Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last name: First name: ID: 10191772 Pag. 1/2 Academic year 2019/2020: 8/9/2020 Compressible-Fluid Dynamics Exercise 1 Consider a symmetric airfoil at zero incidence moving at a uniform velocity V in standard air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)). The freestream
Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last name: First name: ID: 10191772 Pag. 1/2 Academic year 2019/2020: 8/9/2020 Compressible-Fluid Dynamics Exercise 1 Consider a symmetric airfoil at zero incidence moving at a uniform velocity V in standard air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)). The freestream
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Last name: First name: ID: 10191772 Pag. 1/2 Academic year 2019/2020: 8/9/2020 Compressible-Fluid Dynamics Exercise 1 Consider a symmetric airfoil at zero incidence moving at a uniform velocity V in standard air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)). The freestream air pressure and temperature are P¥ = 0.4 atm, T¥ = 40 °C Assuming that the Euler equations can be used to describe the external flow properties, compute the pressure P0 at the stagnation point for the following aifoil velocities V V1 = 141.89 m/s, V2 = 283.77 m/s, V3 = 496.60 m/s, Answer: P0,1 = 0.44662 atm P0,2 = 0.610 atm P0,3 = 1.220 atm Exercise 2 In a one-dimensional compressible flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) under the acoustic approximation, the velocity reads: u(x,t) = - 0.039492 sin (2.22p (x + 394.92 t)) m/s Assuming that the unperturbed value of the pressure is P0 = 1 atm, compute the value of the perturbation pressure P - P0 at x = 9 m and t = 0.002 s. Compute the acoustic wave intensity FE and the maximum particle displacement xM. Answer: P(9, 0.002) - P0 = -9.8660 Pa FE = 114.473 dB xM = 2.86479E-05 mm Exercise 3 Consider the following PDE for the scalar unknown f, !" !" !# + 𝑦 !$ + 4𝑦 % = 0 where x and y are the spatial coordinates. Compute the value of f at x = 10 and y = 11 assuming the initial data f (x,1) = f0(x) = x2 + 3. Answer: f(10, 11) = -179.21 Exercise 4 Consider standard nitrogen (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 2 °C and pressure P0 = 1.9 atm. The gas is at rest in a tube with a circular, constant cross-sectional area. The tube is closed at one end by a moveable piston. The piston is set into motion impulsively at a speed of V1 = 249.374 m/s outwards. Compute the minimum value of the…
Last name: First name: ID: 10500373 Pag. 1/2 Exercise 5 A converging-diverging nozzle nozzle is fed by a reservoir containing air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) P0 = 9.00 bar and T0 = 46.00 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.40. The throat area is At = 11.00 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the massflow rate m discharged by the nozzle for an ambient pressure Pa = 8.8081 bar and the maximum massflow mmax discharged by the nozzle. Answer: Pa,adapted = 6.1696E-01 bar m = 1.60 kg/s mmax = 2.27 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 M1, P1 and R = 287 J/(Kg K)) experiences a gradual expansion as in the δ1 25° picture. The first Mach wave has a slope with respect to the flow of 1 δ1 = 36.0319° and the last one has δ2 = 24.6243°. Compute the upstream Mach number M1, the downstream Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure δ2 5° before the expansion is P1 = 0.50 bar, compute the pressure M2, P2 coefficient cP2 after the expansion, taking state 1 as reference. θ Compute the value of the pressure the pressure coefficient cP2,LIN after the expansion according to the linearized theory, taking state 1 as reference. Answer: M1 = 1.70 M2 = 2.400 |θ| = 18.937° cP2 = -0.32743 bar cP2,LIN = -0.48082 bar
Last name: First name: ID: 10527249 Pag. 1/2 Exercise 5 A converging-diverging nozzle nozzle is fed by a reservoir containing air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) P0 = 15.00 bar and T0 = 40.00 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.30. The throat area is At = 18.00 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the massflow rate m discharged by the nozzle for an ambient pressure Pa = 14.6499 bar and the maximum massflow mmax discharged by the nozzle. Answer: Pa,adapted = 1.1055E-00 bar m = 4.46 kg/s mmax = 6.25 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 M1, P1 and R = 287 J/(Kg K)) experiences a gradual expansion as in the δ1 25° picture. The first Mach wave has a slope with respect to the flow of 1 δ1 = 43.6028° and the last one has δ2 = 26.3878°. Compute the upstream Mach number M1, the downstream Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure δ2 5° before the expansion is P1 = 0.90 bar, compute the pressure M2, P2 coefficient cP2 after the expansion, taking state 1 as reference. θ Compute the value of the pressure the pressure coefficient cP2,LIN after the expansion according to the linearized theory, taking state 1 as reference. Answer: M1 = 1.45 M2 = 2.250 |θ| = 22.580° cP2 = -0.47872 bar cP2,LIN = -0.75067 bar
Last name: First name: ID: 10568134 Pag. 1/2 Exercise 5 A converging-diverging nozzle nozzle is fed by a reservoir containing air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) P0 = 10.00 bar and T0 = 38.00 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.20. The throat area is At = 19.00 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the massflow rate m discharged by the nozzle for an ambient pressure Pa = 9.7435 bar and the maximum massflow mmax discharged by the nozzle. Answer: Pa,adapted = 7.9546E-01 bar m = 3.22 kg/s mmax = 4.41 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 M1, P1 and R = 287 J/(Kg K)) experiences a gradual expansion as in the δ1 25° picture. The first Mach wave has a slope with respect to the flow of 1 δ1 = 36.0319° and the last one has δ2 = 29.1964°. Compute the upstream Mach number M1, the downstream Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure δ2 5° before the expansion is P1 = 0.30 bar, compute the pressure M2, P2 coefficient cP2 after the expansion, taking state 1 as reference. θ Compute the value of the pressure the pressure coefficient cP2,LIN after the expansion according to the linearized theory, taking state 1 as reference. Answer: M1 = 1.70 M2 = 2.050 |θ| = 9.938° cP2 = -0.20584 bar cP2,LIN = -0.25235 bar
Last name: First name: ID: 10755485 Pag. 1/2 Exercise 5 A converging-diverging nozzle nozzle is fed by a reservoir containing air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) P0 = 11.00 bar and T0 = 48.00 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.50. The throat area is At = 16.00 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the massflow rate m discharged by the nozzle for an ambient pressure Pa = 10.7846 bar and the maximum massflow mmax discharged by the nozzle. Answer: Pa,adapted = 7.0386E-01 bar m = 2.74 kg/s mmax = 4.02 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 M1, P1 and R = 287 J/(Kg K)) experiences a gradual expansion as in the δ1 25° picture. The first Mach wave has a slope with respect to the flow of 1 δ1 = 56.4427° and the last one has δ2 = 30.0000°. Compute the upstream Mach number M1, the downstream Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure δ2 5° before the expansion is P1 = 0.10 bar, compute the pressure M2, P2 coefficient cP2 after the expansion, taking state 1 as reference. θ Compute the value of the pressure the pressure coefficient cP2,LIN after the expansion according to the linearized theory, taking state 1 as reference. Answer: M1 = 1.20 M2 = 2.000 |θ| = 22.822° cP2 = -0.68460 bar cP2,LIN = -1.20095 bar
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