1
Chapter 6
Compressible flow
Introduction
2
When is a flow compressible?
Pressure
Fluid element Pathline
A compressible flow differs from an incompressible
one in that significant change happens in the density.
0?
dt
d?
3
Density of a fluid element increases or decreases as
the pressure does in compressible flow
smV /50? smV /100? smV /300?
2
2
1 Vppp
a ?????
%54.1
50
???
?
?
???
? ?
ap
p %17.6
1 0 0
???
?
?
???
? ?
ap
p %5.55
3 0 0
???
?
?
???
? ?
ap
p
RTp ??
Cpk ??
4
When is compressibility important?
When a fluid moves at high speed,density changes
become significant
c
VMa ?
- flow speed
- sound speed
Ma<0.3 incompressible
Ma>0.3 compressible
5
Examples of high speed flow
Ma=0.85 Ma=2.35
Ma=4 Ma=25
6
Flow speed in a typical aeroengine
7
History
?Untill the late 1930s,engineers could ignore the
compressibility of air
?Aerodynamics,a subject to compressible gas and
air flow,makes possible the remarkable progress in
modern aeronautics and astronautics
Ernst Mach Prandtl von Karman
8
Distinctive features of compressible flow
Shock wave Supersonic nozzle
Laval Nozzle
Ma<1 Ma>1
9
Assumptions for compressible flow
TCu v? TCh p?Internal energy u and enthalpy h:
?Adiabatic flow with negligible
heat transfer
?Reversible process with fluid
friction being neglected
isentropic flow
RTp ???Perfect gas
vp CCk /?
Specific heat ration:
RCC vp ??
1?? k
kRC
p 1?? k
RC
v
10
?
dpdhTd s ??
For isentropic flow,
)1/(
1
2
1
2
?
???
?
???
?? kk
T
T
p
p
c ons tpk ?
?
k
p
p
???
?
???
?
?
1
2
1
2
?
?
11
Bernoulli equation for compressible flow
0)()(21 1221222
1
?????? zzgVVdp?
For two arbitrary points 1,2 along a streamline in
steady flow
Cpk ?? ? ? ???
?
???
? ?
?????
???
1
1
2
21
1
1
2
2
1 11 ??
??? ppk kCCk kdp kk
2
2
2
2
2
1
2
1
1
1
2
1
12
1
1 gzV
p
k
kgzVp
k
k ??
????? ??
12
1?? k
kRC
p
c o n s tVpk k ??? 2211 ?
c o n s tVRTk k ??? 2211
TCh p?
c o n stVh ?? 221 *h?
13
The rate of propagation of a pressure pulse of
infinitesimal strength through a still fluid
X
V=0t=0
X+dX
dV c
X
V=0t=dt
ct=Ndt
The speed of sound
14
-c+ΔV
p+Δp
ρ+Δρ
T+ΔT
V=-c
p
ρ
T
Control volume
AcAcVd )())(( ????? ???
Continuity:
Momentum:
)()( cVcAcpAApp ??????? ?
?d
dpc ?2
?T=Const (Newton):
RTc ?
20% lower than
measurement k RTc ?
?Isentropic (Laplace):
cons tpk ??
15
Propagation of pressure disturbance
in a moving fluid
U=cU c-U c+U
U>c
U
ca rc sin??
U?
a?
16
Shock Wave
A Shock wave is a very thin region across which flow
properties change dramatically,its thickness is in the
order of 10-8 m.
Flow across the shock
wave is not an isentropic
process.
17
18
Sonic Boom