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.