6.1 According to Handbook of Chemistry and Physics the composition of air is

Transcription

1 6. Compressible flow 6.1 According to Handbook of Chemistry and Physics the composition of air is From this, compute the gas constant R for air. 6. The figure shows a, Pitot-static tube used for velocity and Mach number measurements. Compute the velocity u and the Mach number M given that the temperature t = 0 C and the pressures are = 80 kpa and p = 10 kpa. Assume subsonic flow in the whole flow field. p1 6.3 An airplane is flying at an atmospheric height where the temperature T = 70 K and the pressure p = 70 kpa. At a point on the wing the pressure p = 63 kpa and the local Mach number M = How large is the speed of the airplane relative to the surrounding atmosphere? Assume that the airspeed is subsonic everywhere around the airplane. 6.4 Consider a blunt body in a flow field where the velocity V = 150 m/s and the temperature = 0 C. How large can the maximum surface temperature of the body be? t 6.5 An airplane is flying at the atmospheric height l0 km with the speed V = 935 km/h relative to the surrounding atmosphere. At this atmospheric height the pressure is kpa and the temperature is 50 C. At a point on the airplane wing the pressure is measured to 5.45 kpa. What is the Mach number and the air speed in this point relative to the airplane? Assume subsonic flow everywhere around the airplane.

2 6.6 Air at the stagnation pressure p 0 = 300 kpa is expanded isentropically through a. Laval nozzle. The pressure in a cross section after the expansion is p = 8 kpa. How large is the cross section area if the cross section at the throat is 10 cm? 6.7 Air is expanded to supersonic flow through a. Laval nozzle. The cross section area in the throat is 15 cm. Estimate the Mach number in a. cross section downstream of the throat where the area is 5.3 cm. 6.8 Helium (γ= 5/3 = 1.67) is expanded isentropically through a Laval nozzle to supersonic flow. How large is the pressure in the throat if the stagnation pressure is = p kpa? 6.9 Air at the stagnation state p 0 = 1000 kpa and T 0 = 300 K is expanded to supersonic speed through a. Laval nozzle. At the throat the cross section area. is 49 cm. Estimate the gas velocity at the exit from the nozzle where the cross section area is 100 cm Air is flowing through a nozzle in such a way that the Mach number = 1 in the throat. In such a case the Mach number M at the exit can be either sub- or supersonic depending on the pressure at the exit from the nozzle. For the nozzle in the figure compute the exit Mach number and pressure p in both these cases. The flow through the nozzle is shock-free. M M 1 A1 Given: = 0.5, = 10 cm, A = 8.4 cm, A* = 7.46 cm and p1 = 170 kpa A perfect, gas is flowing through a nozzle. Show that, if the flow is choked the mass flow through the nozzle is

3 6.1 Air is stored in a reservoir at the pressure MPa and the temperature C. A valve is opened and air is streaming from the reservoir, through the valve to the atmosphere. The minimum cross-section area in the valve is 1 cm. Show that the flow through the valve must be choked and estimate the mass flow through it Air with the stagnation state p0 = 01 kpa and t0 = 0 C is streaming through the nozzle in the figure. At the entrance to the nozzle the flow is known to be subsonic. No shocks are observed in the nozzle between the entrance and the exit. Calculate the mass flow through the nozzle. Given: p1 = 180 kpa, p = 167 kpa, A1 =17.7 cm, A = 14,5 and A = 1, cm t 6.14 The nozzle in a supersonic wind tunnel should be constructed for the Mach number M = 3 in the test section. The mass flow through the tunnel should be kg/s at the stagnation state p0 = 500 kpa and T0 = 300 K. Estimate the cross section area in the nozzle throat and in the test section.

4 6.15 Carbon dioxide (γ= 1.31 and specific gas constant R = 189 J/(kg.K)) is stored in a pressure reservoir at the pressure 500 kpa and temperature 10 C. Some of this carbon dioxide should be transferred to another reservoir where the pressure is 100 kpa through a valve. Estimate the opening cross section area in the valve if the required mass flow is 1 kg/s. Assume adiabatic flow through the valve What is the stagnation pressure needed to get a mass flow of 8 kg/s through a supersonic wind tunnel, when the Mach number in the test section is M =.4. The cross section in the wind tunnel nozzle throat is 0.01 and the stagnation temperature T = 300 K. m A calorically perfect gas is streaming through the straight nozzle in the figure. Show that the force acting on the nozzle from the gas is = p ( 1+ γ M ) A p (1 + γm A F x ) 6.18 Air with stagnation state = 800 kpa and T = 300 K is streaming through a straight nozzle. p0 0 At the nozzle entrance the velocity is subsonic and the cross section area =37 cm. The throat area A t = 6 cm and at the nozzle exit the cross section area A = 47 cm. The velocity at the exit is supersonic and there are no shocks present inside the nozzle. Calculate the force exerted bv the air flow on the nozzle. A A total pressure tube (Pitot tube) is placed in a supersonic airflow. The Mach number M = 1.8 and the pressure p = 60 kpa in the undisturbed flow ahead of the tube. What pressure does the Pitot tube measure?

5 6.0 A bluff body is moving through air with the Mach number M = 1.5. The temperature in the undisturbed air ahead of the body t = - 10 C. Estimate the temperature at the stagnation point on the body. 6.1 The shock wave from an explosion is moving through an undisturbed atmosphere. The atmospheric pressure is 100 kpa and the temperature is 15 C. The shock wave is observed to move with the speed 850 m/s. Estimate the pressure and the temperature directly after the shock has passed. Assume the shock wave to be flat and also that it is moving in the direction of its surface normal. 6. On a Schlieren picture of a plane supersonic flow past a symmetric wedge the angles and are measured. Calculate the free stream Mach number M when β = 44 and θ= A wedge is placed in a homogeneous parallel flow as shown in the figure. Compute the Mach number, the pressure p, the temperature T and the velocity at the wedge surface. Given: M =3, p = 100 kpa, M = 300 K and θ =

6 6.4 One wants to increase the pressure in a supersonic air flown from p1 = 100 kpa to p = 500 kpa through an oblique shock. How large deflection angle θ is needed if the inflow Mach number M 1 =.50? What is the shock angle β? 6.5 A wedge is placed in a, supersonic homogeneous parallel flow of air. In the free stream The Mach number M = 3.0 and the pressure p = 100 kpa. Calculate the maximum possible pressure p on the wedge surface when there should be an attached oblique shock. At which deflection angle θ is this obtained? 6.6 How large deflection angle is needed to expand a homogenous airflow from the Mach M = 1.5 to the Mach number = 1.76? 1 M 6.7 A supersonic air flow with Mach number M 1 = 1.80 bounded by a plane wall is expanded at a convex corner in the wall. The total deflection angle θ = 8. What is the Mach number in the air flow after the expansion? 6.8 A supersonic air flow with Mach number M 1 along a plane wall is deflected the angle θ as shown in the figure. Compute the Mach number M, the velocity v, the pressure p and the temperature T after the deflection. Given: =.00, = 100 kpa, T = 300 K θ = 10. M 1 p One wants to decrease the pressure in an air stream by expanding it past a sharp convex corner. Compute the deflection needed to decrease the pressure from p =00 kpa to p=100 kpa if =1.50. M 6.30 An air stream with the Mach number M 1 = 3.0 is continuously deflected along a concave wall. The total deflector angle θ = 14. What is the Mach number M after the deflection?

7 6.31 One wants to retard a plane supersonic flow from the Mach number M = 3.00 to the Mach number M =.00. This can be done by a continuously concave deflection of a wall. What total deflection angle is needed? 6.3 An air stream with Mach number M 1 and pressure p 1 along a wall is deflected by a wedge. The opening angle of the wedge is θ 1. Further down stream the flow meets a sharp convex corner and is deflected the angle θ. Compute the pressures p and p 3 Given:M 1 = 5.00, p 1 = 100 kpa, θ 1 = 5 and θ = A supersonic airflow with Mach number M 1 along a wall can be compressed either through an oblique shock or through a Prandtl-Meyer compression wave. An oblique shock emanates from a sharp concave corner as shown i figure a). A Prandtl-Meyer compression wave is obtained by continuously deflecting the wall. This is shown in figure b). Compute the Mach numbers M and M 3, the velocities u and u 3, the pressure ratio p /p 3 the temperature ratiot /T 3 and the stagnation pressure ratio p 0 /p 03. Given: M 1 =.8 T 1 =300K θ= Air is streaming through a. duct as shown in the figure. The Mach number M 1 = 3.0. In the duct, there is a wedge generating an oblique shock. This shock is reflected from the upper wall as shown. Ca,lculate the angle α of the reflected shock if the wedge angle θ= 10.

8 6.35 In a supersonic air intake to a jet. engine the air flow must be retarded to subsonic speed. This retardation can be achieved through one or more oblique or normal shocks. The design of the intake duct, can also be made in such a way that, an oblique shock is replaced by a continuous compression. In the figures two possible designs are shown. The designer wants to know which one is to be preferred when the Mach number M 1 = 3.00 and θ= 30. Hence, compute a) the stagnation pressure ratio p 03 /p 01 in the left design and b) b) the stagnation pressure ratio p 05 /p 01 in the right design. (Among other things one wants this stagnation pressure ratio to be as large as possible in a jet engine air intake.) 6.36 A flat plate with chord c is placed in a supersonic free stream. The Mach number in the free stream is M and the angle of attack is α. Compute the pressure coefficients cpu and c pl the upper and lower side of the plate, respectively, the lift-coefficient c l, the drag coefficient c d and the tip moment coefficient c m,x=0 Given: M = 1.60 and α=. 6.37

9 The symmetric wedge profile in the figure is placed without angle of attack in a supersonic free stream. Compute the lift coefficient c l and the drag coefficient c d if the Mach number M = 1.60 and the thickness ratio t/c = 0.1. Answers: J/kg K m/s m/s C , 69 m/s cm kpa m/s ,150 kpa,1.4,6 kpa kg/s kg/s cm, 73 cm cm kpa N kpa C kpa, 343 C ,41 kpa,491 K,838 m/s , kpa, , 90 m/s,18 kpa, 356 K

10 kpa,5kpa , 1.967, m/s, 84.1 m/s, 0.988, 1.053, , , 0.584, 0.11, , ,