Explaining prandtl glauert rule

Question

Question1)(a) State the Prandtl Glauert’s rule and derive the expressions for the coefficients of pressure, lift and moment. In a low speed in compressible flow, the peak coefficient of pressure of an NACA 0009 airfoil is -0.25. Estimate the critical Mach number for this airfoil using Prandt’l Glauret’s rule.

b) Write about quasi-one-dimensional flow. Air flowing in a duct has a velocity of 300 m/s, pressure 1.0 bar and temperature 290 K taking=1.4, R=287 J/kgK. Determine

(a) Stagnation pressure and temperature

(b) Velocity of sound in the dynamic and stagnation conditions.

(c) Stagnation pressure assuming constant density.

Question 2)a) A supersonic flow with Mach number 2.8 and pressure of 1 atm is deflected through an angle such that the Mach number behind the shock is 1.45. Find the angle of deflection and the pressure behind the shock.

b) What is a supercritical airfoil? How is it useful in increasing the drag divergence Mach number?

c) Explain the term area rule. How does it affect the performance of an airplane?

Question 3)a) For the flow field shown in figure 3 determine β_r, M₂ and M₃ if M₁ = 2.0 for β_i = 400.

b) Explain what you understand by linearization.

Question 4)a) Derive velocity potential equation for a two-dimensional, steady, irrotational, isentropic flow.

b) Argon is stored in a reservoir at 300K ; Determine stagnation enthalpy and velocity of sound in it =1.658 and the molecular weight of argon is 39.94.

Question 5)a) At a given point in a flow T = 300 K, P = 1.2 atm and V = 250 m/s. At this point calculate the corresponding values of P_o, To, P*, T* and M*.

b) The flow just upstream of a normal shock is given by P₁ = 1 atm, T₁ = 288 K and M₁ = 2.6. Calculate the following properties at downstream of shock: P₂, T₂, ρ₂, M₂, P_o2 and T_o2.

Question 6)a) Consider a supersonic flow with M = 2, P = 1 atm, T = 288 K. This flow is deflected at a compression corner through 20º. Calculate M, P, T, Po and To behind the resulting shock wave. Take the wave angle to be 53.4º.

b) A supersonic flow with M₁ = 1.5, P₁ = 1 atm and T₁ = 288 K is expanded around a sharp corner through a deflection angle of 15º. Calculate M₂, P₂, T₂, P_o2 and T_o2. Use the Prandtl-Meyer function and Mach number chart here.