Middle East Technical University Department of Aerospace Engineering
AE 342 AERODYNAMICS II Laboratory Report 2
Model Performance and Shock Pattern Analysis 11° Semi Angled Wedge in Supersonic Flow Date
:
Instructor
: Funda Kurtuluş
Submitted by
– Oğuzhan AYDIN : 1942812 – Oğuzhan
1942804 – Ali Ali AVANLIER
ABSTRACT
In this experiment, we aim to calculate pressure coefficients at different mach numbers such as 1.8, 1.9, 2.0, 2.1, 2.2. And we observe oblique shock waves on a 11º semi- angle cone to do this. Pressure coefficients at these Mach numbers were calculated and compared with each other. The pressure coefficient (C_p) is calculated from difference in static pressure values on the model (P_m) and on the wall (P_w) divided the differences normalized by the dynamic pressure. After that, we measure shock angle from photos which are provided by experiment. To conclude, we compare beta_measured vs mach number, beta_theoratical vs mach number and Cp_measured vs mach number, Cp_theoratical vs mach number.
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TABLE OF CONTENTS
ABSTRACT ...................................................................................................................... ii TABLE OF CONTENTS ........................................................................................ ......... iii INTRODUCTION ............................................. .................................................................4 1.1.
Theory ................................................. .................................................................4
1.2.
Experimental Procedure ......................................................................................5
RESULTS ................................................. ..........................................................................7 DISCUSSION AND CONCLUSION ................................................... .............................8 REFERENCES .................................................. ...............................................................15 APPENDICES ................................................... ...............................................................16 1.
APPENDIX A .............................................. ......................................................16
iii
INTRODUCTION
1. In this experiment, we examine oblique shock on the 11 degree semi- angle cone to determine pressure coefficients and shock angles. Also, we examine pressure coefficients and shock waves with two ways as measured and theoretical. During this experiment, we record the gage pressure on the model and on the wall. And then, we use these records in the calculation of pressure coefficient. Also we calculate measured shock wave by using photographs which are provided during this experiment. After all, reasons of errors of measured and theoretical pressure coefficients, and reasons of errors of measured and theoretical shock waves are investigated in discussion part. Also, Mach numbers from photographs are calculated by using θ-β-M graph. All results are examined in conclusion part. 1.1. Theory
2. The pressure coefficient is defined for compressible flow as;
3. =
−
4. Where is model pressure, is wall pressure and is Mach number. 5. According to above formula, we calculate Cp by using provided pressure values and mach numbers. 6. Secondly, we know that = 11° since we work on 11 degree semi-angle cone. From givens photographs, we calculate shock angles for different mach numbers. Since we know shock angles and theta, we use ,, chart to find mach numbers from photos. Also, since we know theoretical mach numbers and theta, we find theoretical shock waves by using again ,, chart as follows:
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7. Table 2. , , Chart.
Also ,, relation is given as follows:
tanθ = 2
sin 1 ( + 2) + 2
This clearly explain why we use ,, chart. As we can see, we cannot calculate shock angles analytically from this above formula. Thus we should use ,, chart. 7.1. Experimental Procedure
The schema of the set-up is shown in Schema 1 below.
Schema 1: Experimental schema
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The procedure: A) Startup: i.
Adjust the mirror and light system so that the model can be observed as clearly as possible on the screen.
ii.
By using the handle next to the test chamber adjust the Mach number value to the desired value.
iii.
Turn the compressor knob to position A in order to let the compressor fill the tanks with air. Observe the pressure and temperature gauges on the control panel so that these parameters do not exceed the limits.
iv.
When the pressure level in the tanks has reached the sufficient level for the experiments to be conducted (≈7 -8 bar), turn the compressor knob to off position. As the pressure will drop quickly, start the experiment immediately.
B) Experiment: i. By carefully opening the large yellow handle on the left hand side of the control panel achieve the start pressure value corresponding to the desired Mach number. Then, try to keep the pressure at the run pressure value indicated in the table below by carefully adjusting the same handle. ii. As soon as the pressure has been stabilized press the remote pressure lock button to freeze the pressure readings and read the model and wall pressure values from the gauges on the control panel. iii. At the same time observe and photograph the shock patterns seen on the screen. v.
Repeat the same procedure for different Mach numbers (M = 1.5, 1.8, 2.0)
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RESULTS
For various alpha degrees, corresponding Mach numbers (theoretical, mea sured and photo) are calculated for each section of the diamond shape. Then from the shock-expansion theory pressure values are computed for each section as well, then pressure coefficients are calculated using those pressures. Note that section 2 is the upper-left side, section 3 is the upper-right side, section 4 is lower-left side and section 5 is lower-right side of the diamond shape. For section 2, measured pressure values are dire ctly used for computing pressure coefficient. All results are tabulated and shown below. Mach_2(theoretical Alpha and measured) Mach_2_photo
Cp_theo_2
Cp_measured_2
Cp_photo_2
5.5
1.780999669
1.690069702
0.422710396668 042
0.1056524035921 82
0.5357523948365 43
7
1.845502448
1.542160196
0.382117909031 704
0.0770950582067 690
0.5516745054459 82
3
1.732475195
2.047954321
0.478029386258 600
0.1660252056448 57
0.5300197231632 70
-3
1.488503412
1.717826058
0.678644268854 015
0.2586652871184 69
0.5611311469426 20
-5.5
1.379107833
1.470216813
0.780356968395 815
0.3320504112897 14
0.6169857399560 80
-7
1.302134924
1.302134924
0.852433997495 208
0.3509169119311 75
0.6328529554246 30
Table 1: section 2
Alpha
Mach_3(theoretical and measured) Mach_3_photo
Cp_theo_3
Cp_measured_3
Cp_photo_3
5.5
2.6344
2.5172
0.140756196
0.530262449
0.561373481
7
2.7196
2.3384
0.123730551
0.535517922
0.589926409
3
2.5712
2.9867
0.160808814
0.549645426
0.416666667
-3
2.2718
2.5487
0.24648542
0.57752455
0.271959852
-5.5
2.1481
2.2504
0.290477938
0.623550684
0.329559614
-7
2.0678
2.0704
0.320424592
0.633769098
0.271959852
Table 2: section 3
Alpha
Mach_4(theoretical and measured) Mach_4_photo
Cp_theo_4
Cp_measured_4
Cp_photo_4
5.5
1.3791078
1.25019051
0.780356968
2.631238387
0.852433997
7
1.3021349
1.253503099
0.852433997
2.889192082
0.880986925
3
1.4885034
1.424747842
0.678644269
2.364023996
0.707727183
-3
1.7324752
1.497179812
0.478029386
1.791575408
0.563020368
-5.5
1.7809997
1.289847902
0.422710397
1.691647644
0.62062013
-7
1.8455024
1.349786204
0.382117909
1.58500514
0.563020368
Table 3:section 4
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Alpha
Mach_5(theoretical and measured) Mach_5_photo
Cp_theo_5
Cp_measured_5 Cp_photo_5
5.5
2.14281
2.0165
0.293440025
1.352035148
1.433098467
7
2.0678
2.0196
0.320424592
1.461353317
1.496039899
3
2.2718
2.1988
0.24648542
1.228678416
1.271364652
-3
2.5712
2.2821
0.160808814
0.975435212
1.103543523
-5.5
2.6344
2.0552
0.140756196
0.950912019
1.230766336
-7
2.7196
2.1177
0.123730551
0.906183115
1.170382356
Table 4: section 4
And then, theoretical, measured and photo Cl, Cd values are calculated using the pressure coefficients. The
results
are
shown
in
Table
5.
Alpha
Cl_theo
Cl_measured
Cl_photo
Cd_theo
Cd_meas
Cd_photo
5.5
0.246828233
1.658013293
1.78891103
0.098839179
0.243090718
0.265732466
7
0.321657314
1.843552972
1.890473883
0.116890231
0.321285082
0.329571495
3
0.13913782
1.432720773
1.50171292
0.080224157
0.148246282
0.157053693
-3
-0.13913782
0.966616769
1.148494516
0.080224157
-0.002261101
-0.0013843
-5.5
-0.245326415
0.843780818
1.247401048
0.098983788
-0.037383661
-0.05216515
-7
-0.321657314
0.752326568
1.122264513
0.116890231
-0.053600707
-0.07826682
Table 5:cl , cd
Also, for each alpha value, those measured, theoretical and photo Cl, Cd values are plotted . See Figure 1 and 2. And, the anglemeter calculations on photos are shown below
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Photo 1: aoa=3
Photo 2: aoa=-3
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Photo 3: aoa=5.5
Photo 4: aoa=-5.5
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Photo 5: aoa=7
Photo 6: aoa=-7
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Figure 1: alpha versus Cl
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Figure 2: alpha versus Cd
DISCUSSION AND CONCLUSION
It is clearly seen from Figure 1 and 2 that the differences between measured and theoretical lift and drag coefficients when the alpha is not zero are quite high. Seems like error is getting bigger when alpha is getting away from zero. On the other hand, measured and photo values are very close to each other. Possible reasons for this error are that the picture is taken from wrong angle (not from perfectly front), and the pressure sensors may not work perfectly, or those pressure measurement includes some real-flow effects which we assume the flow follows the theoretical laws such as inviscid, no heat transfer, ideal gas etc. And since some of those effects are also create some errors in measurements of pressure values , the measured values of Cl and Cd became similar to those obtained by using photos.
Beta values obtained from the photos may not be reliable due t o our measurement technic (angle meter)
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The error in beta cause us to come up with measured Mach numbers with some errors. But the error in Mach numbers is not too much. Also those errors in Mach number cause us to come up with errors in pressure values and so Cp , Cl and Cd values.
All those experimental process errors were expected, but the difference between theoretical Cl,Cd and measured,photo Cl,Cd is not acceptable. In conclusion, for different given Alpha degrees, theoretical and measured values of pressure coefficient and beta angles for shock region (2,4) and theta angles for expansion region(3,5) are calculated. We noticed that the measured,photo and theoretical values are different and the difference is really high in some regions especially for bigger alpha values. The possible reasons for this error is discussed above. Furthermore, the Mach numbers for photo calculations is tried to obtain according to angle beta from the photos taken during experiment.. The result was expected to have some errors but is not acceptable for big alpha degrees. In short, the results have some expected errors which are mentioned in discussion part. Although the measured and photo Cl,Cd values are really close to each other, the result is very different when it comes to theoretical Cl,Cd values especially when absolute value of alpha is getting bigger.
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REFERENCES
Anderson, Fundamentals of Aerodynamics Lab Manuel Appendix C from Anderson, Fundamentals of Aerodynamics
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APPENDICES
1. APPENDIX A
Matlab code: gamma=1.4 mach_inf=2 epsilon=11 theta_1=epsilon-Alphadeg mach_n1_upper=mach_inf.*sind(beta_upper) mach_n2=sqrt(((1+(((gamma1)/2).*((mach_n1_upper).^2)))./((gamma*(mach_n1_upper.^2))-((gamma-1)/2)))) mach_2=mach_n2./sind(beta_upper-theta_1) p01=1.8*100000 p1=p01/((1+(((gamma-1)/2)*(mach_inf^2)))^(gamma/(gamma-1))) p1=101325+p1 p2=p1*(1.+(((2*gamma)/(gamma+1)).*((mach_n1_upper.^2)-1))) p2=101325+p2 cp_theo_1=(2*(p2-p1))/(gamma*p1*mach_inf^2) theta_2=22 p3=p2.*(((1+(((gamma-1)/2).*(mach_2.^2))).^(gamma/(gamma1)))./((1+(((gamma-1)/2).*(mach_3.^2))).^(gamma/(gamma-1)))) p3=101325+p3 cp_theo_2=(2*(p3-p1))/(gamma*p1*mach_inf^2) theta_3=epsilon+Alphadeg mach_n1_lower=mach_inf.*sind(beta_lower) mach_n4=sqrt(((1+(((gamma1)/2).*((mach_n1_lower).^2)))./((gamma*(mach_n1_lower.^2))-((gamma-1)/2)))) mach_4=mach_n4./sind(beta_lower-theta_3) p4=p1*(1.+(((2*gamma)/(gamma+1)).*((mach_n1_lower.^2)-1))) p4=101325+p4 cp_theo_4=(2*(p4-p1))/(gamma*p1*mach_inf^2) theta_4=22 p5=p4.*(((1+(((gamma-1)/2).*(mach_4.^2))).^(gamma/(gamma1)))./((1+(((gamma-1)/2).*(mach_5.^2))).^(gamma/(gamma-1)))) p5=101325+p5 cp_theo_5=(2*(p5-p1))/(gamma*p1*mach_inf^2) p_mpa=100000*p_mbar p_wpa=100000*p_wbar p_wpa=101325+p_wpa p_mpa=101325+p_mpa cp_m_1=(2*(p_mpa-p_wpa))./(gamma*p_wpa*mach_inf.^2) p3_m=p_mpa.*(((1+(((gamma-1)/2).*(mach_2.^2))).^(gamma/(gamma1)))./((1+(((gamma-1)/2).*(mach_3.^2))).^(gamma/(gamma-1)))) p3_m=101325+p3_m cp_m_2=(2*(p3_m-p_wpa))./(gamma*p_wpa*mach_inf.^2) p4_m=p1*(1.+(((2*gamma)/(gamma+1)).*((mach_n1_lower.^2)-1))) p4_m=101325+p4_m cp_m_4=(2*(p4_m-p_wpa))./(gamma*p_wpa*mach_inf.^2) p5_m=p4_m.*(((1+(((gamma-1)/2).*(mach_4.^2))).^(gamma/(gamma1)))./((1+(((gamma-1)/2).*(mach_5.^2))).^(gamma/(gamma-1)))) p5_m=101325+p5_m cp_m_5=(2*(p5_m-p_wpa))./(gamma*p_wpa*mach_inf.^2)
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fx_t=((p2*cosd(79))/(2*cosd(11)))((p3*cosd(79))/(2*cosd(11)))+((p4*cosd(79))/(2*cosd(11)))((p5*cosd(79))/(2*cosd(11))) fy_t=-((p2*sind(79))/(2*cosd(11)))((p3*sind(79))/(2*cosd(11)))+((p4*sind(79))/(2*cosd(11)))+((p5*sind(79))/(2 *cosd(11))) L_t=(fy_t.*cosd(Alphadeg))-(fx_t.*sind(Alphadeg)) D_t=(fy_t.*sind(Alphadeg))+(fx_t.*cosd(Alphadeg)) Cl_t=(2*L_t)./(gamma*p1*mach_inf^2) Cd_t=(2*D_t)./(gamma*p1*mach_inf^2) fx_m=((p_mpa*cosd(79))/(2*cosd(11)))((p3_m*cosd(79))/(2*cosd(11)))+((p4_m*cosd(79))/(2*cosd(11)))((p5_m*cosd(79))/(2*cosd(11))) fy_m=-((p_mpa*sind(79))/(2*cosd(11)))((p3_m*sind(79))/(2*cosd(11)))+((p4_m*sind(79))/(2*cosd(11)))+((p5_m*sind(7 9))/(2*cosd(11))) L_m=(fy_m.*cosd(Alphadeg))-(fx_m.*sind(Alphadeg)) D_m=(fy_m.*sind(Alphadeg))+(fx_m.*cosd(Alphadeg)) Cl_m=(2*L_m)./(gamma*p_wpa*mach_inf^2) Cd_m=(2*D_m)./(gamma*p_wpa*mach_inf^2) mach_n1_upper_photo=mach_inf.*sind(beta_photo_up) mach_n2_photo=sqrt(((1+(((gamma1)/2).*((mach_n1_upper_photo).^2)))./((gamma*(mach_n1_upper_photo.^2))((gamma-1)/2)))) mach_2_photo=mach_n2_photo./sind(beta_photo_up-theta_1) p3_m_photo=p_mpa.*(((1+(((gamma-1)/2).*(mach_2_photo.^2))).^(gamma/(gamma1)))./((1+(((gamma-1)/2).*(mach_3_photo.^2))).^(gamma/(gamma-1)))) p3_m_photo=101325+p3_m_photo cp_m_2_photo=(2*(p3_m_photo-p_wpa))./(gamma*p_wpa*mach_inf.^2) mach_n1_lower_photo=mach_inf.*sind(beta_photo_low) mach_n4_photo=sqrt(((1+(((gamma1)/2).*((mach_n1_lower_photo).^2)))./((gamma*(mach_n1_lower_photo.^2))((gamma-1)/2)))) mach_4_photo=mach_n4_photo./sind(beta_photo_low-theta_3) p4_m_photo=p1*(1.+(((2*gamma)/(gamma+1)).*((mach_n1_lower_photo.^2)-1))) p4_m_photo=101325+p4_m_photo cp_m_4_photo=(2*(p4_m_photo-p1))/(gamma*p1*mach_inf^2) p5_m_photo=p4_m_photo.*(((1+(((gamma1)/2).*(mach_4_photo.^2))).^(gamma/(gamma-1)))./((1+(((gamma1)/2).*(mach_5_photo.^2))).^(gamma/(gamma-1)))) p5_m_photo=101325+p5_m_photo cp_m_5_photo=(2*(p5_m_photo-p_wpa))./(gamma*p_wpa*mach_inf.^2) fx_m_photo=((p_mpa*cosd(79))/(2*cosd(11)))((p3_m_photo*cosd(79))/(2*cosd(11)))+((p4_m_photo*cosd(79))/(2*cosd(11)))((p5_m_photo*cosd(79))/(2*cosd(11))) fy_m_photo=-((p_mpa*sind(79))/(2*cosd(11)))((p3_m_photo*sind(79))/(2*cosd(11)))+((p4_m_photo*sind(79))/(2*cosd(11)))+( (p5_m_photo*sind(79))/(2*cosd(11))) L_m_photo=(fy_m_photo.*cosd(Alphadeg))-(fx_m_photo.*sind(Alphadeg)) D_m_photo=(fy_m_photo.*sind(Alphadeg))+(fx_m_photo.*cosd(Alphadeg)) Cl_m_photo=(2*L_m_photo)./(gamma*p_wpa*mach_inf^2) Cd_m_photo=(2*D_m_photo)./(gamma*p_wpa*mach_inf^2) plot(Alphadeg,Cl_t,Alphadeg,Cl_m,Alphadeg,Cl_m_photo) plot(Alphadeg,Cd_t,Alphadeg,Cd_m,Alphadeg,Cd_m_photo)
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