Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last and first name: ID:10770656 Page 1/2 Academic year 2020/2021: 14/6/2021 Compressible-Fluid Dynamics Exercise 1 A free jet of air (ideal polytropic gas with g = 1.4 and R = 287 J/(Kg K)) enters a reservoir with infinite capacity. The following conditions are observed in the
Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last and first name: ID:10770656 Page 1/2 Academic year 2020/2021: 14/6/2021 Compressible-Fluid Dynamics Exercise 1 A free jet of air (ideal polytropic gas with g = 1.4 and R = 287 J/(Kg K)) enters a reservoir with infinite capacity. The following conditions are observed in the
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Last and first name: ID:10770656 Page 1/2 Academic year 2020/2021: 14/6/2021 Compressible-Fluid Dynamics Exercise 1 A free jet of air (ideal polytropic gas with g = 1.4 and R = 287 J/(Kg K)) enters a reservoir with infinite capacity. The following conditions are observed in the jet at the inlet: r1 = 1.4 kg/m3, u1 = 130 m/s, Et1 = 433720 J/m3 where r is the density, u is the velocity and Et is the total energy per unit volume. The reference value of the internal energy e(T) is e(0 K) = 0 J/kg. A steady condition is established whereby the fluid exits the reservoir at a velocity u2 of 235 m/s. Assuming that the reservoir walls are adiabatic, and that no work is introduced in the fluid system, compute the Mach number M1 at the inlet and the temperature T2 at the outlet. Intermediate results Answers A1.1) M1 = 0.31646 A1.2) T2 = 127.77 °C 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 pressure reads: P(x,t) = 107000 + 149.8 (x + 438.25 t)1/2 Pa Compute the value of the velocity u and the perturbation density r -r0 at x = 6.0 m and t = 0.7 s, where r0 is the unperturbed value of the density. Intermediate results Answers A2.1) u(6.0, 0.7) = -7.7506 m/s A2.2) r(6.0, 0.7) - r0 = 0.013794 kg/m3 Exercise 3 !" $ !"! Consider the following inviscid Burgers equation + = 0, for the scalar unknown u, where x is the spatial coordinate and t is the time. !# % !& The initial data is u(x,0) = 3.70 for x ≤0 u(x,0) = 3.70 - 2.10x for 0 < x < 1 u(x,0) = 1.60 for x≥1 Compute the value of the solution u at x = 1.1548 and t = 0.1905, at x = 2.4667 and t = 0.8086 and at x = 2.8190 and t = 0.8086. Intermediate results Answers A3.1) u(1.1548, 0.1905) = 2.13 A3.2) u(2.4667, 0.8086) = 3.70 A3.3)…
Last name: First name: ID:10778438 Pag. 2/2 Exercise 4 Consider a straight-axis pipe filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = 1.5 bar and temperature T0 = 21 °C. The pipe has a circular, constant cross-sectional area. The pipe is 2 m long and it is closed at the right end B. On the left end A, a piston is set impulsively into motion towards the right with velocity VA = 135 m/s. Neglecting the effects of viscosity and thermal conductivity and assuming one-dimensional flow, compute the pressure PB and the density rB on the right end B after the wave generated by the piston interacted with the end wall B. Intermediate results Answers A4.1) PB = 414.7 kPa A4.2) rB = 3.646 kg/m3 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 = 14.0 bar and T0 = 38.0 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.5. The throat area is At = 19.0 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the mass-flow rate m discharged by the nozzle for an ambient pressure Pa = 13.7259 bar and the maximum mass-flow mmax discharged by the nozzle. Intermediate results Answers A5.1) Pa,adapted = 0.89582 bar A5.2) m = 4.3557 kg/s A5.3) mmax = 6.0950 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) experiences M1, P1 a gradual expansion as in the picture. The first Mach wave has a slope with respect to the flow of δ1 = δ1 25° 43.603° and the last one has δ2 = 27.718°. Compute the upstream Mach number M1, the downstream 1 Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure before the expansion is P1 = 0.40 bar, compute…
Last name: First name: ID:10484374 Pag. 2/2 Exercise 4 Consider a straight-axis pipe filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = 1.8 bar and temperature T0 = 31 °C. The pipe has a circular, constant cross-sectional area. The pipe is 2 m long and it is closed at the right end B. On the left end A, a piston is set impulsively into motion towards the right with velocity VA = 155 m/s. Neglecting the effects of viscosity and thermal conductivity and assuming one-dimensional flow, compute the pressure PB and the density rB on the right end B after the wave generated by the piston interacted with the end wall B. Intermediate results Answers A4.1) PB = 560.9 kPa A4.2) rB = 4.595 kg/m3 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 = 10.0 bar and T0 = 46.0 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.4. The throat area is At = 15.0 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the mass-flow rate m discharged by the nozzle for an ambient pressure Pa = 9.7867 bar and the maximum mass-flow mmax discharged by the nozzle. Intermediate results Answers A5.1) Pa,adapted = 0.68551 bar A5.2) m = 2.4268 kg/s A5.3) mmax = 3.3937 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) experiences M1, P1 a gradual expansion as in the picture. The first Mach wave has a slope with respect to the flow of δ1 = δ1 25° 38.682° and the last one has δ2 = 25.184°. Compute the upstream Mach number M1, the downstream 1 Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure before the expansion is P1 = 0.90 bar, compute the…
Last name: First name: ID:10535321 Pag. 2/2 Exercise 4 Consider a straight-axis pipe filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = 1.2 bar and temperature T0 = 23 °C. The pipe has a circular, constant cross-sectional area. The pipe is 2 m long and it is closed at the right end B. On the left end A, a piston is set impulsively into motion towards the right with velocity VA = 150 m/s. Neglecting the effects of viscosity and thermal conductivity and assuming one-dimensional flow, compute the pressure PB and the density rB on the right end B after the wave generated by the piston interacted with the end wall B. Intermediate results Answers A4.1) PB = 366.5 kPa A4.2) rB = 3.103 kg/m3 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 = 7.0 bar and T0 = 36.0 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.4. The throat area is At = 16.0 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the mass-flow rate m discharged by the nozzle for an ambient pressure Pa = 6.8507 bar and the maximum mass-flow mmax discharged by the nozzle. Intermediate results Answers A5.1) Pa,adapted = 0.47986 bar A5.2) m = 1.8411 kg/s A5.3) mmax = 2.5746 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) experiences M1, P1 a gradual expansion as in the picture. The first Mach wave has a slope with respect to the flow of δ1 = δ1 25° 50.285° and the last one has δ2 = 25.771°. Compute the upstream Mach number M1, the downstream 1 Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure before the expansion is P1 = 0.50 bar, compute the…
Last name: First name: ID:10580521 Pag. 2/2 Exercise 4 Consider a straight-axis pipe filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = 1 bar and temperature T0 = 15 °C. The pipe has a circular, constant cross-sectional area. The pipe is 2 m long and it is closed at the right end B. On the left end A, a piston is set impulsively into motion towards the right with velocity VA = 120 m/s. Neglecting the effects of viscosity and thermal conductivity and assuming one-dimensional flow, compute the pressure PB and the density rB on the right end B after the wave generated by the piston interacted with the end wall B. Intermediate results Answers A4.1) PB = 251.1 kPa A4.2) rB = 2.321 kg/m3 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 = 7.0 bar and T0 = 36.0 °C. The nozzle has an exhaust-to-throat area ratio Ae / At = 2.6. The throat area is At = 11.0 cm2. Determine the ambient pressure in adapted supersonic conditions Pa,adapted, the mass-flow rate m discharged by the nozzle for an ambient pressure Pa = 6.8737 bar and the maximum mass-flow mmax discharged by the nozzle. Intermediate results Answers A5.1) Pa,adapted = 0.41942 bar A5.2) m = 1.2632 kg/s A5.3) mmax = 1.7700 kg/s Exercise 6 A steady, parallel supersonic flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) experiences M1, P1 a gradual expansion as in the picture. The first Mach wave has a slope with respect to the flow of δ1 = δ1 25° 56.443° and the last one has δ2 = 30.000°. Compute the upstream Mach number M1, the downstream 1 Mach number M2 and the flow deflection |θ| (absolute value). Assuming that the pressure before the expansion is P1 = 0.10 bar, compute the…
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