experimental investigation of the speed of a shock wave in a shock tube via pressure transducer...

Experimental Investigation of the Speed of a Shock Wave in a Shock Tube via Pressure Transducer Readings

Rick J. AhlfArizona State University, Tempe, AZ, 85281

The objective of this experiment was to investigate the speed of a shockwave created from the burst pressure across three different diaphragms inside of a shock tube. To accomplish this, a shock tube with an interchangeable diaphragm, two static and two dynamic pressure transducers and an air compressor were utilized to calculate the burst pressure ratio and resulting shockwave speed and shockwave pressure ratio from aluminum foil, 2-mil Mylar plastic and 5-mil Mylar plastic diaphragms. The resulting average shockwave speeds for each of these diaphragms were 378.35, 451.05 and 493.63 m/s, respectively. It was concluded from the data that the higher the burst pressure ratio across the inserted diaphragm, the higher the pressure differential across the incidental shockwave and the higher the speed of that shockwave. The overall objective of the lab was met, and the uncertainties and errors for the experimental parameters were calculated and presented.

Nomenclaturea = Speed of Soundγ = Specific Heat Ratio P = Pressuret = TimeR = Specific Gas Constant of AirT = Temperatureup = Contact Surface Velocityρ = DensityW = Shock Wave Velocityx = Horizontal Distance

I. Introduction

This experiment was carried out in order to measure the speed of a shockwave created by the burst pressure of the various diaphragms inserted into the shock tube, along with the contact surface velocities for the various diaphragm materials. In regards to the experimental setup, “the shock tube consists of a long tube of constant area in which a diaphragm initially separates two bodies of gas at different pressures” (Guggenheim). The driver section of the shock tube would allow for a pressure accumulation on one side of the diaphragm exerting a pressure force that would eventually lead to the bursting of that respective diaphragm. The burst pressure ratio was calculated by measuring the ratio of the static pressure of the driver section of the shock tube at the instant just prior to the bursting of the diaphragm and the static pressure of the driven section at the opposite end of the shock tube following the burst. This burst induces a shock wave towards the section of lower pressure in the shock tube, which moves faster than the local speed of sound inside the tube. The speed of the shockwave could be measured by observing the readings of the two dynamic pressure transducers at each end of the driven section of the shock tube. Knowing the respective measurement time for each of the dynamic pressure transducer readings, one can calculate the difference in time between the first and second reading. The physical distance between the dynamic pressure transducers is a fixed value from the experimental setup, therefore the speed of the shockwave can be calculated with a simple distance over time calculation. Theory behind shockwave phenomena suggests that increasing differential pressures across a diaphragm will lead to an increase in the pressure ratio across the created shockwave.

1Rick Ahlf, Undergraduate, Aerospace Engineering, Email: [email protected]

American Institute of Aeronautics and Astronautics

This increase in pressure ratio across the shockwave will also increase the velocity of the shockwave itself. This relationship was studied in this experiment to confirm this phenomena, by measuring the relationship between the burst pressure ratios, differential pressure ratios across the shockwaves and shockwave velocities for aluminum foil, 2-mil Mylar plastic and 5-mil Mylar plastic diaphragms.

II. Description of Experiment and Procedures

This experiment was performed on the Arizona State University campus inside USEB 174 in Tempe, AZ. The equipment needed to perform this experiment is as follows: a shock tube, two static pressure transducers, two dynamic pressure transducers, a pressurized air source, diaphragm materials (aluminum foil, 2-mil Mylar plastic, and 5-mil Mylar plastic samples) and a data recording program.

To begin this experiment, it is necessary to first measure the ambient pressure of the surrounding air by reading the barometer and measure the ambient air temperature by reading the thermometer in the laboratory, while also recording uncertainties of these devices. The density of the ambient air and its uncertainty can now be calculated using a manipulation of the ideal gas law, as shown through equations 1 and 2.

ρ= PRT

(1)

σρ=±√( 1RT

∗σρ)2

+( −P

R T 2∗σT )2

(2)

The shock tube inside of the laboratory contains two static pressure transducers, with one at the driver section of the tube and one located at the opposite end of the tube in the driven section. Two dynamic pressure transducers are connected to the shock tube in the driven section, spaced apart at a fixed distance of 48.03125 inches. The pressure transducers report data to the computer program at a rate of 40 Hz, or a period of every 0.025 ms. The first step to be taken is the installation of the diaphragm material into the slot separating the driver section and the driven section of the shock tube. Starting with the aluminum foil diaphragm, four bolts are removed from the shock tube at the area of insertion of the diaphragm, allowing for the aluminum foil to be inserted as a barrier between the two sections. The bolts are then reinserted and locked in place with nuts and washers, securing the diaphragm in place. From this point, data collection should be started via the computer program linked to the pressure transducers.

Close the air source inlet valve prior to attaching the air hose. Attach the source hose. Slowly open the air source valve to begin pressurization of the driver section of the shock tube, until the diaphragm audibly bursts. Once the burst has occurred, close the inlet valve and the source valve. Verify that the data collected by the computer program is favorable. Perform this data collection process for the other respective diaphragm materials, the 2-mil Mylar plastic and the 5-mil Mylar plastic. Once the data is fully collected, the data can be analyzed using Matlab computational software. The setup schematic is presented in Fig. 1.



Figure 1. Schematic with Section Definitions for the Shock Tube.

To begin processing of the data collected from the shock tube, the data for all three materials and all trials should be imported into a Matlab script file. The data should then be converted to consistent units of psig (pounds per square inch gauge pressure). To data from the dynamic pressure transducers should be converted from Volts to psig via the conversion factor provided by the manufacturer, 23.29 V/PSI. The data corresponding to the driver section of the shock tube was then analyzed to observe the drop in pressure at the time of the burst of the diaphragm. The data was cut and plotted to capture a snapshot of the instant in time when the diaphragm burst and the pressure difference propagated through the shock tube via the shockwave. The differential pressure across the diaphragm for each material must be determined. The equation for the burst pressure ratio of the various diaphragms is referenced below as equation 3. P4 corresponds to the absolute pressure of the driver side section just before the burst, and P1 corresponds to the absolute pressure of the driven side section at that same instant. The P4 value was measured via the data cursor tool provided in the Matlab graphical analysis window. The pressures obtained from the graph should be converted to absolute pressure before calculation of the ratios can be completed.

Burst Pressure Ratio=P4

P1

(3)

The equation for the shockwave pressure ratio is given below as equation 4. P2 corresponds to the pressure after the shockwave, P1 represents the pressure before the shockwave relative to its motion, and ∆P represents the pressure change across the shockwave. P1 could be obtained graphically using the data cursor tool in the Matlab graphical window and converting the value to absolute pressure, whereas the value ∆P could be calculated by measuring the difference in pressure across the jump seen in the dynamic pressure transducer readings. The reading corresponds to the pressure difference experienced by the dynamic pressure transducer when the shockwave passes by.

Shockwave Pres sure Ratio=P2

P1

=P1+∆ P

P1

(4)

These two experimental ratios were plotted against the theoretical burst pressure ratio obtained via the following formula, labeled equation 5. This equation for the theoretical burst pressure ratio is dependent upon the shockwave pressure ratio, specific heat ratio of air and the difference in speed of sound between the two gases. However, since the gas remains constant, it can be assumed that the speed of sound and the specific heat ratio both remain constant. The Matlab function fzero() was used to solve the equation numerically and the result was plotted on the same plot as the experimental relationship.



P4

P1

=( P2

P1) [1− ( γ 4−1 ) ( a1

a4)(P2

P1

−1)√2 γ1 [2 γ 1+ (γ 1+1 )(P2

P1

−1)] ]−2 γ 4

γ 4−1

(5)

Once these pressure ratios were determined and plotted, the resulting shockwave velocities could be calculated and plotted. The experimental shockwave speed was calculated via the following formula, labeled equation 6. The change in distance was fixed between the two pressure transducers of the shock tube, while the change in time was measured from the plots of the dynamic pressure readings.

W =∆ x∆ t

(6)

The error in the calculation of the shockwave speeds was calculated via equation 7 below. The relevant uncertainties are provided in Table 6.

σW =√( 1∆ t

∗σx )2

+(−∆ x

∆ t 2 ∗σt )2

(7)

To study the effect of the shockwave pressure ratio (P2/P1) on the shockwave speed, the corresponding values were plotted against the theoretical relationship provided by the following formula, labeled equation 8.

W =a√ γ+12 γ (P2

P1

−1)+1(8)

The following, equation 9, allows for the calculation of the local speed of sound, which is dependent only upon the ambient temperature, as the specific heat ratio and gas constant are held constant.

a=√γRT (9)

The average experimental speed of sound for the various diaphragm materials were calculated via the following formula, equation 10, which is dependent upon the shockwave speed and the experimental shockwave pressure difference.

aE=W E

√ γ+12 γ ( P2

P1 E

−1)+1(10)



The contact surface velocity was calculated to observe the relationship between the strength of the shockwave and the contact surface velocity, given here as equation 11.

up=aγ ( P2

P1

−1)(2 γ

γ+1P2

P1

+ γ−1γ+1

)12

(11)

III. Results and Discussion

The following plots were created in Matlab and display the data from the static pressure transducers at the time near the bursting of the various diaphragms, along with the readings from the dynamic pressure transducers that show the propagation of the resulting shockwave through the shock tube. Overall, the pressure reading results were as expected, showing a significant drop in pressure in the driver section after the bursting of the diaphragm, and a sharp increase in the pressure of the driven section. From the static pressure plots, it can be seen that the two lines are converging upon one another as the pressure inside the shock tube equalizes as a result of the absence of the diaphragm barrier separating the two sections. The dynamic pressure plots show the small delay between the readings of the two dynamic pressure transducers as the shockwave travels down the shock tube.



Figure 2. Static Pressure Transducer Readings vs. Time for Aluminum Foil Diaphragm Plot.

Figure 3. Dynamic Pressure Transducer Readings vs. Time for Aluminum Foil Diaphragm Plot.



Figure 4. Static Pressure Transducer Readings vs. Time for 2-mil Mylar Plastic Diaphragm Plot.

Figure 5. Dynamic Pressure Transducer Readings vs. Time for 2-mil Mylar Plastic Diaphragm Plot.



Figure 6. Static Pressure Transducer Readings vs. Time for 5-mil Mylar Plastic Diaphragm Plot.

Figure 7. Dynamic Pressure Transducer Readings vs. Time for 5-mil Mylar Plastic Diaphragm Plot.

Table 1 presents the shockwave speed versus the burst pressure ratio for each diaphragm tested in the experimental setup, along with the corresponding uncertainties and errors in calculation. These values were obtained



using equations 3, 6 and 7. The analytical evaluation of the relative error in calculation showed that all errors were nearly all 1% or less.

Table 1 – Shockwave Speed vs. Burst Pressure Ratio with Errors

Trial #

Diaphragm Material

Burst Pressure Ratio (P4/P1)

Shock Wave Speed (m/s)

Velocity Error (±m/s)

% Relative Error

1 Aluminum Foil 1.51 376.54 2.92 0.781 Aluminum Foil B 1.47 370.82 2.83 0.761 2-mil Mylar 3.72 443.63 4.04 0.911 5-mil Mylar 6.07 488.00 4.89 1.002 Aluminum Foil 1.55 384.86 3.05 0.792 2-mil Mylar 3.75 455.22 4.26 0.942 2-mil Mylar B 3.74 450.18 4.16 0.922 5-mil Mylar 6.20 497.96 5.09 1.023 Aluminum Foil 1.55 377.71 2.93 0.783 2-mil Mylar 3.73 451.85 4.19 0.933 5-mil Mylar 6.16 493.92 5.01 1.013 5-mil Mylar B 6.10 489.96 4.93 1.014 Aluminum Foil 1.51 376.54 2.92 0.784 2-mil Mylar 3.91 451.85 4.19 0.934 2-mil Mylar B 3.75 446.88 4.10 0.924 5-mil Mylar 6.24 491.93 4.97 1.015 Aluminum Foil 1.58 383.65 3.03 0.795 2-mil Mylar 3.80 456.93 4.29 0.945 2-mil Mylar B 3.79 451.85 4.19 0.935 5-mil Mylar 6.33 500.00 5.13 1.03

Further analysis of the data allowed for the plotting of the experimental relationship between the shockwave pressure ratio and burst pressure ratio, which was compared to the theoretical relationship demonstrated through equations 3, 4 and 5 (Fig. 8). The relationship from the experimental data adhered very closely to the theoretical relationship with only small deviations from the theoretical curve.

Finally, the experimental relationship between the shockwave speed was plotted versus the shockwave pressure ratio, and was compared against the theoretical relationship presented in equation 8 (Fig. 9). The experimental data followed the theoretical curve almost perfectly, with all trials for each respective diaphragm falling near one another in individual clusters.



Figure 8. Shockwave Pressure Ratio vs. Burst Pressure Ratio Plot.

Figure 9. Shockwave Speed vs. Shockwave Pressure Ratio Plot.

Table 2 presents the average contact surface velocity for each diaphragm, calculated from the twenty data sets collected from the experimental setup. It can be concluded from the data that a greater contact surface velocity is indicative of a stronger shockwave, meaning the shockwave is traveling at a greater speed. The contact surface velocity is defined as the difference in velocity between the front and rear flows of the propagating shockwave. Similar to the speed of the actual shockwave, the contact surface velocity is dependent upon the specific heat ratio of the air, the differential pressure ratio across the shockwave and the local speed of sound (seen via equation 11). Table 2 – Contact Surface Velocity for the Three Diaphragm Materials

Diaphragm Average Contact Surface Velocity (m/s)



MaterialAluminum Foil 54.54

2-mil Mylar 173.925-mil Mylar 235.40

The ambient air density in the lab and its respective uncertainty was calculated via equations 1 and 2, and yielded a value consistent with the expected value. Table 3 presents the calculated value for ambient air density, along with the ambient properties used to perform the calculation.

Table 3 – Ambient Air Density in the LabLab Properties Value

Ambient Temperature

293.15 K

Barometric Pressure 98.60 kPaGas Constant for Air 287 J/kg-KAmbient Air Density 1.1717 kg/m3

The average experimental speed of sound was calculated for the three diaphragm materials via equation 10, and all values fell within the range for the expected local speed of sound. The results are presented below in Table 4. It may be of interest to note that the experimental speed of sound increased slightly with an increase in shockwave speed and shockwave pressure ratio.

Table 4 – Experimental Speed of Sound for the Three Diaphragm MaterialsDiaphragm

MaterialExperimental Speed of Sound (m/s)

Aluminum Foil 343.982-mil Mylar 334.305-mil Mylar 330.66

In order to gain a better understanding of how the speed of sound deviates with respect to the medium it is traveling through, Table 5 was generated to present the speed of sound in various gases that are included in the composition of air. Nitrogen and air showed to have roughly the same speed of sound, which makes sense because air is comprised of mostly nitrogen gas.

Table 5 – Speed of Sound for the Included GasesGas

γ - Specific Heat Ratio R - Gas Constant (J/kg-K) T – Temperature (K) a – Speed of Sound (m/s)

Air 1.40 287 293.15 343.20H2 1.41 4120 293.15 1304.98He 1.66 2080 293.15 1006.08N2 1.40 297 293.15 349.13

In this experiment, error and uncertainty were considered throughout the results and calculations of the experimental shockwave speed and the ambient air density. Table 6 presents these relevant uncertainties in the various measurements taken throughout the experiment.

Table 6 – Uncertainty ValuesMeasurement Uncertainty



Ambient Temperature ±0.5 K

Barometric Pressure ±0.05 kPaTime ±0.025 ms

Ambient Air Density ±0.0021 kg/m3

Length ±0.00079375 m

IV. Conclusion

Considering the overall outcome of the experiment, it can be concluded that the relationship between the diaphragm burst pressure ratio, the shockwave pressure ratio and the propagation speed of the shockwave through the shock tube correlate well with the theoretical relationships governed by the cited equations. The values obtained from the data do make sense with the theory, and prove the relationships outlined in the lab manual. As the strength of the diaphragm material increases, or in other words, the burst pressure ratio increases, the speed of the resulting shockwave from the burst increases as well. The shockwave pressure ratio also increases with a larger burst pressure ratio. This experiment proved that the shockwave speed did propagate at a speed higher than the ambient speed of sound. In order to generate an even faster shockwave through the shock tube, the ambient speed of sound could be increased by raising the ambient temperature of the gas within the shock tube. These valuable insights could allow for the design of various induced shockwaves in order to manipulate air properties for certain effects on aircraft and other models.

A number of improvements could be made to this experiment, such as improving the sensitivity of the dynamic pressure transducers. This would allow for more precise measurements of the exact instant that the shockwave propagated past the transducer and could allow for a more accurate calculation of the shockwave speed. Performing this experiment inside of a climate-controlled environment would also increase the consistency of the results, since the readings were taken over the course of a few days and at different times throughout the day. Further modifications could include pressure readings and data collection for shock tubes containing gases other than air, thus changing the specific heat ratio and studying the variance of the shockwave properties.

V. Appendices

A. Original Data

Table A-1 –Raw Static and Dynamic Pressure Transducer Measurements (data selected near shock wave formation)Driven Section Pressure

(psia)Driver Side Pressure

(psig)Dynamic Pressure Transducer

1 (V)Dynamic Pressure Transducer

2 (V)

14.4926 72.5372 0.0055 0.0009

14.4699 72.6214 0.0071 0.0024

14.4602 72.5761 0.0064 0.0014

14.4505 72.5761 0.0056 0.0014

14.4861 72.7055 0.0084 0.0008

14.4311 72.5243 0.0084 0.0014

14.4796 72.7185 0.0063 -0.0012



14.4570 72.6343 0.0064 0.0025

14.4343 72.6020 0.0055 0.0017

14.4634 72.5502 0.0063 0.0016

14.4570 72.7314 0.0072 0.0019

14.4375 72.6473 0.0076 0.0016

14.4764 72.5243 0.0063 0.0022

14.4537 72.5178 0.0089 0.0016

14.3922 72.5631 0.0080 0.0027

14.4861 72.5955 0.0072 0.0025

14.4570 72.5825 0.0079 0.0025

14.4311 72.6020 0.0056 0.0027

14.4893 72.6149 0.0056 0.0045

14.4472 72.5372 0.0056 0.0042

14.4311 72.5437 0.0066 0.0009

14.4893 72.5890 0.0071 0.0053

14.4796 72.5113 0.0053 0.0027

14.4181 72.6084 0.0067 0.0042

14.4472 72.5696 0.0074 0.0024

14.4990 72.6408 0.0084 0.0001

14.4764 72.5631 0.0055 0.0009

14.4505 72.6214 0.0056 0.0009

14.5055 72.5502 0.0064 0.0016

14.4926 72.6278 0.0058 0.0012

14.5185 72.5567 0.0050 -0.0010

14.4634 72.5825 0.0059 0.0006

14.4699 72.5567 0.0056 0.0008

14.4699 72.6343 0.0074 0.0014



14.4602 72.5502 0.0059 0.0003

14.4149 72.5825 0.0048 -0.0001

14.4408 72.5567 0.0061 0.0017

14.4440 72.6084 0.0074 0.0008

14.4375 72.6602 0.0072 0.0012

14.4926 72.5890 0.0061 0.0011

14.4699 72.6149 0.0072 0.0025

14.4472 72.6149 0.0061 0.0033

14.4861 72.6214 0.0076 0.0037

14.4634 72.5502 0.0453 0.0032

14.4408 72.6278 0.3605 0.0021

14.4440 72.6149 0.4911 0.0027

14.4537 72.5696 0.4718 0.0006

14.4926 72.6214 0.4440 0.0014

14.4731 72.3819 0.4205 0.0027

14.4990 72.4078 0.4485 -0.0005

14.4570 72.3560 0.4181 0.0009

14.4667 72.2783 0.4094 -0.0004

14.4667 72.1812 0.4251 0.0012

14.4796 72.1424 0.4387 0.0017

14.4375 71.9870 0.4406 0.0006

14.5314 72.0194 0.4222 0.0025

14.4990 71.9158 0.4275 0.0022

14.4570 71.8511 0.4329 0.0017

14.4311 71.6828 0.4482 0.0022

14.4214 71.6504 0.4155 0.0019

14.4796 71.5598 0.4117 0.0008



14.4505 71.6115 0.4267 -0.0007

14.4667 71.4886 0.4341 0.0009

14.4375 71.4044 0.4335 0.0024

14.4505 71.3397 0.4424 0.0022

14.5185 71.3203 0.4498 0.0011

14.4861 71.3073 0.4332 0.0029

14.4634 71.1455 0.4204 0.0019

14.4828 71.1778 0.4322 0.0035

14.4278 71.1390 0.4422 0.0021

14.3890 71.1131 0.4333 -0.0001

14.4375 70.9254 0.4327 0.0008

14.4343 70.9060 0.4356 0.0003

14.4731 70.7571 0.4388 0.0019

14.4343 70.6017 0.4346 0.0032

14.4505 70.5952 0.4458 0.0022

14.4990 70.5111 0.4435 0.0011

14.5055 70.4399 0.4350 -0.0007

14.4570 70.3169 0.4175 0.0001

14.4764 70.1810 0.4207 0.0030

14.4926 70.1486 0.4299 0.0011

14.4278 70.0580 0.4346 0.0017

14.4375 70.0256 0.4353 0.0019

14.4731 69.8120 0.4409 -0.0002

B. Sample Calculations

Calculation of the Shockwave Speed:



W AL=∆ x∆ t

= 1.220 m962.69 ms−959.45 ms

=376.54 m /s

Calculation of the Ambient Density of the Air in the Lab:

ρ= PRT

= 98.60 kPa

(287J

kg−k)(293.15 K )

=0.0012 kg/m3

Calculation of the Speed of Sound in the Shock Tube:

a=√γRT=√(1.4)(287J

kg−K)(293.15 K )=343.20 m /s

C. Complete Matlab Code

%Rick Ahlf%Arizona State University%AEE 362 Lab #1 clc, close, clear allformat compact; format long; %------------Speed of Sound Calculation----------------R = 287.05; %gas constant (J/kg-K)g = 1.4; %specific heat ratio of airT = 293.15; %temperature (K)P = 98600; %pressure (Pa)a = sqrt(g*R*T); %speed of sound (m/s) %------------Density Calculation with Uncertainty------rho = P/(R*T); %density (kg/m^3)dT = 0.5; %temperature error (K)dP = 50; %pressure error (Pa)drho = sqrt(((1./(R.*T)).*dP).^2+((-P./(R.*T.^2)).*dT).^2); %density error (kg/m^3) %------------Aluminum Foil-----------------------------alfoil1 = load('Wed_730_AlFoil.txt'); %aluminum foil dataP1 = (alfoil1(:,1)-14.30); %driven section pressure (psig)P4 = (alfoil1(:,2)); %driver side pressure (psig)P3 = (alfoil1(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (alfoil1(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig)time = linspace(0,0.025*length(alfoil1(:,1)),length(alfoil1(:,1)));figure(1)plot(time(38250:38700),P1(38250:38700),time(38250:38700),P4(38250:38700));title('Static Pressure vs. Time - Aluminum Foil');xlabel('Time (ms)'); ylabel('Static Pressure (psig)'); ylim([0,10])legend('P1 - Static Pressure of Driven Section','P4 - Static Pressure of Driver Section');figure(2)plot(time(38350:38600),P3(38350:38600),time(38350:38600),P2(38350:38600));



title('Dynamic Pressure vs. Time - Aluminum Foil');xlabel('Time (ms)'); ylabel('Dynamic Pressure (psig)'); ylim([0,4.5])legend('P3 - Dynamic Pressure Transducer 1','P2 - Dynamic Pressure Transducer 2'); %------------Shockwave Speed Calculation---------------dtal1 = (962.69-959.45);dxal1 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExpAl1 = (dxal1/dtal1)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic-----------------------mil21 = load('Wed_730_2Mil.txt'); %2Mil dataP1 = (mil21(:,1)-14.30); %driven section pressure (psig)P4 = (mil21(:,2)); %driver side pressure (psig)P3 = (mil21(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil21(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig)figure(3)plot(time(44500:45500),P1(44500:45500),time(44500:45500),P4(44500:45500));title('Static Pressure vs. Time - 2-mil Mylar Plastic');xlabel('Time (ms)'); ylabel('Static Pressure (psig)'); ylim([0,50])legend('P1 - Static Pressure of Driven Section','P4 - Static Pressure of Driver Section');figure(4)plot(time(45000:45300),P3(45000:45300),time(45000:45300),P2(45000:45300));title('Dynamic Pressure vs. Time - 2-mil Mylar Plastic');xlabel('Time (ms)'); ylabel('Dynamic Pressure (psig)'); ylim([0,16])legend('P3 - Dynamic Pressure Transducer 1','P2 - Dynamic Pressure Transducer 2'); %------------Shockwave Speed Calculation---------------dt2mil1 = (1130.34-1127.59);dx2mil1 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil1 = (dx2mil1/dt2mil1)*(1000/12*0.3048); %Shockwave speed (m/s) %------------5-mil Mylar Plastic-----------------------mil51 = load('Wed_730_5Mil.txt'); %5Mil dataP1 = (mil51(:,1)-14.30); %driven section pressure (psig)P4 = (mil51(:,2)); %driver side pressure (psig)P3 = (mil51(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil51(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig)figure(5)plot(time(76500:77500),P1(76500:77500),time(76500:77500),P4(76500:77500));title('Static Pressure vs. Time - 5-mil Mylar Plastic');xlabel('Time (ms)'); ylabel('Static Pressure (psig)'); ylim([0,90])legend('P1 - Static Pressure of Driven Section','P4 - Static Pressure of Driver Section');figure(6)plot(time(76950:77200),P3(76950:77200),time(76950:77200),P2(76950:77200));title('Dynamic Pressure vs. Time - 5-mil Mylar Plastic');xlabel('Time (ms)'); ylabel('Dynamic Pressure (psig)'); ylim([0,25])legend('P3 - Dynamic Pressure Transducer 1','P2 - Dynamic Pressure Transducer 2'); %------------Shockwave Speed Calculation---------------dt5mil1 = (1928.55-1926.05);dx5mil1 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp5mil1 = (dx5mil1/dt5mil1)*(1000/12*0.3048); %Shockwave speed (m/s)

%------------Aluminum Foil B---------------------------alfoilb = load('Wed_730_AlFoil_B.txt'); %aluminum foil dataP1 = (alfoilb(:,1)-14.30); %driven section pressure (psig)P4 = (alfoilb(:,2)); %driver side pressure (psig)P3 = (alfoilb(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (alfoilb(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(15100:15500),P3(15100:15500),time(15100:15500),P2(15100:15500));dtalb = (382.26-378.97);dxalb = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExpAl1b = (dxalb/dtalb)*(1000/12*0.3048); %Shockwave speed (m/s) %------------Aluminum Foil 2---------------------------alfoil2 = load('Fri_1150_AlFoil.txt'); %aluminum foil dataP1 = (alfoil2(:,1)-14.30); %driven section pressure (psig)



P4 = (alfoil2(:,2)); %driver side pressure (psig)P3 = (alfoil2(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (alfoil2(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(52300:52800),P3(52300:52800),time(52300:52800),P2(52300:52800));dtal2 = (1311.52-1308.35);dxal2 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExpAl2 = (dxal2/dtal2)*(1000/12*0.3048); %Shockwave speed (m/s) %------------5-mil Mylar Plastic 2---------------------mil52 = load('Fri_1150_5Mil.txt'); %5Mil dataP1 = (mil52(:,1)-14.30); %driven section pressure (psig)P4 = (mil52(:,2)); %driver side pressure (psig)P3 = (mil52(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil52(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(58500:58800),P3(58500:58800),time(58500:58800),P2(58500:58800));dt5mil2 = (1466.07-1463.62);dx5mil2 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp5mil2 = (dx5mil2/dt5mil2)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 2---------------------mil22 = load('Fri_1150_2Mil.txt'); %2Mil dataP1 = (mil22(:,1)-14.30); %driven section pressure (psig)P4 = (mil22(:,2)); %driver side pressure (psig)P3 = (mil22(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil22(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(1400:1700),P3(1400:1700),time(1400:1700),P2(1400:1700));dt2mil2 = (38.18-35.50);dx2mil2 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil2 = (dx2mil2/dt2mil2)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 2B---------------------mil22b = load('Fri_1150_2Mil_B.txt'); %2Mil dataP1 = (mil22b(:,1)-14.30); %driven section pressure (psig)P4 = (mil22b(:,2)); %driver side pressure (psig)P3 = (mil22b(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil22b(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(66250:66550),P3(66250:66550),time(66250:66550),P2(66250:66550));dt2mil2b = (1659.36-1656.65);dx2mil2b = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil2b = (dx2mil2b/dt2mil2b)*(1000/12*0.3048); %Shockwave speed (m/s) %------------Aluminum Foil 3---------------------alfoil3 = load('Fri_940_AIFoil.txt'); %aluminum foil dataP1 = (alfoil3(:,1)-14.30); %driven section pressure (psig)P4 = (alfoil3(:,2)); %driver side pressure (psig)P3 = (alfoil3(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (alfoil3(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(17400:17700),P3(17400:17700),time(17400:17700),P2(17400:17700));dtal3 = (438.66-435.43);dxal3 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExpAl3 = (dxal3/dtal3)*(1000/12*0.3048); %Shockwave speed (m/s) %------------5-mil Mylar Plastic 3---------------------mil53 = load('fri_940_5mil.txt'); %5Mil dataP1 = (mil53(:,1)-14.30); %driven section pressure (psig)P4 = (mil53(:,2)); %driver side pressure (psig)P3 = (mil53(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil53(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig)



%------------Shockwave Speed Calculation---------------plot(time(38300:38600),P3(38300:38600),time(38300:38600),P2(38300:38600));dt5mil3 = (960.21-957.74);dx5mil3 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp5mil3 = (dx5mil3/dt5mil3)*(1000/12*0.3048); %Shockwave speed (m/s) %------------5-mil Mylar Plastic 3B--------------------mil53b = load('fri_940_5mil_b.txt'); %5Mil dataP1 = (mil53b(:,1)-14.30); %driven section pressure (psig)P4 = (mil53b(:,2)); %driver side pressure (psig)P3 = (mil53b(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil53b(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(10200:10500),P3(10200:10500),time(10200:10500),P2(10200:10500));dt5mil3b = (257.96-255.47);dx5mil3b = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp5mil3b = (dx5mil3b/dt5mil3b)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 3--------------------mil23 = load('fri_940_2mil.txt'); %2Mil dataP1 = (mil23(:,1)-14.30); %driven section pressure (psig)P4 = (mil23(:,2)); %driver side pressure (psig)P3 = (mil23(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil23(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(42800:43100),P3(42800:43100),time(42800:43100),P2(42800:43100));dt2mil3 = (1073.80-1071.10);dx2mil3 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil3 = (dx2mil3/dt2mil3)*(1000/12*0.3048); %Shockwave speed (m/s) %------------Aluminum Foil 4---------------------alfoil4 = load('fri_730_alum.txt'); %aluminum foil dataP1 = (alfoil4(:,1)-14.30); %driven section pressure (psig)P4 = (alfoil4(:,2)); %driver side pressure (psig)P3 = (alfoil4(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (alfoil4(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(20700:21000),P3(20700:21000),time(20700:21000),P2(20700:21000));dtal4 = (522.06-518.82);dxal4 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExpAl4 = (dxal4/dtal4)*(1000/12*0.3048); %Shockwave speed (m/s) %------------5-mil Mylar Plastic 4--------------------mil54 = load('Fri_730_5mil.txt'); %5Mil dataP1 = (mil54(:,1)-14.30); %driven section pressure (psig)P4 = (mil54(:,2)); %driver side pressure (psig)P3 = (mil54(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil54(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(33650:33950),P3(33650:33950),time(33650:33950),P2(33650:33950));dt5mil4 = (844.54-842.06);dx5mil4 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp5mil4 = (dx5mil4/dt5mil4)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 4--------------------mil24 = load('fri_730_2mil.txt'); %2Mil dataP1 = (mil24(:,1)-14.30); %driven section pressure (psig)P4 = (mil24(:,2)); %driver side pressure (psig)P3 = (mil24(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil24(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(66750:67050),P3(66750:67050),time(66750:67050),P2(66750:67050));dt2mil4 = (1672.07-1669.37);dx2mil4 = 48.03125; %distance between DPT1 and DPT2 (inches)



SckspdExp2mil4 = (dx2mil4/dt2mil4)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 4B--------------------mil24b = load('fri_730_2mil_B.txt'); %2Mil dataP1 = (mil24b(:,1)-14.30); %driven section pressure (psig)P4 = (mil24b(:,2)); %driver side pressure (psig)P3 = (mil24b(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil24b(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(53150:53450),P3(53150:53450),time(53150:53450),P2(53150:53450));dt2mil4b = (1332.42-1329.69);dx2mil4b = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil4b = (dx2mil4b/dt2mil4b)*(1000/12*0.3048); %Shockwave speed (m/s) %------------Aluminum Foil 5---------------------alfoil5 = load('Fri_200_AlFoil.txt'); %aluminum foil dataP1 = (alfoil5(:,1)-14.30); %driven section pressure (psig)P4 = (alfoil5(:,2)); %driver side pressure (psig)P3 = (alfoil5(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (alfoil5(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(7600:7900),P3(7600:7900),time(7600:7900),P2(7600:7900));dtal5 = (193.48-190.30);dxal5 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExpAl5 = (dxal5/dtal5)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 5B--------------------mil25b = load('Fri_200_2Mil_B.txt'); %2Mil dataP1 = (mil25b(:,1)-14.30); %driven section pressure (psig)P4 = (mil25b(:,2)); %driver side pressure (psig)P3 = (mil25b(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil25b(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(42800:43100),P3(42800:43100),time(42800:43100),P2(42800:43100));dt2mil5b = (1073.33-1070.63);dx2mil5b = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil5b = (dx2mil5b/dt2mil5b)*(1000/12*0.3048); %Shockwave speed (m/s) %------------2-mil Mylar Plastic 5--------------------mil25 = load('Fri_200_2Mil.txt'); %2Mil dataP1 = (mil25(:,1)-14.30); %driven section pressure (psig)P4 = (mil25(:,2)); %driver side pressure (psig)P3 = (mil25(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil25(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(42620:42920),P3(42620:42920),time(42620:42920),P2(42620:42920));dt2mil5 = (1068.76-1066.09);dx2mil5 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp2mil5 = (dx2mil5/dt2mil5)*(1000/12*0.3048); %Shockwave speed (m/s) %------------5-mil Mylar Plastic 5--------------------mil55 = load('Fri_200_5Mil.txt'); %5Mil dataP1 = (mil55(:,1)-14.30); %driven section pressure (psig)P4 = (mil55(:,2)); %driver side pressure (psig)P3 = (mil55(:,3))/3.378*1000*0.145037738; %dynamic pressure transducer 1 (psig)P2 = (mil55(:,4))/3.378*1000*0.145037738; %dynamic pressure transducer 2 (psig) %------------Shockwave Speed Calculation---------------plot(time(58950:59250),P3(58950:59250),time(58950:59250),P2(58950:59250));dt5mil5 = (1476.69-1474.25);dx5mil5 = 48.03125; %distance between DPT1 and DPT2 (inches)SckspdExp5mil5 = (dx5mil5/dt5mil5)*(1000/12*0.3048); %Shockwave speed (m/s)

%----------------------Plots---------------------------p2p1 = [3.722 2.867 12.48 19.98 3.937 13.87 13.57 20.16 3.194 13.83...



20.18 20.04 3.256 13.92 13.67 21.23 4.042 14.1 14.08 21.42];p4p1 = [7.319 6.691 38.81 72.61 7.934 39.35 39.14 74.41 7.798 39.06 72.93... 73.81 7.358 41.64 39.39 74.96 8.277 40.00 39.92 76.25];%raw MATLAB cursor datap4p1exp = (p4p1+14.3)./14.3; %burst pressure ratiosp2p1exp = (p2p1+14.3)./14.3; %shockwave pressure ratios i = 0;p2p1theo = [0];while i<=8 p4p1theo = i; gamma = 1.4; p2p1i = fzero(@(p2p1) p2p1*(1-((gamma-1)*(p2p1-1))/... (sqrt(2*gamma*(2*gamma+(gamma+1)*(p2p1-1)))))^... (-(2*gamma)/(gamma-1))-p4p1theo, [0,8]); p2p1theo = [p2p1theo,p2p1i]; i = i+0.1;endp4p1theo = linspace(0,8,length(p2p1theo));figure(1)plot(p4p1exp,p2p1exp,'o',p4p1theo,p2p1theo,'-')ylim([0,2.8]); xlim([0,8]);title('Shockwave Pressure Ratio vs. Burst Pressure Ratio');ylabel('Shockwave Pressure Ratio'); xlabel('Burst Pressure Ratio');legend('Experimental','Theoretical'); R = 287.05; %gas constant (J/kg-K)g = 1.4; %specific heat ratio of airT = 293.15; %temperature (K)P = 98600; %pressure (Pa)a = sqrt(g*R*T); %speed of sound (m/s)Wexp = [376.54 370.82 443.63 488 384.86 455.22 450.18 497.96 377.71 451.85... 493.92 489.96 376.54 451.85 446.88 491.93 383.65 456.93 451.85 500];%raw shockwave speed valuesp2p1theo = linspace(0,4,41);Wtheo = a*sqrt((2.4/2.8)*(p2p1theo-1)+1);%figure(2)%plot(p2p1exp,Wexp,'o',p2p1theo,Wtheo,'-')title('Shockwave Speed vs. Shockwave Pressure Ratio');ylabel('Shockwave Speed (m/s)'); xlabel('Shockwave Pressure Ratio');legend('Experimental','Theoretical'); dt = [dtal1 dtalb dt2mil1 dt5mil1 dtal2 dt2mil2 dt2mil2b dt5mil2 dtal3... dt2mil3 dt5mil3 dt5mil3b dtal4 dt2mil4 dt2mil4b dt5mil4 dtal5... dt2mil5 dt2mil5b dt5mil5]/1000; %(sec)dx = dxal1/12/3.28084; %(m)terror = 0.025/1000; %(sec)xerror = 0.00079375; %(m)Werror = sqrt(((1./dt)*xerror).^2+((-dx./(dt.^2))*terror).^2); u_p = (a/1.4).*(p2p1exp-1).*sqrt(((2.8/2.4)./(p2p1exp+(0.4/2.4))));u_p_al = [u_p(1) u_p(2) u_p(5) u_p(9) u_p(13) u_p(17)];csv1 = mean(u_p_al);u_p_2mil = [u_p(3) u_p(6) u_p(7) u_p(10) u_p(14) u_p(15) u_p(18) u_p(19)];csv2 = mean(u_p_2mil);u_p_5mil = [u_p(4) u_p(8) u_p(11) u_p(12) u_p(16) u_p(20)];csv3 = mean(u_p_5mil); aexp = Wexp./sqrt((2.4/2.8).*(p2p1exp-1)+1);sos_al = [aexp(1) aexp(2) aexp(5) aexp(9) aexp(13) aexp(17)];sos1 = mean(sos_al)sos_2mil = [aexp(3) aexp(6) aexp(7) aexp(10) aexp(14) aexp(15) aexp(18) aexp(19)];sos2 = mean(sos_2mil)sos_5mil = [aexp(4) aexp(8) aexp(11) aexp(12) aexp(16) aexp(20)];sos3 = mean(sos_5mil)



D. References

1Kim, H.D., et al. Weak shock reflection from an open end of a tube with a baffle plate. Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 217.6 (2003): 651. Academic Search Premier. EBSCO. Web. 21 Jan. 2010.

2Guggenheim, Daniel, et al. Transient Measurements in a Shock Tube. Georgia Institute of Technology. School of Aerospace Engineering. AE3051 Experimental Fluid Dynamics. Web. Retrieved from: aelabs.gatech.edu/ae3051/AE3051Labshocktube.pdf

