Lab 8: Shockwave Hydrodynamics by Laser Schlieren Photography

Ian M. Rittersdorf NERS 575 Due 20 March 2008

Abstract The measured shockwave was found to have a velocity of 361 [m/s] and a Mach number of 1.0494. Using this value the ratio of pressures, temperatures, and densities of the air before and after the shockwave were measured. These values are 1.11816, 1.03247, and 1.08300, respectively. The shockwave was found to deposit 0.687 [mJ] of energy into the air.

Contents

1 Introduction and Objectives 1

2 Experimental Conﬁguration 1

3 Experimental Results 2

4 Data Analysis and Discussion 2

5 References 3

A Appendix 3 A.1 Analysis of Measurements ...... 3 A.2 Useful Equations ...... 3 A.3 Useful Measurements ...... 4

1 Introduction and Objectives

Schlieren photography is a diagnostic process that can be used to measure the density of a gas. The purpose of this lab is to become familiar with Schlieren photography diagnostics and use the process to measure some parameters of air.

2 Experimental Conﬁguration

1 4 DATA ANALYSIS AND DISCUSSION

Figure 1: Schematic of setup. Figure 1 shows the basic setup of this experiment. A XeCl laser was ﬁred through a plasma arc between two needles. A camera was set up to take pictures of the shockwave after various time delays between 0 and 10 [µs]. The knife edge is setup to eliminate all light from the photo except for diﬀracted light.

3 Experimental Results

The raw data of this experiment was in the form of photographs, which were later analyzed. The photos themselves can be found at the following web address: http://www-personal.umich.edu/∼ianrit/ners575/ners575 thur 130.zip

4 Data Analysis and Discussion

With the aide of NI Vision Assistant, the number of pixels of the diameter of the needles and the diameter of the shockwave was measured at each delay. With the knowledge that the diameter of the needles was 0.6 [mm], the pixel count of the shockwaves could be turned into lengths. Since two pictures were taken at each delay, the average number of pixels were used to make calculations. This data can be seen in Appendix A.1. Using this data, the shockfront can be plotted as a function of time and the speed of the shockwave can be determined.

Figure 2: Shockfront as a function of time. The velocity of the shockwave is the slope of the line on the position vs. time graph. As Figure 2 shows, the speed of the shockwave is 0.361 [mm/µs], or 361 [m/s]. The Mach number is deﬁned as:

v M = wave (4.1) vsound The speed of sound in air is approximately 344 [m/s]. Using this value, we ﬁnd that the Mach number of the measured shockwave is Mx = 1.0494. Since Mx > 1, this is by deﬁnition a true shockwave.

Using the equations in Appendix A.2, the ratios of pressure, temperature, and density before and after the shockwave can be measured.

Ratio Value py 1.11816 px Ty 1.03247 Tx ρy 1.08300 ρx Table 1: Calculated values of pressure, temperature, and density ratios.

2 A APPENDIX

The amount of heat that the shockwave added to the air can be calculated using the following equation:

∆H = Cp (T − T0) (4.2) The temperature in the lab the day of the experiment was 296.48 [K]. Multiplying this by the ratio of temperatures gives the temperature behind the shockwave, Ty. We ﬁnd this value to be 306.1 [K]. Cp is tabulated to be a J J value of 29.12 [ K−mol ]. Using these values with Equation refhcat, an amount of 280.30 [ mol ] were added to the air.

To ﬁnd the amount of energy added to the air, an approximation of the amount of mols in the air needs to be made. Using the radius of the largest shockwave measured, we calculated a shockwave sphere volume of 5.459x10−5 [m3] kg is determined to be the region of interest. Air density is approximately 1.2 [ m3 ] and by multiplying this by the kg g ratio of densities we ﬁnd that ρy, the density behind the shockwave is 1.300 [ m3 ]. Air is known to have 29 [ mol ]. Combining all this reveals that we have 0.00245 [mol] of air in the shockwave. Multiplying this by the value of ∆H gives a value of 0.687 [mJ] deposited in the air, which is close to the value of 1 [mJ] expected.

5 References

[1] F. M. White, Fluid Mechanics, Sixth Edition.

A Appendix A.1 Analysis of Measurements Using the needle diameter of 0.6 [mm], each pixel was determined to have a length of 6.316 [µm]. Needle Width [pixels] Shockwave diameter [pixels] time [micro-sec] Shockwave Radius [mm] 95 298 1.16 0.94105 95 284 1.16 0.89684 95 442 2.16 1.39578 95 420 2.16 1.32631 95 556 3.16 1.75578 95 538 3.16 1.69894 95 690 4.16 2.17894 95 693 4.16 2.18842 95 771 5.16 2.43473 95 801 5.16 2.52947 95 931 6.16 2.94000 95 909 6.16 2.87052 95 1035 7.16 3.26842 95 999 7.16 3.15473 95 1136 8.16 3.58736 95 1140 8.16 3.60000 95 1175 9.16 3.71052 95 1268 9.16 4.00421 95 1316 10.16 4.15578 95 1324 10.16 4.18105

A.2 Useful Equations

py 2k 2 k − 1 = Mx − (A.1) px k + 1 k + 1 py k−1 py T 1 + y = px k+1 px (A.2) py k−1 Tx + px k+1 k+1 ρy p − 1 y = k−1 ρx (A.3) k+1 ρy px − k−1 ρx

3 A APPENDIX A.3 Useful Measurements

A.3 Useful Measurements Measurement Value Air Temperature 296.48 [K] J Cp of Air 29.12 [ K−mol ]