Updated Slides of Compressible Flows
Updated Slides of Compressible Flows
Updated Slides of Compressible Flows
Chapter 9
Frank M. White
9.1. Introduction
• All previous chapters have been concerned with “low-speed’’
or “incompressible’’ flow, i.e., where the fluid velocity is much
less than its speed of sound.
• When a fluid moves at speeds comparable to its speed of
sound, density changes become significant and the flow is
termed compressible.
• Such flows are difficult to obtain in liquids, since high pressures
of order 1000 atm are needed to generate sonic velocities.
• In gases, however, a pressure ratio of only 2:1 will likely cause
sonic flow.
• Thus compressible gas flow is quite common, and this subject
is often called gas dynamics.
• The two most important and distinctive effects of
compressibility on flow are:
– Choking, wherein the duct flow rate is sharply limited by the
sonic condition, and
– Shock waves, which are nearly discontinuous property
changes in a supersonic flow.
• The proper criterion for a nearly incompressible flow was a
small Mach number
• For all real gases, cp, cv , and k vary with temperature but only
moderately; for example, cp of air increases 30 percent as
temperature increases from 0 to 5000°F. therefore it is quite
reasonable to assume constant specific heats.
• The changes in the internal energy û and enthalpy h of a
perfect gas are computed for constant specific heats as:
• This gives:
• And:
• Since a α T1/2,
• Combining:
• Momentum:
• Sound speed:
• Combining Eqs.
• For k = 1.4
It shows that the minimum area which can occur in a given
isentropic duct flow is the sonic, or critical, throat area.
In many flows a critical sonic throat is not actually present, and the
flow in the duct is either entirely subsonic or, more rarely, entirely
supersonic.
• For given stagnation conditions, the maximum possible mass
flow passes through a duct when its throat is at the critical or
sonic condition. Since
• The duct is then said to be choked and can carry no additional
mass flow unless the throat is widened.
• If the throat is constricted further, the mass flow through the
duct must decrease.
• The maximum flow is:
• For k = 1.4
• A common irreversibility
occurring in supersonic
internal or external flows
is the normal-shock wave
• Such shock waves are
very thin (a few
micrometers thick) and
approximate a
discontinuous change in
flow properties.
• The analysis is identical to
that of a fixed pressure
wave.
• Using all our basic one-dimensional steady-flow relations: