Esame completo di Compressible Fluid Dynamics per il corso di Aerospace Engineering presso Politecnico di Milano. Il documento tratta: 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…
Esame completo di Compressible Fluid Dynamics per il corso di Aerospace Engineering presso Politecnico di Milano. Il documento tratta: 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…
Qualità dell’importazione: il testo è stato estratto direttamente dal documento originale.
Passaggi rappresentativi riconosciuti nelle diverse parti del materiale. Il testo completo resta presente nella pagina per la ricerca, mentre l’anteprima compatta rende più semplice la lettura.
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…
Prima pagina del documento.
