity_0mm); % RMS velocity at 0 mm Rms_Velocity_10mm=std(Velocity_10mm); % RMS velocity at 10 mm Rms_Velocity_20mm=std(Velocity_20mm); % RMS velocity at 20 mm Rms_Velocity_30mm=std(Velocity_30mm); % RMS velocity at 30 mm Rms_Velocity_40mm=std(Velocity_40mm); % RMS velocity at 40 mm Rms_Velocity_50mm=std(Velocity_50mm); % RMS velocity at 50 mm Rms_Velocity_60mm=std(Velocity_60mm); % RMS velocity at 60 mm Rms_Velocity_70mm=std(Velocity_70mm); % RMS velocity at 70 mm Rms_Velocity_80mm=std(Velocity_80mm); % RMS velocity at 80 mm Rms_Velocity_90mm=std(Velocity_90mm); % RMS velocity at 90 mm Rms_Velocity_100mm=std(Velocity_100mm); % RMS velocity at 100 mm

Rms_Velocity=[Rms_Velocity_0mm,Rms_Velocity_10mm,Rms_Velocity_20mm,Rms_Velo

city_30mm,Rms_Velocity_40mm,Rms_Velocity_50mm,Rms_Velocity_60mm,Rms_Velocit

y_70mm,Rms_Velocity_80mm,Rms_Velocity_90mm,Rms_Velocity_100mm] wake=(0:10:100) plot(wake, Rms_Velocity) % ploting the RMS velocity along the wake xlabel('Position of Hot Wire Probe along the wake (mm)') ylabel('RMS Velocity (m/s)') title('Variation of RMS Velocity along the wake')

RTI

Data_Velocities=importdata('Velocity Results.xlsx'); % Data input from Axel

to Matlab


Mean_Velocity_0mm=mean(Velocity_0mm); % Mean velocity at 0 mm Mean_Velocity_10mm=mean(Velocity_10mm); % Mean velocity at 10 mm Mean_Velocity_20mm=mean(Velocity_20mm); % Mean velocity at 20 mm Mean_Velocity_30mm=mean(Velocity_30mm); % Mean velocity at 30 mm Mean_Velocity_40mm=mean(Velocity_40mm); % Mean velocity at 40 mm Mean_Velocity_50mm=mean(Velocity_50mm); % Mean velocity at 50 mm Mean_Velocity_60mm=mean(Velocity_60mm); % Mean velocity at 60 mm Mean_Velocity_70mm=mean(Velocity_70mm); % Mean velocity at 70 mm Mean_Velocity_80mm=mean(Velocity_80mm); % Mean velocity at 80 mm Mean_Velocity_90mm=mean(Velocity_90mm); % Mean velocity at 90 mm Mean_Velocity_100mm=mean(Velocity_100mm); % Mean velocity at 100 mm

Rms_Velocity_0mm=std(Velocity_0mm); % RMS velocity at 0 mm Rms_Velocity_10mm=std(Velocity_10mm); % RMS velocity at 10 mm Rms_Velocity_20mm=std(Velocity_20mm); % RMS velocity at 20 mm Rms_Velocity_30mm=std(Velocity_30mm); % RMS velocity at 30 mm Rms_Velocity_40mm=std(Velocity_40mm); % RMS velocity at 40 mm Rms_Velocity_50mm=std(Velocity_50mm); % RMS velocity at 50 mm Rms_Velocity_60mm=std(Velocity_60mm); % RMS velocity at 60 mm Rms_Velocity_70mm=std(Velocity_70mm); % RMS velocity at 70 mm

Rms_Velocity_80mm=std(Velocity_80mm); % RMS velocity at 80 mm Rms_Velocity_90mm=std(Velocity_90mm); % RMS velocity at 90 mm Rms_Velocity_100mm=std(Velocity_100mm); % RMS velocity at 100 mm

Mean_Velocity=[Mean_Velocity_0mm,Mean_Velocity_10mm,Mean_Velocity_20mm,Mean_

Velocity_30mm,Mean_Velocity_40mm,Mean_Velocity_50mm,Mean_Velocity_60mm,Mean_V

elocity_70mm,Mean_Velocity_80mm,Mean_Velocity_90mm,Mean_Velocity_100mm]

Rms_Velocity=[Rms_Velocity_0mm,Rms_Velocity_10mm,Rms_Velocity_20mm,Rms_Veloc

ity_30mm,Rms_Velocity_40mm,Rms_Velocity_50mm,Rms_Velocity_60mm,Rms_Velocity_

70mm,Rms_Velocity_80mm,Rms_Velocity_90mm,Rms_Velocity_100mm]

RTI=(Rms_Velocity./Mean_Velocity)*100 wake= (0:10:100) plot(wake,RTI) % ploting the RTI along the wake xlabel('Position of Hot Wire Probe along the wake (mm)') ylabel('RTI %') title('Variation of RTI along the wake')

Optimum case

%FFT at 700 Hz (optimum value)

Data_Velocities=importdata('700.xlsx') % Data input from Axel to Matlab Velocity_0mm=Data_Velocities(:,1); % velocity data at 0 mm Fs=700; % sampling rate p=1024; t=Fs/p; FFT_0mm=fft(Velocity_0mm); Frequency_0mm=[0:1:512].*t; Magnitude_0mm=abs(FFT_0mm(1:513,:)); plot(Frequency_0mm,Magnitude_0mm) %plotting results axis([5 350 -100 300]) xlabel('Frequency (Hz)') ylabel(' Magnitude (Hz/ W^2)') title( ' Fast Fourier Transform Analysis for Sampling Frequency of 700 Hz')


Data_Velocities=importdata('350.xlsx') % Data input from Axel to Matlab Velocity_0mm=Data_Velocities(:,1); % velocity data at 0 mm Fs=350; % sampling rate p=512;

t=Fs/p; FFT_0mm=fft(Velocity_0mm); Frequency_0mm=[0:1:256].*t; Magnitude_0mm=abs(FFT_0mm(1:257,:)); plot(Frequency_0mm,Magnitude_0mm) %plotting results axis([5 200 -100 600]) xlabel('Frequency (Hz)') ylabel(' Magnitude (Hz/ W^2)') title( ' Fast Fourier Transform Analysis for Sampling Frequency of 350 Hz')


Data_Velocities=importdata('1400.xlsx') % Data input from Axel to Matlab

Velocity_0mm=Data_Velocities(:,1); % velocity data at 0 mm Fs=1400; % sampling rate p=2048; t=Fs/p; FFT_0mm=fft(Velocity_0mm); Frequency_0mm=[0:1:1024].*t; Magnitude_0mm=abs(FFT_0mm(1:1025,:)); plot(Frequency_0mm,Magnitude_0mm) %plotting results axis([5 350 -100 300]) xlabel('Frequency (Hz)') ylabel(' Magnitude (Hz/ W^2)') title( ' Fast Fourier Transform Analysis for Sampling Frequency of 1400

Hz')

Histogram

% histogram for optimum Sampling Rate

Data_Velocities=importdata('700.xlsx') % Data input from Axel to Matlab Velocity_0mm= Data_Velocities(:,1); %velocity data at 0 mm hist(Velocity_0mm,0:0.1:20) axis([0 20 0 50]) xlabel('Velocity (m/s)') ylabel('Velocity Probability Distribution (PDF)') title('Histogram for Sampling Frequency of 700 Hz at 0mm along the Wake')

% histogram for Under Sampling


% histogram for Over Sampling


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Vortex Saad Zafar