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: 1/7/2020 Compressible-Fluid Dynamics Exercise 1 Consider a free jet of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) entering a reservoir with infinite capacity. The following conditions are observed
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: 1/7/2020 Compressible-Fluid Dynamics Exercise 1 Consider a free jet of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) entering a reservoir with infinite capacity. The following conditions are observed
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Last name: First name: ID: 10191772 Pag. 1/2 Academic year 2019/2020: 1/7/2020 Compressible-Fluid Dynamics Exercise 1 Consider a free jet of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) entering a reservoir with infinite capacity. The following conditions are observed in the jet at the inlet: r1 = 1.2 kg/m3, u1 = 140 m/s, Et1 = 390600 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 240 m/s. Assuming that the reservoir walls are adiabatic and no work occurs in the process, compute the Mach number M1 at the inlet and the temperature T2 at the outlet. Answer: M1 = 0.33296 T2 = 147.935 °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) = 103000 + 144.2 sin (2p (x + 409.82 t)) Pa Compute the value of the velocity u and the perturbation density r -r0 at x = 8 s and t = 8 m, where r0 is the unperturbed value of the density. Answer: u(8, 8) = 1.50107E-01 m/s r(8, 8) - r0 = -3.1448E-04 kg/m3 Exercise 3 Consider the following inviscid Burgers equation 𝜕𝑢 1 𝜕𝑢! + =0 𝜕𝑡 2 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time. The initial data is u(x,0) = 3.3 for x ≤0 u(x,0) = 3.3 - 1.5x for 0 < x < 1 u(x,0) = 1.8 for x≥1 compute the value of the solution u at x = 1.3300 and t = 0.2667, at x = 3.0800 and t = 1.0980 and at x = 3.5200 and t = 1.0980. Answer: u(1.3300, 0.2667) = 2.18 u(3.0800, 1.0980) = 3.30 u(3.5200, 1.0980) = 1.80 Exercise 4 Consider a tube filled with still air (polytropic gas with g = 1.4 and R = 287…
Last name: First name: ID: 10532551 Pag. 1/2 Academic year 2019/2020: 1/7/2020 Compressible-Fluid Dynamics Exercise 1 Consider a free jet of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) entering a reservoir with infinite capacity. The following conditions are observed in the jet at the inlet: r1 = 1.2 kg/m3, u1 = 130 m/s, Et1 = 406200 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 230 m/s. Assuming that the reservoir walls are adiabatic and no work occurs in the process, compute the Mach number M1 at the inlet and the temperature T2 at the outlet. Answer: M1 = 0.30238 T2 = 168.931 °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) = 102000 + 142.8 sin (2p (x + 402.40 t)) Pa Compute the value of the velocity u and the perturbation density r -r0 at x = 6 s and t = 6 m, where r0 is the unperturbed value of the density. Answer: u(6, 6) = -2.41980E-01 m/s r(6, 6) - r0 = 5.3032E-04 kg/m3 Exercise 3 Consider the following inviscid Burgers equation 𝜕𝑢 1 𝜕𝑢! + =0 𝜕𝑡 2 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time. The initial data is u(x,0) = 3.2 for x ≤0 u(x,0) = 3.2 - 1.6x for 0 < x < 1 u(x,0) = 1.6 for x≥1 compute the value of the solution u at x = 1.2500 and t = 0.2500, at x = 2.8000 and t = 1.0417 and at x = 3.2000 and t = 1.0417. Answer: u(1.2500, 0.2500) = 2.00 u(2.8000, 1.0417) = 3.20 u(3.2000, 1.0417) = 1.60 Exercise 4 Consider a tube filled with still air (polytropic gas with g = 1.4 and R = 287…
Last name: First name: ID: 10586326 Pag. 1/2 Exercise 5 A reservoir with infinite capacity is filled with carbon dioxide at uniform temperature T0 = 23.5 °C and pressure P0 = 2 atm. Carbon dioxide can be assumed to be an ideal polytropic gas with molecular weight µ = 44.01 g/mol and specific-heat ratio g = 1.297. The fluid discharges isentropically through an orifice with a diameter of 1 mm into an ambient at uniform pressure Pa. At the exit section, compute: a) the mass flux m, the exit pressure Pe and the exit velocity ue if the ambient pressure is Pa = 1.2287 atm. b) the mass flux m, the exit pressure Pe and the exit velocity ue if the ambient pressure is Pa = 0.5463 atm. Answer: a) m = 4.4342E-04 kg/s Pe = 124.49 kPa ue = 227.32 m/s b) m = 4.4826E-04 kg/s Pe = 110.70 kPa ue = 251.57 m/s Exercise 6 Consider a symmetric airfoil, in which the upper and lower surfaces are described by the equations Y+(x) = 0.34 x (1-x) and Y-(x) = - 0.34 x (1-x), respectively, with 0 ≤ x ≤ 1. The airfoil is at zero angle of attack in a uniform flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) with free-stream Mach number M∞= 1.65 and free-stream pressure P∞ = 0.5 kPa and temperature T∞ = 250 K. Using the linearized theory, compute the drag coefficient CD of the airfoil and the flow velocity u in the x direction and the pressure P at points x = 0.7, y = 0.17 and x = 0.6, y = -0.17. Answer: CD = 1.1744E-01 u(0.7, 0.17) = 546.03 m/s P(0.7, 0.17) = 4.1587E-01 kPa u(0.6, -0.17) = 519.80 m/s P(0.6, -0.17) = 5.1147E-01 kPa Exercise 7 The obstacle in the figure is placed in a uniform flow with M∞ P∞ free-stream Mach number M∞= 1.5 and free-stream 2 pressure P∞ = 60 kPa. The angle between the ramp and the 1 free-stream velocity is q = 6.5°. Compute the Mach number M and the…
Last name: First name: ID: Pag. 1/2 Academic year 2019/2020: 1/7/2020 Compressible-Fluid Dynamics Exercise 1 Consider a free jet of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) entering a reservoir with infinite capacity. The following conditions are observed in the jet at the inlet: r1 = 1.1 kg/m3, u1 = 100 m/s, Et1 = 321200 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 200 m/s. Assuming that the reservoir walls are adiabatic and no work occurs in the process, compute the Mach number M1 at the inlet and the temperature T2 at the outlet. Answer: M1 = 0.24944 T2 = 111.917 °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) = 100000 + 140 sin (2p (x + 387.13 t)) Pa Compute the value of the velocity u and the perturbation density r -r0 at x = 0 s and t = 0 m, where r0 is the unperturbed value of the density. Answer: u(0, 0) = 0.00000E+01 m/s r(0, 0) - r0 = 0.0000E+01 kg/m3 Exercise 3 Consider the following inviscid Burgers equation 𝜕𝑢 1 𝜕𝑢! + =0 𝜕𝑡 2 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time. The initial data is u(x,0) = 3 for x ≤0 u(x,0) = 3 - 2x for 0 < x < 1 u(x,0) = 1 for x≥1 compute the value of the solution u at x = 1.0500 and t = 0.2000, at x = 2.1000 and t = 0.8750 and at x = 2.4000 and t = 0.8750. Answer: u(1.0500, 0.2000) = 1.50 u(2.1000, 0.8750) = 3.00 u(2.4000, 0.8750) = 1.00 Exercise 4 Consider a tube filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform…
Last name: First name: ID: Pag. 1/2 Exercise 5 A reservoir with infinite capacity is filled with carbon dioxide at uniform temperature T0 = °C and pressure P0 = atm. Carbon dioxide can be assumed to be an ideal polytropic gas with molecular weight µ = 44.01 g/mol and specific-heat ratio g = 1.297. The fluid discharges isentropically through an orifice with a diameter of 1 mm into an ambient at uniform pressure Pa. At the exit section, compute: a) the mass flux m, the exit pressure Pe and the exit velocity ue if the ambient pressure is Pa = atm. b) the mass flux m, the exit pressure Pe and the exit velocity ue if the ambient pressure is Pa = atm. Answer: a) m = kg/s Pe = kPa ue = m/s b) m = kg/s Pe = kPa ue = m/s Exercise 6 Consider a symmetric airfoil, in which the upper and lower surfaces are described by the equations Y+(x) = x (1-x) and Y-(x) = - x (1-x), respectively, with 0 ≤ x ≤ 1. The airfoil is at zero angle of attack in a uniform flow of air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) with free-stream Mach number M∞= and free-stream pressure P∞ = 0.5 kPa and temperature T∞ = 250 K. Using the linearized theory, compute the drag coefficient CD of the airfoil and the flow velocity u in the x direction and the pressure P at points x = , y = and x = , y = . Answer: CD = u(, ) = m/s P(, ) = kPa u(, ) = m/s P(, ) = kPa Exercise 7 The obstacle in the figure is placed in a uniform flow with M∞ P∞ free-stream Mach number M∞= 1.5 and free-stream 2 pressure P∞ = kPa. The angle between the ramp and the 1 free-stream velocity is q = °. Compute the Mach number M and the pressure P at point 1, located along the ramp surface midway between the two corners, and in point 2 q immediately after the second corner. Answer: M1 = P1 = kPa M2 = P2 = kPa
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