School of Aerospace Engineering. Course Outline

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 1

Flame Response to Harmonic Disturbances Combustion instabilities manifest themselves as narrowband oscillations at natural acoustic modes of combustion chamber Fourier Transform 500 400 300 200 100 0 0 100 200 300 400 500 600 700 Frequency (Hz) 2

Wave Equation: Basic Problem p c 2 p = γ 1 q ( ) tt xx t 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 Flow Instabilities 3

Response of Global Heat Release to Flow Perturbations 0.5 0.4 CH* / CH* o Q /Q o 0.3 0.2 Why does this saturate? Why at this amplitude? 0.1 What factors affect slope of this curve (gain relationship)? 0 0 0.2 0.4 0.6 0.8 u / u o u /u o Q () t = m h da F R flame 4

Analytical Tools/Governing Equations Work within fast chemistry, flamelet approximation and use G- and Z- equations to describe flame dynamics 6

Analytical Tools Z Equation Key assumptions Le=1 assumption flame sheet at Z=Z st surface Imposed flow field Equal diffusivities DYF ρ ( ρd F YF) = ωf Dt ( ( υ + 1) ) DY ( Y ) ( ( + 1) ) = Pr ρ ρd Pr Pr υ Dt ω Pr ( υ + 1) ω ω = F Pr ( 1+ υ ) Add these species equations: ( + Y ( υ + 1) ) DY ( Y Y ) F Pr ρ ρd F Pr υ Dt ( + ( + 1) ) = 0 7

Analytical Tools Z Equation Recall the definition of mixture fraction: Z = Y + F 1 Y ( υ + 1) Pr Yields: DZ ρ ( ρd Z ) = 0 Dt Z + u Z= D Z t ( ) 8

. Premixed Flame Sheets: G-Equation Flame fixed (Lagrangian) coordinate system: D G ( xt, ) = 0 Dt at the flame front Product G>0 Coordinate fixed (Eulerian) coordinate system: G + v F G = 0 t v = u sn F n = G/ G d Reactant G<0 Flame surface G=0 G t G + u sd G G = 0 G + u G = s d G t 9

G-equation for single valued flame Two-dimensional flame front front Position is single valued function, ξ, of the coordinate y Define and substitute Gxyt (,,) x ξ ( yt,) ξ ux u ξ y sd 1 ξ + = + t y y 2 y Reactants ξ (y,t) g x 1 n Products 10

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/ D >>1 or u/s d >>1 limits Z + u Z= D Z t ( ) G + u G = s d G t 11

Governing Equations G-equation only physically meaningful at the flame surface, G=0 Can make the substitution, (,,, ) = ξ (,, ) G xyzt x yzt y Reactants ξ (y,t) g x 1 n Products Z-equation physically meaningful everywhere Cannot make analogous substitution 12

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 13

Excited Bluff Body Flames (Mie Scattering) Increased Amplitude of Forcing 15

Excited Swirl Flame (OH PLIF) 0 45 90 135 315 270 225 180 16

Excited Bluff Body Flames (Line of sight luminosity) 18 m/s 294K 38 m/s 644K 127 m/s 644K 170 m/s 866K 17

Overlay of Instantaneous Flame Edges 18 m/s 294K 38 m/s 644K 127 m/s 644K 170 m/s 866K 18

Quantifying Flame Edge Response Time Series L (x, f 0 ) Power Spectrum 19

Spatial Behavior of Flame Response Strong response at forcing frequency Non-monotonic spatial dependence Convective wavelength: λ c = U 0 /f 0 - distance a disturbance propagates at mean flow speed in one excitation period 20

Flame Wrinkling Characteristics 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 21

Typical Results Other Flames 50 m/s, 644K 1.8m/s, 150hz Magnitude can oscillate with downstream distance 22

Analysis of Flame Dynamics 1. Wrinkle convection and flame relaxation processes 2. Excitation of wrinkles 3. Interference processes 4. Destruction of wrinkles 23

Level Set Equation for Flame Position G-equation : L u L f vf SL 1 L + = + t x x 2 24

Wrinkle Convection Model problem: Step change in axial velocity over the entire domain from u a to u b, both of which exceed s d : u = u u a b t t < 0 0 26

Wrinkle Convection t 1 t 2 t 3 c shock t Flame θ s 1 d = sin ua θ s 1 d = sin ub Flame relaxation process consists of a wave that propagates along the flame in the flow direction. 27

Harmonically Oscillating Bluff Body u 0 s L u t u c,f u t Petersen and Emmons, The Physics of Fluids Vo. 4, No. 4, 1961. 28

Phase Characteristics of Flame Wrinkle Convection speed of Flame wrinkle, u c,f Mean flow velocity, u 0 Disturbance Velocity, u c,v D. Shin et al., Journal of Power and Propulsion, 2011. 29

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 u 0 30

Excitation of Wrinkles on Anchored Flames ( x ) L, t 1 x u n x x 1 x = ( x, t ) dx + u n( x = 0, t = t ) x u 0 x u u u t Linearized solution of G Equation, assume anchored flame t t t Wrinkle convection can be seen from delay term u n u 0 θ L u t s L 32

Excitation of Flame Wrinkles Spatially Uniform Disturbance Field ( x ) L, t 1 x u n x x 1 x = ( x, t ) dx + u n( x = 0, t = t ) x u 0 x u u u t t t t 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! θ ( t) ( t) ( ) 1 s L = cos u t 33

Near Field Behavior- Predictions Can derive analytical formula for nearfield slope for arbritrary velocity field: L u ' ' 1 n = 2 x cos θ u t u n θ u t 34

Comparisons With Data 1 cos 2 u n L = θ u x t 5 m/s, 300K PIV Data S. Shanbhogue et al., Proc of the Comb Inst, 2009. MIE scattering Data 35

Normalized by u Near Field Behavior Increasing amplitude, u Flame starts with small amplitude fluctuations because of attachment L (x=0, t) = 0 Nearfield dynamics are essentially linear in amplitude S. Shanbhogue et al., Proc of the Comb Inst, 2009. 36

Excitation of Flame Wrinkles Spatially Varying Disturbance Field ( x ) L, t 1 x u n x x 1 x = ( x, t ) dx + u n( x = 0, t = t ) x u 0 x u u u t Flame wrinkles generated at all points where disturbance velocity is nonuniform, du /dx 0 Flame disturbance at location x is convolution of disturbances at upstream locations and previous times t Convecting vortex is continuously disturbing flame Vortex convecting at speed of u c,v Flame wrinkle that is excited convects at speed of u t Bechert, D.,Pfizenmaier, E., JFM., 1975. t t 38

Model Problem: Attached Flame Excited by a Harmonically Oscillating, Convecting Disturbance Model problem: flame excited by convecting velocity field, u u n = ε cos(2 π f( t x/ u )) t,0 n cv, Linearized solution: ξ 1 i εn sinθ i2 f ( y/ ( u, tan ) ) i2 f ( y/ ( u,0 sin cv t ) t) Real π θ π t θ = e e u ( ),0 f 2π u,0 cos θ / ucv, 1 t t 39

Solution Characteristics Note interference pattern on flame wrinkling λ int Interference length scale: int ( sin ) λ λ θ = t 1 1 ut u cv, ( ) y / λ sinθ t 40

Interference Patterns D. Shin et al., AIAA Aerospace Science Meeting, 2011. V. Acharya et al., ASME Turbo Expo, 2011. 41

Result emphasizes wave-like, non-local nature of flame response Comparison with Data x peak / λ = c u u cos 0 2 2 cos 1 cv, 2 θ θ Can get multiple maxima/minima if excitation field persists far enough downstream D. Shin et al., Journal of Power and Prop, 2011. 42

Aside: Randomly Oscillating, Convecting Disturbances Space/time coherence of disturbances key to interference patterns Random excitation Example: convecting random disturbances to simulate turbulent flow disturbances or Single frequency excitation 43

Flame Wrinkle Destruction Processes: Flame propagation normal to itself smoothes out flame wrinkles Typical manifestation: vortex rollup of flame Process is amplitude dependent and strongly nonlinear Kinematic Restoration Large amplitude and/or short length scale corrugations smooth out faster L t + u Video : Courtesy Durox, Ducruix & Candel f L x v f = S L L 1+ x 2 Sung & Law, Progress in Energy and Comb Sci, 2000 45

Kinematic Restoration Effects: Oscillating Flame Holder Problem Flame front u 0 ε sin ( ω t) 0 D. Shin & T. Lieuwen, Comb and Flame, 2012. 46

Kinematic Restoration Effects Leads to nonlinear farfield flame dynamics Decay rate is amplitude dependent Numerical Calculation x/λ c Experimental Result 47

Multi- Zone Behavior of Kinematic Restoration Products ξ ( ω0 ) s 2 L,0 t sl,0 t Tangential direction, t 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 Reactants Sung et al., Combustion and Flame, 1996 D. Shin & T. Lieuwen, Comb and Flame, 2012. 48

Flame Wrinkle Destruction Processes: Kinematic Restoration D. Shin & T. Lieuwen, JFM, 2013. 49

Flame Wrinkle Destruction Processes: Flame Stretch in Thermodiffusively Stable Flames Wang, Law, and Lieuwen., Comb and Flame, 2009. Preetham and Lieuwen, JPP, 2010. 50

Flame Stretch Effects ( t ω0 ) ξ, ε σ exp ( σ s t ) L,0 : Normalized Markstein length Linear in amplitude wrinkle destruction process 51

Flame Geometry W I Oxidizer u 0 + u 1 ξ ( xt, ) Z st y W II Fuel u 0 + u 1 x Oxidizer u 0 + u 1 Conditions Over ventilated flame Fuel & oxidizer forced by spatially uniform flow oscillations Will show illustrative solution in Pe>>1 (i.e., W II u 0 >>D ) limit 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. 53

Solution characteristics of Z field 2 ( ) (2 π)sin ( ) iε n n n y 2 x x Z1 = A A cos An exp An 1 exp 2πi StW exp n= 1 2π StW Pe WII PeWII WII ( iωt ) Z x, ξ, t = Zst ( ) 54

Solution characteristics of Z field 2 ( ) (2 π)sin ( ) iε n n n y 2 x x Z1 = A A cos An exp An 1 exp 2πi StW exp n= 1 2π StW Pe WII PeWII WII ( iωt ) Z st Z st ξ 1,n 55

Solution: Space-Time Dynamics of Z st Surface ξ 1,n θ 0 iεu x ξ1, n θ0 π π 2π f ux,0 x,0 ( xt, ) = sin ( x) 1 exp i2 f exp[ i2 f t] Low pass filter characteristic Flame wrinkling only occurs through velocity fluctuations normal to flame Flame wrinkles propagate with axial flow (cause interference) 56

Illustrative Result of Flame Front Dynamics ( / ) x,0 = 3.3 ( L ) f Convective wavelength u f Flame length Oxidizer Fuel L f 57

Illustrative Result of Flame Front Dynamics ( / ) x,0 = 3.3 ( L ) f Convective wavelength u f Flame length Oxidizer Fuel x / L F L f x / L F 58

Illustrative Result of Flame Front Dynamics ( / ) x,0 = 0.5 ( L ) f Convective wavelength u f Flame length Oxidizer Fuel L f 59

Illustrative Result of Flame Front Dynamics ( / ) x,0 = 0.5 ( L ) f Convective wavelength u f Flame length Oxidizer Fuel x / L F L f 0 0.2 0.4 0.6 0.8 x / L 1 F 60

Comparison - similarities Non-premixed iεu x ξ1, n θ π π 2π f u x,0 x,0 ( x, t) = sin ( x) 1 exp i2 f exp[ i2 ft] Premixed iε u x ξ1, n θ π π 2π f u,0 cosθ t x,0 ( x, t) = sin 1 exp i2 f exp[ i2 ft] Similarities between space/time dynamics of premixed and non-premixed flames responding to bulk flow perturbations >Magnitude > Flame Angle > Wave Form 61

Comparison - difference Non-premixed iεu x ξ1, n θ π π 2π f u x,0 x,0 ( x, t) = sin ( x) 1 exp i2 f exp[ i2 ft] Premixed iε u x ξ1, n θ π π 2π f u,0 cosθ t x,0 ( x, t) = sin 1 exp i2 f exp[ i2 ft] Convective wave speeds ξ 1,n u x,0 u t,0 θ s L u x,0 ξ 1,n Non-premixed Premixed 62

Spatially Integrated Heat Release Unsteady heat release Q t m h da () = F R flame Flame surface area (Weighted Area) Mass burning rate (MBR) We ll assume constant composition Flame describing function: F Q u / Q / u 1 0 x,1 x,0 = F + F WA MBR 64

Premixed Flames Spatially integrated heat release: u u Q () t = ρ schrda flame Linearized for constant flame speed, heat of reaction, and density: u u ( ) da s ρ c,1 h 1 R,1 = + Proportional + to flame + area Q t da da da 1 0 0 0 u Q 0 A0 s flame flame c,0 A0 ρ flame 0 A0 h flame R0 A0 At () A o ξ = sinθ W ( y) 1+ y flame 2 dy 65

Premixed Flames W(y) is a geometry dependent weighting factor: Two-dimensional Axisymmetric Cone Axisymmetric Wedge x y L F θ W f W( y ) = 1 W f 2 ( W y) ( πw ) 2π f f ( 2 πw ) 2πy f where: W f = L F tanθ 66

Premixed Flame TF Gain Bulk Flow Excitation St<<1: F =1 St>>1: F ~1/St 67

Why the 1/St Rolloff? Flame position ~1/St iu ε x,0 x ζ1, n( x, t) = sinθ 1 exp i2πst f exp i2π ft 2π f L f,0 [ ] Flame area/unit axial distance: Linearized: da Low pass filter characteristic! ζ = 1+ x 2 dx 2 0 1 da ζ 0 x x dx ζ ζ = 1+ + x ζ 0 1+ x 2 x εsinθexp i2πst f exp i2π ft L f,0 [ ] 68

Why the 1/St Rolloff? Consider spatial integral of traveling wave disturbance: L F x= 0 cos ω x u LF t dx sin t sin t u ω ω = u ω [ ] Traveling Wave 1/St due to interference effects associated with tangential convection of wrinkles 1/St comes from the integration! 69

Premixed Flame Response - Phase Phase rolls off linearly with St (for low St values) Time delayed behavior 180 o phase jumps at nodal locations in the gain 70

Premixed Flame Response - Phase Flame area-velocity relationship for convectively compact flame (low St values): A1() t u1( t τ ) = n A u 0 0 L f Axi-symmetric Wedge: C = + 1 2(1 k C ) 2 3cos θ τ = C u 0 Axi-symmetric Cone: C = ( k + ) 2 1 c 3k cos c 2 θ k C = u u 0 tx,0 Two-dimensional: C = ( k + 1) c 2k cos c 2 θ 71

Nonpremixed Flames-Bulk Flow Excitation Returning to spatially integrated heat release: Q () t = m F hrda flame Linearize the MBR and area terms: Qt () h = m da + m da + m da F,0 0 F,0 1 F,1 0 R flame flame flame Steady State Contribution Area Fluctuation Contribution MBR Fluctuation Contribution 72

F School of Aerospace Engineering Non-Premixed Flames: Role of Area WA Fluctuations = flame flame ( m ) F ( m ) F 0 0 da da weighting For the higher velocity, 1 0 ( m F ) 0 Very strong function of x! Area increases => Premixed Weighted area decreases => Non-premixed y ( m ) 0, F x u/u ref = 1 u/u ref = 1.5 premixed x 73

Weighted Area cont d At low frequencies Area u Weighted Area o o Non-premixed Weighted Area Premixed Area (as weighting is constant) At low frequencies, area and weighted area are out of phase 74

Mass Burning Rate F MBR = flame flame ( m ) F 1 ( m ) F 0 da da 0 0 Non-premixed: 1 ( ) m F 1 1 Z ~ cos θ y Fluctuations in spatial gradients of the mixture fraction Premixed: m ( ) 1 ~ L F s φ Stretch sensitivity of the burning velocity 75

Flame Transfer Functions - Contributions Non-premixed Premixed (weak flame stretch) Mass Burning Rate Heat Release Magnitude Heat Release Magnitude Area Weighted Area Mass Burning Rate St Lf Significant differences in dominant processes controlling heat release oscillations Non-premixed : Mass burning rate Premixed : Area St Lf Magina et al., Proc of the Comb Inst, 2012. 76

1/ St Lf Comparisons of Gain and Phase of FTF Gain F Non premixed Phase F Non premixed 1/ St Lf Magnitude F Premixed Phase (deg) F Premixed ( weak flame stretch) ( weak flame stretch) St Lf St << 1 : ~1 St >> 1 : Non-premixed flames ~1/St St ~ O(1) : Non-premixed flame ~ 1/St 1/2 > Premixed ~ 1/St - At St~0(1), non-premixed flames are more sensitive to flow perturbations St Lf 77

Premixed Flame TF s: More Complex Disturbance Fields Disturbance convecting axially at velocity of U c & k c =U o /U c Axisymmetric wedge flame: v Fn, 1 ( xyt,, ) u t,0 x= ξ, ( yt) = ε cos(2 π f( t x/ u )) = (, ) n c x ξ yt Gain o fcn ( St 2, k c ) o Unity at low St 2 o Gain increases greater than unity o "Nodes" of zero heat release response 78

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 79

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!! 80