fluid_mechanics

11.8 Chapter Summary and Study Guide 637
compressible flow
ideal gas
internal energy
enthalpy
specific heat ratio
entropy
adiabatic
isentropic
Mach number
speed of sound
stagnation pressure
subsonic
sonic
Mach wave
supersonic
Mach cone
transonic flows
hypersonic flows
converging–diverging
duct
throat
temperature–entropy
(T–s) diagram
choked flow
critical state
critical pressure ratio
normal shock wave
oblique shock wave
expansion wave
overexpanded
underexpanded
nonisentropic flow
Fanno flow
Rayleigh flow
may involve substantial fluid density changes at higher speeds. At lower speeds, gas and vapor
density changes are not appreciable and so these flows may be treated as incompressible.
Since fluid density and other fluid property changes are significant in compressible flows,
property relationships are important. An ideal gas, with well-defined fluid property relationships,
is used as an approximation of an actual gas. This profound simplification still allows useful conclusions
to be made about compressible flows.
The Mach number is a key variable in compressible flow theory. Most easily understood as
the ratio of the local speed of flow and the speed of sound in the flowing fluid, it is a measure
of the extent to which the flow is compressible or not. It is used to define categories of compressible
flows which range from subsonic (Mach number less than 1) to supersonic (Mach number
greater than 1). The speed of sound in a truly incompressible fluid is infinite so the Mach
numbers associated with liquid flows are generally low.
The notion of an isentropic or constant entropy flow is introduced. The most important isentropic
flow is one that is adiabatic (no heat transfer to or from the flowing fluid) and frictionless
(zero viscosity). This simplification, like the one associated with approximating real gases with an
ideal gas, leads to useful results including trends associated with accelerating and decelerating
flows through converging, diverging, and converging–diverging flow paths. Phenomena including
flow choking, acceleration in a diverging passage, deceleration in a converging passage, and the
achievement of supersonic flows are discussed.
Three major nonisentropic compressible flows considered in this chapter are Fanno flows,
Rayleigh flows, and flows across normal shock waves. Unusual outcomes include the conclusions
that friction can accelerate a subsonic Fanno flow, heating can result in fluid temperature reduction
in a subsonic Rayleigh flow, and a flow can decelerate from supersonic flow to subsonic
flow across a very small distance. The value of temperature–entropy (T–s) diagramming of flows
to better understand them is demonstrated.
Numerous formulas describing a variety of ideal gas compressible flows are derived. These
formulas can be easily solved with computers. However, to provide the learner with a better grasp
of the details of a compressible flow process, a graphical approach, albeit approximate, is used.
The striking analogy between compressible and open-channel flows leads to a brief discussion
of the usefulness of a ripple tank or water table to simulate compressible flows.
Expansion and compression Mach waves associated with two-dimensional compressible
flows are introduced as is the formation of oblique shock waves from compression Mach waves.
The following checklist provides a study guide for this chapter. When your study of the entire
chapter and end-of-chapter exercises is completed you should be able to
write out the meanings of the terms listed here in the margin and understand each of the
related concepts. These terms are particularly important and are set in italic, bold, and color
type in the text.
estimate the change in ideal gas properties in a compressible flow.
calculate Mach number value for a specific compressible flow.
estimate when a flow may be considered incompressible and when it must be considered
compressible to preserve accuracy.
estimate details of isentropic flows of an ideal gas though converging, diverging, and converging–diverging
passages.
estimate details of nonisentropic Fanno and Rayleigh flows and flows across normal shock
waves.
explain the analogy between compressible and open-channel flows.
Some of the important equations in this chapter are:
Ideal gas equation
of state
r p
RT
(11.1)
Internal energy change ǔ 2 ǔ 1 c v 1T 2 T 1 2
(11.5)
Enthalpy ȟ ǔ p (11.6)
r