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: 15/6/2020 Compressible-Fluid Dynamics Exercise 1 In a compressible two-dimensional flow at thermodynamic equilibrium, the density and velocity fields are r (x,y,t) = f(t) kg/m3, u(x,y,t) = 0.3 x + 24 y m/s,
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: 15/6/2020 Compressible-Fluid Dynamics Exercise 1 In a compressible two-dimensional flow at thermodynamic equilibrium, the density and velocity fields are r (x,y,t) = f(t) kg/m3, u(x,y,t) = 0.3 x + 24 y m/s,
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Last name: First name: ID: 10191772 Pag. 1/2 Academic year 2019/2020: 15/6/2020 Compressible-Fluid Dynamics Exercise 1 In a compressible two-dimensional flow at thermodynamic equilibrium, the density and velocity fields are r (x,y,t) = f(t) kg/m3, u(x,y,t) = 0.3 x + 24 y m/s, v(x,y,t) = 1.6 y + 1.2 m/s respectively, where f(t) is a function of time with f(0) = 1.2 kg/m3. Compute value of the density at x = 1 m and y = 2 m at time t = 1 s and t = 2 s. Answer: r(1,2,1) = 0.17948 kg/m3 r(1,2,2) = 0.02684 kg/m3 Exercise 2 In a one-dimensional flow of diatomic nitrogen (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) in dilute conditions, the flow variables in one field point read r = 1.3 kg/m3, u = 140 m/s, Et = 516425 J/m3 The reference value of the internal energy e(T) is e(0 K) = 0 J/kg. Compute the speed of sound c and the acoustic impedance I Answer: c = 465.80 m/s I = 605.54 kg/(m2 s) Exercise 3 The square wave in the figure has an amplitude r0/106, with r0 unperturbed value of the density, and a wavelength l= 1.8 m. The wave propagates in still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 31 °C and pressure P0 = 1 atm. a) Compute the maximum amplitude of the pressure (PM) and velocity (uM) perturbations. b) Compute the acoustic intensity FE in dB. c) Compute the maximum displacement xM of the fluid particles in the right half-plane. d) Consider an observer located at 2 km from the wave source. After how many seconds (ta) the perturbation will reach the observer? Compute the duration (Dt) of the perturbation at the observer’s location. Answer: PM = 1.4186E-01 Pa uM = 3.4958E-04 m/s FE = 76.954 dB xM = 9.0000E-07 m ta = 5.721 s Dt = 5.1490E-03 s
Last name: First name: ID: 10520798 Pag. 1/2 Academic year 2019/2020: 15/6/2020 Compressible-Fluid Dynamics Exercise 1 In a compressible two-dimensional flow at thermodynamic equilibrium, the density and velocity fields are r (x,y,t) = f(t) kg/m3, u(x,y,t) = 0.9 x + 28 y m/s, v(x,y,t) = 2 y + 1.1 m/s respectively, where f(t) is a function of time with f(0) = 1.1 kg/m3. Compute value of the density at x = 1 m and y = 2 m at time t = 1 s and t = 2 s. Answer: r(1,2,1) = 0.06053 kg/m3 r(1,2,2) = 0.00333 kg/m3 Exercise 2 In a one-dimensional flow of diatomic nitrogen (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) in dilute conditions, the flow variables in one field point read r = 1.9 kg/m3, u = 150 m/s, Et = 757530 J/m3 The reference value of the internal energy e(T) is e(0 K) = 0 J/kg. Compute the speed of sound c and the acoustic impedance I Answer: c = 465.80 m/s I = 885.02 kg/(m2 s) Exercise 3 The square wave in the figure has an amplitude r0/109, with r0 unperturbed value of the density, and a wavelength l= 2 m. The wave propagates in still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 31 °C and pressure P0 = 1 atm. a) Compute the maximum amplitude of the pressure (PM) and velocity (uM) perturbations. b) Compute the acoustic intensity FE in dB. c) Compute the maximum displacement xM of the fluid particles in the right half-plane. d) Consider an observer located at 8 km from the wave source. After how many seconds (ta) the perturbation will reach the observer? Compute the duration (Dt) of the perturbation at the observer’s location. Answer: PM = 1.4186E-04 Pa uM = 3.4958E-07 m/s FE = 16.954 dB xM = 1.0000E-09 m ta = 22.884 s Dt = 5.7211E-03 s
Last name: First name: ID: 10539571 Pag. 1/2 Academic year 2019/2020: 15/6/2020 Compressible-Fluid Dynamics Exercise 1 In a compressible two-dimensional flow at thermodynamic equilibrium, the density and velocity fields are r (x,y,t) = f(t) kg/m3, u(x,y,t) = 0.2 x + 22 y m/s, v(x,y,t) = 1.6 y + 2 m/s respectively, where f(t) is a function of time with f(0) = 2 kg/m3. Compute value of the density at x = 1 m and y = 2 m at time t = 1 s and t = 2 s. Answer: r(1,2,1) = 0.33060 kg/m3 r(1,2,2) = 0.05465 kg/m3 Exercise 2 In a one-dimensional flow of diatomic nitrogen (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) in dilute conditions, the flow variables in one field point read r = 1.2 kg/m3, u = 140 m/s, Et = 425040 J/m3 The reference value of the internal energy e(T) is e(0 K) = 0 J/kg. Compute the speed of sound c and the acoustic impedance I Answer: c = 439.16 m/s I = 527.00 kg/(m2 s) Exercise 3 The square wave in the figure has an amplitude r0/106, with r0 unperturbed value of the density, and a wavelength l= 1.8 m. The wave propagates in still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 27 °C and pressure P0 = 1 atm. a) Compute the maximum amplitude of the pressure (PM) and velocity (uM) perturbations. b) Compute the acoustic intensity FE in dB. c) Compute the maximum displacement xM of the fluid particles in the right half-plane. d) Consider an observer located at 6 km from the wave source. After how many seconds (ta) the perturbation will reach the observer? Compute the duration (Dt) of the perturbation at the observer’s location. Answer: PM = 1.4186E-01 Pa uM = 3.4728E-04 m/s FE = 76.925 dB xM = 9.0000E-07 m ta = 17.277 s Dt = 5.1832E-03 s
Last name: First name: ID: 10582292 Pag. 1/2 Academic year 2019/2020: 15/6/2020 Compressible-Fluid Dynamics Exercise 1 In a compressible two-dimensional flow at thermodynamic equilibrium, the density and velocity fields are r (x,y,t) = f(t) kg/m3, u(x,y,t) = 0.3 x + 23 y m/s, v(x,y,t) = 2 y + 1.3 m/s respectively, where f(t) is a function of time with f(0) = 1.3 kg/m3. Compute value of the density at x = 1 m and y = 2 m at time t = 1 s and t = 2 s. Answer: r(1,2,1) = 0.13034 kg/m3 r(1,2,2) = 0.01307 kg/m3 Exercise 2 In a one-dimensional flow of diatomic nitrogen (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) in dilute conditions, the flow variables in one field point read r = 1.3 kg/m3, u = 150 m/s, Et = 378397.5 J/m3 The reference value of the internal energy e(T) is e(0 K) = 0 J/kg. Compute the speed of sound c and the acoustic impedance I Answer: c = 395.86 m/s I = 514.61 kg/(m2 s) Exercise 3 The square wave in the figure has an amplitude r0/106, with r0 unperturbed value of the density, and a wavelength l= 2 m. The wave propagates in still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at uniform temperature T0 = 21 °C and pressure P0 = 1 atm. a) Compute the maximum amplitude of the pressure (PM) and velocity (uM) perturbations. b) Compute the acoustic intensity FE in dB. c) Compute the maximum displacement xM of the fluid particles in the right half-plane. d) Consider an observer located at 3 km from the wave source. After how many seconds (ta) the perturbation will reach the observer? Compute the duration (Dt) of the perturbation at the observer’s location. Answer: PM = 1.4186E-01 Pa uM = 3.4379E-04 m/s FE = 76.881 dB xM = 1.0000E-06 m ta = 8.726 s Dt = 5.8176E-03 s
Last name: First name: ID: 10755485 Pag. 2/2 Exercise 4 Consider the following PDE for the scalar unknown u, !" !" 𝑥 !# + 2𝑦 !$ = 𝐴𝑢% , 𝑥 > 0, 𝑦 > 0, where x and y are the spatial coordinates and A = 3. Compute the value of u at x = 9 and y = 5, assuming the initial condition u(1,y)=6 y. Answer: u(9,5) = -0.2570 Exercise 5 A normal shock wave with intensity (P2 - P1) / P1 = 5 is moving in a tube filled with still air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at pressure and temperature P1 = 1.9 bar and T1 = 25 °C respectively. The tube is closed on one end and open on the opposite one. The shock is propagating towards the closed end of the tube. Compute the module of the shock velocity in the laboratory frame before (S1) and after (S2) the shock wave interacts with the closed end of the tube. Answer: S1 = 795.74 m/s S2 = -365.61 m/s Exercise 6 A reservoir contains dilute air (polytropic gas with g = 1.4 and R = 287 J/(Kg K)) at temperature T0 = 90 °C and pressure P0 = 860 kPa. Air is discharged through a convergent-divergent nozzle designed to expand up to exit Mach number Me = 2.25, with a throat section At= 0.0081 m2. Compute: a) mass flow rate m in design conditions, b) the exit section area Ae, c) the pressure Pe and the temperature Te on the exit section in design conditions, d) the value of the ambient pressure Psub over which the flow in the nozzle is everywhere subsonic, e) the value of the ambient pressure Pson under which the flow is sonic in the throat but there are no shocks in the divergent part. Answer: m = 14.7747 kg/s Ae = 0.016981124 m2 Pe = 74 kPa Te = -92.70 °C Psub = 808 kPa Pson = 427 kPa Exercise 7 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…
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