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
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
a
VMa
Flow Regime Density Gradient
Shock Waves
Incompressible Negligible None
Subsonic Small None
Transonic Significant First appear
Supersonic Significant Significant
Hypersonic Dominant Dominant
3.0Ma
8.03.0 Ma
2.18.0 Ma
0.32.1 Ma
Ma0.3
Property Changes
v
p
kkk
c
ck
T
T
p
p
;1
2
)1/(
1
2
1
2
1
22
11
22
1
2
1
2
1
2
1
lnln
RT
dTc
p
pR
T
dTcs
p
dpR
T
dhds
T
dp
T
dhds
dpdhTds
vp
For isentropic (Δs=0), constant-heat-capacity conditions
Speed of Sound
0,,, 1111 VTp
VCVTp 2222 ,,,CVTp 1111 ,,,
0,,, 2222 VTp C
C
pressure waveΔx=nλ
1
2
1
211
21112
12211
1
pCCVCp
ApApVACVVmF
CVAVAV
right
Speed of Sound in Materials
2/1
Ka
pK
s
2/12/1
2/12/1
RTp
a
kRTkp
a
Most (perfect) gas conditions
High frequency waves (isothermal rather than isentropic expansion)
Solids and liquids (actually gases as well), where K is bulk modulus
2/1
2
0lim
s
pa
pC
Bulk modulus, not heat capacity ratio
Typical Sound Speeds (STP)Gas ft/s mi/hr m/s Air 1117 762 341 Ar 1038 708 316 C3H6 1009 688 307
C3H8 810 552 247
CH4 1447 987 441
CO 1136 775 346 CO2 869 593 265
H2 4236 2888 1291
H2O 1381 941 421
He 3280 2236 1000 N2 1136 775 346
O2 1061 723 323 238UF6 299 204 91
Solid ft/s mi/hr m/s Aluminum 16896 11520 5150 Beryllium 42290 28834 12890 Brass 11401 7773 3475 Brick 13701 9341 4176 Concrete 10600 7228 3231 Copper 12799 8726 3901 Cork 1312 895 400 Glass 12999 8863 3962 Gold 10630 7248 3240 Iron 19521 13310 5950 Hickory 13189 8992 4020 Ice 10499 7158 3200 Lead 3799 2590 1158 Platinum 10696 7292 3260 Rubber 328 224 100 Steel 19554 13332 5960 Wood 12999 8863 3962
Liquid ft/s mi/hr m/s
Benzene 4340 2959 1323 Carbon Tetrachloride 3080 2100 939 Ethanol 3810 2598 1161 Glycerin 6102 4161 1860 Kerosene 4390 2993 1338 Machine Oil 4240 2891 1292 Mercury 4757 3244 1450 Water, fresh 4888 3333 1490 Water, salt 4990 3402 1521
Generally, sound travels faster in solids than liquids and faster in liquids than gases.
Sound Speed vs. Molecular Speed
2/12/1
2222 33
RTp
ccccc zyx
Molecular theory of gases indicates that the average molecular speed is
Therefore, the average velocity of a molecule (speed in any specified direction) is
RTcRTcc xx 22
3
1
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.
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
Rcc
RTp
vp
)(
)(
Tkc
ck
Tcc
v
p
pp
Good approximation for most conditions far from critical points and at atmospheric pressure or lower.
Ideal Gas
Reasonable approximation for many gases. Generally also assume that the gas is non-dissociating.
Perfect Gas
Gas Flows
T
T
a
kV
k
aT
k
kRTc
T
T
Tc
V
TcV
TchconstV
hV
h
wqgzV
hgzV
h
p
p
p
p
v
02
2
2
0
0
2
2/10max
00
22
2
21
1
2
22
21
21
1
2
11
11
21
2
22
22
Perfect Gas
Mach-Number Relations
)1/(12
)1/(1
00
)1/(2
)1/(
00
2/12
2/1
00
20
2
11
2
11
2
11
2
11
kk
kkkk
Mak
T
T
Mak
T
T
p
p
Mak
T
T
a
a
Mak
T
T
Isentropic Expansion
Isentropic Expansion
Graphical Representation
20
15
10
5
0
stag
natio
n/st
atic
pro
pert
y
1086420
Mach Number
T0/T p0/2000T rho0/100T a0/a
Critical Properties
)1/(1
0
)1/(
0
2/1
0
0
1
2*
1
2*
1
2*
1
2*
k
kk
k
kp
p
ka
a
kT
T
0.8333 for k =1.4 (air)
0.9129 for k =1.4 (air)
0.5283 for k =1.4 (air)
0.6339 for k =1.4 (air)
Blunt Body Flows
Ma = 2.2
Sonic Flows
Ma = 3.0
Ma = 1.7
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
22
2
1
1
0
0
)()()(
V
dp
MaA
dA
V
dV
dadp
VdVdp
A
dA
V
dVd
constxAxVx
Area Changes Differ with Ma
Critical Area
)1(2
1
2
21
21
11
*
*
**
)()()(
k
k
k
Mak
MaA
A
V
V
A
A
constxAxVx
Mass Flow Relationships
2/1
0
02/100max
2/1
0
)1/(1
0max
*6847.0*6847.0)4.1(*
1
2*
1
2****
RT
ApRTAkm
RTk
kA
kVAm
k
2/1
/)1(
0
/2
00
2/10 1
1
2
kkk
p
p
p
p
k
k
p
RT
A
m
Choked flow
All flows
Normal Shock Wave
Shock Waves
)1/(1
21
)1/(
21
21
1,0
2,0
1,0
2,0
1,0
2,0
21
2
212
11
2
2
121
21
1
2
21
212
2
21
1
2
)1(2
1
2)1(
)1(
1
1
)1(22)1(
2)1(
)1(
)1(2
21
)1(21
1
kkk
kkMa
k
Mak
Mak
p
p
T
T
Mak
kkMaMak
T
T
V
V
Mak
Mak
kkMa
MakMa
kkMakp
p
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
x
uuux
uuu
x
uu
xx
uu
x
uu
xx
uu
x
uu
xx
uu
jii
jii
ii
ii
ii
ii
ii
ii
ii
ii
ii
2
432
43
2
21
21
1
1
1
1
1
1
11
11
11 central O(Δx2)
backward O(Δx)
forward O(Δx)
backward O(Δx2)
forward O(Δx2)
First Derivative FV Formulas
2
3
,2
32
3
,2
3
,
,
2/
12/1
212/1
212/1
12/1
2/112/1
12/12/1
12/1
iii
iii
iii
iii
iiii
iiii
iii
uuu
uuu
uuu
uuu
uuuu
uuuu
uuu central O(Δx2)
backward O(Δx)
forward O(Δx)
backward O(Δx2)
forward O(Δx2)
x
uu ii
2/12/1 General Formula
Second Derivative FD Formulas
221
221
211
2
2
2
x
uuux
uuux
uuu
iii
iii
iii
central O(Δx2)
backward O(Δx)
forward O(Δx)
First Derivative FV Formulas
xuuu
xuuu
xuuu
xuuu
xuuu
xuuu
iii
iii
iii
iii
iii
iii
/
,/
/
,/
/
,/
12/1
122/1
212/1
12/1
12/1
12/1 central O(Δx2)
backward O(Δx)
forward O(Δx)
x
uu ii
2/12/1 General Formula
Navier-Stokes: Cartesian Coord.
xxxxx
zx
yx
xx g
z
V
y
V
x
V
x
p
z
VV
y
VV
x
VV
t
V
2
2
2
2
2
2
yyyyy
zy
yy
xy g
z
V
y
V
x
V
y
p
z
VV
y
VV
x
VV
t
V
2
2
2
2
2
2
zzzzz
zz
yz
xz g
z
V
y
V
x
V
z
p
z
VV
y
VV
x
VV
t
V
2
2
2
2
2
2
x component
y component
z component
Outline of CFD model
Stoker: Geometry and Surface Areas
Sup
er h
eate
r #2
Su
pe
r h
ea
ter
#1
Boi
ler
Eco
no.
Secondary air~8 kg/s, 175 ºC
Spreader stokers~9 kg fuel/s
Secondary air~8 kg/s, 175 ºC
Grate air~24 kg/s, 175 ºC
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
y
xz
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.