1. Chapter 12 Compressible Flow
12-1: Stagnation properties
12-2: 1-D isentropic flow
12-3: Isentropic flow through nozzles

2. Objectives:
Reviewing the concept of stagnation state, speed of sound and Mach number for compressible flow.
Relationship for stagnation state and isentropic fluid properties
Effect of area changes (for 1-dimensional) isentropic subsonic and supersonic flows
Concept of shock waves – normal and oblique shock waves
Rayleigh flow (heat transfer effect) and Fanno flow (friction effect)

3. What is compressible flow?
Compressibility:
measure of the change in density that will produced in the fluid by a specified change in pressure.
gases – highly compressible whereas most liquids have very low compressibility.
Changes in pressure due to the changes in velocity in the flow.
pressure changes -> induce density changes,
will later has influence on the flow.

4. What is compressible flow?
These density changes are important. Why???
density changes the temperature.
The temperature changes in the flow due to the kinetic energy, associated with the velocity changes.
Influence the flow.

5. What are the distinctive features of compressible flow?
Shock wave
Choked flow

6. Application of compressible flow?
High speed aircraft
Gas turbines: the flow in the blading and nozzles is compressible
Steam turbines: the flow in the nozzles and blades is compressible
Reciprocating engines: the flow of gases through the valves and in the intake and exhaust system
Combustion chambers: the study of combustion in many cases requires the knowledge of compressible fluid flow
Natural gas transmission lines: calculating the flow through pipelines

7. Assumptions
Gas is continuous, i.e. the motion of individual molecules does not have to be considered, the gas treated as continuous medium.
No chemical changes occur in the flow field.
The gas is perfect
Obeys perfect gas law
Specific heats at constant pressure and constant volume i.e. the gas is calorically perfect.
The gravitational effect on the flow field is negligible.
Magnetic and electrical effect negligible.
The effect of viscosity are negligible.

8. What are the properties at the flow field?
Velocity vector, V
Pressure, P
Density, ρ
Temperature, T

9. So what are the equations involving those 4 variables?
Conservation of mass (continuity equation)
Conservation of momentum (Newton’s law)
Conservation of energy (1st Law of Thermodynamics)
Equation of state

10. Some fundamental aspects of compressible flow
Isentropic flow in a streamtube
Definition of streamtube
The streamlines
Speed of sound
Mach number
Mach waves

13. Mach number : Example
Typical cruising speeds and altitudes for three commercial aircraft are:
Dash 8: cruising speeds: 500km/h at altitude of 4570m
Boeing 747: cruising speeds: 978km/h at altitude of 9150m.
Concorde: cruising speed: 2340km/h at altitude of 16600m.
Find the Mach number of these three aircraft when flying at these cruise conditions. Use properties of the standard atm.

14. Another example
A weak pressure wave (a sound wave) across which the pressure rise is 0.05kPa is travelling down a pipe into air at a temperature of 30˚C and a pressure of 105kPa. Estimate the velocity of the air behind the wave.
Answer: 0.119m/s

15. Mach waves
Consider a small solid body moving relative to a gas. In order for the gas to pass smoothly over the body, disturbances tend to be propagated a head of the body to ‘warn’ the gas of the approach body, i.e because the pressure at the surface of the body is greater that that in the surrounding gas, therefore pressure waves spread out of from the body.

16. Illustration of Mach waves (Figure 3.11 Oosthuizen)

17. Illustration of Mach waves

18. In conclusion… (for the 1st part)
The Mach number, M is a parameter that determines the importance of compressibility effects on a flow.
Incompressible flow: M << 1
Subsonic flow: M < 1
Transonic flow: M approximately equal to 1
Supersonic flow: M > 1
Hypersonic flow: M >> 1

19. One dimensional isentropic flow
Figure 4.2 (Oosthuizen)
Eg. 4.1
Eg. 4.2
Eg. 4.3

20. Example (4.1)
A gas which has molar mass of 39.9 and a specific ratio of 1.67 is discharged from a large chamber in which the pressure is 500kPa and the temperature is 30˚C through a nozzle. Assuming one-dimensional isentropic flow, find:
If the pressure at some section of the nozzle is 80kPa, the Mach no, temperature and velocity at this cross-section.
If the nozzle has a circular cross-section and if its diameter is 12mm at the section discussed in above, the mass flow rate through the nozzle.

21. Example (4.3)
Air flows through a nozzle which has inlet areas of 10m2. if the air has a velocity of 80m/s, a temperature of 28˚C, and a pressure of 700kPa at the inlet section and a pressure of 250kPa at the exit, find the mass flow rate through the nozzle and assuming one-dimensional isentropic flow, the velocity at the exit section of the nozzle.
Figure E4.3

22. Stagnation properties
Definition:
Exist if the flow at any point in a fluid stream was isentropically brought to rest.
To define the stagnation temperature, it is actually only necessary to require that the flow has to be adiabatically brought to rest.
To define stagnation pressure and density, it is requires that the flow be brought to rest isentropically.

23. Stagnation properties
If the entire flow is essentially isentropic and if the velocity is zero at some point in the flow, the the stagnation conditions will be those existing at the zero point as indicated in the figure.
Figure 4.4 Oosthuizen

24. Question to ponder…
A high speed aircraft is cruising in still air. How does the temperature of air at the nose of the aircraft differ from the temperature of air at some location/distance from the aircraft?

25. Answer…. We are to discuss the temperature change from an
airplane’s nose to far away from
the aircraft.
The temperature of the air rises as it approaches the nose because of the stagnation process.
Discussion:
In the frame of reference moving with the aircraft, the air decelerates from high speed to zero at the nose (stagnation point), and this causes the air temperature to rise.

26. Example (4.4)
Air flows over a body. The air flow in the freestream ahead of the body has a Mach no of 0.85 and a static pressure of 80kPa. Find the highest pressure acting on the surface of the body.
Figure E4.4

27. Example (4.5)
A pitot static tube is placed in a subsonic air flow. The static pressure and temperature in the flow are 80kPa and 12˚C respectively. The difference between the pitot and static pressures is measured using a manometer using a manometer and found to be 200mm of mercury. Find the air velocity and the Mach no.
Figure E4.5

28. Example (4.6)
A pitot static tube is placed in a subsonic air flow. The static pressure and temperature in the flow are 96kPa and 27˚C respectively. The difference between the pitot and static pressures is measured using a manometer using a manometer and found to be 32kPa. Find the air velocity
Assuming an incompressible flow
Assuming compressible flow

29. Area changes