A combustion instability model accounting for dynamic - PowerPoint PPT Presentation

A combustion instability model accounting for dynamic flame-flow-acoustic interactions el Assier 1 Xuesong Wu 2 Rapha¨ 1 University of Manchester 2 Imperial College London Analytical methods in thermoacoustics, Keele, September 2014 Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 1 / 37

1 Introduction Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 2 / 37

Combustion instabilities [Photos from Lieuwen, 2005] Applications: Problems: Jet engines Vibrations Rocket engines Structural fatigue Gas turbine engines Increase in fuel consumption Phenomena Self-sustained, large amplitude pressure fluctuations and flame oscillations Generally occurs around the characteristic frequency of the combustor Objective: Understanding the fundamental mechanisms of combustion instabilities Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 2 / 37

Physical mechanisms: Flame-Flow-Acoustic interactions Flame excites the acoustic pressure 1 Acoustic pressure amplifies according to Rayleigh’s criterion, i.e when heat release and acoustic pressure are in phase. Acoustic velocity and acoustic acceleration advect and modulate the 2 flame front. The flame influences the hydrodynamic flow, which in turn influences 3 the flame. Multi-scale, Extremely challenging for DNS! Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 3 / 37

Existing models and comparison The G -equation [see e.g. Dowling, JFM, 1999] Model the advection effect of acoustic on the flame Purely kinematic (no hydrodynamics) Ignores dynamic effect of acoustics Hydrodynamic theory of flames Flame-flow interaction model, no acoustic considerations [Pelc´ e & Clavin, JFM 1982], [Matalon & Matkowsky, JFM 1982] Reduces to Michelson-Sivashinsky equation (M-S) Advection Hydrodynamics (D-L) Flame-acoustics coupling R-T G -equation ✓ ✗ ✓ ✗ M-S ✓ ✓ ✗ ✗ Aim: ✓ ✓ ✓ ✓ Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 4 / 37

The G -equation G -equation � � ∂F � 2 ∂G ∂F ∂t = − u · ∇ G + S u |∇ G | , ∂t = u G − S u 1 + ∂y G = G ( x, y, t ) : level set function representing the flame S u : normal flame speed propagation Write G ( x, y, t ) = x − F ( y, t ) and consider 1D velocity fluctuations u G = u G ( t ) : “u gutter”, acoustic velocity at the flame (1D) Allows for wrinkling of anchored flames... ... but not for freely propagating flames. Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 5 / 37

The Michelson-Sivashinsky equation The M-S equation (2D)  � � 2  ∂ 2 ϕ  ∂ϕ ∂ϕ ∂t = 1 2 I ( ϕ ; y ) + 1 ∂y 2 + 1 , γ 2 ∂y   � I ( ϕ ; y )( k, t ) = | k | � ϕ ( k, t ) , ϕ : flame shape γ : “free” parameter � : Fourier transform in y direction Good for unconfined freely propagating flames Predicts Darrieus-Landau instability: i.e. flames tend to curve Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 6 / 37

Searby’s experiment [ Searby, Combust. Sci. Technol., 1992 ] Setup Results Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 7 / 37

Flame-flow-acoustics in the literature Externally imposed acoustics For example [Markstein & Squire, 1955], [Searby & Rochwerger, JFM 1991], [Clanet & Searby, PRL 1998], [Bychkov, PoF 1999] No consideration of back-action of the flame onto the acoustics Focused on secondary instability Including spontaneous acoustic field [Pelc´ e & Rochwerger, JFM 1992] First mathematical treatment of flow-flame-acoustics interactions Ad-hoc modelling of flame profile as cosine function Focused on primary instability Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 8 / 37

Presentation of the Problem Geometry y x = f ( y, z, t ) Θ −∞ Θ ∞ h ∗ Burnt Unburnt ρ −∞ ρ ∞ x z ℓ ∗ Physical assumptions One-step irreversible chemical reaction Fuel deficient reactant: lean combustion Mixture obey state equation for perfect gas Newtonian compressible fluid Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 9 / 37

Equations to be solved Equations Conservation of mass Conservation of momentum Transport equation governing the diffusion of chemical species Energy conservation State equation Main variables of non-dimensional problem u , ρ , p , θ ρ −∞ , θ −∞ , U L M : Mach number δ : flame thickness q : heat release q = ( θ ∞ − θ −∞ ) /θ −∞ β : activation energy M a : Markstein number Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 10 / 37

Change of coordinate, flame frame of reference x = f ( y, z, t ) y Non-dimensionalised coordinates y = π Unburnt z Burnt 2 π x y = − π 2 πℓ ∗ h ∗ x = 2 πℓ ∗ x = 0 h ∗ Change of variable ( x, y, z, t ) → ( ξ, η, ζ, τ ) ξ = x − f ( y, z, t ) , η = y , ζ = z and τ = t η η = π   Unburnt ( ξ < 0) Burnt ( ξ > 0) ∂ u = u e ξ + v ζ   2 π ∂η �   ξ ∇ =   ∂ ∂ζ η = − π ξ = 0 Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 11 / 37

2 Asymptotic analysis Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 12 / 37

Different scalings of the problem Preheat zone O ( δ ) Reaction zone O ( δ/β ) η η = π u a , p a U, V, P Acoustic zone O (1 /M ) ζ 2 π ξ O (1 /M ) Acoustic zone U, V, P u a , p a η = − π ξ = 0 O (1) Hydrodynamic zone Large-Activation-Energy : β ≫ 1 Thin flame : δ ≪ 1 Low Mach Number: M ≪ 1 β ≪ δ ≪ 1 ≪ 1 δ [Hydro. theo. flames] M [Wu et al, JFM 2003] Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 12 / 37

Acoustic zone Stretch of variable � ξ = Mξ Acoustic Equations  ∂p a ∂τ + ∂u a   = 0  ∂ � ρ = R to first order in δ ξ R∂u a ∂τ + ∂p a   = 0  ∂ ˜ ξ Acoustic Jumps (weakly nonlinear)  f = F to first order in δ � p a � +  − = 0 � � 2 Acoustic-flame coupling − = J a ( τ ) = q �  � u a � + ∇ F 2 Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 13 / 37

Hydrodynamic zone 1/2: ( u, v , p ) = ( U, V , P ) + O ( δ 2 ) Linearising the hydrodynamic equations and the jumps to second order in δ leads to: Hydrodynamic equations Jump conditions  � F, V − �  [ U ] + ∂U = J U   ∂ξ + ˜ ∇ · V = 0 ,  −      � F, V − �    [ V ] + = J V R∂U ∂τ + ∂U − ∂P −  = ∂ξ + δ Pr ∆ U,   � �   ∂ξ  [ P ] + F, U − , V − , B a ( τ ) = J P   −  R∂ V ∂τ + ∂ V  − ˜  = ∇ P + δ Pr ∆ V , ∂ξ U − = U (0 − , η, ζ, τ ) Weakly nonlinear Flame equation � � ∂F U − − V − · ˜ F ) 2 / 2 + δ M a ˜ ∇ 2 F + δ E F = ∇ F − ( ∇ V − , F ∂τ � ∂p a � + � � + − R∂u a Acoustic-flow coupling B a ( τ ) = = ∂ � ∂τ ξ Flow-flame coupling − − Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 14 / 37

What now? Using Fourier analysis, this whole system can be simplified. Making some simplifications, the equations can be reduced to The G -equation The M-S equation Objective Retaining key terms to allow for a simple model accounting for the three-way coupling physics of the problem. Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 15 / 37

Hydrodynamic zone 2/2: ( u, v , p ) = ( U, V , P ) + O ( δ ) Considering only the leading order in δ , and partially linearising the flame equation leads to: Hydrodynamic equations Jump conditions  ∂U  ∂ξ + �  = 0  ∇ · V  [ U ] +   = 0   − � �     [ V ] + � R∂U ∂τ + ∂U − ∂P = − q ∇ F = − � �  ∂ξ ∂ξ    qG   [ P ] + = B a ( τ ) +   − F  R∂ V ∂τ + ∂ V −  1 + q  − � = ∇ P ∂ξ Weakly nonlinear Flame equation � � 2 ∂F U − − 1 � + δ M a � ∇ 2 F ∇ F = ∂τ 2 � ∂p a � + � � + − R∂u a Acoustic-flow coupling B a ( τ ) = = ∂ � ∂τ ξ Flow-flame coupling − − Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 16 / 37

3 Steady state solutions and linear instability Rapha¨ el Assier (University of Manchester) Modelling combustion instabilities Keele, Sept 16, 2014 17 / 37