Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last name: First name: ID: 10395927 Pag. 1/2 Academic year 2019/2020: 27/1/2021 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), the velocity fields
Full exam for Compressible Fluid Dynamics in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Last name: First name: ID: 10395927 Pag. 1/2 Academic year 2019/2020: 27/1/2021 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), the velocity fields
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Last name: First name: ID: 10395927 Pag. 1/2 Academic year 2019/2020: 27/1/2021 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), the velocity fields u(x,y) and v(x,y) in the x and y directions, respectively, are r (x,y) = 0.9 x kg/m3, u (x,y) = 0.3 xy v (x,y) = 11 y + 1 m/s Compute the derivative of the pressure with respect to the x and y variables at point (x,y) = (0.6,1) Answer: ¶ P /¶ x = A1.1 kPa/m ¶ P /¶ y = A1.2 kPa/m 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 = 411.36 m/s. The unperturbed value of the pressure is P0 = 101480 Pa. The perturbation velocity reads: u(x,t) = 0.41136 (x - 411.36t) + 0.017885 (x + 411.36t)2 m/s Compute the value of the perturbation pressure P’(x,t) and of the perturbation density r’ (x,t) at x = 1.1 m and t = 0.06 s. Answer: P’(1.1,0.06) = A2.1 Pa r’(1.1,0.06) = A2.2 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u(x,t) 𝜕𝑢 𝜕𝑢! +𝐴 =0 𝜕𝑡 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = 4. The initial data is u(x,0) = 7 for x < 0 u(x,0) = 6 for x ≥ 0 compute the value of the solution u at time t = 0.7 and at points x1 = 34.58, and x2 = 38.22 and x3 = 40.04 Answer: u (34.58,0.7) = A3.1 u (38.22,0.7) = A3.2 u (40.04,0.7) = A3.3 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 = 27 °C, with a circular, constant cross-sectional area. The tube is 2 m long and it is closed…
Last name: First name: ID: Pag. 1/2 Academic year 2019/2020: 27/1/2021 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), the velocity fields u(x,y) and v(x,y) in the x and y directions, respectively, are r (x,y) = x kg/m3, u (x,y) = xy v (x,y) = y + 1 m/s Compute the derivative of the pressure with respect to the x and y variables at point (x,y) = (,) Answer: ¶ P /¶ x = A1.1 kPa/m ¶ P /¶ y = A1.2 kPa/m 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 unperturbed value of the pressure is P0 = Pa. The perturbation velocity reads: u(x,t) = (x - t) + (x + t)2 m/s Compute the value of the perturbation pressure P’(x,t) and of the perturbation density r’ (x,t) at x = m and t = s. Answer: P’(,) = A2.1 Pa r’(,) = A2.2 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u(x,t) 𝜕𝑢 𝜕𝑢! +𝐴 =0 𝜕𝑡 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = . The initial data is u(x,0) = for x < 0 u(x,0) = for x ≥ 0 compute the value of the solution u at time t = and at points x1 = , and x2 = and x3 = Answer: u (,) = A3.1 u (,) = A3.2 u (,) = A3.3 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 outwards, away from the end wall B. Neglecting the effects of…
Last name: First name: ID: Pag. 1/2 Exercise 5 A reservoir with infinite capacity is filled with Helium at uniform temperature T0 = °C and pressure P0 = atm. Methane can be assumed to be an ideal polytropic gas with molecular weight µ = 4.01 g/mol and specific-heat ratio of monoatomic gases. 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 = A5.1 kg/s Pe = A5.2 kPa ue = A5.3 m/s b) m = A5.4 kg/s Pe = A5.5 kPa ue = A5.6 m/s Exercise 6 Consider a two-dimensional, uniform steady supersonic flow of dilute M∞ P∞ air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) with Mach number M∞ = 2.5 past the geometry in the figure, where q = °. Free stream pressure and temperature are P¥ = bar and T¥ = C°, respectively. By neglecting the effects of viscosity and thermal conductivity, compute the fluid velocity u and the pressure at point A, A using the non-linear and the linearized theory. q Answer: Linearized theory uA = A6.1 m/s PA = A6.2 kPa Non-linear theory: uA = A6.3 m/s PA = A6.4 kPa
Last name: First name: ID: Pag. 1/2 Academic year 2019/2020: 27/1/2021 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), the velocity fields u(x,y) and v(x,y) in the x and y directions, respectively, are r (x,y) = x kg/m3, u (x,y) = xy v (x,y) = y + 1 m/s Compute the derivative of the pressure with respect to the x and y variables at point (x,y) = (,) Answer: ¶ P /¶ x = A1.1 kPa/m ¶ P /¶ y = A1.2 kPa/m 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 unperturbed value of the pressure is P0 = Pa. The perturbation velocity reads: u(x,t) = (x - t) + (x + t)2 m/s Compute the value of the perturbation pressure P’(x,t) and of the perturbation density r’ (x,t) at x = m and t = s. Answer: P’(,) = A2.1 Pa r’(,) = A2.2 kg/m3 Exercise 3 Consider the following partial differential equation for the scalar quantity u(x,t) 𝜕𝑢 𝜕𝑢! +𝐴 =0 𝜕𝑡 𝜕𝑥 for the scalar unknown u, where x is the spatial coordinate and t is the time and where A = . The initial data is u(x,0) = for x < 0 u(x,0) = for x ≥ 0 compute the value of the solution u at time t = and at points x1 = , and x2 = and x3 = Answer: u (,) = A3.1 u (,) = A3.2 u (,) = A3.3 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 outwards, away from the end wall B. Neglecting the effects of…
Last name: First name: ID: Pag. 1/2 Exercise 5 A reservoir with infinite capacity is filled with Helium at uniform temperature T0 = °C and pressure P0 = atm. Methane can be assumed to be an ideal polytropic gas with molecular weight µ = 4.01 g/mol and specific-heat ratio of monoatomic gases. 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 = A5.1 kg/s Pe = A5.2 kPa ue = A5.3 m/s b) m = A5.4 kg/s Pe = A5.5 kPa ue = A5.6 m/s Exercise 6 Consider a two-dimensional, uniform steady supersonic flow of dilute M∞ P∞ air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) with Mach number M∞ = 2.5 past the geometry in the figure, where q = °. Free stream pressure and temperature are P¥ = bar and T¥ = C°, respectively. By neglecting the effects of viscosity and thermal conductivity, compute the fluid velocity u and the pressure at point A, A using the non-linear and the linearized theory. q Answer: Linearized theory uA = A6.1 m/s PA = A6.2 kPa Non-linear theory: uA = A6.3 m/s PA = A6.4 kPa
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