AE4451 Winter 2002

Review Test

 

1 sheet of your handwriting allowed.

 

(Answers in italics)

 

1. (28 points)

Plot the following quantities along the axis of  a converging-diverging nozzle, starting from the reservoir, where the flow is not moving at any appreciable speed, going  through the throat, and downstream, through a normal shock and out to the nozzle exit. On the plot, also show the level of the outside pressure for reference.

Stagnation temperature: constant – unless the Mach number is so high that you have to consider changes in specific heats across the shock.

Stagnation enthalpy: constant – adiabatic process, no work extracted.

Stagnation pressure: constant until the shock (isentropic expansion), drops across the shock and remains constant at that level (isentropic deceleration)  thereafter.

Static temperature: decreases (from the stagnation value in the reservoir) through the nozzle until the shock, shoots up across the shock; increases towards the stagnation value as the flow slows down after the shock.

Static pressure: decreases (from the stagnation value in the reservoir) through the nozzle until the shock, shoots up across the shock; increases towards the new stagnation value as the flow slows down after the shock. Much more than the temperature. The final value of static temperature at the exit must be equal to the outside (ambient) pressure because the region downstream of the shock is subsonic – information can propagate upstream in the nozzle up to the shock, causing the pressure at the nozzle exit to equalize with the outside pressure.

Mach number: increases from 0 at the reservoir to 1.0 at the throat, increases further in the supersonic region in the divergent part of the nozzle (otherwise there can’t be a shock). Drops across the shock to a subsonic value, then decreases further further downstream in the nozzle.

Flow velocity: increases from 0 at the reservoir , and until the shock; drops across the shock, and further decreases downstream of the shock as the area increases in subsonic flow in the divergent part of the nozzle.

 

 

2.       (30 points)

·         Given that the composition of air is 79% diatomic nitrogen, 20%diatomic oxygen and 1% rare gases with a molecular weight of 44, and the ratio of specific heats for this air is 1.3, find the specific heats of air at constant pressure and at constant volume.

Answer: Molecular weight comes out to be 28.97 – gas constant becomes 287. Specific heat at constant pressure is    , which comes out to be  1244 m2 s-1 K-1

·         A stagnation probe is placed in an air flow where the velocity is 200 m/s, static temperature is 500K, and static pressure is 1 atmosphere. What is the static enthalpy of the flow? What is the stagnation enthalpy?

Answer: ;       which works out to be 

50,250 + 200*200/2 = 522,250 m2 s-2

·         Calculate the Mach angle at Mach 2.5

Answer:  23.58 degrees.

·         Calculate the speed of sound at the surface of the planet Xylon (pressure: 0.1 million Newtons per square meter) where the atmosphere is 100% Xenon and the temperature is 400K.

Answer:     which is 206.25 m/s

3.       (12 points)

Plot the section lift coefficient as a function of angle of attack for a 2-D, low-speed, symmetric airfoil. Also plot the lift coefficient versus angle of attack for a 3-D rectangular wing with a symmetric section (incompressible flow). What is the slope of this line? Why, physically, are the two slopes similar / different? What happens when the angle of attack gets large?

Both are straight lines passing through (0,0) since the airfoil section is symmetric; the 2-D lift curve slope is less than or equal to  per radian;  the 3-D lift curve slope is lower because tip losses and downwash increase as angle of attack increases. At high angle of attack, the airfoil and the 3-D wing both stall.   As angle of attack keeps on increasing, the lift coefficient does increase again, reaching a maximum somewhere around 45 degrees (still much less than the peak reached at best L/D). The drag coefficient  is also very high at high angle of attack.

4.       (30 points)

A shock “reflects”  as a shock from a solid wall (or plane of symmetry) and as an expansion from a free surface.

      Kutta condition – stagnation point at the trailing edge.

Same as the conditions at the edge of a supersonic jet exiting a nozzle – slip line, velocity vectors parallel on either side of the slip line, pressure same across the slip line.

In supersonic case, there is no information passing between the upper and lower surfaces, hence no mechanism for making each adjust to the other.