HW#6 Gas Dynamics - Modern Compressible Flow ME627]H.W#6...

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Unformatted text preview: Modern Compressible Flow ME627 H.W#6[ . Consider an incident normal shock wave that reflects from the end wall of a shock tube. The air in the driven section of the shock tube (ahead of the incident wave) is at 0.01 and 300 . The pressure ratio across the incident shock is 1050. With the use of Eq. (7.23), calculate a) The reflected shock wave velocity relative to the tube b) The pressure and temperature behind the reflected shock up w u1 0 wR up u5 0 Known:‐ 0.01 , 300 , 1050 Find:‐ a) b) , Solution:‐ a) From Table A.2, for Tariq A. Khamlaj 1050, ⟹ 1 30. Modern Compressible Flow ME627 H.W#6[ 287 √1.4 Also, from Table A.2, for, 30 347.2 ≅ 10416 / 30 ⟹ 175.9 175.9 175.9 √1.4 1050 300 ≅ 347.2 / 300 287 ⟹ 0.01 10416 0.379. 52770 52770 1050 4605 / 10.5 0.379 4605 8671 / From equation (7.23) 1 1 1 30 1 899 2 1 0.8 2.4 1 1 899 1 1.4 1 900 1 0.4423 Thus, 0.4423 Solving the quadratic, 0.4423 0 , we have: √ 2 1 4 1 4 0.4423 2 0.4423 2.63972 (We throw away the negative root); Hence, is the Mach number of the gas ahead of the reflected wave relative to the wave. Tariq A. Khamlaj 2 Modern Compressible Flow ME627 H.W#6[ 2.638 b) From Table A.2, for 4605 8671 3477 / we have pressure ratio across reflected shock: 2.63972 , 2.28 7.9631 ⁄ ≅ 2.28 2.28 52770 7.9631 10.5 ⁄ 7.9631 120300 83.613 From table A.2 . . . ⁄T 2.238 2.27931 2.29 Note: The temperatures increase across the incident shock 52470 . The temperatures increase across the reflected shock 67530 . Even larger than that across the incident shock. So the reflected shock is a useful mechanism for obtaining high temperature in a gas, and many shock tubes are designed to use the very hot slug of gas behind the reflected shock at the end wall as the test gas. . Consider a blunt‐nosed aerodynamic model mounted inside the driven section of a shock tube. The axis of the model is aligned parallel to the axis of the shock tube, and the nose of the model faces towards the on‐coming incident shock wave. The driven gas is air initially at a temperature and pressure of 300 0.1 , respectively. After the diaphragm is broken, an incident shock wave with a pressure ratio 40.4 propagates into the driven section. of ⁄ a) Calculate the pressure and temperature at the nose of the model. b) Shortly after the incident shock sweeps by the model. Calculate the pressure and temperature at the nose of the model after the reflected shock sweeps by the model. Tariq A. Khamlaj 3 ⁄ 7.72 7.9631 9.026 Modern Compressible Flow ME627 H.W#6[ p1,T1, 1, a1, 1 p4 ,T4 , 4 , a4 , 4 p4 p1 up w p4 p3 p2 p1 Known:‐ ⁄ 40.4 , 0.1 Find:‐ a) b) and , Solution:‐ a) Tariq A. Khamlaj 4 , 300 Modern Compressible Flow ME627 H.W#6[ From Table A.2 for ⁄ 40.4, 5.9, 7.709 7.709 40.4 40.4 0.405, 300 0.1 4.04 287 300 √1.4 287 2313 5.9 347.2 2047 0.405 1657 964 7.709. 2313 √1.4 ⁄ 347.2 / 964 / 2047 / 964 1657 / 1.72 1.72: From Table A.1, for 5.087 , 1.592 5.087 5.087 4.04 1.592 1.592 20.6 2313 3683 Let the temperature and pressure at the nose of the body be denoted by and respectively. Across the bow shock wave, 3683 ⁄ 1.72, From Table A.2 for Hence: 0.8474 0.8474 , 0.8474 20.6 17.5 b) From equation (7.23) for 1 Tariq A. Khamlaj 5.9 1 1 2 5 1 1 1 1 Modern Compressible Flow ME627 H.W#6[ 1 0.8 2.4 5.9 1 33.81 33.81 1.4 1 34.81 0.475962 Thus, 0.475962 ≅ 2.5 ⟹ up 1 4 √ 2 ∴ 0.475962 0 1 4 0.475962 2 0.475962 . 2, 2.137 , 7.125 wR u5 0 Since 0, there is no flow around the model, and the pressure and temperature at the nose are simply and respectively. 7.125 2.137 7.125 4.04 2.137 2313 28.785 4942.881 . For the shock tube in Prob. 7.10, the lengths of the driver and driven sections are 3 and 9 m, respectively. On graph paper, plot the wave diagram (x‐t diagram) showing the wave motion in the shock tube, including the incident and reflected shock waves. The contact surface and the incident and reflected expansion waves. To construct the non‐simple region of the reflected expansion wave, use the method of characteristics as outlined in Sec. 7.6. Use at least four characteristic lines to define the incident expansion wave, as shown in Fig. 7.16. Tariq A. Khamlaj 6 Modern Compressible Flow ME627 H.W#6[ Known:‐ Solution:‐ From problem 7.10 we calculate the Mach number by using equation (7.94) by trail and error: 1 1 2 2 ⁄ ⁄ ⁄ 1 ⁄ 1 7.94 1 4.5 substitute into equation (7.94) yields: Assume 4.5 1 Assume Tariq A. Khamlaj 0.4 1 3.5 2.8 2.8 2.4 3.5 3 into equation (7.94) yields: 7 7.748 5 Modern Compressible Flow ME627 H.W#6[ 3 1 0.4 1 2 2.8 2.8 11.3797 2.4 2 5 2 into equation (7.94) yields: Assume 2 1 0.4 1 1 2.8 2.8 4.342 2.4 1 5 2.2 into equation (7.94) yields: Assume 2.2 1 0.4 1 1.2 2.8 2.8 5.3987 2.4 1.2 5 2.15 into equation (7.94) yields: Assume 2.15 1 0.4 1 1.15 2.8 2.8 5.1202 5 5.0114 2.4 1.15 5 2.13 into equation (7.94) yields: Assume 2.13 1 0.4 1 1.13 2.8 2.8 2.4 1.13 ∴ 2.13 ⟹ 1.4. From table A.2 2.13 1 1 5 . 0.426 . . From Table A.2 ⁄ 1.225, 0.7397. 1.225 1.225 √1.4 √1.4 287 1.4 486 Tariq A. Khamlaj 287 300 300 376.5 347.2 , 8 376.5 347.2 / 388.9 / 486 / ⁄ 2.12 2.13 2.186 Modern Compressible Flow ME627 H.W#6[ Thus, 486 0.7397 388.9 The contact surface is moving at the velocity surface 198 198 / . Hence, for the contact , Thus, 388.9 1.352 198 328 / Hence, for the reflected shock wave 328 / Consider the expansion waves The strength of the incident expansion wave is / 0.426 0.7835 . 0.426. Hence 0.7835 0.7835 √1.4 300 287 235.1 235.1 307.3 / From Eq. (7.84) 2 1 2 1 0.4 1 0.5746 0.5746 307.3 347.2 347.2 0.5746 200 / For the head of the incident expansion wave, 200 Also, from Eq. (7.74) for Tariq A. Khamlaj 307.3 characteristics 9 107.3 / Modern Compressible Flow ME627 H.W#6[ 2 1 And for characteristics 2 1 . . / / . . / 0 347.2 347.2 1736 70 140 200 333.3 319.2 307.3 263.2 179.2 107.3 1596 1456 1337 1596, For point e: 0. (Boundary condition) 1 0.2 0 2 1/2 Average slope of line 1 2 – 263 1596 0 319.2 / – 319.2 291.2 / 1596 / For points a, b, c and d: 2 0 347.2 0.4 1736 / 1 0 2 347.2 70 1 70 2 333.2 140 319.2 431.2 / 1 140 2 319.2 200 307.3 483.3 / 200 Tariq A. Khamlaj 10 333.2 307.3 375.2 / 507.3 / Modern Compressible Flow ME627 H.W#6[ For point f: 1596 1456 1 0.1 1596 4 1 1596 2 1 2 1 0 2 1 70 2 1456 319.2 70 / 1456 70 347.2 / 305.2 140 305.2 305.2 / 319.2 207.2 / For point g: 1596 1337 1 4 1 2 1 70 2 0.1 1596 1 1596 2 305.2 1337 1337 129.5 293.2 / 129.5 / 293.2 399 / 422.7 / 1 129.5 2 293.2 200 307.3 For point h: 1456 0 Tariq A. Khamlaj 11 135.5 / Modern Compressible Flow ME627 H.W#6[ 1 0.2 0 2 2 1 1 0 2 129.5 2 291.2 0.4 70 291.2 / 1456 305.2 0 1456 / 263.2 / For point i: 1456 / 1337 / 1 0.1 1456 4 1 2 1337 279.3 / 1 1456 2 1 0 2 1 59.5 2 291.2 279.3 1337 59.5 / 59.5 279.3 315 / 129.5 293.2 59.5 279.3 191.8 / 338.8 / For point j: 1337 0 1 0.2 0 2 1 0 2 267.4 59.5 1337 267.4 / 279.3 267.4 / Tariq A. Khamlaj 12 243.6 / ...
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