1. Compressible Flow Introduction
Objectives:
Indicate when compressibility effects are important.
Classify flows with Mach Number.
Introduce equations for adiabatic, isentropic flows.
Larry Baxter
Ch En 374

2. Flow Classifications Flow Regime Density Gradient Shock Waves
Incompressible
Negligible
None
Subsonic
Small
Transonic
Significant
First appear
Supersonic
Hypersonic
Dominant

3. Property Changes
For isentropic (Δs=0), constant-heat-capacity conditions

4. Speed of Sound C C
pressure wave
Δx=nλ

5. Speed of Sound in Materials
Most (perfect) gas conditions
High frequency waves (isothermal rather than isentropic expansion)
Solids and liquids (actually gases as well), where K is bulk modulus
Bulk modulus, not heat capacity ratio

6. Typical Sound Speeds (STP)
Generally, sound travels faster in solids than liquids and faster in liquids than gases.

7. Sound Speed vs. Molecular Speed
Molecular theory of gases indicates that the average molecular speed is
Therefore, the average velocity of a molecule (speed in any specified direction) is
In the case of a sound wave, molecules don’t have time to adjust their temperatures to the rapid change in pressure, so their temperature changes slightly inside the wave. If this change is completely adiabatic – generally a good assumption – the specific heat ratio accounts for the temperature change. Thus, the speed of sound is identically equal to the speed at which molecules travel in any one direction under conditions of a propagating wave.

8. Sound Travels in Longitudinal Waves
Light, cello strings, and surfing waves are transverse waves.
Sound travels in a longitudinal or compression wave.

9. Ideal and Perfect Gases
Ideal Gas
Good approximation for most conditions far from critical points and at atmospheric pressure or lower.
Perfect Gas
Reasonable approximation for many gases. Generally also assume that the gas is non-dissociating.

10. Gas Flows
Perfect Gas

11. Mach-Number Relations
Isentropic Expansion
Isentropic Expansion

12. Graphical Representation

13. Critical Properties 0.8333 for k =1.4 (air) 0.9129 for k =1.4 (air)

14. Blunt Body Flows
Ma = 2.2

15. Sonic Flows
Ma = 1.7
Ma = 3.0

16. Compressible Flow Essentials
Know what a Mach number is and the regimes of flow as indicated by the Mach number. (Mach number is ratio of velocity to the speed of sound at the same conditions. Mach numbers of 0.3, 0.8, 1.2, and 3 separate incompressible, subsonic, transonic, supersonic, and hypersonic regimes, respectively).
Know how pressure, temperature, density, and velocity change across a normal shock wave. (First three all increase in direction of decreasing velocity, with pressure increasing the most. Velocity decreases from supersonic to subsonic value, with post-shock velocity decreasing as pre-shock velocity increases).

17. Supersonic vs. Subsonic Flows

18. Area Changes Differ with Ma

19. Critical Area

20. Mass Flow Relationships
Choked flow
All flows

21. Normal Shock Wave

22. Shock Waves

23. Normal Shock Wave

24. Nozzle Performance

25. Compressible Flow Essentials
Be able to explain on a molecular level the origin of the changes in pressure, temperature and density. (Molecules collide into one another or a surface, exchanging kinetic energy for pressure or temperature. Ideal gas law still applies to give relationship between density, pressure, and temperature).
Know how streamlining designs differ for compressible flows compared to incompressible flows. (Leading edges are relatively sharp edges rather than rounded corners and heat dissipation is a major issue).

26. Three Classes of CFD Finite Difference Finite Volume Finite Element
Original and still widely used formulation for CFD describes flow fields as values of velocity vectors at discrete points.
Finite Volume
Close cousin to finite difference, but discrete points represent average values of velocities in a volume rather than at a point.
Finite Element
Most commonly used for heat transfer and stress calculations in solid bodies rather than fluid mechanics (because of stability issues).
Much easier to describe general/complex geometries than FD/FV techniques.
Solves for dependent variable (velocity, temperature, stress) with variations across element by minimizing an objective function

27. First Derivative FD Formulas
central O(Δx2)
backward O(Δx)
forward O(Δx)
backward O(Δx2)
forward O(Δx2)

28. First Derivative FV Formulas
General Formula
central O(Δx2)
backward O(Δx)
forward O(Δx)
backward O(Δx2)
forward O(Δx2)

29. Second Derivative FD Formulas
central O(Δx2)
backward O(Δx)
forward O(Δx)

30. First Derivative FV Formulas
General Formula
central O(Δx2)
backward O(Δx)
forward O(Δx)

31. Navier-Stokes: Cartesian Coord.
x component
y component
z component

32. Outline of CFD model

33. Stoker: Geometry and Surface Areas
Super heater #2: 194 m2 / ft2
Super heater #1: 364 m2 / ft2
Boiler
Boiler Bank: m2 / ft2
Super heater #1
Super heater #2
Economizer: m2 / ft2
Econo. y
Secondary air
~8 kg/s, 175 ºC
Secondary air x
~8 kg/s, 175 ºC z
Spreader stokers
Grate air
~9 kg fuel/s
~24 kg/s, 175 ºC

34. Computational mesh

35. Cloud (Particle) Trajectories

36. Oxygen Mass Fraction Contours

37. Velocity and Heat Release Vary

38. Initial Deposition Rates Vary

39. Temporal Deposition Variation

40. Gas Temperature Field

41. CFD Essentials
Know the distinguishing characteristics of finite difference, finite volume, and finite element approaches to numerical methods differ.
Know where to find (in these notes) common algebraic approximations for first and second derivatives for FD and FV approaches and the accuracy of the approximation.
Know (conceptually) how the algebraic approximations are substituted into the partial differential equations and how these are then solved.
Recognize that entire careers are dedicated to small fractions of CFD problem solving because of issues of convergence, stability, non-uniform grids, turbulence, etc.