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

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

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

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

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

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

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 )

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

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 .

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

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

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

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

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 ?

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

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

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

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

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 

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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).

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

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

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 · · · · · ·· · ·