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Internal Combustion Engines I:

Gas Turbines

Tim Lieuwen

Affiliation:

Professor

School of Aerospace Engineering

Georgia Institute of Technology

Email: [email protected]

Ph. 404-894-3041

2012 Princeton-CEFRC Summer School on Combustion

Course Length: 6 hrs

June 25 – 26, 2012

mailto:[email protected]

CEFRC Summer School, Copyright T. Lieuwen, 2012

Course Outline

• Introduction

• Flashback and Flameholding

• Flame Stabilization and Blowoff

• Combustion Instabilities

• Flame Dynamics

2


Course Outline

• Introduction

– Constraints, metrics, future outlook




• Flame Dynamics

3

CEFRC Summer School, Copyright T. Lieuwen, 2012 4

• Conclusions – Combustor has little effect upon cycle efficiency (e.g. fuel –> kilowatts) or

specific power

– Combustor does however have important impacts on

• Realizability of certain cycles

– E.g., steam addition, water addition, EGR, etc.

• Engine operational limits and transient response

• Emissions from plant

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40

Pressure Ratio

Therm

al E

ffic

iency Microturbine

Heavy

frame Gas

Turbine

Aeroengine• Example: Ideal Brayton Cycle – th = 1- (Pr)-( -1)/

• Pr = compressor pressure ratio

• = Cp/Cv, ratio of specific heats

Role of Combustor within Larger

Energy System


Combustor Performance Metrics

5

• What are important combustor performance parameters?

– Operability • Blow out

• Combustion instability

• Flash back

• Autoignition

– Low pollutant emissions

– Fuel flexibility

– Good turndown

– Transient response

Fuel

Air


Blowoff

Combustion

Instabilities

Turndown

Emissions

NOX, CO, CO2

Cost/

Complexity

Tradeoffs and Challenges

6

7

• Source: L.

Witherspoon and

A. Pocengal,

Power Engineering

October 2008

Alternative Fuel Compositions


Natural Gas Composition Variability

8

Source: C. Carson, Rolls Royce Canada


Operability issues have caused

significant problems in deployment of

low NOX technologies

9

• Power – Example: Broken part

replacement largest non-fuel

related cost for F class gas

turbines

• Industrial

• Residential – Example: issues in EU with

deployment of low NOX water

heaters, burners

Goy et al., in Combustion instabilities in gas turbine engines: operational

experience, fundamental mechanisms, and modeling,

T. Lieuwen and V. Yang, Editors. 2005. p. 163-175.


Financial Times

Power in Latin America 23 July 99, Issue 49

Daggers Drawn over Nehuenco

“The Patience of Chile’s Colbun power company has finally run out over the continued non-

performance of the Siemens-built Nehuenco generating plant. Exasperated by repeated

break-downs at the new plant and under pressure from increasingly reluctant insurers – (and

with lawsuits looking likely) – the generator announced that it will not accept the $140m

combined-cycle plant - built and delivered by the Germany equipment manufacturer.

Siemens, together with Italy’s Ansaldo, took the turnkey contract for the 350 MW plant in 1996

and should have had it in service by May of last year. The startup was delayed till January.

Since then matters have worsened. There have been two major breakdowns and, says

Colbun, there have been no satisfactory explanations.

The trouble could not have come worse for Colbun. The manly hydroelectric generator, which

is controlled by a consortium made up of Belgium’s Tractebel, Spain’s Iberdrola and the local

Matte and Yaconi-Santa Cruz groups, has been crippled by severe drought in Chile, which has

slashed its output and thrown it back – without Nehuenco – onto a prohibitively expensive spot

market.”


Combustion Instabilities

12

• Single largest issue associated with

development of low NOX GT’s

• Designs make systems susceptible to

large amplitude acoustic pulsations


Turndown

13

• Operational flexibility has been substantially crimped in

low NOX technologies

• Significant number of combined cycle plants being cycled

on and off daily

27 27.5 28 28.5 29 29.5 30 30.50

20

40

60

80

100

Time (Days)

No

rma

lize

d L

oa

d (

%)


Blowoff

14

• Low NOX designs make

flame stabilization more

problematic

Industry Advisory

June 26, 2008

Background:

On Tuesday February 26th, 2008, the FRCC Bulk Power

System experienced a system disturbance initiated by a138 kV transmission system fault that remained on the system for approximately 1.7 seconds. The fault and subsequent delayed clearing led to the loss of approximately 2,300 MW of load concentrated in South Florida along with the loss of approximately 4,300 MW of generation within the Region. Approximately 2,200 MW of under-frequency load shedding subsequently operated and was scattered across the peninsular part of Florida.

Indications are that six combustion turbine (CT) generators within the Region that were operating in a lean-burn mode (used for reducing emissions) tripped offline as result of a phenomenon known as “turbine combustor lean blowout.” As the CT generators accelerated in response to the frequency excursion, the direct-coupled turbine compressors forced more air into their associated combustion chambers at the same time as the governor speed control function reduced fuel input in response to the increase in speed. This resulted in what is known as a CT “blowout,” or loss of flame, causing the units to trip offline.


Autoignition

15

• Liquid fuels

• Higher hydrocarbons in natural gas

• Poor control of dewpoint

Images:

• B. Igoe, Siemens

• Petersen et. al. “Ignition of Methane Based Fuel Blends at Gas Turbine Pressures”, ASME 2005-68517


Emissions

16

• NOX – Reactions with

nitrogen in air and/or fuel

• CO – Incomplete or rich

combustion

• UHC – Incomplete

combustion

• SOX – sulfur in fuel

• Particulates (soot,

smoke)

• CO2 and H20? – Major

project of hydrocarbon

combustion


• Rich flames – large amounts formed

due to insufficient oxygen to react

fuel to CO2

• Lean flames – incomplete

combustion

• Slow CO conversion

• High CO levels formed in flame

that relax slowly to equilibrium

• Low power, low temperature

operation

• Limits turndown range

• Unburned hydrocarbons (UHC) also

associated with incomplete

combustion

0

1 0

2 0

3 0

4 0

0 .3 5 0 .4 0 .4 5 0 .5 0 .5 5 0 .6 0 .6 5

E q uiv a le nc e ra tioC

O a

t 1

5%

ex

ce

ss

O2,

pp

mv

1 0 0 ps i

2 0 0 ps i

2 8 0 ps i

From A. Kendrick, et al,

ASME-GT-2000-0008

Kinetically

controlled

Equilibrium

controlled

Pollutant Trends, CO and UHC


Pollutant Trends, NOX

18

• Primarily formed at high temperatures (>1800 K), due to

reaction of atmospheric oxygen and nitrogen

– Water/steam injection used to cool flame in nonpremixed combustors

– Fuel lean operation to minimize flame temperature is a standard

strategy in DLN combustors

Source: A. Lefebvre, “Gas Turbine Combustion”


Combustor Configurations

Nonpremixed

19

• Water/steam injection used for NOX control



Combustor Configurations

Dry, Low NOX (DLN) Systems

20

• Premixed operation

– If liquid fueled, must prevaporize

fuel (lean, premixed,

prevaporized, LPP)

• Almost all air goes through

front end of combustor for fuel

lean operation – little available

for cooling

• Multiple nozzles required for

turndown



Combustion challenges in a CO2

constrained world

21

• CO2 emissions set by fuel and cycle choice

– Sets combustion configuration and challenges

• Combustion research areas:

– Pre-combustion carbon capture

– Post-combustion carbon capture

– Bio-fuels (near zero net CO2 emitting fuels)


Pre-combustion Carbon Capture

• Carbon removed prior to

combustion, producing high H2

fuel stream

– IGCC

• High H2 introduces significant

combustion issues

– VERY high flame speed – causes

flashback

• Warranties generally limit H2 <5%

by volume

– Plants burning high H2 fuels use

older, high NOx technology

80% H2 20% CH4 flashback at 281 K, 1 atm,

nozzle velocity of 58.7 m/s, and Φ = 0.426

22


Post Combustion Carbon Capture

23

• Sequesterable stream

preferably composed primarily

of CO2 and H2O

– Oxy-combustion

• Control flame temperature by

diluting oxygen with recycled

steam or CO2

– Exhaust gas recirculation

Kimberlina Power Plant


Significant Issues associated with generating a sequesterable

exhaust

24

• Air:

– O2/N2 ratio fixed

– Stoichiometry varied to control flame temperature

– Emissions:

• NOX a major pollutant

• CO to a lesser extent

Component Canyon Reef Weyburn pipeline

CO2 CO H2O

H2S N2 O2

CH4 Hydrocarbon Temperature

Pressure

>95% -

No free water < 0.489 m-3 in the vapour phase

<1500 ppm 4%

<10ppm (weight) -

<5% <49°C

-

96% 0.1%

<20ppm

0.9% <300ppm <50ppm

0.7% - -

15.2 MPa

Table 1. Specifications for two CO2 transport pipelines for EOR

• Oxy-System:

– CO2/O2 ratio varied to control

flame temperature

– Stoichiometry close to 1

– Emissions:

• Near zero NOx emissions

• CO and O2 emissions


Challenges: Emissions

25

• Emissions:

– CO: high CO2 levels

lead to orders of

magnitude increase in

exhaust CO

– O2: normally, a major

exhaust effluent;

requires operating

slightly rich to minimize

O2 ppm

CO ppm


Concluding Remarks

26

• Many exciting challenges associated with

– Fuel flexibility

– “Clean Air Act” emissions and CO2

– Operational flexibility

– Reliability

– Low cost

1. Overview and Basic Equations

Section I: Flow Disturbances in Combustors

2. Decomposition and Evolution of Disturbances

3. Hydrodynamic Flow Stability I: Introduction

4. Hydrodynamic Flow Stability II: Common Combustor Flow Fields

5. Acoustic Wave Propagation I: Basic Concepts

6. Acoustic Wave Propagation II: Heat Release, Complex Geometry, and Mean

Flow Effects

Section II: Reacting Flow Dynamics

7.Flame-Flow Interactions

8. Ignition

9. Internal Flame Processes

Section III: Transient and Time-Stationary Combustor Phenomenon

10. Flame Stabilization, Flashback, and Blowoff

11. Forced Response I - Flamelet Dynamics

12. Force Response II- Heat Release Dynamics

UNSTEADY COMBUSTOR PHYSICS Tim Lieuwen

Cambridge Press, to appear in 2012

~430 pages, with exercises

27

1

Flashback and Flameholding

2

Course Outline

• Introduction


– Boundary Layer Flashback

– Core Flow Flashback and Combustion Induced Vortex Breakdown



• Flame Dynamics


3

• Flashback:

– Upstream propagation of a premixed flame into a region not

designed for the flame to exist

– Occurs when the laminar and/or turbulent flame speed exceeds

the local flow velocity

• Reference flow speed and burning velocity?

• Flameholding:

– Flame stabilizes in an undesired region of the combustor after a

flashback/autoignition event

– Problem has hysteretic elements

• Wall temperature effects

• Boundary layer and swirl flow stability effects



4

• Flashback in the boundary layer

• Flame propagation into core flow

– We’ll focus on swirl flows

• Combustion instabilities

– Strong acoustic pulsations lead to

nearly reverse flow

• Note: p’/p~u’/c=Mu’/u

• i.e. u’/u=(1/M)p’/p

• Significance of above mechanisms is

a strong function of:

– Fuel composition

– Operating conditions

– Fluid mechanics

Flashback and Flameholding Mechanisms

Kröner et al. CST 2007

Heeger et al. Exp. In Fluids 2010

Show video

5


Boundary layer flashback

6

• Neglects effects of

– Heat release (changes approach flow)

– Stretch (changes burning velocity)

• Flashback occurs if flame speed exceeds

flow velocity at distance, from the wall

– Expanding velocity in a Taylor series,

establish flashback condition:

– Assuming, , define flashback

Karlovitz number

Boundary Layer Flashback-Classical Treatment

q

0

x

y

u

y

flame

xu

q

( ) ( )u

x q d qu y s y

( ) 1

u

u qx

x q q u

d

g

guu y

y s

q F

u F

u

d

gK a

s


7

• Flashback Karlovitz number

approach is well validated for open

flames, such as Bunsen burners

– Performed detailed kinetics

calculations to determine flame

speed and thickness for several

data sets

– Shows how prior burning

velocity, flame thickness

tendencies can be used to

understand tendencies

• Pressure

• Preheat temperature

• Stoichiometry

Boundary Layer Flashback

100

1000

0.5 0.75 1 1.25 1.5

Cri

tica

l vel

oci

ty g

radie

nt

(sec

-1)

Increasing uT

2000

gu

(1/s

)

CH4-air

0

0.25

0.5

0.75

1

0.5 0.75 1 1.25 1.5

0,0

u

uF

dg

s

CH4-air (solid) and C3H8-air (dashed)

Data for figures obtained from:

Grumer Ind. & Eng. Chem. 1954

Dugger Ind. & Eng. Chem 1955 CEFRC Summer School, Copyright T. Lieuwen, 2012

8

• Turbulent Boundary Layers

– Multi-zoned

• Near wall laminar

sublayer,

– Basic scaling developed for

laminar flows holds if:

– Most literature data shows

– Significant space-time

variation during flashback

• Images suggest flame

interactions with boundary

layer instabilities

Boundary Layer Flashback

x qu y

,u tu r b u le n tg

,u la m in a rg

0

x

q

y

u

y

C. Eichler Exp. In Fluids 2012

, ,3

u tu r b u le n t u la m in a rg g

Show video


9

• Flame bulging into reactants

– Approach flow decelerates

– Streamlines diverge

– Adverse pressure gradient

• Implications:

– Boundary layers – adverse

pressure gradients lead to

separation

– Swirl flows – adverse pressure

gradients can lead to vortex

breakdown

– Triple flames – flame can

propagate into region with velocity

that is higher than flame speed

– Flame stability – flame

spontaneously develops wrinkles

Coupled Effects of Flame Curvature and Gas

Expansion

, 0

u

xu

, 0

b

xu

2

k

A

Solid thick contours: positive pressure fluctuations

-1 -0.8 -0.6 -0.4 -0.2 0-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Pressure

Velocity

kxCEFRC Summer School, Copyright T. Lieuwen, 2012

10

• Important implications for

– Scaling velocity gradients in shear layers

– Flame stretch rates

– Shear layer instability frequencies – acoustic sensitivities

Heat Conduction Influences on Boundary Layers

m a xu u

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Reacting

Nonreacting

yD

u


11

• Heat release modifies approach

flow

• Stretch modifies burning velocity

Heat Release and Stretch Effects

Unconfined Confined

Inner lip

C. Eichler Turbo Expo 2011


12

• Particularly important in explaining

flameholding phenomenon

• Once a flashback event has

occurred, difficult to expel flame

from combustor

• Leading point of advancing

flashback event subject to positive

curvature

• Effect of gas expansion due to

heat release on local flow velocity

Heat Release and Stretch Effects

0

25000

50000

75000

100000

125000

150000

0.25 0.5 0.75 1

confined

unconfined

1

0

0.5

1.5

510

1/s

ug

Unconfined Confined

Inner lip



13

• Leading point of advancing

flashback event subject to

positive curvature

– For , this can cause:

• can be a significant

underestimate of flame

speed

Stretch Effects

q

0

x

y

u

y

flame

xu

Positively curved flame

0M a

, 0( )

u u

d q ds y s

, 0u

ds


14

• Gas expansion across a

curved flame alters the

approach flow

– Resulting adverse

pressure gradient ahead

of flame decelerates flow

• In extreme cases, can

cause boundary layer

separation

• Approach flow “sucks”

flame back into nozzle

Heat Release Effects

Flame Separated flow region

Reactants

2 m

m

Reactants

1

10

4 10

12

,,

uco

nfined

uunco

nfined

gg

1

10

1 10

,,

uconf

ined

uunco

nfined

gg

dM a

b uT T

-0.9 -0.7 -0.4-0.5

Figures:


Heeger et al. Exp. In Fluids 2010 CEFRC Summer School, Copyright T. Lieuwen, 2012

15


Core Flow Flashback and Combustion Induced Vortex Breakdown

16

• The degree of swirl in the

flow, S, has profound

influences on the flow

structure

• Most prominent feature of

high swirl number flows is

the occurrence of “vortex

breakdown”, which is

manifested as a stagnation

point followed by reverse

flow

Flow Stability and Vortex Breakdown

Stagnation

points

Billant et al., JFM, 1998

Sarpkaya, JFM, 1971


17

• The flow does not instantaneously rotate

about the geometric centerline

• The location of zero azimuthal velocity is

referred to as the “precessing vortex core”

(PVC)

– The frequency of rotation of the

precessing vortex core scales with a

Strouhal number based on axial flow

velocity and diameter

– Leads to a helical pattern in

instantaneous axial flow velocity

– Important to differentiate the PVC from

the other helical shear flow structures

which may also be present

Prominent Features of Swirling Flows

with Vortex Breakdown: Precessing Vortex Core

Region

of

reverse

flow

Region of

highest forward

axial velocity

PVC

center

PVC center

Azimuthal

Velocity

Path of PVC

center

Region

of

reverse

flow

Region of

highest forward

axial velocity

PVC

center

PVC center

Azimuthal

Velocity

Path of PVC

center

Source: Syred, Prog. Energy and Comb. Sci., 2006 CEFRC Summer School, Copyright T. Lieuwen, 2012

18

• Shear layers exist in both span- and streamwise directions

– Can be axisymmetric or helical

Prominent Features of Swirling Flows:

Shear Layer Instability

Huang and Yang, Proc. Comb. Inst., 2005 CEFRC Summer School, Copyright T. Lieuwen, 2012

19

• Vortex breakdown can be

described as a “fold catastrophe”

– Bifurcation of the possible

steady state solutions to the

Navier-Stokes equations

• In high Re flows, there is an

intermediate swirl number range

where flow is bi-stable and

hysteretic

– i.e., either vortex breakdown

or no vortex breakdown flow

state possible

Flow Stability and Vortex Breakdown

Source: Lopez, Physics of Fluids, 1994

Sw

irl N

um

ber

Vortex Core Size/Nozzle Radius

Vortex

Breakdown

Either

No

Breakdown


20

• Vortex breakdown can be predicted for given velocity profile

– “Q-vortex" velocity profile:

Flow Stability and Vortex Breakdown: Example

calculation

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

, 0 , 0x bu u

, 0 , 0br u u

cr r

, 0 2

, 0

2 51 e x p ( ( ) )

1 4

x

b c

u r

u r

2

, 0

, 0

5(1 e x p ( ( ) ) )

4

( / ) (1 e x p ( 5 / 4 ) )

v c

b c

r

r u S r

u r r

Axial and azimuthal velocity profiles used for

vortex breakdown calculation, using Sv=0.71 for uθ,0plot.

, 0 , 0

, 0 , 0

a b

a b

u u

u u


21

Flow Stability and Vortex Breakdown: Example

Calculation


0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.2 0.4 0.6 0.8

rc /a

vS

Breakdown

No breakdown

1.0

0

0.5

1

1.5

2

2.5

-0.5 -0.25 0 0.25 0.5

No breakdown

Breakdown

χ

vS

Wake Jet

Following Z. Rusak

22

• Vortex breakdown – flame interaction

– Can occur even if flame speed everywhere less than flow speed

– Gas expansion across a curved flame:

1. Adverse pressure gradient & radial divergence imposed on reactants

2. Low/negative velocity region generated upstream of flame

3. Flame advances further into reactants

4. Location of vortex breakdown region advances upstream

– Due to bi-stable nature of vortex breakdown boundaries

• CIVB itself not necessarily bi-stable

Core Flow Flame Propagation

C IV B

z

r

fU

C o m b u s t io n C h a m b e r

M ix in g T u b e

S ta b l e F la m e

S w ir l G e n e r a to r

A B

Image reproduced from T. Sattelmayer


Flame Anchoring -1



Flame Stabilization and Blowoff

Natarajan et al. Combustion and Flame 2007

Flame Anchoring -2



Course Outline

• Introduction



– Introductory Concepts

– Flame Stabilization in Shear Layers

– Flame Stabilization by Stagnation Points


• Flame Dynamics

Flame Anchoring -3



INTRODUCTORY CONCEPTS

Flame Anchoring -4




• Flame stabilization requires:

– A point where the local flame

speed and flow velocity

match:

d

us u

Figures:

Natarajan et al. Combustion and Flame 2007

Petersson et al. Applied Optics 2007

• Typically found in regions with:

– Low flow velocity

» Aerodynamically decelerated

regions (VBB)

– High shear

» Locations of flow separation

(ISL & OSL)

Flame Anchoring -5



Premixed Flame Stabilization: Basic Effects

• Flame stabilization:

– Balance combustion wave propagation

with flow velocity

• Burning velocity, edge speed,

autoignition front?

• Suggests that stable flames are rare

– However, flames have self-stabilizing

mechanisms

• Shear layer stabilized: Upstream flame

propagation increases wall quenching

• Aerodynamic stabilization – velocity

profiles

q

abc

u

ds

xu

Flow

0

25

50

75

100

125

150

0.55 0.7 0.85 1 1.15 1.25

Blow-off in

N2

CO2

Flashback

air

cms

u

Lewis & Von Elbe 1987

Flame Anchoring -6



Multiple Anchoring

• Complex flows can provide multiple anchor locations

I: VB

B

II: VB

B/IS

R

III: ISR

IV: O

SR

Flame Anchoring -7




• Stabilization locations determine location/spatial

distribution of flame

– Flame Shape

– Flame Length

• Combustor operability, durability, and emissions

directly tied to these fundamental characteristics

affecting:

– Heat loadings to combustor hardware

– Combustion instability boundaries

– Blowoff limits

(d)(b) (d)(b)(a)(a) (c)(c)

Flame Anchoring -8



Course Outline

• Introduction







• Flame Dynamics

Flame Anchoring -9



Stretch Effects on Shear Layer Flames

• Flame will extinguish

when flame stretch

rate exceeds ext

– As expected, higher

flow velocities result

in flame extinction

occurring at higher

values of ext

Flame Anchoring -10



Shear Layer Stabilized Flames

• In high speed flows, although locally low velocities

exist within the shear layer, flame extinction

typically leads to liftoff/blowoff

– Limited by the amount of flame stretch which they

can withstand before extinction

– e.g, 50 m/s jet with 1 mm shear layer thickness

shear~duz/dx ~ 50 103 s-1

• Much greater than typical extinction strain rates,

ext

• How is flame stretch related to flow strain in a

shear layer?

Flame Anchoring -11



Sources of Flame Stretch

1. Flame curvature

2. Unsteadiness in flame and

flow

3. Hydrodynamic strain:

– For reference, fluid strain rate

given by tensor:

Q. Zhang et al. J. Eng. for Gas Turbines & Power 2010

1

2

ji

i j

j i

uuS

x x

,

i

s s t r a in i j i j

j

un n

x

flow

xn

zn

x

z

flame

n

Flame Anchoring -12



Flame Stretch Due to Fluid Strain

• General expression can be reduced by

assuming 2-D flow and incompressibility

(upstream of flame):

2 2

( )

uu u

xz z

s x z x z

uu un n n n

z z x

Normal Strain

Contribution

Shear Strain

Contribution

flow

xn

zn

x

z

flame

n

Flame Anchoring -13



Shearing flow

velocity profile

Decelerating flow

velocity profile

(a) (b)

Shear Strain Contribution

,

u u

x z

s s h e a r x z

u un n

z x

• Flame strain occurs due to

variations in tangential velocity

• Leads to positive stretch


flow

xn

zn

x

z

flame

n

Flame Anchoring -14




Shearing flow

velocity profile

Decelerating flow

velocity profile

(a) (b) 2 2

,

u

z

s n o r m a l x z

un n

z

Normal Strain Contribution

• Jet flows typically decelerate

producing normal strain

• Leads to negative stretch flow

xn

zn

x

z

flame

n

Flame Anchoring -15



2 2

,

u

z

s n o r m a l x z

un n

z

,

u u

x z

s s h e a r x z

u un n

z x

n

zn

xn

For high speed flows:

21 ( )

xn O

3

zn O

x

u

z

uu

z

u

x

,

u

z

s s h e a r

u

x,

u

z

s n o r m a l

u

z

1

u u

z du s

&

Also assume:

Expressions for shear and normal stretch can be simplified as follows:


flow

xn

zn

x

z

flame

n

Flame Anchoring -16



Stretch from Shear Strain

•

• If then

,

u

z

s s h e a r

u

x

u

d

u

z

s

u,

u u

d z

s s h e a r u

z

s u

u x

sh

u

dshearss~

,

Stretch rate due to shear ~indep. of u ?

increasing u shear but

but u can influence shear layer: sh u-1/2

Shearing flow

velocity profile

flow

xn

zn

x

z

flame

n

Flame Anchoring -17



Stretch from Normal Strain

• Obtain traditional flow

time scaling approach Decelerating flow

velocity profile

z

uu

z

normals~

,

char

u

z

L

u~

flow

xn

zn

x

z

flame

n

Flame Anchoring -18



Total Stretch from Strain

• Total stretch due to strain

Opposite signs if decelerating flow

which term dominates - small ?

x

u

z

uu

z

u

z

s

x

u

z

uu

z

u

xifeven ??vs.

x

u

z

uu

z

u

x

Shearing flow

velocity profile

Decelerating flow

velocity profile

Flame Anchoring -19



Flame Stretch Distribution in Swirl Flame

• Dimensionless value

of 1= 4,125 1/s

Zhang, Q., Shanbogue, S., Shreekrishna,O’Connor, J., Lieuwen, T., “Strain Characteristics Near the Flame

Attachment Point in a Swirling Flow”, Combustion Science and Technology, Vol. 83, 2011, 665-685.

• Shows

dominance of

deceleration

term in first

10 mm

• Shear term

dominates

farther

downstream

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Axial Location (mm)

No

rmalized

Str

ain

Rate

(1/s

)

-nr

2* u/ r

-nz

2* v/ z

-nr*n

z* u/ z

-nr*n

z* v/ r

s

2

2

x x

z z

x z x

x z z

s

n u x

n u z

n n u z

n n u x

Flame Anchoring -20



Piloting or Flow Recirculation Effects

• Flame stabilization can be enhanced through:

– Pilot flames

– Recirculation zones

• Transport hot products to the attachment point of a flame

Flame Anchoring -21



Dilution/Liftoff Effects • At high dilution/preheating levels, the flame does not

"extinguish"

– Increases in reactant temperature are equivalent to a reduction in

dimensionless activation energy

– Example: calculation of CH4/air flame stagnating against hot

products with indicated temperature

0

5

10

15

20

25

0 200 400 600

[cm

/s]

Strain Rate [1/s]

1450 K1400 K

1350 K

u cs

-5

0

5

10

15

20

0 200 400 600

[cm

/s]

Strain Rate [1/s]

1450 K

1400 K

1350 Ku ds

(cm

/s)

u cs

(cm

/s)

u ds

Strain rate (1/s) Strain rate (1/s)

Flame Anchoring -22



Blowoff of Bluff Body Stabilized Flames

• Stages of blowoff

– Stage 1: Flame is continuous but

marked by local extinction events

– Stage 2: Changes in wake dynamics as

large scale structures become visible

– Stage 3: Blowoff of flame

• Da Approach:

– Scaling captures onset of flame

extinction events

– Ability to capture blowoff dependent

on link between extinction events and

blowoff physics

Nair J. Prop. Power 2007

Images courtesy of D. R. Noble

Flame Anchoring -23



Course Outline

• Introduction







• Flame Dynamics

Flame Anchoring -24



Flame Anchoring Locations and Flame

Shapes in Swirling Flows

(d) Centerbody & Outer Nozzle (c) IRZ & Outer Nozzle

Flame Anchoring -25



Flame Anchoring Locations and Flame

Shapes in Annular Nozzle Geometries

From Kumar and Lieuwen

• Flame stabilizes in front of stagnation

point of vortex breakdown bubble

– Stagnation point apparently precesses,

probably also moves up and down

– i.e., flame anchoring position highly

unsteady, in contrast to stabilization at

edges/corners

• Under what circumstances can such

flames exist?

– Not always observed; flames may

blowoff directly without reverting to a

“free floating” configuration

– Flow must have interior stagnation

point

Flame Anchoring -26



Vortex Breakdown in Annular Geometries • Nature of centerbody wake/ VBB changes with geometry, swirl #,

and Reynolds #

Swirl number/ Centerbody Diameter

Recirculation zone with vortex tube

Recirculation zone with bubble-like breakdown above

Merged recirculation zones

Sheen et al., Phys. Fluids, 1996

Flame Anchoring -27



• Nature of centerbody wake/ VBB changes with geometry, swirl #,

heat release parameter, and Reynolds #

Vortex Breakdown in Annular Geometries

Flame Anchoring -28



• Nature of centerbody wake/ VBB changes with geometry, swirl #,

and Reynolds #

Vortex Breakdown in Annular Geometries

Flame Anchoring -29



Flame Stabilization Influenced by

Downstream Boundary Conditions

• Fluid rotation introduces

inertial wave

propagation mechanism

– “sub-” and “supercritical”

flow distinction

• Exit boundary condition

has significant influence

on vortex breakdown

bubble topology

1

Disturbances in Combustors:

Propagation, Amplification, and other Characteristics

2

• Introduction




– Motivation

– Disturbance propagation, amplification, and stability

– Acoustic Wave Propagation Primer

– Unsteady Heat Release Effects and Thermoacoustic Instability

• Flame Dynamics

Course Outline


3

Motivation – Combustion Instabilities

• Large amplitude

acoustic oscillations

driven by heat release

oscillations

• Oscillations occur at

specific frequencies,

associated with resonant modes of

combustor

Heat

release Acoustics

Video courtesy of S. Menon


4

Resonant Modes – You can try this at home


• Helmholtz Mode

• 190 Hz

• Longitudinal Modes

• 1,225 Hz

• 1,775 Hz

• Transverse Modes

• 3,719 Hz

• 10,661 Hz

Slide courtesy of R. Mihata, Alta Solutions

5

Key Problem: Flame Sensitive to Acoustic Waves

Video from Ecole Centrale – 75 Hz, Courtesy of S. Candel


6

Key Problem: Combustor System Sensitive to Acoustic Waves


Rubens Tube

http://www.youtube.com/watch?v=HpovwbPGEoo

7

Thermo-acoustics


• Rijke Tube (heated

gauze in tube)

• Self-excited oscillations

in cryogenic tubes

• Thermo-acoustic

refrigerators/heat pumps

Purdue’s Thermoacoustic Refrigerator

8

Liquid Rockets

• F-1 Engine

– Used on Saturn V

– Largest thrust engine

developed by U.S

– Problem overcome

with over 2000 (out of

3200) full scale tests

From Liquid Propellant Rocket

Combustion Instability, Ed. Harrje and

Reardon, NASA Publication SP-194

Injector face destroyed by combustion instability, Source: D.

Talley


9

Ramjets and Afterburners


• Systems prone to

damage because of light

construction

– Damage to flame holders,

spray bars

• Ramjets: un-starting of

inlet shock

Mig-29 with RD-33 Engines

“Moskit” Ramjet Powered Missile

Images courtesy of E. Lubarsky

10

Solid Rockets

• Adverse effects –thrust oscillations, mean pressure changes, changes

in burning rates

From Blomshield, AIAA Paper #2001-3875


• Examples:

– SERGEANT Theater ballistic

missile – tangential instabilities

generated roll torques so strong

that outside of motor case was

scored due to rotation in restraints

– Minuteman missile –USAF

experienced 5 flight failures in

1968 during test due to loss of

flight control because of severe

vibrations

– Space shuttle booster- 1-3 psi

oscillations (1 psi = 33,000 pounds

of thrust)

11

• Introduction




– Motivation




• Flame Dynamics

Course Outline


12

Small Amplitude Propagation in Uniform, Inviscid Flows

• Assume infinitesimal perturbations superposed upon a spatially

homogenous background flow.

• We will also introduce two additional assumptions:

(1) The gas is non-reacting and calorically perfect, implying that the specific heats are constants.

(2) Neglect viscous and thermal transport.

Decomposition Approach

– Decompose variables into the sum of a base and fluctuating

component; e.g.,

(2.9)

0 1

0 1

0 1

( , ) ( , )

( , ) ( , )

( , ) ( , )

p x t p p x t

x t x t

u x t u u x t


13

Summary of Disturbance Equations

• Vorticity: (2.18)

• Acoustic: (2.20)

• Entropy:

0 10

D

D t

2

2 20 1

0 120

D pc p

D t

0 10

D

D t

s(2.27)


14

Canonical Decomposition

• Further decompose perturbations by their origin

(2.28)

• Examples:

– vorticity fluctuations induced by entropy fluctuations.

– pressure fluctuations induced by vorticty fluctuations.

• Dynamical equations are linear and can be decomposed into

subsystems

1 1 1 1

1 1 1 1

1 1 1 1p p p p

s

s

s

s s s s

1

s

1p


15

Oscillations from Vorticity

• Oscillations associated with vorticity mode:

(2.29)

(2.30)

(2.31)

– Vorticity, and induced velocity, fluctuations are convected by the mean flow.

– Vorticity fluctuations induce no fluctuations in pressure, entropy,

temperature, density, or dilatation.

0 10

D

D t

1 1 1 10p T

s

0 10

D u

D t


16

Oscillations from Acoustics

• Oscillations associated with acoustic mode:

(2.32)

• The density, temperature, and pressure fluctuations are locally and

algebraically related through their isentropic relations

2

2 20 1

020

1

D pc p

D t

1 10

s

01

1 12 2

0 0

pcp

Tc c

1

0 1

D up

D t

(2.33)

(2.34)

(2.35)


17

Oscillations from Entropy

• Oscillations associated with entropy mode:

(2.37)

(2.38)

• Entropy oscillations do not excite vorticity, velocity, pressure, or

dilatational disturbances

• They do excite density and temperature perturbations

0 1

1 1 10

Dp u

D t

s

s s s

s

0 0

1 1 1

0p

Tc T

s s ss


18

Comments on the Decomposition

• In a homogeneous, uniform flow, these three disturbance modes

propagate independently in the linear approximation.

• three modes are decoupled within the approximations of this

analysis - vortical, entropy, and acoustically induced fluctuations are

completely independent of each other.

• For example, velocity fluctuations induced by vorticity and acoustic

disturbances, and are independent of each other and each

propagates as if the other were not there.

• Moreover, there are no sources or sinks of any of these disturbance

modes. Once created, they propagate with constant amplitude.

1u

1u


19

Comments on Length Scales

• Acoustic disturbances propagate at the speed of sound.

• Vorticity and entropy disturbances convect at bulk flow velocity, .

• Acoustic properties vary over an acoustic length scale, given by

• Entropy and vorticity modes vary over a convective length scale,

given by .

• Entropy and vortical mode “wavelength” is shorter than the acoustic

wavelength by a factor equal to the mean flow Mach number.

0u

0c f

0cu f

0 0cu c M


20

Comments on Acoustic Disturbances

• Acoustic disturbances, being true waves, reflect off boundaries, are

refracted at property changes, and diffract around obstacles.

FlameAcoustic

wave

Image of instantaneous pressure field and flame front of a sound wave incident upon a

turbulent flame from the left at three successive times. Courtesy of D. Thévenin.


21

Reactants

Transverse plane wave

Refracted

acoustic

wave

Comments on Acoustic Disturbances

• PIV data shows example of

– refraction in a combustor environment

– Simultaneous presence of acoustic and vortical disturbances

Illustration of acoustic

refraction effects. Data courtesy of J. O'Connor


22

Example: Effects of Simultaneous Acoustic and

Vortical Disturbances

• Consider superposition of two disturbances with different phase

speeds:

• For simplicity, assume

• Velocity field oscillates harmonically at each point as cos(wt)

• Amplitude of these oscillations varies spatially as due to

interference

1 1 , 1 ,

0 , 0

, c o s c o s

x

x xu x t u u t t

c uw w

= A A

A A A

0 0 0 0

1

0 0 0 0

, 2 c o s c o s2 2

c u c uu x t x t x

c u c uw

A

0 0

0 0

c o s2

c ux

c u


23

Some Data Illustrating This Effect

• Fit parameter:

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

x/c

|un|/U

0

MeasurementsFit-u1,+u1,Ω

u1,only

u1, Ω only

u

u

0 0.5 1 1.5-5

-4

-3

-2

-1

0

1

x/c

u

n/

Measurements

Fit-u1,+u1,Ω

u1,only

u1, Ω only

0 .6

A A


24

Modal Coupling Processes

• We just showed that the small amplitude canonical disturbance

modes propagate independently within the fluid domain in a

homogeneous, inviscid flow.

• These modes couple with each other from:

– Boundaries

• e.g., acoustic wave impinging obliquely on wall excites vorticity and entropy

– Regions of flow inhomogeneity

• e.g., acoustic wave propagating through shear flow generates vorticity

• Accelerating an entropy disturbance generates an acoustic wave

– Nonlinearities

• e.g., large amplitude vortical disturbances generate acoustic waves (jet

noise)


25

Disturbance Energy

26

Energy Density and Energy Flux Associated with

Disturbance Fields

• Although the time average of the disturbance fields may be zero,

they nonetheless contain non-zero time averaged energy and lead

to energy flux whose time average is also non-zero.

– Ex:

• Consider the energy equation.

(2.50)

(2.51)

EI

t

V

V A V

dE d V I n d A d

d t

1 1

1

2k in

E u u


27

Acoustic Energy Equation

• Consider the acoustic energy equation, assuming:

– Entropy and vorticity fluctuations are zero

– Zero mean velocity, homogenous flow.

– Combustion process is isomolar

(2.52)

(2.53)

(2.54)

(2.55)

2

1

0 1 1 1 1 1 12

0 0 0

11 1

2 2

pu u p u p q

t c p

1 1

0

1p q

p

1 1I p u

2

1

0 1 1 2

0 0

1 1

2 2

pE u u

c


28

Time Average of Products

• Time average of product of two fluctuating quantities depends on

their relative phasing

• Example 1

s in s in c o s2

t tw w

s in s int tw w

t

t

t

o0

o9 0

o1 8 0


29

Comments on Acoustic Energy

• The energy density, , is a linear superposition of the kinetic

energy associated with unsteady motions, and potential energy

associated with the isentropic compression of an elastic gas.

• The flux term, , reflects the familiar “pumping work” done by

pressure forces on a system.

• The source term, , shows that unsteady heat release can add or

remove energy from the acoustic field, depending upon its phasing

with the acoustic pressure.

E

1 1p u


30

Rayleigh’s Criterion

• Rayleigh’s Criterion states that unsteady heat addition locally adds

energy to the acoustic field when the phases between the pressure

and heat release oscillations is within ninety degrees of each other.

• Conversely, when these oscillations are out of phase, the heat

addition oscillations damp the acoustic field.

• Figure illustrates that the highest pressure amplitudes are observed

at conditions where the pressure and heat release are in phase.

/p p

-180

-135

-90

-45

0

45

90

135

180

0.00 0.04 0.08 0.12

ph

ase

dif

fere

nce

(p

c'-O

H*')

pc'/pc,mean

p q

Data courtesy of K. Kim and D. Santavicca.

31

Nonlinear Behavior

32

Linear and Nonlinear Stability of Disturbances

• The disturbances which have been analyzed arise because of

underlying instabilities, either in the local flow profile or to the

coupled flame-combustor acoustic systems (such as thermoacoustic

instabilities).

• Consider a more general study of stability concepts by considering

the time evolution of a disturbance

FA and FD denote processes responsible for amplification and

damping of the disturbance, respectively.

( )

DA

d tF F

d t

A(2.63)


33

Linear Stable/Unstable Systems

• In a linearly stable/unstable system, infinitesimally small

disturbances decay/grow, respectively.

• To illustrate these points, we can expand the functions FA and FD around their A=0 values in a Taylor’s series:

(2.64)

(2.65)

(2.66)

Am

pli

fica

tio

n/D

amp

ing

FA

AALC

1

1

εD

εA

FD

,A A A N LF F A

,D D D N LF F A

0

A

A

F

AA


34

Comments on Linear Behavior

• The amplification and damping curves intersect at the origin,

indicating that a zero amplitude oscillation is a potential equilibrium

point.

• However, this equilibrium point is unstable, since and any

small disturbance makes FA larger than FD resulting in further growth

of the disturbance.

• If the A=0 point is an example of an “attractor” in that

disturbances are drawn toward it

• Linearized solution:

(2.67)

A D

1( ) ( 0 ) e x p (( ) )

DAt t t A A

A D


35

Comments on Nonlinear Behavior

• This linearized solution may be a reasonable approximation to the

system dynamics if the system is linearly stable (unless the initial excitation, A(t=0) is large).

• However, it is only valid for small time intervals when the system is

unstable, as disturbance amplitudes cannot increase indefinitely.

• In this situation, the amplitude dependence of system

amplification/damping is needed to describe the system dynamics.

• The steady state amplitude is stable at ALC=0 because amplitude

perturbations to the left (right) causes FA to become larger (smaller)

than FD, thus causing the amplitude to increase (decrease).


36

Example of Stable Orbits: Limit Cycles

• In many other problems, is used

to describe the amplitude of a

fluctuating disturbance; for example,

• Limit cycle is example of orbit that

encircles an unstable fixed point – A stable limit cycle will be pulled back into

this attracting, periodic orbit even when it is

slightly perturbed.

(2.69) 0( ) ( ) c o s ( )p t p t tw A

( )tAp

t

( )p t


37

Variation in System Behavior with Change in

Parameter

• Consider a situation where some combustor parameter is

systematically varied in such a way that A increases while D remains

constant.

Am

pli

fica

tio

n/D

amp

ing

A


38

Supercritical Bifurcations

• The A=D condition separates two regions of fundamentally different

dynamics and is referred to as a supercritical bifurcation point.

– Note smooth monotonic variation of limit cycle amplitude with parameter

Stable

Unstable

Am

pli

tud

e,A

εA- εD

00

Pressure

amplitude

Nozzle velocity, m/s

p

p

0

0.005

0.01

0.015

0.02

18 21 24 27 30


39

Nonlinearly Unstable Systems

• Small amplitude disturbances decay

in a nonlinearly unstable system, but

disturbances with amplitudes exceeding a critical value, Ac, will

grow.

• This type of instability is sometimes

referred to as subcritical.

• Other examples:

– hydrodynamic stability of shear flows

without inflection points

– Certain kinds of thermoacoustic instabilities in combustors.

• historically referred to as “triggering”

in rocket instabilities


40

Subcritical Instability

• Figure provides example of amplitude

dependence of FA and FD that

produces subcritical instability.

• In this case, the system has three

equilibrium points where the amplification and dissipation curves

intersect.

• All disturbances with amplitudes A<AT return to the stable solution A=0 and disturbances with

amplitudes A>AT grow until their

amplitude attains the value A=ALC.

• Consequently, two stable solutions

exist at this operating condition. The

one observed at any point in time will depend upon the history of the

system.

Am

pli

fica

tio

n/D

amp

ing

AALC

1εD

FD

1εA

FA

AT


41

Variation in System Behavior with Change in

Parameter

Am

pli

fica

tio

n/D

amp

ing

A


42

Subcritical Bifurcation Diagram A

mp

litu

de,

A

A-D

StableUnstable

AT

0

0

0.005

0.01

0.015

13 13.5 14 14.5 15 15.5

Nozzle velocity, m/s

p

p

Pressure

amplitude

• If a system parameter is monotonically increased to change the sign

of from a negative to a positive value, the system’s amplitude jumps discontinuously from A=0 to A= ALC at

.

• Note hysteresis as well

A D

0A D


43

• Introduction




– Motivation




• Flame Dynamics

Course Outline


44

Acoustic Wave Propagation Primer

45

Overview 1 of 2

• Acoustic Disturbances:

– Propagate energy and information through the medium without requiring

bulk advection of the actual flow particles.

– Details of the time averaged flow has relatively minor influences on the

acoustic wave field (except in higher Mach number flows).

– Acoustic field largely controlled by the boundaries and sound speed

field.

• Vortical disturbances

– Propagate with the local flow field.

– Highly sensitive to the flow details.

– No analogue in the acoustic problem to the hydrodynamic stability

problem.


46

Overview 2 of 2

• Some distinctives of the acoustics problem:

– Sound waves reflect off of boundaries and refract around bends or other

obstacles.

• Vortical and entropy disturbances advect out of the domain where they are

excited.

– An acoustic disturbance in any region of the system will make itself felt

in every other region of the flow.

– Wave reflections cause the system to have natural acoustic modes;

oscillations at a multiplicity of discrete frequencies.


47

Traveling and Standing Waves

• The acoustic wave equation for a homogeneous medium with no

unsteady heat release is a linearized equation describing the

propagation of small amplitude disturbances.

• We will first assume a one-dimensional domain and neglect mean

flow, and therefore consider the wave equation:

(5.1)

2 2

21 1

02 20

p pc

t x


48

Harmonic Oscillations

• We will use complex notation for harmonic disturbances. In this

case, we write the unsteady pressure and velocity as

(5.13)

(5.14)

• As such, the one-dimensional acoustic field is given by:

(5.15)

1 1ˆR e a l ( , , ) e x pp p x y z i tw

1 1ˆR e a l ( , , ) e x pu u x y z i tw

1R e a l e x p e x p e x pp i k x i k x i tw A B

,1

0 0

1R e a l e x p e x p e x p

xu ik x i k x i t

cw

A B (5.16)


49

Time Response of Harmonically Oscillating

Traveling Waves

• The disturbance field consists of space-time harmonic disturbances

propagating with unchanged shape at the sound speed.

• An alternative way to visualize these results is to write the pressure

in terms of amplitude and phase as:

(5.17) 1 1ˆ ˆ e x pp x p x i x

p1

x

t

Temporal variation of harmonically

varying pressure for rightward moving wave


50

Amplitude and Phase of Traveling Wave

Disturbances

• The magnitude of the disturbance

stays constant but the phase

decreases/increases linearly with axial

distance.

• The slope of these lines are .

• Harmonic disturbances propagating

with a constant phase speed have a

linearly varying axial phase

dependence, whose slope is inversely

proportional to the disturbance phase

speed.

BA

leftward moving

rightward moving

1p̂

x

x

0/ cw

0/ cw

1

1θ

Spatial amplitude/phase

variation of harmonically varying acoustic disturbances.

0/

dc

d x

w


51

Standing Waves

• Consider next the superposition of a left and rightward traveling wave of equal amplitudes, A=B, assuming without loss of generality

that A and B are real.

(5.18)

• Such a disturbance field is referred to as a “standing wave”.

• Observations:

– amplitude of the oscillations is not spatially constant, as it was for a single

traveling wave.

– phase does not vary linearly with x, but has a constant phase, except across the

nodes where it jumps 180 degrees.

– pressure and velocity have a 90 degree phase difference, as opposed to being

in-phase for a single plane wave.

1( , ) 2 c o s c o sp x t k x tw A

,1

0 0

12 s in s in

xu k x t

cw

A (5.19)


52

Behavior of Standing Waves

Spatial dependence of pressure (solid) and velocity (dashed) in a

standing wave.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

p1

ρ0c0ux,1

kx/2π

1

1,5

2,8 2,4

3,7

3

4,6

5

6,8

7

(a)

(degrees)

180o

kx/2π

270o

90o

0o

(c)

kx/2π

1p̂

0 0 ,1ˆ

xc u

2A(b)


53

One Dimensional Natural Modes

• Consider a duct of length L with rigid boundaries at both ends,

. Applying the boundary condition at x=0 implies that A=B, leading to:

(5.62)

(5.64)

• These natural frequencies are integer multiples of each other, i.e.,

f2=2f1, f3=3f1, etc.

(5.65)

(5.66)

,1 ,1

0 , , 0x x

u x t u x L t

,1

0 0

1R e a l 2 s in e x p

xu i k x i t

cw

A

nk

L

0 0

2 2 2n

k c n cf

L

w

1

( , ) 2 c o s c o sn x

p x t tL

w

A

,1

0 0

2( , ) s in s in

x

n xu x t t

c L

w

A

(5.63)


54

• Introduction




– Motivation




• Flame Dynamics

Course Outline


55

Unsteady Heat Release Effects

• Oscillations in heat release generate acoustic waves.

– For unconfined flames, this is manifested as broadband noise emitted

by turbulent flames.

– For confined flames, these oscillations generally manifest themselves

as discrete tones at the natural acoustic modes of the system.

• The fundamental mechanism for sound generation is the unsteady

gas expansion as the mixture reacts.

• To illustrate, consider the wave equation with unsteady heat release

(57)

• The two acoustic source terms describe sound wave production

associated with unsteady gas expansion, due to either heat release

(first term) or non isomolar combustion (second term).

2

2 21 1

0 1 02

1

( 1)p q n

c p pt nt


56

Unsteady Heat Release Effects – Confined

Flames

• Unsteady heat release from an acoustically compact flame induces

a jump in unsteady acoustic velocity, but no change in pressure:

(61)

(62)

• Unsteady heat release causes both amplification/damping and shifts

in phase of sound waves traversing the flame zone.

– The relative significance of these two effects depends upon the

relative phase of the unsteady pressure and heat release.

• The first effect is typically the most important and can cause systems with

unsteady heat release to exhibit self-excited oscillations.

• The second effect causes shifts in natural frequencies of the system.

1 1b ap p

,1 ,1 1

0

1 1

x b x au u Q

A p


57

Combustor Stability Model Problem

• Assume rigid and pressure release boundary conditions at x=0 and

x=L, respectively, and that the flame is acoustically compact and

located at x=LF .

• Unsteady pressure and velocity in the two regions are given by:

Region I: (63)

Region II: (64)

0 , 0 ,

,

1,

I I F I I F

I I F I I F

i k x L ik x L i t

I I I I I I

i k x L ik x L i t

I I I I I I

I I I I

p x t e e e

u x t e e ec

w

w

A B

A B

,1

0 0

,

1,

I F I F

I F I F

i k x L ik x L i t

I I I

i k x L ik x L i t

x I I I

I I

p x t e e e

u x t e e ec

w

w

A B

A B

ux,1=0 p1=0

L

I II

Heat release

zone

a b

LF


58

Matching Conditions

• Applying the boundary/matching conditions leads to the following

algebraic equations:

(Left BC)

(Right BC)

(Pressure Matching)

(Velocity Matching)

6)

0I F I Fi k L ik L

I Ie e

A - B

I I I I I I A B A B

1

2

0 , 0 ,

, , 1I I F I F

I I

Qu L t u L t

c

0

I I F I I Fik L L ik L L

I I I Ie e

A B


59

Unsteady Heat Release Model

• Model for unsteady heat release is the most challenging aspect of combustion instability prediction

• We will used a “velocity coupled” flame response model

• Assumes that the unsteady heat release is proportional to the unsteady flow velocity, multiplied by the gain factor, n, and delayed in time by the time delay, τ.

• This time delay could originate from, for example, the convection time associated with a vortex that is excited by the sound waves.

2

0 , 0 ,

1 1,

1

I I

F

cQ A u x L t

n


60

Heat Release Modeling Simplifications

• Solving the four boundary/matching conditions leads to:

• In order to obtain analytic solutions, we will next assume that the

flame is located at the midpoint of the duct, i.e., L=2LF and that the

temperature jump across the flame is negligible.

(72)

2

c o s s in 2i

k L e k Lw

n

0 , 0 ,

0 , 0 ,

c o s c o s

0

1 s in s in

I I I I

I F I I F

I I

i

I F I I F

ck L k L L

c

e k L k L Lw

n


61

Approximate Solution for Small Gain Values

• We can expand the solution around the solution by looking at

a Taylor series in wavenumber k in the limit .

0

( 2 1)

2

nk L

n

0n

0 2

0

0 0

e x p1

2 s in

ik k O

k L k L

w

nn

n n

nn

< < 1n


62

Complex Wavenumbers

• Wavenumber and frequency have an imaginary component,.

Considering the time component and expanding ω:

(75)

• Real component is the frequency of combustor response.

• Imaginary component corresponds to exponential growth or decay in

time and space. Thus, ωi>0 corresponds to a linearly unstable

situation, referred to as combustion instability.

(77)

e x p ( ) e x p ( ) e x p ( )r i

i t i t tw w w

0

0

c o s ( )1 ( 1 )

( 2 1 )

n

r

n

w w w

nn

n

0 0s in ( )

( 1)2

n

i

c

L

w w

nn


63

Rayleigh’s Criterion

• Rayleigh's criterion: Energy is added to the acoustic field when the

product of the time averaged unsteady pressure and heat release is

greater than zero.

1 1 0 0

2 12 , ( ) s in c o s s in ( )

2 2

np x L t Q t t t

w w

n n

n

1

1 1 0( / 2 , ) ( ) 0 1 s in 0

n

p x L t Q t w

n

1 1 0

2 1( / 2 , ) ( ) s in s in

4 2

np x L t Q t

w

n

n


64

Stability of Combustor Modes

• Unstable 1/4 wave mode (n=1)

• Unstable 3/4 wave mode (n=2)

• Sign of alternates with time

delay – Important implications on why instability prediction

is so difficult- no monotonic dependence upon

underlying parameters

S U S1/4 wave

mode

1/2 1 3/2 0

3/4 wave

mode

Frequency

ωn=0

S S S S UUUUU

1 / 4

T

( 2 1)n

n

1 / 4

1 / 2m m

T

1 / 4

1 / 2

3 3

m m T

1 1p Q

• Largest frequency shifts occur at the values where oscillations are not

amplified and that the center of instability bands coincides with points

of no frequency shift.


65

Thermo-acoustic Instability Trends

• Two parameters, the heat release

time delay, τ, and acoustic period, T, control instability conditions.

• Data clearly illustrate the non-

monotonic variation of instability

amplitude with .

Air

Fuel

LFIShorter

flame

LFL

Longer

flameFlame stabilized

at centerbody

Air

FuelLSO

Flame stabilized

downstream

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

τ /T

( )

p

p s i

Data illustrating variation of instability amplitude with normalized

time delay. Image courtesy of D. Santavicca

T


66

Understanding Time Delays

• Data from variable length combustor

Measured instability

amplitude (in psi) of combustor as a function of fuel/air ratio and combustor

length. Data courtesy of D. Santavicca.

0.60 0.65 0.70 0.75

2075200019251850

25

30

35

40

45

50

Adiabatic flame temperature, K

Equivalence ratio

Com

bust

or

length

, in

4

3

32

1

1

2Decreasing

Period, T

Decreasing time delay, τ


67

More Instability Trends 1 of 2

• Note non-monotonic variation

of instability amplitude with

axial injector location, due to

the more fundamental variation

of fuel convection time delay, τ.

• Biggest change in frequency is

observed near the stability

boundary.

Measured dependence of instability

amplitude and frequency upon axial location of fuel injector. Data obtained from

Lovett and Uznanski.

0

1

2

3

4

5

6

6 8 10 12 14 16 18 20 22 24 26

Fuel injector location, LFI

p

6 8 10 12 14 16 18 20 22 24 26

Fuel injector location, LFI

0

30

60

90

120

150

180

Fre

quen

cy, H

z


68

More Instability Trends 2 of 2

Measured dependence of the excited instability mode

amplitude upon the mean velocity in the combustor inlet. Obtained from measurements by the author.

0 500 1000Frequency (Hz)

=18m/s

p̂

u

Frequency (Hz)

=25m/s

0 500 1000

u

0 500 1000Frequency (Hz)

=36m/su

0

0.02

0.04

0.06

0.08

15 20 25 30 35 40

Premixer Velocity (m/s)

430 Hz

630 Hz

p̂


69 Normalized time, f0t

p

p

2500 50000

-0.015

-0.01

-0.005

0.0

0.005

0.01

0.015

Limit Cycles and Nonlinear Behavior

• As amplitudes grow nonlinear effects grow in significance and the

system is attracted to a new orbit in phase space, typically a limit

cycle.

• This limit cycle oscillation can consist of relatively simple oscillations

at some nearly constant amplitude, but in real combustors the

amplitude more commonly "breathes" up and down in a somewhat

random or quasi-periodic fashion.


70

Combustion Process and Gas Dynamic

Nonlinearities

• Gas dynamic nonlinearities introduced by nonlinearities present in

Navier-Stokes equations

• Combustion process nonlinearities are introduced by the nonlinear

dependence of the heat release oscillations upon the acoustic

disturbance amplitude.

Dependence of unsteady heat

release magnitude and phase upon velocity disturbance amplitude.

Graph generated from data

obtained by Bellows et al.

0

30

60

90

120

150

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Q

Q

x xu u

Ph

ase,

deg

rees


71

Boundary Induced Nonlinearities

• The nonlinearities in processes that occur at or near the combustor

boundaries also affect the combustor dynamics as they are

introduced into the analysis of the problem through nonlinear

boundary conditions.

• Such nonlinearities are caused by, e.g., flow separation at sharp

edges or rapid expansions, which cause stagnation pressure losses

and a corresponding transfer of acoustic energy into vorticity.

• These nonlinearities become significant when .

• Also, wave reflection and transmission processes through choked

and unchoked nozzles become amplitude dependent at large

amplitudes.

1u u



Dynamics of Harmonically Excited

Flames


Course Outline

• Introduction




• Flame Dynamics

– General characteristics of excited flames

– Analysis of flame dynamics

– Global heat release response and Flame Transfer Functions



Flame Response to Harmonic

Disturbances

• Combustion instabilities

manifest themselves as

narrowband oscillations at

natural acoustic modes of

combustion chamber

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0

F r e q u e n c y ( H z )

Fo

urie

r T

ra

ns

fo

rm

3 CEFRC Summer School, Copyright T. Lieuwen, 2012


Basic Problem • Wave Equation:

• Key issue – combustion response

How to relate q’ to variables p’, u’, and etc., in order

to solve problem

21

t t x x tp c p q



Flashback - Heat Release Modeling

Slide from Last Session • Model for unsteady heat release is the most challenging aspect of

combustion instability prediction

• We will used a “velocity coupled” flame response model

• Assumes that the unsteady heat release is proportional to the unsteady flow velocity, multiplied by the gain factor, n, and

delayed in time by the time delay, τ.

• This time delay could originate from, for example, the

convection time associated with a vortex that is excited by the

sound waves.

2

0 , 0 ,

1 1,

1

I I

F

cQ A u x L tn



0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

u / uo

CH

* / C

H* o

u’/uo

Q’/

Qo

Response of Global Heat Release to

Flow Perturbations

What factors affect

slope of this curve (gain

relationship) ?

Why does this saturate?

Why at this amplitude?

In this unit, we’ll discuss in detail what flames do when harmonically

excited and where curves like these come from (we’ll also discuss

conditions under which n- model is decent approximation) 6



Course Outline

• Introduction




• Flame Dynamics






Increased

Amplitude of Forcing

Excited Bluff Body Flames (Mie Scattering)



Excited Swirl Flame - Attached (OH PLIF)

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

50

100

150

200

250

0° 45° 90° 135°

315° 270° 225° 180°



Excited Swirl Flame – Not Attached (OH PLIF)

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

50

100

150

200

250

50

100

150

200

250

300

350

0° 45° 90° 135°

315° 270° 225° 180°



18 m/s

294K

38 m/s

644K

127 m/s

644K

170 m/s

866K

Excited Bluff Body Flames (Line of sight luminosity)



Overlay of Instantaneous Flame Edges

18 m/s

294K

38 m/s

644K

127 m/s

644K

170 m/s

866K



Time

Series

Power

Spectrum

L’(x, f0)

Quantifying Flame Edge Response



Convective wavelength:

λc= U0/f0

- distance a disturbance

propagates at mean flow

speed in one excitation

period

Spatial Behavior of Flame Response



Spatial Behavior of Flame Response

• Strong response at forcing frequency

‒ Non-monotonic spatial dependence



1. Low amplitude flame fluctuation near attachment point, with subsequent growth downstream

2. Peak in amplitude of fluctuation, L’=L’peak

3. Decay in amplitude of flame response farther downstream

4. Approximately linear phase-frequency dependence

Flame Wrinkling Characteristics



Typical Results – Other Flames

1.8m/s, 150hz

• Magnitude can oscillate with downstream distance

50 m/s, 644K



Course Outline

• Introduction




• Flame Dynamics






Analysis of Flame Dynamics

1. Wrinkle convection and flame

relaxation processes

2. Excitation of wrinkles

3. Interference processes

4. Destruction of wrinkles


G-equation :

2

1f f L

L L Lu v S

t x x

Level Set Equation for Flame Position



Wrinkle Convection

0

0

a

b

u tu

u t

Model problem:

Step change in axial velocity over the entire domain from ua to

ub, both of which exceed sd:



Wrinkle Convection

t1 t3t2

• Flame relaxation process consists of a “wave” that propagates along

the flame in the flow direction.

Flame

cshock t

1s in

d

b

s

u

1s in

d

a

s

u



Harmonically Oscillating Bluff Body

24 CEFRC Summer School, Copyright T. Lieuwen, 2012 Petersen and Emmons, The Physics of Fluids Vo. 4, No. 4, 1961

u0

sL

ut ut

uc,f


Phase Characteristics of Flame Wrinkle

Convection speed of

Flame wrinkle, uc,f

Disturbance Velocity, uc,v

Shin et al., Journal of Power and Propulsion, 2011

Mean flow velocity, u0



Harmonically Oscillating Bluff Body

• Linearized, constant burning

velocity formulation:

– Excite flame wrinkle with

spatially constant amplitude

– Phase: linearly varies

• Wrinkle convection is

controlling process responsible

for low pass filter character of

global flame response

u0



Excitation of Wrinkles on Anchored Flames

• Linearized solution of G

Equation, assume anchored flame

• Wrinkle convection can be seen

from delay term

u0

sL

ut

un’

L’

0

, 1 1( , ) ( 0 , )

x

tt

n

n

t t

L x

u

t x x xx t d x x t t

x xu

uu u

u



Excitation of Flame Wrinkles –

Spatially Uniform Disturbance Field

• Wave generated at attachment point

(x=0), convects downstream

• If excitation velocity is spatially

uniform, flame response exclusively

controlled by flame anchoring

“boundary condition”

– Kinetic /diffusive/heat loss effects,

though not explicitly shown here, are

very important!

1c o s

Ls t

tu t

0

, 1 1( , ) ( 0 , )

x

tt

n

n

t t

L x

u


x xu

uu u

u




Spatially Uniform Disturbance Field

• Wave generated at attachment point

(x=0), convects downstream

• If excitation velocity is spatially

uniform, flame response exclusively

controlled by flame anchoring

“boundary condition”

– Kinetic /diffusive/heat loss effects,

though not explicitly shown here, are

very important!

1c o s

Ls t

tu t

0

, 1 1( , ) ( 0 , )

x

tt

n

n

t t

L x

u


x xu

uu u

u



Near Field Behavior- Predictions

• Can derive analytical formula

for nearfield slope for

arbritrary velocity field:

nu

tu

2

'

' 1

c o s

n

tx u

L u



PIV Data MIE scattering Data

Comparisons With Data

2

1

c o s

n

t

u L

u x

5 m/s, 300K

Shanbhogue et al., Proceedings of the Combustion Institute, 2009. 32



• Flame starts with small

amplitude fluctuations

because of attachment

L’(x=0, t) = 0

• Nearfield dynamics are

essentially linear in

amplitude

Increasing

amplitude, u’

Near Field Behavior

Normalized by u’

Shanbhogue et al., Proceedings of the Combustion Institute, 2009. 33




Spatially Varying Disturbance Field

• Flame wrinkles generated at all points where disturbance velocity is non-uniform, du’/dx ≠0

– Net flame disturbance at location x is convolution of disturbances at upstream locations and previous times

• Convecting vortex is continuously disturbing flame

– Vortex convecting at speed of uc,v

– Flame wrinkle that is excited convects at speed of ut

Bechert , D. ,Pfizenmaier, E., JFM., 1975

0

, 1 1( , ) ( 0 , )

x

tt

n

n

t t

L x

u


x xu

uu u

u



Model Problem: Attached Flame Excited by a

Harmonically Oscillating, Convecting Disturbance

,

, 0

c o s ( 2 ( / ) )n

n c v

uf t x u

u t

,0,2 / s in2 / ta n

1

,0 ,0 ,

s inR e a l

2 c o s / 1

c vi f y u ti f y u t

n

c v

ie e

u f u u

t

t t

• Model problem: flame excited by convecting velocity field,

• Linearized solution:



Solution Characteristics

• Note interference pattern on

flame wrinkling

• Interference length scale:

,

1s in

| 1 |in t

c vu u

t

t

/ s inyt

in t



Interference Patterns

10

cos

ft

0 1 2 30

0.02

0.04

0.06

0.08

x/(tcos )

|1(f

0)|

/(tc

os

)

c o sxt

0 1 2 30

0.02

0.04

0.06

0.08

0.1

10

cos

ft

c o sxt

Shin et al., AIAA Aerospace Science Meeting, 2011 Acharya et al., ASME Turbo Expo, 2011



• Result emphasizes

“wave-like”, non-local

nature of flame response

• Can get multiple

maxima/minima if

excitation field persists

far enough downstream

Comparison with Data

2

20

,

c o s/

2 c o s 1

p e a k c

c v

x

u

u

Shin et al., Journal of Power and Propulsion, 2011 39



Aside: Randomly Oscillating,

Convecting Disturbances

• Space/time coherence of

disturbances key to

interference patterns

• Example: convecting

random disturbances to

simulate turbulent flow

disturbances 0 2 4 6 8 10

0

1

2

3

4

1 1Lt

, 0u fttor

Single frequency

excitation

Random

excitation

1/2

2

1/

ref



• Flame propagation normal to

itself smoothes out flame

wrinkles

Flame Wrinkle Destruction Processes:

Kinematic Restoration





wrinkles

• Typical manifestation: vortex

rollup of flame



Sung & Law, Progress in Energy and Combustion Science, 2000 43





wrinkles

• Typical manifestation: vortex

rollup of flame

• Process is amplitude dependent

and strongly nonlinear

– Large amplitude and/or short

length scale corrugations

smooth out faster

2

1x

LSv

x

Lu

t

L

Lff





Kinematic Restoration Effects:

Oscillating Flame Holder Problem

u0

0s in t

Flame front

Shin & Lieuwen, AIAA 50th Aerospace Science Meeting, 2012 45



• Leads to nonlinear farfield flame

dynamics

• Decay rate is amplitude dependent

Numerical Calculation Experimental Result

Kinematic Restoration Effects

x/ c



Two Zone Behavior

• Near flame holder

– Higher amplitudes and shorter wavelengths decay faster

• Farther downstream

– Flame position independent of wrinkling magnitude

– Flame position is determined by the leading points

• Only a function of wrinkling wavelength 47


T a n g e n tia l d ir e c tio n , t

0|

|

, 0

2

Ls t

0 .1 3 ~ 0 .6 3Five simulations with

Reactants

Products

,0Ls t

Sung et al., Combustion and Flame, 1996

Shin & Lieuwen, AIAA 50th Aerospace Science Meeting, 2012




Shin & Lieuwen, Central State Section Spring Technical Meeting, Apr 12 48 CEFRC Summer School, Copyright T. Lieuwen, 2012


Flame Wrinkle Destruction Processes: Flame Stretch in Thermodiffusively Stable Flames

Wang, Law, and Lieuwen., Combustion and Flame, 2009

Preetham and Lieuwen, JPP, 2010 49



Flame Stretch Effects

0

, 0

| , |

e x pL

s

tt

: N o r m a liz e d M a rk s te in le n g th

Linear in amplitude

wrinkle destruction process



Course Outline

• Introduction




• Flame Dynamics



– Global heat release response and flame transfer functions



Transfer Function


Definition of transfer functions V

F ,p

F ,and F :

1 0

1 0

ˆ

ˆp

Q QF

p p

1 0

1 0

ˆ

ˆV

Q QF

u u

1 0

1 0

ˆ

ˆ

Q QF

Complex numbers have

o Magnitude: relative magnitude ratio

o Phase: relative phase difference

The total unsteady heat release is the superposition ( by linear approximation):

1 1 1 1

0 0 0 0

ˆ ˆˆ ˆ

V P

Q u pF F F

Q u p

12.3

12.5

12.4

12.6

Acoustic disturbance:

Velocity disturbance:

Fuel/Air ratio disturbance:


Velocity Coupled Linear Flame Response

Flame Response to low frequency velocity fluctuations

Assumptions

o Constant burning velocity

o Area fluctuation is the only mechanism for the unsteady heat release

Take the solution in Eq. (11.53), then substitute it into Eq. (11.212) for flame area

Axisymmetric wedge flame:

222 /2

2 2 22 2

2

( , ) 1 1 2 22 (1 )

ci S t ki S tc

V c c c

c

kF S t k k i S t e k i S t e

k S t12.15

o Depends on two parameters, 2

S t and c

k , ( see Sec. 11.1.2)

, 1

,

, 0,

, ,c o s ( 2 ( / ) )

F n

n c x y t

x y t

v x y tf t x u

u t



Velocity Coupled Linear Flame

Response: Gain

Gain

o Unity at low 2

S t

1% fluctuation in velocity leads 1%

fluctuation in heat release

o Gain increases greater than unity

Due to constructive interference between

wrinkles excited by the convecting flow

disturbance and those propagating along the

flame

o "Nodes" of zero heat release response

Flame is responding locally

Due to destructive interference between

oscillations at different parts of the flame

(a) (b)

10-1

100

101

10-1

100

St2

|F|

F

,V c

F k

, 0 .5V c

F k

2S t

F

0 5 10 15 20-1080

-720

-360

0

2S t

F

,V c

F k

, 0 .5V c

F k

21 ,

cS t k

F

21 , 0 .5

cS t k

Dependence of the linear flame transfer

function on the Strouhal number for

axisymmetric wedge flames showing (a) gain

and (b) phase (methane-air, 0=0.9, and mean

flame angle =30 degrees).



Velocity Coupled Linear Flame

Response: Phase

( )1

2

Vd F

d f

Phase

o Rolls off nearly linearly with frequency

o 180 degree phase jumps at nodal locations

in the gain

o The slope of phase

Assumption: velocity and area

disturbances are related by a time-

delayed as:

1 1

0 0

( ) ( )

V

A t u t

A un

Then, slope of the phase and the time

delay are:

12.16

(a) (b)

10-1

100

101

10-1

100

St2

|F|

F

,V c

F k

, 0 .5V c

F k

2S t

F

0 5 10 15 20-1080

-720

-360

0

2S t

F

,V c

F k

, 0 .5V c

F k

21 ,

cS t k

F

21 , 0 .5

cS t k

Dependence of the linear flame transfer

function on the Strouhal number for

axisymmetric wedge flames showing (a) gain

and (b) phase (methane-air, 0=0.9, and mean

flame angle =30 degrees).



Measured Transfer Functions

(a) (b)

0

0.5

1

1.5

2

2.5

3

3.5

0 60 120 180 240 300 360

Frequency (Hz)

0 .6 5

0 .8

VF

-1080

-720

-360

0

0 60 120 180 240 300 360

Frequency (Hz)

0 .8

0 .6 5

(degre

es)

VF

Measured (a) gain and (b) phase response of a flame to upstream flow velocity

perturbations; reproduced from Jones et al. [88].



Model n Flame area-velocity relationship in time domain

o Obtained by inverse Fourier transform

o Applies for 2

1S t (i.e., a convectively compact flame)

o In terms of a simple n model (see Exercise 2.7):

1 1

0 0

( ) ( )

V

A t u t

A un

For axisymmetric conical flame (see Exercise 12.3) :

1Vn and

1

2

2 (1 )

3 c o s

c F

o

k L

u

Shows that n heat release model is a rigorous approximation of the heat release

dynamics in the low2

S t limit

o Time delay,

Time taken for the mean flow to convect some fractional distance of the flame length

Equivalent to replacing the distributed flame by a concentrated source at this location

1 22 1 3 c o s

c Fk L for the axisymmetric wedge flame

12.20



That’s All!


• Introduction




• Flame Dynamics

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