Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last name: First name: ID: 866172 Pag. 1/2 Academic year 2019/2020: 11/1/2021 Compressible-Fluid Dynamics Exercise 1 Consider a steady, two-dimensional compressible flows, with uniform entropy and total enthalpy per unit mass. The density field r(x,y) and the velocity field
Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last name: First name: ID: 866172 Pag. 1/2 Academic year 2019/2020: 11/1/2021 Compressible-Fluid Dynamics Exercise 1 Consider a steady, two-dimensional compressible flows, with uniform entropy and total enthalpy per unit mass. The density field r(x,y) and the velocity field
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Last name: First name: ID: 866172 Pag. 1/2 Academic year 2019/2020: 11/1/2021 Compressible-Fluid Dynamics Exercise 1 Consider a steady, two-dimensional compressible flows, with uniform entropy and total enthalpy per unit mass. The density field r(x,y) and the velocity field u(x,y) in the x direction are given by: r (x,y) = 0.4 x2 + 0.8 kg/m3, u (x,y) = 3 xy2 m/s Compute the derivative with respect to the x variable of the y component of the velocity field v(x,y) at point (x,y) = (0.7,0.7) Answer: ¶ v /¶ x = 2.94 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 = 0.3 m and t = 0.07 s. Answer: u’(0.3, 0.07) = -32.6328 m/s r’(0.3, 0.07) = -0.024284 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u 𝜕𝑢 𝜕u + 𝐴𝑡 ! + 𝐵𝑥 = 0 𝜕𝑡 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = 0.9 and B = 1.8. The initial data is u(x,0) = 0.3 x2 compute the value of the solution u at x = 0.8 and t = 0.8 and at x = 1 and t = 1 Answer: u (0.8, 0.8) = -0.86076 u (1, 1) = -1.24800 Exercise 4 Consider a tube filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = 2 bar and temperature T0 = 23 °C, with a circular, constant cross-sectional area. The tube 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…
Last name: First name: ID: 945193 Pag. 1/2 Academic year 2019/2020: 11/1/2021 Compressible-Fluid Dynamics Exercise 1 Consider a steady, two-dimensional compressible flows, with uniform entropy and total enthalpy per unit mass. The density field r(x,y) and the velocity field u(x,y) in the x direction are given by: r (x,y) = 0.5 x2 + 1 kg/m3, u (x,y) = 3 xy2 m/s Compute the derivative with respect to the x variable of the y component of the velocity field v(x,y) at point (x,y) = (0.6,0.5) Answer: ¶ v /¶ x = 1.80 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 = 397.45 m/s. The pressure reads: P(x,t) = 101200 + 141.68 (3x - 397.45t) Pa Compute the value of the perturbation velocity u’(x,t) and the perturbation density r’ (x,t) at x = 0.3 m and t = 0.06 s. Answer: u’(0.3, 0.06) = -28.3149 m/s r’(0.3, 0.06) = -0.020581 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u 𝜕𝑢 𝜕u + 𝐴𝑡 ! + 𝐵𝑥 = 0 𝜕𝑡 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = 1.2 and B = 2. The initial data is u(x,0) = 0.3 x2 compute the value of the solution u at x = 0.7 and t = 0.6 and at x = 0.9 and t = 0.8 Answer: u (0.7, 0.6) = -0.64929 u (0.9, 0.8) = -1.04925 Exercise 4 Consider a tube filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = 1.4 bar and temperature T0 = 21 °C, with a circular, constant cross-sectional area. The tube 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 = 170 m/s. Neglecting the effects of viscosity…
Last name: First name: ID: Pag. 1/2 Exercise 5 A reservoir with infinite capacity is filled with methane at uniform temperature T0 = °C and pressure P0 = atm. Methane can be assumed to be an ideal polytropic gas with molecular weight µ = 16.04 g/mol and specific-heat ratio g = 1.31. The fluid discharges isentropically through a converging nozzle with a diameter of 1 cm into an ambient at uniform pressure Pa. At the exit section, by neglecting the effects of viscosity and thermal conductivity, 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 two-dimensional, uniform steady supersonic flow of dilute air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) with Mach number M∞ = 2 around a flat plate at ° of incidence. Free stream pressure and temperature are P¥ = bar and T¥ = C°, respectively. By neglecting the effects of viscosity and thermal conductivity, compute the pressure Ptop on the top side and the pressure Pbot on the bottom sides of the plate, respectively, and the temperature Ttop on the top side and the temprature Tbot on the bottom sides of the plate, respectively, using the non-linear and the linearized theory. Answer: Linearized theory Ptop = kPa Pbot = kPa Ttop = °C Tbot = °C Non-linear theory: Ptop = kPa Pbot = kPa Ttop = °C Tbot = °C
Last name: First name: ID: Pag. 1/2 Academic year 2019/2020: 11/1/2021 Compressible-Fluid Dynamics Exercise 1 Consider a steady, two-dimensional compressible flows, with uniform entropy and total enthalpy per unit mass. The density field r(x,y) and the velocity field u(x,y) in the x direction are given by: r (x,y) = x2 + kg/m3, u (x,y) = xy2 m/s Compute the derivative with respect to the x variable of the y component of the velocity field v(x,y) at point (x,y) = (,) Answer: ¶ v /¶ x = 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 = m/s. The pressure reads: P(x,t) = + (3x - t) Pa Compute the value of the perturbation velocity u’(x,t) and the perturbation density r’ (x,t) at x = m and t = s. Answer: u’(, ) = m/s r’(, ) = kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u 𝜕𝑢 𝜕u + 𝐴𝑡 ! + 𝐵𝑥 = 0 𝜕𝑡 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = and B = . The initial data is u(x,0) = x2 compute the value of the solution u at x = and t = and at x = and t = Answer: u (, ) = u (, ) = Exercise 4 Consider a tube filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform pressure P0 = bar and temperature T0 = °C, with a circular, constant cross-sectional area. The tube 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 = 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…
Last name: First name: ID: Pag. 1/2 Exercise 5 A reservoir with infinite capacity is filled with methane at uniform temperature T0 = °C and pressure P0 = atm. Methane can be assumed to be an ideal polytropic gas with molecular weight µ = 16.04 g/mol and specific-heat ratio g = 1.31. The fluid discharges isentropically through a converging nozzle with a diameter of 1 cm into an ambient at uniform pressure Pa. At the exit section, by neglecting the effects of viscosity and thermal conductivity, 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 two-dimensional, uniform steady supersonic flow of dilute air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) with Mach number M∞ = 2 around a flat plate at ° of incidence. Free stream pressure and temperature are P¥ = bar and T¥ = C°, respectively. By neglecting the effects of viscosity and thermal conductivity, compute the pressure Ptop on the top side and the pressure Pbot on the bottom sides of the plate, respectively, and the temperature Ttop on the top side and the temprature Tbot on the bottom sides of the plate, respectively, using the non-linear and the linearized theory. Answer: Linearized theory Ptop = kPa Pbot = kPa Ttop = °C Tbot = °C Non-linear theory: Ptop = kPa Pbot = kPa Ttop = °C Tbot = °C
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