Compressible Flow Introduction
Objectives:
1. Indicate when compressibility effects are important.
2. Classify flows with Mach Number.
3. Introduce equations for adiabatic, isentropic flows.
Larry Baxter
Ch En 374
Flow Classifications
V
Ma 
a
Flow Regime
Ma  0.3
Density
Gradient
Incompressible Negligible
Shock Waves
None
0.3  Ma  0.8 Subsonic
Small
None
0.8  Ma  1.2 Transonic
Significant
First appear
1.2  Ma  3.0 Supersonic
Significant
Significant
Dominant
Dominant
3.0  Ma
Hypersonic
Property Changes
dp
dh dp
Tds  dh 
 ds 


T T
2
2
2
dh
dp
1 ds  1 T  R 1 p
2
s  
1
c p dT
T
p2
cv dT
2
 R ln

 R ln
p1 1 T
1
2
For isentropic (Δs=0), constant-heat-capacity conditions
p2  T2 
  
p1  T1 
k /( k 1)
k
cp
 2 
   ; k 
cv
 1 
Speed of Sound
C
p1, 1,T1,V1  0
p2, 2,T2,V2  0
p1, 1,T1,V1  C
C
p2, 2,T2,V2  C  V
pressure wave
Δx=nλ

1AV1   2 AV2  V  C
1  
 Fright  m V2  V1   1AC V   p1A  p2 A

p   
2
1 

p  1CV  1C
C 
1  
 
1 
2
Speed of Sound in Materials
 p
p
lim C 
a
 
 0


2
1/ 2
 kp 
a   
  
1/ 2
 p
a   

 kRT 
1/ 2
1/ 2



s
Most (perfect) gas conditions
High frequency waves (isothermal
rather than isentropic expansion)
 RT 
1/ 2
1/ 2
K 
p
K
 a   
 s

Solids and liquids (actually gases
as well), where K is bulk modulus
Bulk modulus, not heat capacity ratio
Typical Sound Speeds (STP)
Gas
Air
Ar
C3H6
1117
1038
1009
mi/hr
762
708
688
C3H8
810
552
247
CH4
1447
987
441
CO
CO2
1136
869
775
593
346
265
H2
4236
2888
1291
H2O
1381
941
421
He
N2
3280
1136
2236
775
1000
346
O2
1061
723
323
238
299
204
91
UF6
ft/s
Liquid
Benzene
Carbon Tetrachloride
Ethanol
Glycerin
Kerosene
Machine Oil
Mercury
Water, fresh
Water, salt
ft/s
4340
3080
3810
6102
4390
4240
4757
4888
4990
m/s
341
316
307
mi/hr
2959
2100
2598
4161
2993
2891
3244
3333
3402
m/s
1323
939
1161
1860
1338
1292
1450
1490
1521
Solid
Aluminum
Beryllium
Brass
Brick
Concrete
Copper
Cork
Glass
Gold
Iron
Hickory
Ice
Lead
Platinum
Rubber
Steel
Wood
ft/s
16896
42290
11401
13701
10600
12799
1312
12999
10630
19521
13189
10499
3799
10696
328
19554
12999
mi/hr
11520
28834
7773
9341
7228
8726
895
8863
7248
13310
8992
7158
2590
7292
224
13332
8863
m/s
5150
12890
3475
4176
3231
3901
400
3962
3240
5950
4020
3200
1158
3260
100
5960
3962
Generally, sound travels
faster in solids than liquids
and faster in liquids than
gases.
Sound Speed vs. Molecular Speed
Molecular theory of gases indicates that the average
molecular speed is
1/ 2
 3p 
1/ 2
2
2
2
2
  3RT 
c  c x  c y  c z c  
  
Therefore, the average velocity of a molecule (speed in any
specified direction) is
1 2
c  c  RT  c x  RT
3
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.
2
x
Sound Travels in Longitudinal Waves
Light, cello strings, and surfing waves are transverse waves.
Sound travels in a longitudinal or compression wave.
Ideal and Perfect Gases
Ideal Gas
p  RT
c p  cv  R
Good approximation for most conditions far
from critical points and at atmospheric
pressure or lower.
Perfect Gas
c p  c p (T )
Reasonable approximation for many gases.
Generally also assume that the gas is noncp
k
 k (T ) dissociating.
cv
Gas Flows
V12
V22
h1 
 gz1  h2 
 gz2  q  w v
2
2
V12
V22
h1 
 h2 
 const  h0  c pT0
2
2
1/ 2
Perfect Gas
Vmax  2c pT0 
T0
V2
1

2c pT0 T
2
k
a


c pT   R
T 
k 1
 k  1
V 2 k  1 T0
1

2
2a
T
Mach-Number Relations
T0
k 1 2
 1
Ma
T
2
1/ 2
a0  T0 
 
a T 
p0  T0 
 
p T 
 k 1 2
 1 
Ma 
2


k /( k 1)
1/( k 1)
 0  T0 
 
 T 
1/ 2
 k 1 2
 1 
Ma 
2


 k 1 2
 1 
Ma 
2


k /( k 1)
Isentropic
Expansion
1 /( k 1)
Isentropic
Expansion
Graphical Representation
stagnation/static property
20
T0/T
p0/2000T
rho0/100T
a0/a
15
10
5
0
0
2
6
4
Mach Number
8
10
Critical Properties
T*
2

T0 k  1
a*  2 

a0  k  1
0.8333 for k =1.4 (air)
1/ 2
p*  2 

p0  k  1
0.9129 for k =1.4 (air)
k /( k 1)
*  2 

 0  k  1
0.5283 for k =1.4 (air)
1/( k 1)
0.6339 for k =1.4 (air)
Blunt Body Flows
Ma = 2.2
Sonic Flows
Ma = 1.7
Ma = 3.0
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).
Supersonic vs. Subsonic Flows
 ( x )V ( x )A( x )  const
d dV dA


0

V
A
dp

 VdV  0
dp  a d
2
dV dA
1
dp


2
V
A 1  Ma
V 2
Area Changes Differ with Ma
Critical Area
 ( x )V ( x )A( x )  const
A
 V*

A*  * V
  k  1 2 
1 
Ma 

A
1
2 




A * Ma 
 k  1





 2 
k 1
2( k 1)
Mass Flow Relationships
Choked flow
1/( k 1)
1/ 2
 2 

m *max   * A * V *  0 

 k  1
 2k

A*
RT0 
 k 1

0.6847 p0 A *
1/ 2

m *max (k  1.4)  0.6847 A * 0 RT0  
1/ 2
RT0 
All flows
1/ 2
 2k  p 

m RT0 
 

A
p0
 k  1  p0 
2/k
  p
1   
  p0 
( k 1) / k
1/ 2
 

 
Normal Shock Wave
Shock Waves


p2
1

2kMa12  (k  1)
p1 k  1
Ma22 
k  1Ma12  2
2kMa12  (k  1)
2
(k  1)Ma12
V1


2
1 (k  1)Ma1  2 V2


T2
2kMa12  (k  1)
2
 (k  1)Ma1  2
T1
k  12 Ma12
T0,2
1
T0,1
0,2 p0,2  (k  1)Ma 


 0,1 p0,1  (k  1)Ma12  2 
2
1
k /( k 1)


k 1


2
2
kMa

(
k

1
)
1


1/( k 1)
Normal Shock Wave
Nozzle Performance
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).
Three Classes of CFD
• Finite Difference
• 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
First Derivative FD Formulas
ui 1  ui 1 ui 1  ui 1

x i 1  x i 1
2x
central O(Δx2)
ui  ui 1 ui  ui 1

x i  x i 1
x
backward O(Δx)
ui 1  ui ui 1  ui

x i 1  x i
x
forward O(Δx)
3ui  4ui 1  u j 2
2x
 3ui  4ui 1  u j  2
2x
backward O(Δx2)
forward O(Δx2)
First Derivative FV Formulas
ui 1/ 2  ui 1/ 2
x
General Formula
u i 1/ 2  u i  u i 1  / 2
central O(Δx2)
u i 1/ 2  u i , u i 1/ 2  u i 1
backward O(Δx)
u i 1/ 2  u i 1, u i 1/ 2  u i
forward O(Δx)
u i 1/ 2 
u i 1/ 2 
u i 1/ 2 
u i 1/ 2 
3u i  u i 1
,
2
3u i 1  u i 2
2
3u i 1  u i  2
,
2
3u i  u i 1
2
backward O(Δx2)
forward O(Δx2)
Second Derivative FD Formulas
ui 1  2ui  ui 1
2
x
ui  2ui 1  ui 2
2
x
ui  2ui 1  ui  2
x 2
central O(Δx2)
backward O(Δx)
forward O(Δx)
First Derivative FV Formulas
ui 1/ 2  ui 1/ 2
x
General Formula
ui 1/ 2  ui 1  ui  / x,
ui 1/ 2  ui  ui 1  / x
central O(Δx2)
ui 1/ 2  ui  ui 1  / x,
backward O(Δx)
ui 1/ 2  ui 1  ui 2  / x
ui 1/ 2  ui  2  ui 1  / x,
ui 1/ 2  ui 1  ui  / x
forward O(Δx)
Navier-Stokes: Cartesian Coord.
x component
  2Vx  2Vx  2Vx 
 Vx
Vx
Vx
Vx 
p
  g x
  
 
 Vx
 Vy
 Vz
   2 

2
2 
x
y
z 
x
y
z 
 t
 x
y component
  2Vy  2Vy  2Vy
Vy
Vy
Vy 
 Vy
p
  
 
 Vx
 Vy
 Vz
  2 

2
2

x
y
z 
y

x

y

z
 t


  g y


z component
  2Vz  2Vz  2Vz 
 Vz
Vz
Vz
Vz 
p
  g z
  
 
 Vx
 Vy
 Vz
   2 

2
2 
x
y
z 
z
y
z 
 t
 x
Outline of CFD model
y
Econo.
Boiler
Secondary air
~8 kg/s, 175 ºC
Secondary air
~8 kg/s, 175 ºC
x
z
Super heater #1
Super heater #2: 194 m2 / 2090 ft2
Super heater #1: 364 m2 / 3920 ft2
Boiler Bank:
1181 m2 / 12700 ft2
Economizer:
330 m2 / 3550 ft2
Super heater #2
Stoker: Geometry and Surface
Areas
Spreader stokers
~9 kg fuel/s
Grate air
~24 kg/s, 175 ºC
Computational mesh
Cloud (Particle) Trajectories
Oxygen Mass Fraction Contours
Velocity and Heat Release Vary
Initial Deposition Rates Vary
Temporal Deposition Variation
Gas Temperature Field
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.