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```									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    Density       Shock Waves
Incompressible Negligible    None
Ma  0.3
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

3.0  Ma      Hypersonic     Dominant      Dominant
Property Changes
dpdh dp
Tds  dh     ds    
        T T
2        2              2
dh      dp
 ds   T  R  p
1      1       1
2
c p dT            p2
2
cv dT        2
s                 R ln            R ln
1
T             p1 1 T             1

For isentropic (Δs=0), constant-heat-capacity conditions
k /( k 1)           k
p2  T2                          2    cp
                             ;k 
p1  T1 
 
 
 1     cv
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   
p  1CV  1C    2
C 
2
1 
       
1             1 

Speed of Sound in Materials
1/ 2
p     p 
lim C 2
a    
 0         
   s
1/ 2
 kp 
a             kRT            Most (perfect) gas conditions
1/ 2
  
 
1/ 2
 p                             High frequency waves (isothermal
a           RT 
1/ 2
                              rather than isentropic expansion)
 

1/ 2
p      K                     Solids and liquids (actually gases
K      a 

 s                          as well), where K is bulk modulus
Bulk modulus, not heat capacity ratio
Typical Sound Speeds (STP)
Gas       ft/s          mi/hr         m/s
Air              1117        762            341               Solid       ft/s        mi/hr      m/s
Ar               1038        708            316               Aluminum       16896     11520        5150
C3H6             1009        688            307               Beryllium      42290     28834      12890
Brass          11401       7773       3475
C3H8             810           552          247
Brick          13701       9341       4176
CH4              1447          987          441               Concrete       10600       7228       3231
CO               1136          775          346               Copper         12799       8726       3901
CO2               869          593          265               Cork             1312        895       400
H2               4236      2888         1291                  Glass          12999       8863       3962
H2O              1381          941          421               Gold           10630       7248       3240
Iron           19521     13310        5950
He               3280      2236         1000
Hickory        13189       8992       4020
N2               1136       775          346
Ice            10499       7158       3200
O2               1061          723          323               Lead             3799      2590       1158
238
UF6           299           204          91                Platinum       10696       7292       3260
Rubber            328        224       100
Liquid                      ft/s          mi/hr       m/s         Steel          19554     13332        5960
Benzene                            4340      2959        1323     Wood           12999       8863       3962
Carbon Tetrachloride               3080      2100         939
Ethanol                            3810      2598        1161
Glycerin                           6102      4161        1860   Generally, sound travels
Kerosene                           4390      2993        1338
Machine Oil                        4240      2891        1292
faster in solids than liquids
Mercury                            4757      3244        1450   and faster in liquids than
Water, fresh                       4888      3333        1490
Water, salt                        4990      3402        1521   gases.
Sound Speed vs. Molecular Speed
Molecular theory of gases indicates that the average
molecular speed is
1/ 2
 3p 
c  c x  cy  cz c        3RT 
2    2    2   2                    1/ 2
  
    
Therefore, the average velocity of a molecule (speed in any
specified direction) is
1 2
c  c  RT  c x  RT
2
x
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.
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           Good approximation for most conditions far
from critical points and at atmospheric
c p  cv  R      pressure or lower.

Perfect Gas
c p  c p (T )  Reasonable approximation for many gases.
Generally also assume that the gas is non-
cp
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
Vmax  2c pT0 
1/ 2
Perfect Gas
V2     T0
1       
2c pT0 T
     k       a2
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                      1/ 2
a0  T0             k 1 2
              1    Ma 
a T                   2    
k /( k 1)                     k /( k 1)
p0  T0                    k 1 2                    Isentropic
                     1    Ma 
p T                          2                      Expansion
1 /( k 1)                     1 /( k 1)
 0  T0                  k 1 2                    Isentropic
                   1    Ma 
 T                         2                      Expansion
Graphical Representation

20       T0/T
p0/2000T
rho0/100T
a0/a
stagnation/static property

15

10

5

0
0               2   4                 6   8   10
Mach Number
Critical Properties

T*   2
                         0.8333 for k =1.4 (air)
T0 k  1
1/ 2
a*  2 
                         0.9129 for k =1.4 (air)
a0  k  1

k /( k 1)
p*  2 

p0  k  1
0.5283 for k =1.4 (air)

1 /( k 1)
*  2 
                       0.6339 for k =1.4 (air)
 0  k  1

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
            
V   A 1  Ma 2
V 2
Area Changes Differ with Ma
Critical Area

 ( x )V ( x )A( x )  const
A        V*

A*  * V
k 1
  k  1 2              2( k 1)

1   2 Ma 
          
A    1
                  
A * Ma       k  1 
            
      2     
Mass Flow Relationships
Choked flow
1/( k 1)                           1/ 2
 *max   * A * V *  0  2 
m                                  A*
 2k       
RT0 
 k  1           k 1     
(k  1.4)  0.6847A * 0 RT0  
                                       1/ 2    0.6847p0 A *
m *max
RT0 1/ 2

All flows
1/ 2
 RT0 1/ 2  2k  p                p                    
2/k               ( k 1) / k
m                                                             
                    1                     
A   p0        k  1  p0 
 
p 
  0                    
                                               
Normal Shock Wave
Shock Waves
p2

1
p1 k  1

2kMa1  (k  1)
2

Ma2 
2     k  1Ma12  2
2kMa1  (k  1)
2

2   (k  1)Ma12
V1
             
1 (k  1)Ma1  2 V2
2

                
2kMa1  (k  1)
2
T2
 (k  1)Ma1  2
2

T1                   k  12 Ma12
T0,2
1
T0,1
k /( k 1)                         1 /( k 1)
 0,2 p0,2  (k  1)Ma     2
     k 1        
                    1
 0,1 p0,1  (k  1)Ma1  2 
2




2kMa1  (k  1) 
2
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
relatively sharp edges rather than rounded
corners and heat dissipation is a major issue).
Three Classes of CFD
• Finite Difference                    • Finite Element
• Original and still widely used       • Most commonly used for heat
formulation for CFD describes          transfer and stress
flow fields as values of               calculations in solid bodies
velocity vectors at discrete           rather than fluid mechanics
points.                                (because of stability issues).
• Finite Volume                           • Much easier to describe
• Close cousin to finite                 general/complex geometries
difference, but discrete points        than FD/FV techniques.
represent average values of          • Solves for dependent variable
velocities in a volume rather          (velocity, temperature, stress)
than at a point.                       with variations across element
by minimizing an objective
function
First Derivative FD Formulas
ui 1  ui 1 ui 1  ui 1
             central O(Δx2)
x i 1  x i 1   2x
ui  ui 1 ui  ui 1
                backward O(Δx)
x i  x i 1   x
ui 1  ui ui 1  ui
                forward O(Δx)
x i 1  x i   x
3ui  4ui 1  u j 2
backward O(Δx2)
2x
 3ui  4ui 1  u j  2      forward O(Δx2)
2x
First Derivative FV Formulas
ui 1/ 2  ui 1/ 2   General Formula
x
ui 1/ 2  ui  ui 1  / 2             central O(Δx2)
ui 1/ 2  u i , ui 1/ 2  ui 1        backward O(Δx)
ui 1/ 2  u i 1, ui 1/ 2  u i        forward O(Δx)
3u i  u i 1
ui 1/ 2                  ,
2                     backward O(Δx2)
3ui 1  ui 2
ui 1/ 2 
2
3u i 1  ui  2
ui 1/ 2                     ,
2
forward O(Δx2)
3ui  ui 1
ui 1/ 2 
2
Second Derivative FD Formulas

ui 1  2ui  ui 1
central O(Δx2)
x  2

ui  2ui 1  ui 2
backward O(Δx)
x 2

ui  2ui 1  ui  2
forward O(Δx)
x 2
First Derivative FV Formulas
ui 1/ 2  ui 1/ 2   General Formula
x

ui 1/ 2  ui 1  ui  / x,
central O(Δx2)
ui 1/ 2  ui  ui 1  / x

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
 Vx     Vx     Vx     Vx     p       2Vx  2Vx  2Vx 
 t  Vx x  Vy y  Vz z    x    x 2  y 2  z 2   g x
                                                            
                                                            

y component
 Vy     Vy     Vy     Vy     p       2Vy  2Vy  2Vy    
                                                                 g y
 t  Vx x  Vy y  Vz z    y    x 2  y 2  z 2
                                  
                                                              

z component
 Vz     Vz     Vz     Vz     p       2Vz  2Vz  2Vz 
 t  Vx x  Vy y  Vz z    z    x 2  y 2  z 2   g z
                                                            
                                                            
Outline of CFD model
Stoker: Geometry and Surface
Areas

Super heater #2: 194 m2 / 2090 ft2

Super heater #1
Super heater #2
Super heater #1: 364 m2 / 3920 ft2

Boiler
Boiler Bank:    1181 m2 / 12700 ft2
Economizer:      330 m2 / 3550 ft2

Econo.
y              Secondary air
~8 kg/s, 175 ºC

Secondary air
x                                                                    ~8 kg/s, 175 ºC
z
~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.

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