Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last and first name: ID:10395927 Page 1/2 Academic year 2020/2021: 7/9/2021 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
Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last and first name: ID:10395927 Page 1/2 Academic year 2020/2021: 7/9/2021 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
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Last and first name: ID:10395927 Page 1/2 Academic year 2020/2021: 7/9/2021 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¥ = 0.9 atm and T¥ = -10 °C, respectively. 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 airfoil velocities V: V1 = 130.07 m/s, V2 = 260.13 m/s, V3 = 455.23 m/s. Intermediate results Answers A1.1) P0,1 = 1.005 atm A1.2) P0,2 = 1.372 atm A1.3) P0,2 = 2.744 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 unperturbed value of the speed of sound is c0 = 407.44 m/s. The pressure reads: P(x,t) = 101400 + 141.96 (3x - 407.44t) Pa Compute the value of the perturbation velocity u’(x,t) and the perturbation density r’ (x,t) at x = 1.10 m and t = 0.06 s. Intermediate results Answers A2.1) u’(1.10, 0.06) = -29.4325 m/s A2.2) r’(1.10, 0.06) = -0.018083 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u !" !u + %# ! + (' = 0 !# !' for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = 2.4 and B = 1.3. The initial data is u(x,0) = 1.1 x2 compute the value of the solution u at x = 0.7 and t = 1.1 and at x = 0.9 and t = 1.3 Intermediate results Answers A3.1) u (0.7, 1.1) = 0.28738 A3.2) u (0.9, 1.3) = 1.51578
Last and first name: ID:10539948 Page 1/2 Academic year 2020/2021: 7/9/2021 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¥ = 1 atm and T¥ = 10 °C, respectively. 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 airfoil velocities V: V1 = 134.92 m/s, V2 = 269.84 m/s, V3 = 472.22 m/s. Intermediate results Answers A1.1) P0,1 = 1.117 atm A1.2) P0,2 = 1.524 atm A1.3) P0,2 = 3.049 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 unperturbed value of the speed of sound is c0 = 409.89 m/s. The pressure reads: P(x,t) = 101450 + 142.03 (3x - 409.89t) Pa Compute the value of the perturbation velocity u’(x,t) and the perturbation density r’ (x,t) at x = 1.10 m and t = 0.10 s. Intermediate results Answers A2.1) u’(1.10, 0.10) = -49.9529 m/s A2.2) r’(1.10, 0.10) = -0.031861 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u !" !u + %# ! + (' = 0 !# !' for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = 2.7 and B = 1.5. The initial data is u(x,0) = 1.1 x2 compute the value of the solution u at x = 1.1 and t = 0.5 and at x = 1.3 and t = 0.7 Intermediate results Answers A3.1) u (1.1, 0.5) = 0.31095 A3.2) u (1.3, 0.7) = -0.04096
Last and first name: ID:10583075 Page 2/2 Exercise 4 Consider standard air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 14 °C and pressure P0 = 1.9 atm. The gas is at rest in a pipe with a circular, constant cross-sectional area. The pipe is closed at one end by a moveable piston and it is 10 km long. The piston is set into motion impulsively at a speed of V1 = 254.75 m/s outwards. Compute the minimum value of the pressure Pm,1 and the maximum value of the fluid velocity uM,1 within the tube after 1 s. Compute the same quantities in the case V2 = 2038.03 m/s. Intermediate results Answers A4.1) Pm,1 = 0.60910 atm A4.2) |uM,1 | = 254.754 m/s A4.3) Pm,2 = 0.0 atm A4.4) |uM,2 | = 1698.36 m/s Exercise 5 A reservoir with infinite capacity is filled with carbon dioxide at uniform temperature T0 = 23.0 °C and pressure P0 = 5 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 = 3.6388 atm. b) the mass flux m, the exit pressure Pe and the exit velocity ue if the ambient pressure is Pa = 1.3657 atm. Intermediate results Answers a) A5.1) m = 0.00103088 kg/s A5.2) Pe = 3.6388 atm A5.3) ue = 185.19 m/s b) A5.4) m = 0.00112159 kg/s A5.5) Pe = 2.7314 atm A5.6) ue = 251.36 m/s Exercise 6 Consider a symmetric airfoil, in which the upper and lower surfaces are described by the equations Y+(x) = 0.32 x (1-x) and Y-(x) = - 0.32 x (1-x), respectively, with 0 ≤ x ≤ 1. The airfoil is at zero angle of attack in a uniform flow of air…
Last and first name: ID:10665222 Page 1/2 Academic year 2020/2021: 7/9/2021 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 and T¥ = -10 °C, respectively. 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 airfoil velocities V: V1 = 130.07 m/s, V2 = 260.13 m/s, V3 = 455.23 m/s. Intermediate results Answers A1.1) P0,1 = 0.447 atm A1.2) P0,2 = 0.610 atm A1.3) P0,2 = 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 unperturbed value of the speed of sound is c0 = 394.92 m/s. The pressure reads: P(x,t) = 101150 + 141.61 (3x - 394.92t) Pa Compute the value of the perturbation velocity u’(x,t) and the perturbation density r’ (x,t) at x = 0.40 m and t = 0.06 s. Intermediate results Answers A2.1) u’(0.40, 0.06) = -27.9146 m/s A2.2) r’(0.40, 0.06) = -0.020425 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u !" !u + %# ! + (' = 0 !# !' for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = 0.9 and B = 1.3. The initial data is u(x,0) = 0.4 x2 compute the value of the solution u at x = 0.7 and t = 0.8 and at x = 0.9 and t = 1 Intermediate results Answers A3.1) u (0.7, 0.8) = -0.48877 A3.2) u (0.9, 1) = -0.73350
Last and first name: ID: Page 2/2 Exercise 4 Consider standard air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 120 °C and pressure P0 = atm. The gas is at rest in a pipe with a circular, constant cross-sectional area. The pipe is closed at one end by a moveable piston and it is 10 km long. The piston is set into motion impulsively at a speed of V1 = m/s outwards. Compute the minimum value of the pressure Pm,1 and the maximum value of the fluid velocity uM,1 within the tube after 1 s. Compute the same quantities in the case V2 = m/s. Intermediate results Answers A4.1) Pm,1 = atm A4.2) |uM,1 | = m/s A4.3) Pm,2 = atm A4.4) |uM,2 | = m/s 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. Intermediate results Answers a) A5.1) m = kg/s A5.2) Pe = atm A5.3) ue = m/s b) A5.4) m = kg/s A5.5) Pe = atm A5.6) 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…
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