Appendix a Isentropic Flow Table

Appendix A Isentropic Flow Table

TableA.1 displayes the isentropic ﬂow properties for γ = 1.4.

Table A.1 Isentropic ﬂow properties for γ = 1.4 ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 0.02 1.00E+00 1.00E+00 1.00E+00 2.89E+01 1.20E+00 1.89E+00 1.58E+00 0.04 1.00E+00 1.00E+00 1.00E+00 1.45E+01 1.20E+00 1.89E+00 1.58E+00 0.06 1.00E+00 1.00E+00 1.00E+00 9.67E+00 1.20E+00 1.89E+00 1.57E+00 0.08 1.00E+00 1.00E+00 1.00E+00 7.26E+00 1.20E+00 1.88E+00 1.57E+00 0.1 1.00E+00 1.01E+00 1.01E+00 5.82E+00 1.20E+00 1.88E+00 1.57E+00 0.12 1.00E+00 1.01E+00 1.01E+00 4.86E+00 1.20E+00 1.87E+00 1.57E+00 0.14 1.00E+00 1.01E+00 1.01E+00 4.18E+00 1.20E+00 1.87E+00 1.56E+00 0.16 1.01E+00 1.02E+00 1.01E+00 3.67E+00 1.19E+00 1.86E+00 1.56E+00 0.18 1.01E+00 1.02E+00 1.02E+00 3.28E+00 1.19E+00 1.85E+00 1.55E+00 0.2 1.01E+00 1.03E+00 1.02E+00 2.96E+00 1.19E+00 1.84E+00 1.55E+00 0.22 1.01E+00 1.03E+00 1.02E+00 2.71E+00 1.19E+00 1.83E+00 1.54E+00 0.24 1.01E+00 1.04E+00 1.03E+00 2.50E+00 1.19E+00 1.82E+00 1.53E+00 0.26 1.01E+00 1.05E+00 1.03E+00 2.32E+00 1.18E+00 1.81E+00 1.53E+00 0.28 1.02E+00 1.06E+00 1.04E+00 2.17E+00 1.18E+00 1.79E+00 1.52E+00 0.3 1.02E+00 1.06E+00 1.05E+00 2.04E+00 1.18E+00 1.78E+00 1.51E+00 0.32 1.02E+00 1.07E+00 1.05E+00 1.92E+00 1.18E+00 1.76E+00 1.50E+00 0.34 1.02E+00 1.08E+00 1.06E+00 1.82E+00 1.17E+00 1.75E+00 1.49E+00 0.36 1.03E+00 1.09E+00 1.07E+00 1.74E+00 1.17E+00 1.73E+00 1.48E+00 0.38 1.03E+00 1.10E+00 1.07E+00 1.66E+00 1.17E+00 1.71E+00 1.47E+00 0.4 1.03E+00 1.12E+00 1.08E+00 1.59E+00 1.16E+00 1.70E+00 1.46E+00 0.42 1.04E+00 1.13E+00 1.09E+00 1.53E+00 1.16E+00 1.68E+00 1.45E+00 0.44 1.04E+00 1.14E+00 1.10E+00 1.47E+00 1.16E+00 1.66E+00 1.43E+00 0.46 1.04E+00 1.16E+00 1.11E+00 1.42E+00 1.15E+00 1.64E+00 1.42E+00 (continued)

© Springer Science+Business Media Dordrecht 2015 513 R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics, Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4 514 Appendix A: Isentropic Flow Table

Table A.1 (continued) ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 0.48 1.05E+00 1.17E+00 1.12E+00 1.38E+00 1.15E+00 1.62E+00 1.41E+00 0.5 1.05E+00 1.19E+00 1.13E+00 1.34E+00 1.14E+00 1.60E+00 1.40E+00 0.52 1.05E+00 1.20E+00 1.14E+00 1.30E+00 1.14E+00 1.57E+00 1.38E+00 0.54 1.06E+00 1.22E+00 1.15E+00 1.27E+00 1.13E+00 1.55E+00 1.37E+00 0.56 1.06E+00 1.24E+00 1.16E+00 1.24E+00 1.13E+00 1.53E+00 1.35E+00 0.58 1.07E+00 1.26E+00 1.18E+00 1.21E+00 1.12E+00 1.51E+00 1.34E+00 0.6 1.07E+00 1.28E+00 1.19E+00 1.19E+00 1.12E+00 1.48E+00 1.33E+00 0.62 1.08E+00 1.30E+00 1.20E+00 1.17E+00 1.11E+00 1.46E+00 1.31E+00 0.64 1.08E+00 1.32E+00 1.22E+00 1.15E+00 1.11E+00 1.44E+00 1.30E+00 0.66 1.09E+00 1.34E+00 1.23E+00 1.13E+00 1.10E+00 1.41E+00 1.28E+00 0.68 1.09E+00 1.36E+00 1.25E+00 1.11E+00 1.10E+00 1.39E+00 1.26E+00 0.7 1.10E+00 1.39E+00 1.26E+00 1.09E+00 1.09E+00 1.36E+00 1.25E+00 0.72 1.10E+00 1.41E+00 1.28E+00 1.08E+00 1.09E+00 1.34E+00 1.23E+00 0.74 1.11E+00 1.44E+00 1.30E+00 1.07E+00 1.08E+00 1.32E+00 1.22E+00 0.76 1.12E+00 1.47E+00 1.31E+00 1.06E+00 1.08E+00 1.29E+00 1.20E+00 0.78 1.12E+00 1.49E+00 1.33E+00 1.05E+00 1.07E+00 1.27E+00 1.18E+00 0.8 1.13E+00 1.52E+00 1.35E+00 1.04E+00 1.06E+00 1.24E+00 1.17E+00 0.82 1.13E+00 1.56E+00 1.37E+00 1.03E+00 1.06E+00 1.22E+00 1.15E+00 0.84 1.14E+00 1.59E+00 1.39E+00 1.02E+00 1.05E+00 1.19E+00 1.13E+00 0.86 1.15E+00 1.62E+00 1.41E+00 1.02E+00 1.05E+00 1.17E+00 1.12E+00 0.88 1.15E+00 1.66E+00 1.43E+00 1.01E+00 1.04E+00 1.14E+00 1.10E+00 0.9 1.16E+00 1.69E+00 1.46E+00 1.01E+00 1.03E+00 1.12E+00 1.08E+00 0.92 1.17E+00 1.73E+00 1.48E+00 1.01E+00 1.03E+00 1.10E+00 1.07E+00 0.94 1.18E+00 1.77E+00 1.50E+00 1.00E+00 1.02E+00 1.07E+00 1.05E+00 0.96 1.18E+00 1.81E+00 1.53E+00 1.00E+00 1.01E+00 1.05E+00 1.03E+00 0.98 1.19E+00 1.85E+00 1.55E+00 1.00E+00 1.01E+00 1.02E+00 1.02E+00 1 1.20E+00 1.89E+00 1.58E+00 1.00E+00 1.00E+00 1.00E+00 1.00E+00 1.02 1.21E+00 1.94E+00 1.60E+00 1.00E+00 9.93E-01 9.77E-01 9.83E-01 1.04 1.22E+00 1.98E+00 1.63E+00 1.00E+00 9.87E-01 9.54E-01 9.67E-01 1.06 1.22E+00 2.03E+00 1.66E+00 1.00E+00 9.80E-01 9.31E-01 9.50E-01 1.08 1.23E+00 2.08E+00 1.69E+00 1.01E+00 9.73E-01 9.09E-01 9.34E-01 1.1 1.24E+00 2.14E+00 1.72E+00 1.01E+00 9.66E-01 8.87E-01 9.18E-01 1.12 1.25E+00 2.19E+00 1.75E+00 1.01E+00 9.59E-01 8.65E-01 9.01E-01 1.14 1.26E+00 2.24E+00 1.78E+00 1.02E+00 9.52E-01 8.43E-01 8.85E-01 1.16 1.27E+00 2.30E+00 1.81E+00 1.02E+00 9.46E-01 8.22E-01 8.69E-01 1.18 1.28E+00 2.36E+00 1.85E+00 1.02E+00 9.39E-01 8.01E-01 8.54E-01 1.2 1.29E+00 2.43E+00 1.88E+00 1.03E+00 9.32E-01 7.81E-01 8.38E-01 1.22 1.30E+00 2.49E+00 1.92E+00 1.04E+00 9.25E-01 7.60E-01 8.22E-01 1.24 1.31E+00 2.56E+00 1.95E+00 1.04E+00 9.18E-01 7.41E-01 8.07E-01 (continued) Appendix A: Isentropic Flow Table 515

Table A.1 (continued) ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 1.26 1.32E+00 2.63E+00 1.99E+00 1.05E+00 9.11E-01 7.21E-01 7.92E-01 1.28 1.33E+00 2.70E+00 2.03E+00 1.06E+00 9.04E-01 7.02E-01 7.77E-01 1.3 1.34E+00 2.77E+00 2.07E+00 1.07E+00 8.97E-01 6.83E-01 7.62E-01 1.32 1.35E+00 2.85E+00 2.11E+00 1.08E+00 8.90E-01 6.65E-01 7.47E-01 1.34 1.36E+00 2.93E+00 2.15E+00 1.08E+00 8.83E-01 6.47E-01 7.33E-01 1.36 1.37E+00 3.01E+00 2.20E+00 1.09E+00 8.76E-01 6.29E-01 7.18E-01 1.38 1.38E+00 3.09E+00 2.24E+00 1.10E+00 8.69E-01 6.12E-01 7.04E-01 1.4 1.39E+00 3.18E+00 2.29E+00 1.11E+00 8.62E-01 5.95E-01 6.90E-01 1.42 1.40E+00 3.27E+00 2.33E+00 1.13E+00 8.55E-01 5.78E-01 6.76E-01 1.44 1.41E+00 3.37E+00 2.38E+00 1.14E+00 8.48E-01 5.62E-01 6.63E-01 1.46 1.43E+00 3.47E+00 2.43E+00 1.15E+00 8.41E-01 5.46E-01 6.49E-01 1.48 1.44E+00 3.57E+00 2.48E+00 1.16E+00 8.34E-01 5.31E-01 6.36E-01 1.5 1.45E+00 3.67E+00 2.53E+00 1.18E+00 8.28E-01 5.16E-01 6.23E-01 1.52 1.46E+00 3.78E+00 2.58E+00 1.19E+00 8.21E-01 5.01E-01 6.10E-01 1.54 1.47E+00 3.89E+00 2.64E+00 1.20E+00 8.14E-01 4.86E-01 5.98E-01 1.56 1.49E+00 4.01E+00 2.70E+00 1.22E+00 8.07E-01 4.72E-01 5.85E-01 1.58 1.50E+00 4.13E+00 2.75E+00 1.23E+00 8.00E-01 4.59E-01 5.73E-01 1.6 1.51E+00 4.25E+00 2.81E+00 1.25E+00 7.94E-01 4.45E-01 5.61E-01 1.62 1.52E+00 4.38E+00 2.87E+00 1.27E+00 7.87E-01 4.32E-01 5.49E-01 1.64 1.54E+00 4.51E+00 2.93E+00 1.28E+00 7.80E-01 4.20E-01 5.38E-01 1.66 1.55E+00 4.65E+00 3.00E+00 1.30E+00 7.74E-01 4.07E-01 5.26E-01 1.68 1.56E+00 4.79E+00 3.06E+00 1.32E+00 7.67E-01 3.95E-01 5.15E-01 1.7 1.58E+00 4.94E+00 3.13E+00 1.34E+00 7.60E-01 3.84E-01 5.04E-01 1.72 1.59E+00 5.09E+00 3.20E+00 1.36E+00 7.54E-01 3.72E-01 4.94E-01 1.74 1.61E+00 5.24E+00 3.27E+00 1.38E+00 7.47E-01 3.61E-01 4.83E-01 1.76 1.62E+00 5.41E+00 3.34E+00 1.40E+00 7.41E-01 3.50E-01 4.73E-01 1.78 1.63E+00 5.57E+00 3.41E+00 1.42E+00 7.35E-01 3.40E-01 4.62E-01 1.8 1.65E+00 5.75E+00 3.49E+00 1.44E+00 7.28E-01 3.29E-01 4.52E-01 1.82 1.66E+00 5.92E+00 3.56E+00 1.46E+00 7.22E-01 3.20E-01 4.43E-01 1.84 1.68E+00 6.11E+00 3.64E+00 1.48E+00 7.16E-01 3.10E-01 4.33E-01 1.86 1.69E+00 6.30E+00 3.72E+00 1.51E+00 7.09E-01 3.00E-01 4.24E-01 1.88 1.71E+00 6.50E+00 3.81E+00 1.53E+00 7.03E-01 2.91E-01 4.14E-01 1.9 1.72E+00 6.70E+00 3.89E+00 1.56E+00 6.97E-01 2.83E-01 4.05E-01 1.92 1.74E+00 6.91E+00 3.98E+00 1.58E+00 6.91E-01 2.74E-01 3.97E-01 1.94 1.75E+00 7.13E+00 4.07E+00 1.61E+00 6.85E-01 2.66E-01 3.88E-01 1.96 1.77E+00 7.35E+00 4.16E+00 1.63E+00 6.79E-01 2.57E-01 3.79E-01 1.98 1.78E+00 7.58E+00 4.25E+00 1.66E+00 6.73E-01 2.50E-01 3.71E-01 2 1.80E+00 7.82E+00 4.35E+00 1.69E+00 6.67E-01 2.42E-01 3.63E-01 2.02 1.82E+00 8.07E+00 4.44E+00 1.72E+00 6.61E-01 2.35E-01 3.55E-01 (continued) 516 Appendix A: Isentropic Flow Table

Table A.1 (continued) ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 2.04 1.83E+00 8.33E+00 4.54E+00 1.75E+00 6.55E-01 2.27E-01 3.47E-01 2.06 1.85E+00 8.59E+00 4.65E+00 1.78E+00 6.49E-01 2.20E-01 3.39E-01 2.08 1.87E+00 8.86E+00 4.75E+00 1.81E+00 6.43E-01 2.14E-01 3.32E-01 2.1 1.88E+00 9.14E+00 4.86E+00 1.84E+00 6.38E-01 2.07E-01 3.25E-01 2.12 1.90E+00 9.44E+00 4.97E+00 1.87E+00 6.32E-01 2.01E-01 3.17E-01 2.14 1.92E+00 9.73E+00 5.08E+00 1.90E+00 6.26E-01 1.94E-01 3.10E-01 2.16 1.93E+00 1.00E+01 5.20E+00 1.94E+00 6.21E-01 1.88E-01 3.04E-01 2.18 1.95E+00 1.04E+01 5.31E+00 1.97E+00 6.15E-01 1.83E-01 2.97E-01 2.2 1.97E+00 1.07E+01 5.43E+00 2.01E+00 6.10E-01 1.77E-01 2.90E-01 2.22 1.99E+00 1.10E+01 5.56E+00 2.04E+00 6.04E-01 1.72E-01 2.84E-01 2.24 2.00E+00 1.14E+01 5.68E+00 2.08E+00 5.99E-01 1.66E-01 2.78E-01 2.26 2.02E+00 1.17E+01 5.81E+00 2.12E+00 5.94E-01 1.61E-01 2.71E-01 2.28 2.04E+00 1.21E+01 5.94E+00 2.15E+00 5.88E-01 1.56E-01 2.65E-01 2.3 2.06E+00 1.25E+01 6.08E+00 2.19E+00 5.83E-01 1.51E-01 2.60E-01 2.32 2.08E+00 1.29E+01 6.21E+00 2.23E+00 5.78E-01 1.47E-01 2.54E-01 2.34 2.10E+00 1.33E+01 6.35E+00 2.27E+00 5.73E-01 1.42E-01 2.48E-01 2.36 2.11E+00 1.37E+01 6.50E+00 2.32E+00 5.68E-01 1.38E-01 2.43E-01 2.38 2.13E+00 1.42E+01 6.64E+00 2.36E+00 5.63E-01 1.34E-01 2.37E-01 2.4 2.15E+00 1.46E+01 6.79E+00 2.40E+00 5.58E-01 1.29E-01 2.32E-01 2.42 2.17E+00 1.51E+01 6.95E+00 2.45E+00 5.53E-01 1.26E-01 2.27E-01 2.44 2.19E+00 1.56E+01 7.10E+00 2.49E+00 5.48E-01 1.22E-01 2.22E-01 2.46 2.21E+00 1.61E+01 7.26E+00 2.54E+00 5.43E-01 1.18E-01 2.17E-01 2.48 2.23E+00 1.66E+01 7.43E+00 2.59E+00 5.38E-01 1.14E-01 2.12E-01 2.5 2.25E+00 1.71E+01 7.59E+00 2.64E+00 5.33E-01 1.11E-01 2.08E-01 2.52 2.27E+00 1.76E+01 7.76E+00 2.69E+00 5.29E-01 1.07E-01 2.03E-01 2.54 2.29E+00 1.82E+01 7.94E+00 2.74E+00 5.24E-01 1.04E-01 1.99E-01 2.56 2.31E+00 1.88E+01 8.12E+00 2.79E+00 5.19E-01 1.01E-01 1.94E-01 2.58 2.33E+00 1.93E+01 8.30E+00 2.84E+00 5.15E-01 9.78E-02 1.90E-01 2.6 2.35E+00 2.00E+01 8.48E+00 2.90E+00 5.10E-01 9.49E-02 1.86E-01 2.62 2.37E+00 2.06E+01 8.67E+00 2.95E+00 5.06E-01 9.20E-02 1.82E-01 2.64 2.39E+00 2.12E+01 8.87E+00 3.01E+00 5.01E-01 8.92E-02 1.78E-01 2.66 2.42E+00 2.19E+01 9.06E+00 3.06E+00 4.97E-01 8.65E-02 1.74E-01 2.68 2.44E+00 2.26E+01 9.27E+00 3.12E+00 4.93E-01 8.38E-02 1.70E-01 2.7 2.46E+00 2.33E+01 9.47E+00 3.18E+00 4.88E-01 8.13E-02 1.67E-01 2.72 2.48E+00 2.40E+01 9.68E+00 3.24E+00 4.84E-01 7.88E-02 1.63E-01 2.74 2.50E+00 2.48E+01 9.90E+00 3.31E+00 4.80E-01 7.65E-02 1.59E-01 2.76 2.52E+00 2.55E+01 1.01E+01 3.37E+00 4.76E-01 7.42E-02 1.56E-01 2.78 2.55E+00 2.63E+01 1.03E+01 3.43E+00 4.71E-01 7.19E-02 1.53E-01 2.8 2.57E+00 2.71E+01 1.06E+01 3.50E+00 4.67E-01 6.98E-02 1.49E-01 (continued) Appendix A: Isentropic Flow Table 517

Table A.1 (continued) ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 2.82 2.59E+00 2.80E+01 1.08E+01 3.57E+00 4.63E-01 6.77E-02 1.46E-01 2.84 2.61E+00 2.88E+01 1.10E+01 3.64E+00 4.59E-01 6.56E-02 1.43E-01 2.86 2.64E+00 2.97E+01 1.13E+01 3.71E+00 4.55E-01 6.37E-02 1.40E-01 2.88 2.66E+00 3.07E+01 1.15E+01 3.78E+00 4.51E-01 6.18E-02 1.37E-01 2.9 2.68E+00 3.16E+01 1.18E+01 3.85E+00 4.47E-01 5.99E-02 1.34E-01 2.92 2.71E+00 3.26E+01 1.20E+01 3.92E+00 4.44E-01 5.81E-02 1.31E-01 2.94 2.73E+00 3.36E+01 1.23E+01 4.00E+00 4.40E-01 5.64E-02 1.28E-01 2.96 2.75E+00 3.46E+01 1.26E+01 4.08E+00 4.36E-01 5.47E-02 1.26E-01 2.98 2.78E+00 3.56E+01 1.28E+01 4.15E+00 4.32E-01 5.31E-02 1.23E-01 3 2.80E+00 3.67E+01 1.31E+01 4.23E+00 4.29E-01 5.15E-02 1.20E-01 3.05 2.86E+00 3.96E+01 1.38E+01 4.44E+00 4.20E-01 4.78E-02 1.14E-01 3.1 2.92E+00 4.26E+01 1.46E+01 4.66E+00 4.11E-01 4.44E-02 1.08E-01 3.15 2.98E+00 4.59E+01 1.54E+01 4.88E+00 4.02E-01 4.12E-02 1.03E-01 3.2 3.05E+00 4.94E+01 1.62E+01 5.12E+00 3.94E-01 3.83E-02 9.73E-02 3.25 3.11E+00 5.32E+01 1.71E+01 5.37E+00 3.86E-01 3.56E-02 9.23E-02 3.3 3.18E+00 5.72E+01 1.80E+01 5.63E+00 3.78E-01 3.31E-02 8.76E-02 3.35 3.24E+00 6.15E+01 1.90E+01 5.90E+00 3.70E-01 3.08E-02 8.32E-02 3.4 3.31E+00 6.61E+01 2.00E+01 6.18E+00 3.62E-01 2.86E-02 7.90E-02 3.45 3.38E+00 7.10E+01 2.10E+01 6.48E+00 3.55E-01 2.67E-02 7.51E-02 3.5 3.45E+00 7.63E+01 2.21E+01 6.79E+00 3.48E-01 2.48E-02 7.14E-02 3.55 3.52E+00 8.19E+01 2.33E+01 7.11E+00 3.41E-01 2.31E-02 6.78E-02 3.6 3.59E+00 8.78E+01 2.45E+01 7.45E+00 3.34E-01 2.16E-02 6.45E-02 3.65 3.66E+00 9.42E+01 2.57E+01 7.80E+00 3.27E-01 2.01E-02 6.14E-02 3.7 3.74E+00 1.01E+02 2.70E+01 8.17E+00 3.21E-01 1.87E-02 5.84E-02 3.75 3.81E+00 1.08E+02 2.84E+01 8.55E+00 3.15E-01 1.75E-02 5.56E-02 3.8 3.89E+00 1.16E+02 2.98E+01 8.95E+00 3.09E-01 1.63E-02 5.29E-02 3.85 3.96E+00 1.24E+02 3.13E+01 9.37E+00 3.03E-01 1.53E-02 5.04E-02 3.9 4.04E+00 1.33E+02 3.28E+01 9.80E+00 2.97E-01 1.43E-02 4.80E-02 3.95 4.12E+00 1.42E+02 3.45E+01 1.03E+01 2.91E-01 1.33E-02 4.58E-02 4 4.20E+00 1.52E+02 3.62E+01 1.07E+01 2.86E-01 1.25E-02 4.36E-02 4.05 4.28E+00 1.62E+02 3.79E+01 1.12E+01 2.80E-01 1.17E-02 4.16E-02 4.1 4.36E+00 1.73E+02 3.97E+01 1.17E+01 2.75E-01 1.09E-02 3.97E-02 4.15 4.44E+00 1.85E+02 4.16E+01 1.22E+01 2.70E-01 1.02E-02 3.79E-02 4.2 4.53E+00 1.98E+02 4.36E+01 1.28E+01 2.65E-01 9.58E-03 3.62E-02 4.25 4.61E+00 2.11E+02 4.57E+01 1.34E+01 2.60E-01 8.98E-03 3.45E-02 4.3 4.70E+00 2.25E+02 4.78E+01 1.40E+01 2.55E-01 8.42E-03 3.30E-02 4.35 4.78E+00 2.40E+02 5.01E+01 1.46E+01 2.51E-01 7.90E-03 3.15E-02 4.4 4.87E+00 2.55E+02 5.24E+01 1.52E+01 2.46E-01 7.42E-03 3.01E-02 4.45 4.96E+00 2.72E+02 5.48E+01 1.59E+01 2.42E-01 6.96E-03 2.88E-02 (continued) 518 Appendix A: Isentropic Flow Table

Table A.1 (continued) ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 4.5 5.05E+00 2.89E+02 5.73E+01 1.66E+01 2.38E-01 6.54E-03 2.75E-02 4.55 5.14E+00 3.08E+02 5.99E+01 1.73E+01 2.33E-01 6.15E-03 2.63E-02 4.6 5.23E+00 3.28E+02 6.26E+01 1.80E+01 2.29E-01 5.78E-03 2.52E-02 4.65 5.32E+00 3.48E+02 6.54E+01 1.88E+01 2.25E-01 5.43E-03 2.41E-02 4.7 5.42E+00 3.70E+02 6.83E+01 1.96E+01 2.21E-01 5.11E-03 2.31E-02 4.75 5.51E+00 3.93E+02 7.13E+01 2.04E+01 2.18E-01 4.81E-03 2.21E-02 4.8 5.61E+00 4.18E+02 7.45E+01 2.13E+01 2.14E-01 4.53E-03 2.12E-02 4.85 5.70E+00 4.43E+02 7.77E+01 2.21E+01 2.10E-01 4.27E-03 2.03E-02 4.9 5.80E+00 4.70E+02 8.11E+01 2.31E+01 2.07E-01 4.02E-03 1.95E-02 4.95 5.90E+00 4.99E+02 8.46E+01 2.40E+01 2.03E-01 3.79E-03 1.87E-02 5 6.00E+00 5.29E+02 8.82E+01 2.50E+01 2.00E-01 3.58E-03 1.79E-02 5.1 6.20E+00 5.94E+02 9.58E+01 2.71E+01 1.93E-01 3.19E-03 1.65E-02 5.2 6.41E+00 6.66E+02 1.04E+02 2.93E+01 1.87E-01 2.84E-03 1.52E-02 5.3 6.62E+00 7.46E+02 1.13E+02 3.16E+01 1.81E-01 2.54E-03 1.40E-02 5.4 6.83E+00 8.34E+02 1.22E+02 3.42E+01 1.76E-01 2.27E-03 1.29E-02 5.5 7.05E+00 9.30E+02 1.32E+02 3.69E+01 1.70E-01 2.03E-03 1.20E-02 5.6 7.27E+00 1.04E+03 1.43E+02 3.97E+01 1.65E-01 1.83E-03 1.11E-02 5.7 7.50E+00 1.15E+03 1.54E+02 4.28E+01 1.60E-01 1.64E-03 1.02E-02 5.8 7.73E+00 1.28E+03 1.66E+02 4.60E+01 1.55E-01 1.48E-03 9.50E-03 5.9 7.96E+00 1.42E+03 1.79E+02 4.95E+01 1.51E-01 1.33E-03 8.82E-03 6 8.20E+00 1.58E+03 1.93E+02 5.32E+01 1.46E-01 1.20E-03 8.19E-03 6.1 8.44E+00 1.75E+03 2.07E+02 5.71E+01 1.42E-01 1.08E-03 7.62E-03 6.2 8.69E+00 1.93E+03 2.22E+02 6.12E+01 1.38E-01 9.79E-04 7.09E-03 6.3 8.94E+00 2.13E+03 2.39E+02 6.56E+01 1.34E-01 8.87E-04 6.60E-03 6.4 9.19E+00 2.35E+03 2.56E+02 7.02E+01 1.31E-01 8.04E-04 6.16E-03 6.5 9.45E+00 2.59E+03 2.75E+02 7.51E+01 1.27E-01 7.30E-04 5.75E-03 6.6 9.71E+00 2.85E+03 2.94E+02 8.03E+01 1.24E-01 6.63E-04 5.37E-03 6.7 9.98E+00 3.14E+03 3.14E+02 8.58E+01 1.20E-01 6.03E-04 5.02E-03 6.8 1.02E+01 3.45E+03 3.36E+02 9.16E+01 1.17E-01 5.49E-04 4.69E-03 6.9 1.05E+01 3.78E+03 3.59E+02 9.77E+01 1.14E-01 5.01E-04 4.39E-03 7 1.08E+01 4.14E+03 3.83E+02 1.04E+02 1.11E-01 4.57E-04 4.12E-03 7.1 1.11E+01 4.53E+03 4.09E+02 1.11E+02 1.08E-01 4.18E-04 3.86E-03 7.2 1.14E+01 4.95E+03 4.36E+02 1.18E+02 1.06E-01 3.82E-04 3.62E-03 7.3 1.17E+01 5.41E+03 4.64E+02 1.26E+02 1.03E-01 3.50E-04 3.40E-03 7.4 1.20E+01 5.90E+03 4.94E+02 1.34E+02 1.00E-01 3.21E-04 3.19E-03 7.5 1.23E+01 6.43E+03 5.25E+02 1.42E+02 9.80E-02 2.94E-04 3.00E-03 7.6 1.26E+01 7.01E+03 5.58E+02 1.51E+02 9.56E-02 2.70E-04 2.83E-03 7.7 1.29E+01 7.62E+03 5.93E+02 1.60E+02 9.33E-02 2.48E-04 2.66E-03 7.8 1.32E+01 8.29E+03 6.29E+02 1.69E+02 9.11E-02 2.28E-04 2.51E-03 (continued) Appendix A: Isentropic Flow Table 519

Table A.1 (continued) ∗ ∗ ∗ ∗ M Tt /T pt /p ρt /ρ A/A T/T p/p ρ/ρ 7.9 1.35E+01 9.00E+03 6.67E+02 1.80E+02 8.90E-02 2.10E-04 2.36E-03 8 1.38E+01 9.76E+03 7.07E+02 1.90E+02 8.70E-02 1.94E-04 2.23E-03 8.1 1.41E+01 1.06E+04 7.49E+02 2.01E+02 8.50E-02 1.79E-04 2.10E-03 8.2 1.44E+01 1.15E+04 7.93E+02 2.13E+02 8.31E-02 1.65E-04 1.99E-03 8.3 1.48E+01 1.24E+04 8.40E+02 2.25E+02 8.12E-02 1.53E-04 1.88E-03 8.4 1.51E+01 1.34E+04 8.88E+02 2.38E+02 7.94E-02 1.41E-04 1.78E-03 8.5 1.55E+01 1.45E+04 9.38E+02 2.51E+02 7.77E-02 1.31E-04 1.68E-03 8.6 1.58E+01 1.57E+04 9.91E+02 2.65E+02 7.60E-02 1.21E-04 1.59E-03 8.7 1.61E+01 1.69E+04 1.05E+03 2.80E+02 7.44E-02 1.12E-04 1.51E-03 8.8 1.65E+01 1.82E+04 1.10E+03 2.95E+02 7.28E-02 1.04E-04 1.43E-03 8.9 1.68E+01 1.96E+04 1.16E+03 3.11E+02 7.13E-02 9.66E-05 1.36E-03 9 1.72E+01 2.11E+04 1.23E+03 3.27E+02 6.98E-02 8.97E-05 1.29E-03 9.1 1.76E+01 2.27E+04 1.29E+03 3.44E+02 6.83E-02 8.34E-05 1.22E-03 9.2 1.79E+01 2.44E+04 1.36E+03 3.62E+02 6.69E-02 7.76E-05 1.16E-03 9.3 1.83E+01 2.62E+04 1.43E+03 3.81E+02 6.56E-02 7.22E-05 1.10E-03 9.4 1.87E+01 2.81E+04 1.51E+03 4.01E+02 6.43E-02 6.73E-05 1.05E-03 9.5 1.91E+01 3.02E+04 1.58E+03 4.21E+02 6.30E-02 6.27E-05 9.96E-04 9.6 1.94E+01 3.23E+04 1.66E+03 4.42E+02 6.18E-02 5.85E-05 9.48E-04 9.7 1.98E+01 3.47E+04 1.75E+03 4.64E+02 6.06E-02 5.46E-05 9.02E-04 9.8 2.02E+01 3.71E+04 1.84E+03 4.87E+02 5.94E-02 5.10E-05 8.59E-04 9.9 2.06E+01 3.97E+04 1.93E+03 5.11E+02 5.82E-02 4.77E-05 8.19E-04 10 2.10E+01 4.24E+04 2.02E+03 5.36E+02 5.71E-02 4.46E-05 7.81E-04 Appendix B Normal Shock Table

TableB.1 shows the normal shock table for γ = 1.4.

Table B.1 Normal shock table for γ = 1.4

M1 M2 T2/T1 p2/p1 ρ2/ρ1 pt2/pt1 Pt2/p1 s/R 1 1 1 1 1 1 1.8929 0 1.02 0.9805 1.01E+00 1.05E+00 1.03E+00 1.00E+00 1.94E+00 9.96E-06 1.04 0.962 1.03E+00 1.10E+00 1.07E+00 1.00E+00 1.98E+00 7.67E-05 1.06 0.9444 1.04E+00 1.14E+00 1.10E+00 1.00E+00 2.03E+00 2.49E-04 1.08 0.9277 1.05E+00 1.19E+00 1.13E+00 9.99E-01 2.08E+00 5.69E-04 1.1 0.9118 1.06E+00 1.25E+00 1.17E+00 9.99E-01 2.13E+00 1.07E-03 1.12 0.8966 1.08E+00 1.30E+00 1.20E+00 9.98E-01 2.19E+00 1.79E-03 1.14 0.882 1.09E+00 1.35E+00 1.24E+00 9.97E-01 2.24E+00 2.74E-03 1.16 0.8682 1.10E+00 1.40E+00 1.27E+00 9.96E-01 2.29E+00 3.96E-03 1.18 0.8549 1.12E+00 1.46E+00 1.31E+00 9.95E-01 2.35E+00 5.45E-03 1.2 0.8422 1.13E+00 1.51E+00 1.34E+00 9.93E-01 2.41E+00 7.23E-03 1.22 0.83 1.14E+00 1.57E+00 1.38E+00 9.91E-01 2.47E+00 9.31E-03 1.24 0.8183 1.15E+00 1.63E+00 1.41E+00 9.88E-01 2.53E+00 1.17E-02 1.26 0.8071 1.17E+00 1.69E+00 1.45E+00 9.86E-01 2.59E+00 1.44E-02 1.28 0.7963 1.18E+00 1.74E+00 1.48E+00 9.83E-01 2.65E+00 1.75E-02 1.3 0.786 1.19E+00 1.81E+00 1.52E+00 9.79E-01 2.71E+00 2.08E-02 1.32 0.776 1.20E+00 1.87E+00 1.55E+00 9.76E-01 2.78E+00 2.45E-02 1.34 0.7664 1.22E+00 1.93E+00 1.59E+00 9.72E-01 2.84E+00 2.86E-02 1.36 0.7572 1.23E+00 1.99E+00 1.62E+00 9.68E-01 2.91E+00 3.30E-02 1.38 0.7483 1.24E+00 2.06E+00 1.65E+00 9.63E-01 2.98E+00 3.77E-02 1.4 0.7397 1.25E+00 2.12E+00 1.69E+00 9.58E-01 3.05E+00 4.27E-02 1.42 0.7314 1.27E+00 2.19E+00 1.72E+00 9.53E-01 3.12E+00 4.81E-02 1.44 0.7235 1.28E+00 2.25E+00 1.76E+00 9.48E-01 3.19E+00 5.38E-02 (continued)

© Springer Science+Business Media Dordrecht 2015 521 R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics, Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4 522 Appendix B: Normal Shock Table

Table B.1 (continued)

M1 M2 T2/T1 p2/p1 ρ2/ρ1 pt2/pt1 Pt2/p1 s/R 1.46 0.7157 1.29E+00 2.32E+00 1.79E+00 9.42E-01 3.26E+00 5.98E-02 1.48 0.7083 1.31E+00 2.39E+00 1.83E+00 9.36E-01 3.34E+00 6.61E-02 1.5 0.7011 1.32E+00 2.46E+00 1.86E+00 9.30E-01 3.41E+00 7.28E-02 1.52 0.6941 1.33E+00 2.53E+00 1.90E+00 9.23E-01 3.49E+00 7.98E-02 1.54 0.6874 1.35E+00 2.60E+00 1.93E+00 9.17E-01 3.57E+00 8.71E-02 1.56 0.6809 1.36E+00 2.67E+00 1.96E+00 9.10E-01 3.65E+00 9.46E-02 1.58 0.6746 1.37E+00 2.75E+00 2.00E+00 9.03E-01 3.72E+00 1.03E-01 1.6 0.6684 1.39E+00 2.82E+00 2.03E+00 8.95E-01 3.81E+00 1.11E-01 1.62 0.6625 1.40E+00 2.90E+00 2.07E+00 8.88E-01 3.89E+00 1.19E-01 1.64 0.6568 1.42E+00 2.97E+00 2.10E+00 8.80E-01 3.97E+00 1.28E-01 1.66 0.6512 1.43E+00 3.05E+00 2.13E+00 8.72E-01 4.05E+00 1.37E-01 1.68 0.6458 1.44E+00 3.13E+00 2.16E+00 8.64E-01 4.14E+00 1.46E-01 1.7 0.6405 1.46E+00 3.21E+00 2.20E+00 8.56E-01 4.22E+00 1.56E-01 1.72 0.6355 1.47E+00 3.28E+00 2.23E+00 8.47E-01 4.31E+00 1.66E-01 1.74 0.6305 1.49E+00 3.37E+00 2.26E+00 8.39E-01 4.40E+00 1.76E-01 1.76 0.6257 1.50E+00 3.45E+00 2.30E+00 8.30E-01 4.49E+00 1.86E-01 1.78 0.621 1.52E+00 3.53E+00 2.33E+00 8.22E-01 4.58E+00 1.97E-01 1.8 0.6165 1.53E+00 3.61E+00 2.36E+00 8.13E-01 4.67E+00 2.07E-01 1.82 0.6121 1.55E+00 3.70E+00 2.39E+00 8.04E-01 4.76E+00 2.18E-01 1.84 0.6078 1.56E+00 3.78E+00 2.42E+00 7.95E-01 4.86E+00 2.30E-01 1.86 0.6036 1.58E+00 3.87E+00 2.45E+00 7.86E-01 4.95E+00 2.41E-01 1.88 0.5996 1.59E+00 3.96E+00 2.48E+00 7.77E-01 5.05E+00 2.53E-01 1.9 0.5956 1.61E+00 4.05E+00 2.52E+00 7.67E-01 5.14E+00 2.65E-01 1.92 0.5918 1.62E+00 4.13E+00 2.55E+00 7.58E-01 5.24E+00 2.77E-01 1.94 0.588 1.64E+00 4.22E+00 2.58E+00 7.49E-01 5.34E+00 2.89E-01 1.96 0.5844 1.66E+00 4.32E+00 2.61E+00 7.40E-01 5.44E+00 3.02E-01 1.98 0.5808 1.67E+00 4.41E+00 2.64E+00 7.30E-01 5.54E+00 3.14E-01 2 0.5774 1.69E+00 4.50E+00 2.67E+00 7.21E-01 5.64E+00 3.27E-01 2.02 0.574 1.70E+00 4.59E+00 2.70E+00 7.12E-01 5.74E+00 3.40E-01 2.04 0.5707 1.72E+00 4.69E+00 2.73E+00 7.02E-01 5.85E+00 3.54E-01 2.06 0.5675 1.74E+00 4.78E+00 2.75E+00 6.93E-01 5.95E+00 3.67E-01 2.08 0.5643 1.75E+00 4.88E+00 2.78E+00 6.84E-01 6.06E+00 3.81E-01 2.1 0.5613 1.77E+00 4.98E+00 2.81E+00 6.74E-01 6.17E+00 3.94E-01 2.12 0.5583 1.79E+00 5.08E+00 2.84E+00 6.65E-01 6.27E+00 4.08E-01 2.14 0.5554 1.80E+00 5.18E+00 2.87E+00 6.56E-01 6.38E+00 4.22E-01 2.16 0.5525 1.82E+00 5.28E+00 2.90E+00 6.46E-01 6.49E+00 4.36E-01 2.18 0.5498 1.84E+00 5.38E+00 2.92E+00 6.37E-01 6.60E+00 4.51E-01 2.2 0.5471 1.86E+00 5.48E+00 2.95E+00 6.28E-01 6.72E+00 4.65E-01 2.22 0.5444 1.87E+00 5.58E+00 2.98E+00 6.19E-01 6.83E+00 4.80E-01 (continued) Appendix B: Normal Shock Table 523

Table B.1 (continued)

M1 M2 T2/T1 p2/p1 ρ2/ρ1 pt2/pt1 Pt2/p1 s/R 2.24 0.5418 1.89E+00 5.69E+00 3.01E+00 6.10E-01 6.94E+00 4.94E-01 2.26 0.5393 1.91E+00 5.79E+00 3.03E+00 6.01E-01 7.06E+00 5.09E-01 2.28 0.5368 1.93E+00 5.90E+00 3.06E+00 5.92E-01 7.18E+00 5.24E-01 2.3 0.5344 1.95E+00 6.01E+00 3.08E+00 5.83E-01 7.29E+00 5.39E-01 2.32 0.5321 1.97E+00 6.11E+00 3.11E+00 5.75E-01 7.41E+00 5.54E-01 2.34 0.5297 1.98E+00 6.22E+00 3.14E+00 5.66E-01 7.53E+00 5.70E-01 2.36 0.5275 2.00E+00 6.33E+00 3.16E+00 5.57E-01 7.65E+00 5.85E-01 2.38 0.5253 2.02E+00 6.44E+00 3.19E+00 5.49E-01 7.77E+00 6.00E-01 2.4 0.5231 2.04E+00 6.55E+00 3.21E+00 5.40E-01 7.90E+00 6.16E-01 2.42 0.521 2.06E+00 6.67E+00 3.24E+00 5.32E-01 8.02E+00 6.32E-01 2.44 0.5189 2.08E+00 6.78E+00 3.26E+00 5.23E-01 8.15E+00 6.47E-01 2.46 0.5169 2.10E+00 6.89E+00 3.29E+00 5.15E-01 8.27E+00 6.63E-01 2.48 0.5149 2.12E+00 7.01E+00 3.31E+00 5.07E-01 8.40E+00 6.79E-01 2.5 0.513 2.14E+00 7.13E+00 3.33E+00 4.99E-01 8.53E+00 6.95E-01 2.52 0.5111 2.16E+00 7.24E+00 3.36E+00 4.91E-01 8.66E+00 7.11E-01 2.54 0.5092 2.18E+00 7.36E+00 3.38E+00 4.83E-01 8.79E+00 7.27E-01 2.56 0.5074 2.20E+00 7.48E+00 3.40E+00 4.75E-01 8.92E+00 7.44E-01 2.58 0.5056 2.22E+00 7.60E+00 3.43E+00 4.68E-01 9.05E+00 7.60E-01 2.6 0.5039 2.24E+00 7.72E+00 3.45E+00 4.60E-01 9.18E+00 7.76E-01 2.62 0.5022 2.26E+00 7.84E+00 3.47E+00 4.53E-01 9.32E+00 7.93E-01 2.64 0.5005 2.28E+00 7.96E+00 3.49E+00 4.45E-01 9.45E+00 8.09E-01 2.66 0.4988 2.30E+00 8.09E+00 3.52E+00 4.38E-01 9.59E+00 8.26E-01 2.68 0.4972 2.32E+00 8.21E+00 3.54E+00 4.31E-01 9.72E+00 8.42E-01 2.7 0.4956 2.34E+00 8.34E+00 3.56E+00 4.24E-01 9.86E+00 8.59E-01 2.72 0.4941 2.36E+00 8.46E+00 3.58E+00 4.17E-01 1.00E+01 8.76E-01 2.74 0.4926 2.39E+00 8.59E+00 3.60E+00 4.10E-01 1.01E+01 8.92E-01 2.76 0.4911 2.41E+00 8.72E+00 3.62E+00 4.03E-01 1.03E+01 9.09E-01 2.78 0.4896 2.43E+00 8.85E+00 3.64E+00 3.96E-01 1.04E+01 9.26E-01 2.8 0.4882 2.45E+00 8.98E+00 3.66E+00 3.89E-01 1.06E+01 9.43E-01 2.82 0.4868 2.47E+00 9.11E+00 3.68E+00 3.83E-01 1.07E+01 9.60E-01 2.84 0.4854 2.50E+00 9.24E+00 3.70E+00 3.76E-01 1.09E+01 9.77E-01 2.86 0.484 2.52E+00 9.38E+00 3.72E+00 3.70E-01 1.10E+01 9.94E-01 2.88 0.4827 2.54E+00 9.51E+00 3.74E+00 3.64E-01 1.12E+01 1.01E+00 2.9 0.4814 2.56E+00 9.65E+00 3.76E+00 3.58E-01 1.13E+01 1.03E+00 2.92 0.4801 2.59E+00 9.78E+00 3.78E+00 3.52E-01 1.15E+01 1.05E+00 2.94 0.4788 2.61E+00 9.92E+00 3.80E+00 3.46E-01 1.16E+01 1.06E+00 2.96 0.4776 2.63E+00 1.01E+01 3.82E+00 3.40E-01 1.18E+01 1.08E+00 2.98 0.4764 2.66E+00 1.02E+01 3.84E+00 3.34E-01 1.19E+01 1.10E+00 (continued) 524 Appendix B: Normal Shock Table

Table B.1 (continued)

M1 M2 T2/T1 p2/p1 ρ2/ρ1 pt2/pt1 Pt2/p1 s/R 3 0.4752 2.68E+00 1.03E+01 3.86E+00 3.28E-01 1.21E+01 1.11E+00 3.02 0.474 2.70E+00 1.05E+01 3.88E+00 3.23E-01 1.22E+01 1.13E+00 3.04 0.4729 2.73E+00 1.06E+01 3.89E+00 3.17E-01 1.24E+01 1.15E+00 3.06 0.4717 2.75E+00 1.08E+01 3.91E+00 3.12E-01 1.25E+01 1.17E+00 3.08 0.4706 2.77E+00 1.09E+01 3.93E+00 3.06E-01 1.27E+01 1.18E+00 3.1 0.4695 2.80E+00 1.10E+01 3.95E+00 3.01E-01 1.28E+01 1.20E+00 3.12 0.4685 2.82E+00 1.12E+01 3.96E+00 2.96E-01 1.30E+01 1.22E+00 3.14 0.4674 2.85E+00 1.13E+01 3.98E+00 2.91E-01 1.32E+01 1.23E+00 3.16 0.4664 2.87E+00 1.15E+01 4.00E+00 2.86E-01 1.33E+01 1.25E+00 3.18 0.4654 2.90E+00 1.16E+01 4.01E+00 2.81E-01 1.35E+01 1.27E+00 3.2 0.4643 2.92E+00 1.18E+01 4.03E+00 2.76E-01 1.37E+01 1.29E+00 3.22 0.4634 2.95E+00 1.19E+01 4.05E+00 2.71E-01 1.38E+01 1.30E+00 3.24 0.4624 2.97E+00 1.21E+01 4.06E+00 2.67E-01 1.40E+01 1.32E+00 3.26 0.4614 3.00E+00 1.22E+01 4.08E+00 2.62E-01 1.42E+01 1.34E+00 3.28 0.4605 3.02E+00 1.24E+01 4.10E+00 2.58E-01 1.43E+01 1.36E+00 3.3 0.4596 3.05E+00 1.25E+01 4.11E+00 2.53E-01 1.45E+01 1.37E+00 3.32 0.4587 3.08E+00 1.27E+01 4.13E+00 2.49E-01 1.47E+01 1.39E+00 3.34 0.4578 3.10E+00 1.28E+01 4.14E+00 2.45E-01 1.48E+01 1.41E+00 3.36 0.4569 3.13E+00 1.30E+01 4.16E+00 2.40E-01 1.50E+01 1.43E+00 3.38 0.456 3.15E+00 1.32E+01 4.17E+00 2.36E-01 1.52E+01 1.44E+00 3.4 0.4552 3.18E+00 1.33E+01 4.19E+00 2.32E-01 1.54E+01 1.46E+00 3.42 0.4544 3.21E+00 1.35E+01 4.20E+00 2.28E-01 1.55E+01 1.48E+00 3.44 0.4535 3.23E+00 1.36E+01 4.22E+00 2.24E-01 1.57E+01 1.49E+00 3.46 0.4527 3.26E+00 1.38E+01 4.23E+00 2.20E-01 1.59E+01 1.51E+00 3.48 0.4519 3.29E+00 1.40E+01 4.25E+00 2.17E-01 1.61E+01 1.53E+00 3.5 0.4512 3.32E+00 1.41E+01 4.26E+00 2.13E-01 1.62E+01 1.55E+00 3.52 0.4504 3.34E+00 1.43E+01 4.27E+00 2.09E-01 1.64E+01 1.56E+00 3.54 0.4496 3.37E+00 1.45E+01 4.29E+00 2.06E-01 1.66E+01 1.58E+00 3.56 0.4489 3.40E+00 1.46E+01 4.30E+00 2.02E-01 1.68E+01 1.60E+00 3.58 0.4481 3.43E+00 1.48E+01 4.32E+00 1.99E-01 1.70E+01 1.62E+00 3.6 0.4474 3.45E+00 1.50E+01 4.33E+00 1.95E-01 1.72E+01 1.63E+00 3.62 0.4467 3.48E+00 1.51E+01 4.34E+00 1.92E-01 1.73E+01 1.65E+00 3.64 0.446 3.51E+00 1.53E+01 4.36E+00 1.89E-01 1.75E+01 1.67E+00 3.66 0.4453 3.54E+00 1.55E+01 4.37E+00 1.85E-01 1.77E+01 1.68E+00 3.68 0.4446 3.57E+00 1.56E+01 4.38E+00 1.82E-01 1.79E+01 1.70E+00 3.7 0.4439 3.60E+00 1.58E+01 4.39E+00 1.79E-01 1.81E+01 1.72E+00 3.72 0.4433 3.63E+00 1.60E+01 4.41E+00 1.76E-01 1.83E+01 1.74E+00 3.74 0.4426 3.65E+00 1.62E+01 4.42E+00 1.73E-01 1.85E+01 1.75E+00 3.76 0.442 3.68E+00 1.63E+01 4.43E+00 1.70E-01 1.87E+01 1.77E+00 (continued) Appendix B: Normal Shock Table 525

Table B.1 (continued)

M1 M2 T2/T1 p2/p1 ρ2/ρ1 pt2/pt1 Pt2/p1 s/R 3.78 0.4414 3.71E+00 1.65E+01 4.44E+00 1.67E-01 1.89E+01 1.79E+00 3.8 0.4407 3.74E+00 1.67E+01 4.46E+00 1.64E-01 1.91E+01 1.81E+00 3.82 0.4401 3.77E+00 1.69E+01 4.47E+00 1.62E-01 1.93E+01 1.82E+00 3.84 0.4395 3.80E+00 1.70E+01 4.48E+00 1.59E-01 1.95E+01 1.84E+00 3.86 0.4389 3.83E+00 1.72E+01 4.49E+00 1.56E-01 1.97E+01 1.86E+00 3.88 0.4383 3.86E+00 1.74E+01 4.50E+00 1.54E-01 1.99E+01 1.87E+00 3.9 0.4377 3.89E+00 1.76E+01 4.52E+00 1.51E-01 2.01E+01 1.89E+00 3.92 0.4372 3.92E+00 1.78E+01 4.53E+00 1.48E-01 2.03E+01 1.91E+00 3.94 0.4366 3.95E+00 1.79E+01 4.54E+00 1.46E-01 2.05E+01 1.92E+00 3.96 0.436 3.98E+00 1.81E+01 4.55E+00 1.44E-01 2.07E+01 1.94E+00 3.98 0.4355 4.02E+00 1.83E+01 4.56E+00 1.41E-01 2.09E+01 1.96E+00 4 0.435 4.05E+00 1.85E+01 4.57E+00 1.39E-01 2.11E+01 1.98E+00 4.1 0.4324 4.20E+00 1.94E+01 4.62E+00 1.28E-01 2.21E+01 2.06E+00 4.2 0.4299 4.37E+00 2.04E+01 4.67E+00 1.17E-01 2.32E+01 2.14E+00 4.3 0.4277 4.53E+00 2.14E+01 4.72E+00 1.08E-01 2.43E+01 2.23E+00 4.4 0.4255 4.70E+00 2.24E+01 4.77E+00 9.95E-02 2.54E+01 2.31E+00 4.5 0.4236 4.88E+00 2.35E+01 4.81E+00 9.17E-02 2.65E+01 2.39E+00 4.6 0.4217 5.05E+00 2.45E+01 4.85E+00 8.46E-02 2.77E+01 2.47E+00 4.7 0.4199 5.23E+00 2.56E+01 4.89E+00 7.81E-02 2.89E+01 2.55E+00 4.8 0.4183 5.42E+00 2.67E+01 4.93E+00 7.21E-02 3.01E+01 2.63E+00 4.9 0.4167 5.61E+00 2.78E+01 4.97E+00 6.67E-02 3.14E+01 2.71E+00 5 0.4152 5.80E+00 2.90E+01 5.00E+00 6.17E-02 3.27E+01 2.79E+00 5.1 0.4138 6.00E+00 3.02E+01 5.03E+00 5.72E-02 3.40E+01 2.86E+00 5.2 0.4125 6.20E+00 3.14E+01 5.06E+00 5.30E-02 3.53E+01 2.94E+00 5.3 0.4113 6.40E+00 3.26E+01 5.09E+00 4.91E-02 3.66E+01 3.01E+00 5.4 0.4101 6.61E+00 3.39E+01 5.12E+00 4.56E-02 3.80E+01 3.09E+00 5.5 0.409 6.82E+00 3.51E+01 5.15E+00 4.24E-02 3.94E+01 3.16E+00 5.6 0.4079 7.04E+00 3.64E+01 5.17E+00 3.94E-02 4.08E+01 3.23E+00 5.7 0.4069 7.26E+00 3.77E+01 5.20E+00 3.66E-02 4.23E+01 3.31E+00 5.8 0.4059 7.48E+00 3.91E+01 5.22E+00 3.41E-02 4.38E+01 3.38E+00 5.9 0.405 7.71E+00 4.04E+01 5.25E+00 3.18E-02 4.53E+01 3.45E+00 6 0.4042 7.94E+00 4.18E+01 5.27E+00 2.97E-02 4.68E+01 3.52E+00 6.1 0.4033 8.18E+00 4.32E+01 5.29E+00 2.77E-02 4.84E+01 3.59E+00 6.2 0.4025 8.42E+00 4.47E+01 5.31E+00 2.58E-02 5.00E+01 3.66E+00 6.3 0.4018 8.66E+00 4.61E+01 5.33E+00 2.42E-02 5.16E+01 3.72E+00 6.4 0.4011 8.91E+00 4.76E+01 5.35E+00 2.26E-02 5.32E+01 3.79E+00 6.5 0.4004 9.16E+00 4.91E+01 5.37E+00 2.11E-02 5.49E+01 3.86E+00 6.6 0.3997 9.41E+00 5.07E+01 5.38E+00 1.98E-02 5.65E+01 3.92E+00 6.7 0.3991 9.67E+00 5.22E+01 5.40E+00 1.86E-02 5.83E+01 3.99E+00 (continued) 526 Appendix B: Normal Shock Table

Table B.1 (continued)

M1 M2 T2/T1 p2/p1 ρ2/ρ1 pt2/pt1 Pt2/p1 s/R 6.8 0.3985 9.93E+00 5.38E+01 5.41E+00 1.74E-02 6.00E+01 4.05E+00 6.9 0.3979 1.02E+01 5.54E+01 5.43E+00 1.63E-02 6.18E+01 4.11E+00 7 0.3974 1.05E+01 5.70E+01 5.44E+00 1.54E-02 6.36E+01 4.18E+00 7.1 0.3968 1.07E+01 5.86E+01 5.46E+00 1.44E-02 6.54E+01 4.24E+00 7.2 0.3963 1.10E+01 6.03E+01 5.47E+00 1.36E-02 6.72E+01 4.30E+00 7.3 0.3958 1.13E+01 6.20E+01 5.49E+00 1.28E-02 6.91E+01 4.36E+00 7.4 0.3954 1.16E+01 6.37E+01 5.50E+00 1.20E-02 7.10E+01 4.42E+00 7.5 0.3949 1.19E+01 6.55E+01 5.51E+00 1.13E-02 7.29E+01 4.48E+00 7.6 0.3945 1.22E+01 6.72E+01 5.52E+00 1.07E-02 7.48E+01 4.54E+00 7.7 0.3941 1.25E+01 6.90E+01 5.53E+00 1.01E-02 7.68E+01 4.60E+00 7.8 0.3937 1.28E+01 7.08E+01 5.54E+00 9.51E-03 7.88E+01 4.66E+00 7.9 0.3933 1.31E+01 7.26E+01 5.56E+00 8.98E-03 8.08E+01 4.71E+00 8 0.3929 1.34E+01 7.45E+01 5.57E+00 8.49E-03 8.29E+01 4.77E+00 8.1 0.3925 1.37E+01 7.64E+01 5.58E+00 8.03E-03 8.49E+01 4.83E+00 8.2 0.3922 1.40E+01 7.83E+01 5.58E+00 7.59E-03 8.70E+01 4.88E+00 8.3 0.3918 1.43E+01 8.02E+01 5.59E+00 7.19E-03 8.92E+01 4.94E+00 8.4 0.3915 1.47E+01 8.22E+01 5.60E+00 6.81E-03 9.13E+01 4.99E+00 8.5 0.3912 1.50E+01 8.41E+01 5.61E+00 6.45E-03 9.35E+01 5.04E+00 8.6 0.3909 1.53E+01 8.61E+01 5.62E+00 6.11E-03 9.57E+01 5.10E+00 8.7 0.3906 1.57E+01 8.81E+01 5.63E+00 5.80E-03 9.79E+01 5.15E+00 8.8 0.3903 1.60E+01 9.02E+01 5.64E+00 5.50E-03 1.00E+02 5.20E+00 8.9 0.3901 1.63E+01 9.22E+01 5.64E+00 5.23E-03 1.02E+02 5.25E+00 9 0.3898 1.67E+01 9.43E+01 5.65E+00 4.96E-03 1.05E+02 5.31E+00 9.1 0.3895 1.70E+01 9.64E+01 5.66E+00 4.72E-03 1.07E+02 5.36E+00 9.2 0.3893 1.74E+01 9.86E+01 5.67E+00 4.49E-03 1.09E+02 5.41E+00 9.3 0.3891 1.78E+01 1.01E+02 5.67E+00 4.27E-03 1.12E+02 5.46E+00 9.4 0.3888 1.81E+01 1.03E+02 5.68E+00 4.06E-03 1.14E+02 5.51E+00 9.5 0.3886 1.85E+01 1.05E+02 5.69E+00 3.87E-03 1.17E+02 5.56E+00 9.6 0.3884 1.89E+01 1.07E+02 5.69E+00 3.68E-03 1.19E+02 5.60E+00 9.7 0.3882 1.92E+01 1.10E+02 5.70E+00 3.51E-03 1.22E+02 5.65E+00 9.8 0.388 1.96E+01 1.12E+02 5.70E+00 3.35E-03 1.24E+02 5.70E+00 9.9 0.3878 2.00E+01 1.14E+02 5.71E+00 3.19E-03 1.27E+02 5.75E+00 Appendix C Prandtl-Meyer Function

TableC.1 shows the Prandtl-Meyer fucntion and Mach angle for γ = 1.4.

Table C.1 Prandtl-Meyer function and Mach angle for γ = 1.4 M ν (deg) μ (deg) M ν (deg) μ (deg) M ν (deg) μ (deg) 1.0 0.00 90.00 3.6 60.09 16.13 6.4 87.56 8.99 1.1 1.34 65.38 3.7 61.60 15.68 6.6 88.76 8.71 1.2 3.56 56.44 3.8 63.05 15.26 6.8 89.90 8.46 1.3 6.17 50.28 3.9 64.44 14.86 7.0 90.97 8.21 1.4 8.99 45.58 4.0 65.79 14.48 7.5 93.44 7.66 1.5 11.91 41.81 4.1 67.08 14.12 8.0 95.63 7.18 1.6 14.86 38.68 4.2 68.33 13.77 8.5 97.57 6.76 1.7 17.81 36.03 4.3 69.54 13.45 9.0 99.32 6.38 1.8 20.73 33.75 4.4 70.71 13.14 9.5 100.89 6.04 1.9 23.59 31.76 4.5 71.83 12.84 10.0 102.32 5.74 2.0 26.38 30.00 4.6 72.92 12.56 10.5 103.61 5.47 2.1 29.10 28.44 4.7 73.97 12.28 11.0 104.80 5.22 2.2 31.73 27.04 4.8 74.99 12.02 11.5 105.88 4.99 2.3 34.28 25.77 4.9 75.97 11.78 12.0 106.88 4.78 2.4 36.75 24.62 5.0 76.92 11.54 12.5 107.80 4.59 2.5 39.12 23.58 5.1 77.84 11.31 13.0 108.65 4.41 2.6 41.42 22.62 5.2 78.73 11.09 13.5 109.44 4.25 2.7 43.62 21.74 5.3 79.60 10.88 14.0 110.18 4.10 2.8 45.75 20.92 5.4 80.43 10.67 14.5 110.87 3.95 2.9 47.79 20.17 5.5 81.25 10.48 15.0 111.51 3.82 (continued)

© Springer Science+Business Media Dordrecht 2015 527 R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics, Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4 528 Appendix C: Prandtl-Meyer Function

Table C.1 (continued) M ν (deg) μ (deg) M ν (deg) μ (deg) M ν (deg) μ (deg) 3.0 49.76 19.47 5.6 82.03 10.29 15.5 112.11 3.70 3.1 51.65 18.82 5.7 82.80 10.10 16.0 112.68 3.58 3.2 53.47 18.21 5.8 83.54 9.93 16.5 113.21 3.47 3.3 55.22 17.64 5.9 84.26 9.76 17.0 113.71 3.37 3.4 56.91 17.10 6.0 84.96 9.59 17.5 114.18 3.28 3.5 58.53 16.60 6.2 86.30 9.28 18.0 114.63 3.18 Partial Answers to Selected Problems

Chapter 1

1.1 (a) a = 296 m/s (b) R|L/D=16 = 5350 km, R|L/D=19 = 6353 km, R|L/D=22 = 7356 km. 1.2

Chapter 2

2.3 14.1m/s. 2.4 β = π + 1 ϕ = π + 1 ( , ) = (a) eigenvalues: n L n 2 ; eigenfunctions: n sin L n 2 x ; u x t ∞ ( (ω ) + (ω )) β ω = β n=0 an cos nt bn sin nt sin n x, with n nc. ∞ ( , ) = 2 + 1 × + 1 (b) u x t n=0 2 sin n 2 t sin n 2 x . π n+ 1 2 ( , ) = t + t x − 6 5t 5x (c) u x t cos 2 4sin 2 sin 2 5 sin 2 sin 2 . 2.7 (a) u = 1/2t2 (b) 1/2(x2 + t2 + 1) − xt − x − t

© Springer Science+Business Media Dordrecht 2015 529 R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics, Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4 530 Partial Answers to Selected Problems ∞ − λ2 π 2.10 u(x, t) = D ϕ e k nt ϕ = (λ x) λ = n + 1 n=0 n n , with n sin n , n L 2 , and = L ( )ϕ Dn 0 f x ndx.

= 2 (−1)n 2.11 Dn π ( 1 + )2 . 2 n

2.12 classiﬁcation: parabolic. 2.14 (a) (2, 7, −2) (b) −61 (c) (53, −22, −24) (d) (⎡−53, 22, 24) ⎤ −8 −4 −14 (e) ⎣ 20 10 35 ⎦ . −36 −18 −63

2.15 (a) φ = x2/2 + y2 + 3z2/2 (b) φ = xyz (c) φ = yex + z (d) does not exist. 2.16 (a) 6 (b) 0 (c) yex (d) 0. 2.17 (a) 0 (b) 0 (c) 0 (d) (0, 0, 3). 2.18

γ = 1.4, cv = 716J/kg/K, e = 160 kJ/k, h = 224 kJ/kg, and T = 223 K. 2.20 37.3kJ/kg. Partial Answers to Selected Problems 531

2.27 (b) ∂u 2 ∂v 2 ∂w 2 ∂v ∂u 2 ∂w ∂v 2 Φ = μ 2 + + + + + + ∂x ∂y ∂z ∂x ∂y ∂y ∂z ∂ ∂ 2 ∂ ∂ ∂ 2 + u + w − 2 u + v + w ∂z ∂x 3 ∂x ∂y ∂z

2.29 (a) p = (γ − 1)ρe (b) q =−k ∇e cv / 3 2 + (c) q =−cp μ e T0 110 ∇e; not conservative. Pr 0 cvT0 e + 110cv

Chapter 3

3.1 −1.704

3.3 cl = 0.756; The circulation is 51.2% higher than the circulation in the incom- pressible limit 3.7 = . (a) cl2 0 9615 (b) Cp min =−2.248 (c) c / c = 1.32 lα M∞=0.65 lα M∞=0 3.9

(a) Mmax = 1.372 (b) C p,crit =−2.133 (c) C p,stag = 1.064 =− . (d) C p0,min 3 031 3.11

(a) MA = 1.0909 3.13 (b) u = 178.18m/s (c) C p = 0.53 3.15

(a) cd,w = 0.0569 (b) h/ = 0.0256 532 Partial Answers to Selected Problems

3.17 C p,cone = 0.1486 3.19 τ = 6.79 %; C =−0.2396; C , =−0.18786; 2 p2 p crit M∞=0.9 C , =−0.30199 p crit M∞=0.85 3.21

(a) pA/p∞ = 1.56 (b) q∞/p∞ = 2.8 3.23 (a) K = 07368 (b) τ2 = 0.0177 = . (c) C pmin,2 0 06

3.25 cl2 = 0.1587 3.27

(a) Mmax = 1.43 (b) C p,stag = 1.117 3.29

(a) M∞2 = 0.937 ◦ (b) α2 = 3 = . (c) 10 079 (d) CL2/CL1 = 0.6299 (e) CD2/CD1 = 0.3149 (f) Cm2/Cm1 = 0.6299 ◦ 3.31 δmax = 20 ◦ 3.33 σc = 78.11

Chapter 4

4.1

(a) p1/p∞ = 2.971 (b) p2/p∞ = 1.991 (c) cd = 0.02755 4.3

(a) p1 = 20.793 kPa (b) p1 = 11.923 kPa (c) p1 = 9.953 kPa Partial Answers to Selected Problems 533

4.5 pt,2 = 53.38 kPa; pt,3 = 6.02 kPa; pt,4 = 32.83 kPa 4.7 ◦ (a) θw = 32.15 (b) V2/V1 = 1.264

4.9 cl = 0.65, cd = 0.31 4.11 (a) t/c = 0.08816 (b) C p2 = 0.1061, C p3 = 0.4236, C p4 =−0.1531, C p5 =−0.1537 (c) ca = 0.03687 4.13

(a) C p1 = C p4 = 0.433, C p2 = C p5 =−0.1099, C p3 = C p6 = 0.1124 (b) cd = 0.0545 4.15

(a) M2 = 3.4 (b) θw = 17.79 4.17

(a) θLE = 23.38 (b) p2/p1 = 0.213 (c) p3/p1 = 0.2419 (d) θTE = 16.02 4.19

(a) t1/c = 0.1063, t2/c = 0.1246 (b) C p1 = 0.2239, C p2 = 1.3896, C p3 =−0.2151, C p4 =−0.1322 (c) cn = 0.1652 (d) ca = 0.0609 (e) L/D = 2.46 (f) cm,LE =−0.0995 4.21

(a) a1 = 260 m/s ∗ = . (b) M2 0 78 4.23

(a) pnose/p∞ = 1.350 (b) pA/p∞ ≈ 0.735

4.25 C p,N = 0.821, C p,A = 0.250 534 Partial Answers to Selected Problems

4.27 (a) R¯ = 6.5 (b) δ0 = 22.6 (c) C pp = 0.05, C pv = 0.20 (d) CD = 0.118 4.31 (a) σ = 48◦ (b) Mc = 0.8 = . (c) CDp 0 65 4.33

(a) C p1 = 0, C p2 = 0.837 (b) cd = 0.4835 (c) cl = 0.8374

4.35 M3 = 2.85 4.39 (a) |α| > 3◦ (b) α =±2.5◦

Chapter 5

5.1 M2 ≈ 2.2, dy/dx = 0.869 5.3

No K− K+ θ (deg) ν (deg) M μ(deg) 1 −0.14 −29.86 −15 14.86 1.6 36.68 2 20.73 −20.73 0 20.73 1.8 33.75 3 29.86 0.14 15 14.86 1.6 36.68 4 20.73 −29.86 −4.565 25.295 1.96 30.68 5 29.86 −20.73 4.565 25.295 1.96 30.68 6 20.73 −29.86 −4.565 25.295 1.96 30.68 7 29.86 −29.86 0 29.86 2.1 28.44 8 29.86 −20.73 4.565 25.295 1.96 30.68

5.5 ◦ (a) θw,max = 22.875 (c) dy/dx =−2.37 (d) dy/dx = 0.3823 Partial Answers to Selected Problems 535

5.7

No K− K+ θ (deg) ν (deg) M μ(deg) 1 26.38 −26.38 0 26.38 2 36.68 2 24.73 −16.73 4 20.73 1.8 33.75 3 19.91 −3.91 8 11.91 1.5 41.81 4 24.73 −26.38 −0.825 25.555 1.98 30.33 5 19.91 −16.73 1.59 18.32 1.72 35.55 6 24.73 −24.73 0 24.73 1.94 31.02 7 19.91 −26.38 −3.235 23.145 1.88 32.13 8 19.91 −16.73 1.59 18.32 1.72 35.55 9 19.91 −24.73 −2.41 22.32 1.86 32.52 10 19.91 −26.38 −3.235 23.145 1.88 32.13 11 19.91 −19.91 0 19.91 1.78 34.18 12 19.91 −24.73 −2.41 22.32 1.86 32.52

5.13

Parameter Initial data line 1 2 3 4 5 x (m) 0.966 0.991 1.000 0.991 0.966 y (m) −0.259 −0.131 0.000 0.131 0.259 θ (deg) −15 −7.5 0 7.5 15 μ (deg) −0.265 −0.131 0.000 0.131 0.265 ν (deg) 26.38 26.38 26.38 26.38 26.38 M 2 2 2 2 2

5.15

(a) M2 = 2.1 (b) M3 = 2 ( / ) =− . ( / ) = . (c) dy dx RRW 0 536, dy dx LRW 0 582 (d) p3/p1 = 1 5.17

3.2 3 2.8 y 2.6 2.4 2.2 2 0 5 10 15 x 536 Partial Answers to Selected Problems

5.19

(a) M2 = 2.85 (b) M3 = 4.05 (c) A3/A1 = 6.645

Chapter 6

6.2

= ρ C ( − ) ≈ ρ C ( − ) (a) see text on p. 275: D D u2 u1 u2 dy V∞ D V∞ u2 dy 1 − = ρ( 2 − 2) ≈ ρ ∞( ∞ − ) (b) pt∞ pt2 2 V∞ u2 V V u2 (c) Using a Pitot rake the total pressure deﬁciency over the wake can be measured. 6.4 Notional pressure distributions:

- C c p a

b 0

0 x/l 1

6.6 (b) 2.40 (c) 1.69+ 1.76i (d) 2.23 6.8

(c)

6 (~)

min 5

4 be area, S/S u 3

Relative stream t 1 0 0.5 1 1.5 2 Mach number, M (~) Partial Answers to Selected Problems 537

6.10 16V x x 2 3 (a) S(x) = (4 − 4 ) 2 3lπ l l 2 3 ( ) = ( x − x ) 4 (b) R x Rmax 4 l 4 l 6.11

V = π 1 2( ) = π = π (a) 0 R x dx 30 and Rmax 16 . = . (b) CDwave 0 461 =− (c) CDwave 11 % 6.15 (a) 5.12, 3.47, 2.60 [m/s/m] (b) 5.13, 3.47, 2.60 [mN/m] (c) 10.3·10−6,6.94·10−6,5.20·10−6 6.16

x 10−3 3

2.9 (~) f 2.8

2.7

2.6

2.5 Friction coefficient, C

2.4 0 0.5 1 1.5 Mach number, M (~)

6.17

(a)

−0.4 (~) p

C −0.2

0 0 2 4 6 8 10 postion, x (m) 538 Partial Answers to Selected Problems

(b) (x/c)separation = 0.54 (c, d)

x 10−3 2 θ *

(m) δ

* 1 δ , θ

0 0 2 4 6 postion, x (m)

(e)

10 ) 2 5 (N/m w τ

0 0 2 4 6 postion, x (m)

6.18

(a) (x/c)separation = 0.72 (b) (x/c)separation = 0.54 (c) (x/c)separation = 0.30

6.23 + = /κ ( / ) + /κ 1 + = . Rewrite (6.121)tou 1 ln y k 1 ln 1/k + 0.3 B. Substitute B 5 0 and evaluate for k+ →∞. 6.24

(a) cf = 0.00320 (b) τw = 19.6N/m (∂ /∂ ) = . m/s (c) u y w 1 07 μm

6.25 6 (a) Rel = 6.70, 20.1, 67.0, 134, 335·10 (b) C f = 0.0145, 0.0136, 0.0129, 0.0126, 0.0123 Partial Answers to Selected Problems 539

(c)

x 10−3 3.5

3 f C

2.5

2 0 1 2 3 4 Re 8 l x 10

6.26

(a)

x 10−3 7 (−) f 6

3 Local friction coefficient, c 2 0 1 2 3 x−position (m)

(b)

x 10−3 4 (~) f 3.5

2.5 Friction coefficient, C

2 0 0.2 0.4 0.6 0.8 1 sand grain height, k (m) x 10−4 540 Partial Answers to Selected Problems

Chapter 7

7.1

Mdd increases from 0.80 to 0.83. 7.10

(a) M0m ISA = 0.197 √ (b) C ∗ =−27.7 pM = 2 (c) Vstall = 72.4m/s. Mstall = 0.217 √ (d) C ∗ =−22.8 pM = 2 √ (e) Since C ∗ > C it is likely that the suction peak is lower in reality than pM = 2 pmin we have anticipated here and that the maximum lift coefﬁcient is therefore also lower. 7.12 (a) 7.7km (b) 40◦ 7.14 (a) ϕ = 0.5433, 1.096, 1.783, 2.083 ms. (b) amplitude= 0.7505, 0.4064, 0.3068, 0.2990 (c) 1 de (~)

u 0.8

0.6 mp amplit u 0.4

Shock j 0.2 5 4 3 2 1 0

1.5

hase shift (rad) 0.5 P

0 5 4 3 2 1 0 dM /dx (1/m) 1

7.15 (a) ϕ = 7, 20, 40, 44 (deg). (b) amplitude= 0.44, 0.45, 0.32, 0.32 Partial Answers to Selected Problems 541

(c)

0.5 de (~)

u 0.45

0.4 mp amplit u 0.35 Shock j 0 20 40 60 80

10 hase shift (rad) P 5

0 0 20 40 60 80 frequency (Hz)

Chapter 8

8.2

(a) Mdd = 0.70 (b) Mdd = 0.77 8.6 As long as the drag divergence of the wing is governed by section characteristics ◦ (such as for wings with 35 sweep or less) a rather rough estimate of Mdd is possible with the method of (8.29), but that an upper limit exists for Mdd in the neighborhood of 0.95 caused by three-dimensional sonic-ﬂow conditions (onset of wave drag). This boundary cannot be altered by a further increase in sweep or changes in section other than by their contribution to a change in the longitudinal distribution of cross- sectional area. 8.13 (a) b = 30 m (b) croot = 9.6m,ctip = 2.4m ◦ (c) le = 32.5 (d) ρ = 0.412 kg/m3, μ = 14.7 · 10−6 kg/(ms) − = ctip croot + (e) c b y croot 542 Partial Answers to Selected Problems

(f)

250

200

150 (~) al θ

Re 100

0 0 10 20 30 y (m) 8.17 (a) θ =−3.33◦ (b) θ =−2.5◦ (c) θ =−3.31◦ (d) Most sensitive to changes in span (b). 8.18 (a) dθ/db =−1.45 deg/m (b) dθ/d(EI) = 7.15 · 10−6 deg/(Nm2) (c) dθ/d =−0.509 deg/deg (d) Most sensitive to changes in sweep angle (). ( / )4 = l bs 2 / bs = = ( / ) = 8.20 Use 8EI , L 2 4 k , bs 2 , and L lbs. 8.21

θ( ) = y m¯ η η (a) Use y 0 GJd

-0.5

-0.4

-0.3 (rad) θ -0.2 average

-0.1

0 0 0.5 1 y (m) Partial Answers to Selected Problems 543

/ (b) Use θ = 1 bs 2 θ(y)dy to ﬁnd θ = 1/24 rad average bs /2 0 average (c) Use kT θaverage =¯mbs/2 8.22 (b)

Λ = 0.6 0 15 30 0.4 45

0.2 ) g Λ = Λ 0 = (de 0 15 0 15 4530

α, θ, Γ −0.2 30 45

−0.4 α θ Γ

0 0.2 0.4 0.6 0.8 Mach number, M (~)

8.24 (b)

0.6

h = 0 0.4 3 6 9 )

g 0.2 (de h = 0 9 6

α, θ, Γ 3 h = 0 −0.2 9 6 3 0 −0.4 α θ Γ

0 0.2 0.4 0.6 0.8 Mach number, M (~) Glossary

Latin symbols

2 A Cross-sectional area or axial force√ (m ,N) a Ambient speed of sound (a = γ RT)(m/s2) ∼ Aspect ratio ( ) B Non-dimensional constant in law of the wake (6.109)(∼) b Wing span (m) C Characteristic curve in physical space (N/A) c Chord length, wave propagation speed, or specific heat capacity in a soild (m,m/s,J/kg/K) ca Section axial force coefficient (∼) CD Three-dimensional drag coefficient or dissipation coefficient (∼, ∼) cd Section drag coefficient (∼) = 1 l ∼ C f Friction coefficient C f c 0 c f dx ( ) c f Local friction coefficient (∼) CL Lift coefficient (∼) cl Section lift coefficient (∼) cm Section moment coefficient (∼) cn Section normal force coefficient (∼) C p Pressure coefficient (∼) cp Specific heat at constant pressure (J/kg/K) cv Specific heat at constant volume (J/kg/K) D Drag (N) E Young’s modulus (N/m2) e Internal energy, Oswald factor or normalized distance between elastic axis and aerodynamic center (J/kg, ∼, ∼) 3 Et Total energy (J/m ) F Fineness ratio of a body (∼) G Shear modulus (N/m2) g Gravitational constant (m/s2)

H Total enthalpy per unit mass, altitude, or shape factor (J/kg, m, ∼) h Enthalpy per unit mass or height (J/kg, m) I Second moment of area or mass moment of inertia (m4,kgm2) K Transonic similarity parameter or constant of integration (∼, rad) k Heat transfer coefficient, sand grain height, or spring stiffness (J/m/s/K, m, Nm/rad) k¯ Specific kinetic energy of turbulence (J/kg) L Lift (N) l Characteristic length (m) M Mach number (∼) m Moment (Nm) N Normal force or logarithm of maximum amplification factor in linear stability theory for laminar boundary layers (N, ∼) n Load factor (∼) P Steady-state pressure (N/m2) p Pressure (N/m2) Pr Prandtl number (Pr = 0.71 for air) (∼) Q Heat per unit mass or volume flow rate from a source (J/kg, m3/s) q Heat flux (J/s/m2) R Range, radius, or gas constant (R = 287.05 for air) (m, m, J/kg/K) r Radius or radius of gyration (m, m) ρVx Re Reynolds number Rex = μ (∼) S Surface area or stress parameter (m2, ∼) s Entropy or normalized static unbalance (J/kg/K, ∼) 2 Sij Rate of the mean strain tensor (1/s ) T Temperature (K) t Time or thickness (s, m) Tu Turbulence (%) U Velocity or steady-state velocity (m/s, m/s) u Excitation or cartesian velocity component (m, m/s) + + u Normalized velocity inside a√ boundary layer (u = u/uτ )(∼) uτ Wall friction velocity (uτ = τw/ρ)(m/s) V Velocity (m/s) V Volume (m 3) v Specific volume (v = 1/ρ) or cartesian velocity component (m3/kg, m/s) W Nondimensional velocity (W = V/Vmax)(∼) w Specific work, cartesian velocity component or vertical displacement of a beam (J/kg, m/s, m) x, y, z Cartesian coordinates (m) + + y Normalized distance from the wall inside a boundary layer (y = yuτ /ν) (m) Glossary 547

Greek Symbols

α Thermal diffusivity: α = k/(ρcp) or angle of attack (1/s, rad) β Prandtl-Glauert compressibility correction factor in subsonic flow (β = 2 1 − M∞), oblique shock wave angle, or angle between wall shear stress vector and inviscid streamline (∼, rad, rad) Circulation or dihedral angle (m2/s, rad) γ Ratio of specific heats (γ = 1.4 for air) (∼) δ Boundary layer thickness, shock thickness, cone angle, or flap deflection angle (m, m, rad, rad) δ∗ Displacement thickness (m) δij Kronecker delta (δij = 1ifi = j, δij = 0ifi = j)(∼) ε Dissipation rate of turbulent energy (J/m3/s) ζ Length scale (m) η Position on the semispan: η = 2y/b (∼) η, ξ, ζ Generalized coordinate system (∼) θ Local body angle, momentum thickness, or twist angle (rad, m, rad) κ Von Kármán constant: κ = 0.41 (∼) Sweep angle (rad) 2 λ Compressibility correction factor in supersonic flow (λ = M∞ − 1), 2 λ = θ due pressure gradient parameter in laminar boundary layer ( ν dx ), measure for local friction coefficient in a turbulent boundary layer (λ = 2/c f ), wing taper ratio (λ = cT /cC ), or temperature lapse rate (λ = −0.0065 K/m in troposphere of ISA) (∼, ∼, ∼, ∼,K/m) μ Dynamic (shear) viscosity or Mach angle (kg/m/s, rad) μ Bulk (volume) viscosity (kg/m/s) μ¯ Normalized mass (μ¯ = m/ρv)(∼) ν Kinematic viscosity coefficient (ν = μ/ρ) or Prandtl-Meyer angle (m2/s, rad) ξ Flap deflection angle (rad) Coles’ wake parameter (∼) ρ Density (kg/m3) σ Conical shock angle (rad) τ Shear stress or thickness ratio (N/m2, ∼) Φ Velocity potential, azimuth angle, or dissipation function (m2/s, rad, J/s) φ Span efficiency factor or perturbed velocity potential (∼,m2/s) ˆ ˆ φ Disturbance velocity potential (φ = φ/V∞)(m) ϕ Rotation of a beam (rad) ω Frequency (rad/s) Vectors ∇ ∇ ≡ ∂ + ∂ + ∂ Differential operator, ∂x i ∂y j ∂z k (1/m) ζ Vorticity: ζ = ∇ × V (1/s) τ Shear stress tensor (N/m2) 548 Glossary

E, F, G Steady vector terms in the governing equations (N/A) f Body force per unit mass (N/kg) F Force acting on a body (N) g Gravitational acceleration constant (m/s2) i, j, k Orthogonal unit item in cartesian coordinate system (∼) n Surface normal vector (∼) q Heat transfer per unit area (J/s/m2) S Surface vector (m2) U Unsteady vector term in governing equations (N/A) V Velocity vector (m/s) v Perturbation velocity vector (V∞v = V − V∞i)(∼) Subscripts

0 Stagnation ∞ Free stream conditions α Angle of attack θ Polar angle Φ Azimuth angle al Attachment line aw Adiabatic wall b Bending C Center crit Critical condition, i.e. ML = 1 d Drag dd Drag divergence e Boundary layer edge f Friction or ﬂutter h Horizontal tailplane or bending i, j Indices: 1,2,3 i Incomnpressible or induced k Kinematic L local l Lower or lift LE Leading edge m Moment min Minimum N Nose n Normal p Pressure r Radius s Surface, static or structural t Total T Turbulent, thermal, tip, or torsion th Thickness or throat Glossary 549 tr Transition turb Turbulent u Upper w Wall, wave, or wing Wet Wetted Supscripts

* Critical condition, i.e. ML = 1 + Normalized distance or speed within the boundary layer Acronyms AGARD Advisory Group for Aerospace Research and Development AIAA American Institute of Aeronautics and Astronautics CFD Computational Fluid Dynamics CS Certiﬁcation Speciﬁcation DC Douglas Company DNS Direct Numerical Simulations DoF Degrees of Freedom DSMA Douglas Santa Monica Airfoil ETW European Transonic Windtunnel FAR Federal Aviation Regulation FTC Fundamental Theorem of Calculus ISA International Standard Atmosphere KD Kinetic Diagram LCO Limit Cycle Oscillation LHS Left Hand Side LRW Left Running Wave MD McDonnell Douglas MiG Mikoyan-and-Gurevich MLN Minimum Length Nozzle MOC Method of Characteristics NACA National Advisory Committee for Aeronautics NASA National Aeronautics and Space Administration NLF Natural Laminar Flow NLR Nationaal Lucht-en Ruimtevaartlaboratorium NS Navier-Stokes NTF National Transonic Facility ODE Ordinary Differential Equation P-M Prandtl-Meyer PDE Partial Differential Equation RAE Royal Aircraft Establishment RAF Royal Air Force RANS Reynolds Averaged Navier Stokes Equations RHS Right Hand Side 550 Glossary

RMS Root Mean Square ROM Reduced Order Model RRW Right Running Wave SC Supercritical SFC Speciﬁc Fuel Consumption SI Système International SWBLI Shock Wave Boundary Layer Interaction USAF United States Air Force WWII World War II Index

A B Adiabatic, 46 Blasius equation, 327 Aerodynamic Boundary layer, 303 center, 482 blending layer, 333 damping, 492 confluence, 404 stiffness, 492 crossflow, 461 twist, 476 laminar, 303 Aeroelastic merging layer, 333 penalty, 489 overlap layer, 333 tailoring, 476 separation, 302, 304, 470 Aeroelastic penalty, 475 shock wave interaction, 319 Aeroelasticity, 474 thickness, 305, 306 dynamic, 474 transition, 343, 372, 464 static, 474 turbulent, 313 Aeroservoelasticity, 495 Boussinesq assumption, 67 Aft loading, 376 Bréguet range formula, 2 Aileron Buffet, 405 high-speed, 488 onset boundary, 405 reversal, 481 transonic, 405, 412 Airfoil, 368 bi-convex, 173, 198 bi-convex parabolic, 200 C circular-arc, 173 Calorically-prefect gas, 150 hexagonal, 198, 200 Camber, 369 laminar flow, 372 line, 369 natural-laminar-flow, 374 Canonical form, 35 ‘peaky’, 382 Center of mass, 491 shock-Free, 379 Characteristic sonic rooftop, 390 coordinates, 24 supercritical, 375 curves, 35 Angle-of-attack Characteristic Mach number, 150, 151 effective, 476 Characteristics, 211 Area rule, 12 Chord supersonic, 13 length, 369 transonic, 288 Clauser’s equilibrium parameter, 335 Attachment line, 465 Coles, 333 instability, 465 Coles’ wake parameter, 333, 362 © Springer Science+Business Media Dordrecht 2015 551 R. Vos and S. Farokhi, Introduction to Transonic Aerodynamics, Fluid Mechanics and Its Applications 110, DOI 10.1007/978-94-017-9747-4 552 Index

Compressible laminar boundary layer, 308 E Conical flowfield, 163 Eigenfunction, 32 Conical shock, 146, 163 Eigenvalue, 32 Conical shock charts, 164 Energy Conservation form, 33 equation, 56 Conservation principle, 52 kinetic, 56 Conservative form, 33 specific, 49 Constant of motion, 149–151, 155, 162 Enthalpy, 44 Continuity equation, 51, 52 specific, 49 Coordinated turn, 386 Entropy, 48 Critical Mach number, 4 Equation of state, 44 Critical pressure coefficient, 9 Equations of fluid motion, 51 Crocco’s eqaution, 70 Equations of motion, 33 Equilibrium thickness, 147 Crocco’s method, 309 Eulerian approach, 34 Crocco’s theorem, 42 Crossflow, 461 instability, 468 F vortex, 468 Fatigue, 475 Curl, see vector field First law of thermodynamics, 46 Flap leading-edge, 403 D trailing-edge, 403 D’Alembert, 22, 24 Flow Density, 44 reversal, 310 Design condition, 390 reversed, 312 Determinant, 27 Flutter, 491 Direct numerical simulations, 62 hard, 498 Displacement thickness, 71, 307 soft, 498 Dissipation function, 58 speed, 495 Form drag, 458 Divergence, see vector field Fourier series, 29 Divergence theorem, 42, 78 sine series, 32 Domain of dependence, 28, 217 Fourier’s law, 57 Dorsal fin, 474 Fourier’s law of heat conduction, 30 Drag Friction coefficient, 303, 305 bucket, 374 Froude, 317 counts, 374 Full potential equation, 74 creep, 3 Fundamental theorem in calculus, 27, 30 divergence, 3, 324, 435 Fundamental Theorem of Calculus, 27 friction, 302 Fuselage inteference, 349 “coke bottle”, 300 interference, 349 pressure, 169 rise, 354 G wave, 169, 287, 290, 291 Görtler vortices, 469 Drag coefficient, 18 Gas constant, 23, 44 induced, 18 Gauss, 42 zero-lift, 18 Geometric twist, 476 Drag divergence, 375 Glauert Drag divergence Mach number, 3, 375, 377 variable, 297, 359 Dynamic effect, 419 Gradient, see scalar field Dynamically scaled model, 497 Gradient theorem, 43 Index 553

H Mach angle, 146, 157 Heat equation, 29, 34 Mach cone, 161, 183 Helmbold equation, 485 Mach number, 51 High-lift devices, 403 characteristic, 150, 151 Hirst, 333 critical, 399, 435 Hydraulically rough surface, 339 drag divergence, 3, 435 Hydraulically smooth surface, 339 effect on maximum lift coefficient, 399 maximum local ∼ over convex profile, 402 I Mach tuck, 422 Instability Mach wave, 146 centrifugal, 469 Maximum take-off weight, 18 Interaction Mean-free path, 147 strong, 321, 323 Mixing length, 67 weak, 321 Momentum integral equation, 327 Internal energy, 44 Momentum theory, 291 Isentropic, 49 Momentum thickness, 307 recompression, 380 Isentropic process, 48 Isentropic relations, 50 N Isobaric process, 46 Natural frequency, 474 Isothermal process, 46 Navier-Stokes equations, 22, 61 Near-field analysis, 376 Newton, 53, 305 K Newtonian fluid, 55 Küchemann, 13 Nonconservative form, 34 Küchemann carrots, 301 Normal shock Küchemann tip, 457 thickness, 149 Kelvin-Stokes theorem, 43 Kronecker delta, 55 Kutta condition, 391 O Oblique flow, 430 Oblique shock charts, 157 L Ogive nose, 188 Laminar separation bubble, 395 Order of PDE, 23 Laminar sublayer, 331 Oswald factor, 446 Laminar-flow control, 343 Outer product, 39 Law of the wake, 333 Law of the wall, 333 Leading edge P radius, 369 Partial differential equations, 22 Leading-edge contamination, 465 elliptic, 35 Leading-edge root extensions, 474 hyperbolic, 35 Leading-edge vortex, 472 parabolic, 35 Left-running waves, 165 Perfect gas law, 44 Limit cycle oscillation, 491 Poiseuille, 305 Linear momentum equation, 53 Polytropic compression, 46 Linear stability theory, 346 Power law, 315 Load alleviation, 495 Prandtl’s relation, 151, 152 Prandtl-Meyer expansion theory, 169 Prandtl-Meyer expansion waves, 147, 168, M 175 Mach Prandtl-Meyer function, 168 tuck, 386 Pressure, 44 554 Index

Pressure coefficient, 282 lambda, 323 minimum, 401 oblique, 155, 160 Pressure distribution, 282 recompression, 319 Pressure gradient thickness, 148, 149 adverse, 304, 312 wave, 147 favorable, 310 Shock wave unfavorable, 312 boundary layer interaction, 319 Principle of stationary local Mach numbers, Shock-expansion theory, 145, 200 192 Slat, 403 Slip line, 323 Slipstream, 175 R Sonic domain, 4 Range parameter, 2 Sonic line, 321, 379 Ratio of specific heats, 23 Sound pressure, 23 Rayleigh circulation criterion, 469 Spalding, 334 Receptivity, 468 Span efficiency factor, 446, 454 Reflection Specific fuel consumption, 2 from a boundary of constant pressure, Specific heat 379 constant pressure, 44 from a solid boundary, 380 constant volume, 44 Region of influence, 36, 217, 282, 286 ratio of ∼, 44 Reversible process, 48 Stability Reynolds decomposition, 62 aircraft longitudinal, 480 Reynolds number, 51 Stall, 303, 304, 393 critical, 313, 344, 345 high-speed, 405 cut-off, 340 hysteresis, 393 effect on maximum lift coefficient, 397 leading-edge, 395 transition, 345 low-speed, 393 Reynolds stress, 63 qualification of ∼, 394 Reynolds-Averaged Navier-Stokes equa- thin-airfoil, 395 tions, 62 trailing-edge, 396 Right-running waves, 165 State principle of thermodynamics, 44 Root effect, 454, 499 State property, 48 Roughness, 338 Static unbalance, 491 Runge-Kutta, 163 Stokes flow, 339 Runge-Kutta method, 337 Stokes’ theorem, see Kelvin-Stokes theorem Strakes, 474 S Strong interaction, 321 Scalar field, 38 Strong solution, 157 gradient of a ∼, 40 Subsonic domain, 4 Sears-Haack body, 298 Substantial derivative, 52 Second law of motion, 53 Supercritical condition, 376 Second law of thermodynamics, 48 Supersonic Separation, 470 bubble, 379 shock-induced, 406 Supersonic domain, 4 bubble, 323 Supersonic tongue, 322 point, 312 Sutherland’s equation, 45 trailing edge, 324 Sweep, 428 Shadow zone, 184 theory, 430, 431 Shock bow, 5 conical, 161, 163, 183 T detached, 160 Tangent ogive, 188 Index 555

Taper, 432 curl of a, 42 Taylor-Maccoll equation, 162, 163 divergence of a, 41 Temperature, 44 Velocity Thermodynamic pressure, 54 distribution in boundary layer, 331 Thermodynamics, 43 gradient, 305 Thickness, 369 Velocity potential function, 41 to-chord ratio, 369 Viscosity, 55 Three-dimensional relieving effect, 163, 165 bulk, 55 Thwaites’ method, 328 dynamic, 305 Tip effect, 454, 499 kinematic, 306 Tollmien Schlichting waves, 464 second coefficient of ∼, 55 Trailing edge Viscous sublayer, 331 thickness, 390 Von Kármán Transition, 343, 395, 464 constant, 333 Transonic, 1 momentum integral relation, 327 Transonic dip, 493 ogive, 299 Transonic flow definition, 4 Trefftz plane, 293 Triple point, 323 W Tuck under, 386, 422 Wave cancelation, 223 Turbulence Wave envelope, 168 spots, 313 Wave equation, 22 Weak interaction, 321 V Weak solution, 157 Vector Wetted area, 302 addition, 38 Whitcomb, 288 cross product, 39 area rule, 288 direction, 38 Wing dot product, 38 oblique, 434 inner product, 38 sheared, 434 magnitude, 38 yawed, 434 product, 39 scalar product, 38 subtraction, 38 Z Vector algebra, 38 Zone of action, 154, 155 Vector field, 38, 40 Zone of silence, 154, 155