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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign The Effect of Background Turbulence on the Propagation of Large-scale Flames Moshe Matalon Turbulent Mixing and Beyond The Abdus Salam International Center for Theoretical Physics Trieste, Italy August 18-26, 2007 University of Illinois at Urbana-Champaign Department of Mechanical Science and Engineering

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University of Illinois at Urbana-Champaign A planar front is unconditionally unstable. Darrieus-Landau (hydrodynamic) instability Burned b Unburned u Darrieus,1938; Landau, 1944 flame speed The instability results from thermal expansion > 1 Dominant in large-scale flames, where diffusion influence plays a limited role. u / b

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign cellular (diffusive-thermal) lean hydrogen-air flame Le eff < 1 inherently stable lean butane-air flame Le eff > 1 R.A. Strehlow, 1969

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Slightly rich propane-air expanding flame in low-intensity turbulent premixture Palm-Leis & Strehlow 1969

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Wrinkled methane-air inverted conical flame Sattler, Knaus & Gouldin, 2002 burned gas unburned gas

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Hydrodynamic theory Mach number << 1 typically ~ thin flame: l f /L << 1 typically l f = D th /S L ~ cm one-step overall reaction Fuel + Oxidizer Products burned O( ) unburned n Clavin & Williams (JFM, 1982), Matalon & Matkowsky (JFM, 1982), Matalon, Cui & Bechtold (JFM, 2003) thermal expansion, temperature-dependent transport, differential & preferential diffusion, equivalence ratio, different reaction orders, radiative losses internal flame structure heat conduction, species diffusion, viscous dissipation resolved on the diffusion scale Multi-scale approach jump conditions across the flame - asymptotic matching

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Across the flame front: Flame speed: Equations: L = Markstein length ~ l f Flame stretch ( curvature & strain )

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Markstein number: Bechtold & Matalon, C&F 2001

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Planar Flames – Linear theory Pelce & Clavin 1982; Matalon & Matkoswky 1982, Frankel & Sivashinsky 1983 DL instability > 0 Heat conduction stabilizing Viscous effects stabilizing Species diffusion stabilizing in lean C n H m /air mixtures rich H 2 /air mixtures Short waves are also unstable Hydrodynamic instability is enhanced by diffusion effects Diffusive-thermal instability Short wavelength disturbances are stabilized by diffusion Stable plane flame may result ( when long wavelength disturbances are stabilized by gravity or confinement ). The coefficients DL, B 1, B 2, B 3 > 0 depend only on thermal expansion .

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign R r = R(t) [1 + A(t) S n ( )] Spherically Expanding Flames – Linear theory ( R >> l f ) Istratov & Librovich 1969; Bechtold & Matalon, 1987, Addabbo,Bechtold & Matalon 2002 hydrodynamic instability The coefficients , Q 1, Q 2,, Q 3 > 0 depend on thermal expansion and on the wavenumber n growth-rate : Diffusion Short wavelength disturbances are stabilized by diffusion Stable spherical flame results when R < R c Hydrodynamically unstable flame when R > R c Short waves are also unstable Hydrodynamic instability is enhanced by diffusion effects Diffusive-thermal instability

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Computed for (lean propane/air) = 5.9, (Le eff - 1) = 4.93 and ~ T 1/2

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Propagation in a spherical bomb (nearly-constant pressure) -- C.K. Law rich hydrocarbon-air cellular flame ( diffusive-thermal instability ) forms shortly after ignition R c ~ l f Le eff < 1 lean hydrocarbon-air flame is initially smooth cells form spontaneously when R > R c Le eff > 1 hydrodynamic instability is enhanced by increasing the pressure, i.e. reducing the stabilizing influence of diffusion

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Cellular instability: diffusive-thermal depends strongly on the mixture composition ( observed in mixtures deficient in the more mobile reactant, or Le < 1 ) characteristic cell size ~ wavelength of the most amplified disturbance of the linear theory Hydrodynamic Instability diffusion acts to stabilize the short wavelength disturbances ( Le > 1 ) scale of wrinkling - typically much larger than the size of cellular flames; controlled by the overall size of the system. propagation speed significantly larger than normal burning velocity Experiments of large flames ( 5 – 10 m ) using lean hydrocarbon-air mixtures show a rough surface with corrugations ~ 10 cm propagating at a speed ~ 1.6 S L (independent of the fuel). Lind & Whitson, 1977; Bradley et al., 2001

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign -L-L unburned burned n 0 L 2L x y Weakly-nonlinear Michelson-Sivashinsky equation nominally planar flame small thermal expansion - 1 << 1 y = -S L t – (x,t) diffusion effect stabilizing for destabilizing effect of thermal expansion (DL) depends on a single parameter = L /( -1)L On a finite domain 0 < x < L with periodic BCs, the MS equation possesses exact solutions (pole solutions) Thual et al What’s beyond Linearity ?

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign For any value of L, there exists one and only one stable pole solution The cusp-like solution propagates steadily at a speed = S L + ( - 1) 2 U The nonlinear development of a hydrodynamically unstable planar flame converges to a stable pole-solution. Vaynblat & Matalon, 2000 -1 = ( -1)L/ L 0 L

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Formation of a single-cusp structure (a cell) starting from arbitrary ICs. The dotted curve is the corresponding pole solution (N = 8) Solution of the MS equation for = 0.005

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Fully nonlinear hydrodynamic model accounts for thermal expansion - 1 = O(1) Mach number << 1 one-step overall chemical reaction thin flame: l f /L << 1 Across the flame front: Flame speed: Equations: curvature L = Markstein length ~ l f burned unburned n

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign burned unburned n = 0 Numerical procedure (Rastigejev & Matalon, 2005): smoothing discontinuities source terms in NS equations zero Mach number variable-density NS solver level-set method proper evaluation of v* on the Lagrangian mesh representing the flame front is accomplished using an “immersed boundary” method. SfSf

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign = = = 0 time evolution of an initial cosine perturbation of a planar front Linear development = 6 A = A 0 exp( t) ~ DL k - 1 k 2 Instability: < (4 ) -1

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign t =0 Hydrodynamic model = L /( -1)L = 0.005, =6

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign t =0 Hydrodynamic model = L /( -1)L = , =6

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Long-time behavior: steadily-propagating structure = L /( -1)L = MS equ. Larger amplitude sharper folds

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Incremental increase in propagation speed 1/ = ( -1)L/ L

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Incremental increase in speed increases linearly with

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign What happens when we decrease = L /( -1)L ? i.e. increase L (cell size)

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign t =0 Hydrodynamic model = L /( -1)L = 0.001, =6

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Hydrodynamic model = L /( -1)L = , =6 t =0

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Solution of the MS equation for = 0.004

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign N = < < 160 N = < < 1000 no wrinkles were observed when N = 8192 reducing the “aliasing” effect (setting to zero the coefficients of the 7680 higher frequencies, thus using effectively 512 modes) Solution of the MS equation for = 0.004

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign N = 8192 Fourier modes x = 0.25 x = 0.5 x = 0.75 level of noise at several representative locations smooth steadily-propagating profile Solution of the MS equation for = 0.004

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign abc d

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Michelson & Sivashinsky, Two-dimensional flame surface: is also a solution of the MS equation. The increase in speed U is higher than for the corresponding one-dimensional case. 1.Solution of MS with Neumann BCs exhibits similar (but richer) dynamic Denet, 2006

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign For moderate values of L - the evolving structure propagates steadily at a speed larger than the laminar flame speed. The incremental increase U varies linearly with . For large values of L - the solution, which is practically independent of the Markstein number L, is sensitive to background noise (numerical noise, weak background turbulence); small-scale wrinkles appear sporadically on the flame front, are amplified as a result of the hydrodynamic instability and cause a significant increase in speed. Alors que la prédiction d’une turbulence spontanée et essentielle …, il est paradoxel que l’introduction de la viscosité … ne paraisse pas assurer la stabilité Darrieus, 1938 Instability of the discontinuity surface should lead to turbulence Landau, 1944 The hydrodynamic instability leads to corrugated structures of relatively large wavelength L (much larger than the most amplified disturbance predicted by the linear theory).

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign E N D

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign Long-time behavior: steadily propagating cusp-like structure = L /( -1)L = 0.02 = L /( -1)L = y = - [ 1 + ( - 1) 2 U ] S L t - ( - 1) (x) Perturbed amplitude exact pole-solution numerical simulation

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Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign · · · The unburned gas velocity v * at the flame front is determined by interpolating values at the rectangular grid points to the Lagrangian mesh representing the flame front surface. Evaluation of v * at the flame front · · · · · ·· · ·

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