school of aerospace engineering course outline · school of aerospace engineeringgoverning...

School of Aerospace Engineering

Course Outline A) Introduction and Outlook B) Flame Aerodynamics and Flashback C) Flame Stretch, Edge Flames, and Flame Stabilization Concepts D) Disturbance Propagation and Generation in Reacting Flows E) Flame Response to Harmonic Excitation

• Governing Equations • Premixed Flame Dynamics

– General characteristics of excited flames

– Wrinkle convection and flame relaxation processes

– Excitation of wrinkles – Interference processes – Destruction of wrinkles

• Non Premixed Flame Dynamics

• Global heat release response and Flame Transfer Functions

Copyright © T. Lieuwen 2013 Unauthorized Reproduction Prohibited 1


Flame Response to Harmonic Disturbances

• Combustion instabilities

manifest themselves as narrowband oscillations at natural acoustic modes of combustion chamber

0

100

200

300

400

500

0 100 200 300 400 500 600 700

Frequency (Hz)

Four

ier T

rans

form

2 Copyright © T. Lieuwen 2013 Unauthorized Reproduction Prohibited


Basic Problem • Wave Equation:

• Key issue – combustion

response How to relate q’ to variables p’,

u’, and etc., in order to solve problem

Focus of this talk is on sensitivity of heat release to flow disturbances

( )2 1tt xx tp c p qγ′ ′ ′− = −

3

Flow Instabilities

Copyright © T. Lieuwen 2013 Unauthorized Reproduction Prohibited


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’/Q

o

Response of Global Heat Release to Flow Perturbations

What factors affect slope of this curve (gain relationship) ?

4

Why does this saturate? Why at this amplitude?

( ) F Rflame

Q t m dA′′= ∫

hCopyright © T. Lieuwen 2013 Unauthorized Reproduction Prohibited


Analytical Tools/Governing Equations • Work within fast chemistry, flamelet approximation

and use G- and Z- equations to describe flame dynamics



Analytical Tools – Z Equation • Key assumptions

– Le=1 assumption – flame sheet at Z=Zst surface

7

( )FF F F

DY YDt

ρ ρ ω− ∇ ⋅ ∇ = D

( ) ( )Pr PrPr Pr

( 1)( ( 1) )

( 1)D Y

YDt

υ ωρ ρ υυ

+− ∇ ⋅ ∇ + =

+

D

– Imposed flow field – Equal diffusivities

( )Pr

1Fωω υ− = +

Add these species equations:

( ) ( )PrPr

( 1)( ( 1) ) 0F

F

D Y YY Y

Dtυ

ρ ρ υ+ +

− ∇ ⋅ ∇ + + =D



Analytical Tools – Z Equation

( )ut

∂+ ⋅∇ = ∇ ⋅ ∇

∂Z Z ZD

8

• Recall the definition of mixture fraction:

• Yields:

Pr1

( 1)FY Yυ

= ++

Z

( ) 0DDt

ρ ρ− ∇ ⋅ ∇ =Z ZD



Premixed Flame Sheets: G-Equation

ProductG>0

ReactantG<0

Flame surfaceG=0

.

( )

, 0at the flame front

D G x tDt

=

0FG v Gt

∂ + ⋅∇ =∂

dFv u s n= −

/ | |n G G= ∇ ∇

Flame fixed (Lagrangian) coordinate system:

Coordinate fixed (Eulerian) coordinate system:

9

0| |d

G Gu s Gt G

∂ ∇+ − ⋅∇ =∂ ∇

| |dG u G s Gt

∂ + ⋅∇ = ∇∂



G-equation for single valued flame front

Two-dimensional flame front Position is single valued function, ξ , of the coordinate y Define and substitute ( , , ) ( , )G x y t x y tξ≡ − ξ (y,t)

x

y

Reactants

Productsg

1

n

2

1x y du u st y yξ ξ ξ

∂ ∂ ∂− + = − +∂ ∂ ∂



Governing Equations • Left side:

– Same convection operator – Wrinkles created on surface by

fluctuations normal to iso- G or Z surfaces

• Right side: – Non-premixed flame –

diffusion operator, linear – Premixed flame – flame

propagation, nonlinear – Right side of both equations

becomes negligible in Pe = uL/ >>1 or u/sd>>1 limits

( )ut

∂+ ⋅∇ = ∇ ⋅ ∇

∂Z Z ZD

| |dG u G s Gt

∂ + ⋅∇ = ∇∂

11

D



Governing Equations • G-equation only physically

meaningful at the flame surface, G=0 – Can make the substitution,

• Z-equation physically meaningful everywhere – Cannot make analogous

substitution

( ) ( ), , , , ,G x y z t x y z tξ= −

ξ (y,t)

x

y

Reactants

Productsg

1

n



Governing Equations • Reflects fundamental

difference in problem physics

• Premixed flame sheet only influenced by flow velocity at flame

• Non-premixed flame sheet influenced by flow disturbances everywhere



Increased

Amplitude of Forcing

Excited Bluff Body Flames (Mie Scattering)



Excited Swirl Flame (OH PLIF)

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 • 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



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 LL L Lu v St x x

∂ ∂ ∂ + − = + ∂ ∂ ∂

Level Set Equation for Flame Position



Wrinkle Convection

00

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

1sin d

b

su

θ − =

1sin d

a

su

θ − =



Harmonically Oscillating Bluff Body

28

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

D. 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



θ

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

nn

t t

L xu

t x x xx t dx x t tx x

uuu 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!

( ) ( )( )

1cos Ls tt

u tθ −

=

( )0

, 1 1( , ) ( 0, )x

tt

nn

t t

L xu


uuu u

u′∂ ′∂ −′ ′′

′= − + ⋅ = = −∂ ∂∫



Near Field Behavior- Predictions

• Can derive analytical formula for nearfield slope for arbritrary velocity field:

nu′tu

θ

2

'' 1cos

n

tx uL u

θ∂

=∂



PIV Data MIE scattering Data

Comparisons With Data

2

1cos

n

t

u Lu xθ′ ′∂

=∂

5 m/s, 300K

S. Shanbhogue et al., Proc of the Comb Inst, 2009. 35



• 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’

S. Shanbhogue et al., Proc of the Comb Inst, 2009.



Excitation of Flame Wrinkles – Spatially Varying Disturbance Field

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

– 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

nn

t t

L xu


uuu u

u′∂ ′∂ −′ ′′

′= − + ⋅ = = −∂ ∂∫



Model Problem: Attached Flame Excited by a Harmonically Oscillating, Convecting Disturbance

,,0

cos(2 ( / ))nn c v

u f t x uu

ε π′

= −t

( )( )( ) ( )( ),0, 2 / sin2 / tan1

,0 ,0 ,

sinReal2 cos / 1

c v i f y u ti f y u tn

c v

i e eu f u u

π θπ θξ ε θπ θ

−− − ⋅ = × − −

t

t t

• Model problem: flame excited by convecting velocity field,

• Linearized solution:



Solution Characteristics

• Note interference pattern on flame wrinkling

• Interference length scale:

( ),

1sin| 1 |int

c vu uλ λ θ =

−tt

( )/ siny λ θt

intλ



Interference Patterns

D. Shin et al., AIAA Aerospace Science Meeting, 2011. V. 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

,

cos/2 cos 1

peak c

c v

xuu

θλθ

=

⋅ −

D. Shin et al., Journal of Power and Prop, 2011.



Aside: Randomly Oscillating, Convecting Disturbances

• Space/time coherence of disturbances key to interference patterns

• Example: convecting random disturbances to simulate turbulent flow disturbances or

Single frequencyexcitation

Randomexcitation



• Flame propagation normal to itself smoothes out flame 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

1

∂∂

+=

−

∂∂

+∂∂

xLSv

xLu

tL

Lff

Video : Courtesy Durox, Ducruix & Candel

Flame Wrinkle Destruction Processes: Kinematic Restoration

Sung & Law, Progress in Energy and Comb Sci, 2000



Kinematic Restoration Effects: Oscillating Flame Holder Problem

u0

( )0sin tε ω⋅

Flame front

D. Shin & T. Lieuwen, Comb and Flame, 2012.



• Leads to nonlinear farfield flame dynamics

• Decay rate is amplitude dependent

Numerical Calculation Experimental Result

Kinematic Restoration Effects

x/λc



Tangential direction, t

()

0|

|ξ

ω

,0

2

Ls

t

Multi- Zone Behavior of Kinematic Restoration

• Near flame holder – Higher amplitudes and shorter wavelengths decay faster

• Farther downstream – Flame position independent of wrinkling magnitude – Flame position only a function of wrinkling wavelength – is determined by the leading points

48

Reactants

Products

,0Ls t⋅

Sung et al., Combustion and Flame, 1996

D. Shin & T. Lieuwen, Comb and Flame, 2012.



Flame Wrinkle Destruction Processes: Kinematic Restoration

D. Shin & T. Lieuwen, JFM , 2013.



Flame Wrinkle Destruction Processes: Flame Stretch in Thermodiffusively Stable Flames

Wang, Law, and Lieuwen., Comb and Flame, 2009. Preetham and Lieuwen, JPP, 2010.



Flame Stretch Effects

( ) ( )0,0

| , |exp Ls

ξ ωσ

ε≈ −

tt

: Normalized Markstein lengthσ

Linear in amplitude wrinkle destruction process


School of Aerospace Engineering Flame Geometry

• Conditions – Over ventilated flame – Fuel & oxidizer forced by spatially uniform flow

oscillations – Will show illustrative solution in Pe>>1 (i.e.,

WIIu0>>D ) limit

WI

WII x

y

( ),x tξ

Zst

Oxidizer u0 + u1

Oxidizer u0 + u1

Fuel u0 + u1

53

K. Balasubramanian, R. Sujith, Comb sci and tech, 2008. M. Tyagi, S. Chakravarthy, R. Sujith, Comb Theory and

Modelling, 2007. N. Magina et al., Proc of the Comb Inst, 2012. Copyright © T. Lieuwen 2013 Unauthorized Reproduction Prohibited


Solution characteristics of Z field ( ) ( ) ( )

22

11

(2 )sincos exp 1 exp 2 exp

2n n

n n Wn W II II II

i n y x xi St i tSt Pe W PeW W

ε ππ ω

π

∞

=

= − − −

∑A AZ A A

( ), , stx tξ =Z Z



( ) ( ) ( )2

21

1

(2 )sincos exp 1 exp 2 exp

2n n

n n Wn W II II II

i n y x xi St i tSt Pe W PeW W

ε ππ ω

π

∞

=

= − − −

∑A AZ A A

Solution characteristics of Z field

stZ1,nξ

stZ



Solution: Space-Time Dynamics of Zst Surface

( ) [ ],01, 0

,0

sin ( ) 1 exp 2 exp 22

, xn

x

i u xx i f i f tf u

x t εξ θ π π

π

= − − ⋅

Flame wrinkling only occurs through velocity fluctuations normal to flame

Low pass filter characteristic

Flame wrinkles propagate with axial flow (cause interference)

1,nξ

0θ



Illustrative Result of Flame Front Dynamics

( )( )

,0 /3.3x

f

Convective wavelength u fFlame length L

=

Lf

Oxidizer

Fuel



( )( )

,0 /3.3x

f


=

x / LF

Lf

Oxidizer

Fuel

x / LF 58




( )( )

,0 /0.5x

f


=

Lf

Oxidizer

Fuel




( )( )

,0 /0.5x

f


=

x / LF

0 0.2 0.4 0.6 0.8 1

Lf

Oxidizer

Fuel

x / LF



School of Aerospace Engineering Comparison - similarities

• Non-premixed

( ) [ ],01,

,0sin ( ) 1 exp 2 exp 2

2, x

nx

i u xx i f i ftf

tu

x εξ θ π π

π

= − −

( ) [ ],01,

,0, sin 1 exp 2 exp 2

2 cosx

ni u xx t i f i ft

f uε

ξ θ π ππ θ

= ⋅ − − t

• Premixed

Similarities between space/time dynamics of premixed and non-premixed flames responding to bulk flow perturbations

>Magnitude

> Flame Angle

> Wave Form



( ) [ ],01,

,0sin ( ) 1 exp 2 exp 2

2, x

nx

i u xx i f i ftf

tu

x εξ θ π π

π

= − −

( ) [ ],01,

,0, sin 1 exp 2 exp 2

2 cosx

ni u xx t i f i ft

f uε

ξ θ π ππ θ

= ⋅ − − t

Comparison - difference

• Non-premixed

• Premixed

ux,0

sL ,0u t

1,nξ

θ ux,0

Non-premixed Premixed

Convective wave speeds

ξ1,n



Spatially Integrated Heat Release

– Flame surface area (Weighted Area) – Mass burning rate (MBR) – We’ll assume constant composition

( ) F Rflame

Q t m dA′′= ∫

h

• Flame describing function:

1 0

,1 ,0

//x x

Q Qu u

≡

F

• Unsteady heat release

WA MBR= +F F


School of Aerospace Engineering Premixed Flames • Spatially integrated heat release:

• Linearized for constant flame speed, heat of reaction, and density:

65

( ) u uc R

flame

Q t s dAρ= ∫ h

( ) ,1 ,11 0 0 01

0 0 ,0 0 0 0 0 0

u uc Ruc Rflame flame flame flame

sQ t dA dA dAdAQ A s A A A

ρρ

= + ⋅ + ⋅ + ⋅∫ ∫ ∫ ∫

hh

( )2

( ) sin 1o flame

A t y dyA y

ξθ ∂= + ∂

∫ W

Proportional to flame area


School of Aerospace Engineering Premixed Flames • W(y) is a geometry dependent weighting factor:

where:

66

fW

FLθ

y

x

Axisymmetric Cone

tanθ=f FW L

( )22 fy Wπ π( ) ( )22 f fW y Wπ π−1 fW( )W y =

Two-dimensional Axisymmetric Wedge



Premixed Flame TF Gain– Bulk Flow Excitation

• St<<1: =1 • St>>1: ~1/St

67

FF



Why the 1/St Rolloff?

• Flame position ~1/St

• Flame area/unit axial distance:

• Linearized:

68

[ ],01,

,0

( , ) sin 1 exp 2 exp 22

xn f

f

i u xx t i St i ftf L

εζ θ π π

π

= − −

2

1dA dxxζ∂ = + ∂

0 120

20

1

1

dA x xdx x

x

ζ ζζ

ζ

∂ ∂∂ ∂ ∂= + + ∂ ∂ + ∂

Low pass filter characteristic!

[ ],0

sin exp 2 exp 2ff

xi St i ftL

ε θ π π

∝ −



Why the 1/St Rolloff?

• Consider spatial integral of traveling wave disturbance:

• 1/St comes from the integration!

69

[ ]0

cos sin sinFL

F

x

x u Lt dx t tu u

ω ω ωω=

− = − − − ∫

Traveling Wave 1/St due to interference effects associated with tangential convection of wrinkles



Premixed Flame Response - Phase

• Phase rolls off linearly with St (for low St values) – Time delayed behavior

• 180o phase jumps at nodal locations in the gain 70



Premixed Flame Response - Phase

71

Flame area-velocity relationship for convectively compact flame (low St values):

1 1

0 0

( ) ( )A t u tA u

τ−= n

Axi-symmetric Wedge:

0

fLC

uτ =

Axi-symmetric Cone:

Two-dimensional:

1

2

2(1 )3cos

CkCθ

−+=

( )2

2 13 cos

c

c

kC

k θ+

=

( )2

12 cos

c

c

kC

k θ+

=

0

,0C

tx

uku

=



Nonpremixed Flames-Bulk Flow Excitation

72

• Returning to spatially integrated heat release:

• Linearize the MBR and area terms:

( ) F Rflame

Q t m dA′′= ∫

h

Area Fluctuation Contribution

MBR Fluctuation Contribution

,0 0 ,0 1 ,1 0( )

F F FR flame flame flame

Q t m dA m dA m dA′′ ′′ ′′= + +∫ ∫ ∫

h

Steady State Contribution



( )

( )

10

00

Fflame

WA

Fflame

m dA

m dA

′′

=′′

∫

∫

F

Non-Premixed Flames: Role of Area Fluctuations

weighting

( )0Fm′′

x

x

y

For the higher velocity, • Area increases • Weighted area decreases

Very strong function of x!

u/uref = 1 u/uref = 1.5

( )0,Fm premixed′′

=> Premixed => Non-premixed



o Non-premixed • Weighted Area

o Premixed • Area (as weighting is

constant)

Weighted Area cont’d

u’

Area

Weighted Area

At low frequencies, area and weighted area are out of phase

At low frequencies



– Non-premixed:

• Fluctuations in spatial gradients of the mixture fraction

( )

( )

01

00

Fflame

MBR

Fflame

m dA

m dA

′′

=′′

∫

∫

F

Mass Burning Rate

– Premixed:

• Stretch sensitivity of the burning velocity

( )1~ L

Fsmφ

∂′′∂

( ) 11

1~cosFm

yθ∂′′∂

Z



Flame Transfer Functions - Contributions

Non-premixed

Weighted Area

Mass Burning Rate

Heat Release

StLf

Mag

nitu

de

Area

Mass Burning Rate

Heat Release

StLf

Mag

nitu

de

Significant differences in dominant processes controlling heat release oscillations

• Non-premixed : Mass burning rate • Premixed : Area

Premixed (weak flame stretch)

76

Magina et al., Proc of the Comb Inst, 2012.



Comparisons of Gain and Phase of FTF

Non premixed−F

1 /fLSt

fLSt

Mag

nitu

de

1 /fLSt

( )

Premixed

weak flame stretch

F

Non premixed−∠F

Phas

e (d

eg)

fLSt

( )Premixed

weak flame stretch

∠F

Gain Phase

St << 1 : ~1 St >> 1 : Non-premixed flames ~1/St St ~ O(1) : Non-premixed flame ~ 1/St1/2

- At St~0(1), non-premixed flames are more sensitive to flow perturbations

> Premixed ~ 1/St



• Gain o fcn ( St2, kc)

o Unity at low 2St o Gain increases greater than unity o "Nodes" of zero heat release response

Premixed Flame TF’s: More Complex Disturbance Fields

Disturbance convecting axially at velocity of Uc & kc=Uo/Uc Axisymmetric wedge flame:

( )( )

( ), 1

,,0 ,

, ,cos(2 ( / ))F n

n c x y tx y t

v x y tf t x u

u ξξ

ε π=

=

= −t



Closing Remarks • Flame response exhibits “wavelike”, non-local

behavior due to wrinkle convection, leading to:

• maxima/minima in gain curves, interference phenomenon, etc. • 1/f behavior in transfer functions

• Premixed flame wrinkles controlled by different processes in different regions

• Role of area, weighted area, mass burning rate are quite different for premixed and non-premixed flames



Summary

• Many great fundamental problems in gas turbine combustion – Edge flames – Reacting swirl flows – Differential diffusion effects on turbulent flames – Turbulent flames in highly preheated flows – Response of flames to harmonic disturbances – …and many more!!


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