Dynamics of Front Propagation in Narrow Channels Mohammed Abid, Jamal A. Sharif, Paul D. Ronney Department of Aerospace & Mechanical Engineering University of Southern California Los Angeles, CA

2 Motivation Premixed gas flame instabilities & buoyancy effects not well understood due to  Large  - baroclinic production of vorticity , , D increase ≈ 25x across flame  3d hydrodynamics with large Re  Effects most pronounced near extinction, but couple strongly to flame stretch & heat losses Practical applications  Accidental explosions - laminar flame  wrinkled flame  turbulent flame  detonation  Industrial furnaces - large Grashof #  Fire safety in spacecraft

3 Approach n Hele-Shaw flow  Flow between closely-spaced parallel plates  Described by linear 2-D equation (Darcy’s law)  1000's of references  Practical application - flame propagation in cylinder crevice volumes Premixed-gas flames, PLUS propagating chemical fronts in aqueous solution   << 1 - No baroclinicity, Bousinnesq approximation valid   T ≈ 3 K - no change in transport properties  Not affected by heat loss  Large Schmidt # - front stays "thin” even at high Re  Simpler chemistry than gaseous flames

4 Types of flame instabilities Thermal expansion (Darrieus-Landau, DL) Rayleigh-Taylor (buoyancy-driven, RT) Viscous fingering (Saffman-Taylor, ST) (flow in narrow channels when viscous fluid displaced by less viscous fluid) Diffusive-thermal (Lewis number) Joulin & Sivashinsky (1994) - combined effects of DL, ST, RT & heat loss (but no diffusive-thermal effect - no damping at small )

5 Objectives Study behavior of flame propagation in Hele-Shaw cells  Wrinkling characteristics  Propagation rates  Buoyancy effects Compare variable-density flames & nearly constant-density aqueous chemical fronts

6 Chemical considerations Iodate-hydrosulfite system IO H e -  I H 2 O S 2 O H 2 O  6 e H SO IO S 2 O H 2 O  I SO H + Autocatalytic in H + Simple solutions Non-toxic "Lightning fast" (up to 0.05 cm/sec)

7 Apparatus (gaseous flames) Aluminum frame sandwiched between Lexan windows 40 cm x 60 cm x 1.27 cm test section CH 4 & C 3 H 8 fuel, N 2 & CO 2 diluent - affects Le, Peclet # Upward, horizontal, downward orientation Spark ignition (1 or 3 locations)

8 Apparatus (liquid flames) 25cm x 20cm; 0.04cm < W < 0.2cm; 0.2 < Gr w < 25 Color-changing or fluorescent pH indicators Initiate at a point using acid solution

9 Results - gaseous flames - qualitative n Orientation effects u Horizontal propagation - large wavelength wrinkle fills cell u Upward propagation - more pronounced large wrinkle u Downward propagation - globally flat front (buoyancy suppresses large-scale wrinkles); oscillatory modes, transverse waves u Consistent with Joulin-Sivashinsky predictions n Large-scale wrinkling observed even at high Le; small scale wrinkling suppressed at high Le For practical range of conditions, buoyancy & diffusive-thermal effects cannot prevent wrinkling due to viscous fingering & thermal expansion For practical range of conditions, buoyancy & diffusive-thermal effects cannot prevent wrinkling due to viscous fingering & thermal expansion Evidence of preferred wavelengths, but selection mechanism unclear (DT + ?)

10 Results - gas flames - propagation rates Propagation rate (S T ) always larger than S L 3-stage propagation  Thermal expansion - most rapid  Quasi-steady  Near-end-wall - slowest - large-scale wrinkling suppressed

11 Results - gas flames - orientation effect Horizontal - S T /S L ≈ independent of Pe = S L w/  ) n Upward - S T /S L  as Pe  (decreasing benefit of buoyancy); highest propagation rates n Downward - S T /S L  as Pe  (decreasing penalty of buoyancy); lowest propagation rates n S T /S L converges to ≈ constant value at large Pe

12 Results - gas flames - Lewis # effect n S T /S L generally higher at lower Le n CH 4 -air (Le ≈ 1) - S T /S L ≈ independent of Pe n C 3 H 8 -air (Le ≈ 1.7) - S T /S L  as Pe  n CH 4 -O 2 -CO 2 (Le ≈ 0.7) - S T /S L  as Pe  n S T /S L convergence at large Pe uncertain

13 Results - liquid flames - fingering FINGERING (???) for upward propagation No wrinkling for downward or horizontal propagation - unlike gaseous flames Wavelength ( ) nearly independent of S L & width

14 Results - fingering - continued n Saffman & Taylor (1958) n No wavelength selection without surface tension!  µ = 0,  ≠ 0: independent of U (~S L ) & K (~w 2 ) but depends on cell angle  - consistent with experiments

15 Results - fingering - continued Zhu (1998) - wavelength selection due to curvature (Markstein) effect on S L :  =max = 48π D/gw 2 cos(  )  - much too small (≈0.1 mm) & not correct effect of w,  Surface tension issues  Can miscible interfaces have a surface tension?  = 1 cm   ≈ 5 x dyne/cm ≈ 7 x  water-air  Is this  reasonable?  Davis (1988): miscible systems  ~ C/  ;  = interface thickness, C = 2 x dyne   ≈ 7 x dyne/cm  If  ~ 1/ ,  = D/S L for chemical front,  water-air = 70 dyne/cm,  water-air ≈ cm,  liquid flame ≈ 7 x dyne/cm F  Probably reasonable value - need rotating drop or capillary wave experiment to check  Unlike a diffusing passive interface between miscible fluids, chemical fronts have constant thickness  constant 

16 Results - liquid flames - propagation rates Wrinkled fronts propagate quasi-steadily with rate S T >> S L Can S T be related to a “turbulence intensity”? Estimated buoyant velocity u b ≈  max /k  max

17 Results - liquid flames - propagation rates S T of rising fronts in Hele-Shaw cells (K = w 2 /12) consistent with Yakhot (1988) renormalization-group model for Huygens’ propagation with U = u’/S L (!)

18 Results - liquid flames - propagation rates 5 different flows no adjustable parameters Data on S T /S L in 5 different flows consistent with Yakhot’s model with no adjustable parameters

19 Conclusions Flame propagation in quasi-2D Hele-Shaw cells reveals effects of  Thermal expansion  Viscous fingering  Buoyancy  Lewis number  Surface tension (!?) Fronts in Hele-Shaw cells enable rational comparisons of models & experiments Flame propagation in cylinder crevice volumes may be quite different from expectations based on unconfined flame experiments Rich dynamics observed even for aqueous 2d system much simpler than freely propagating gaseous flames