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1 AME Spring Lecture 7 Turbulent combustion (Lecture 1) Motivation (Lecture 1) Motivation (Lecture 1) Basics of turbulence (Lecture 1) Basics of turbulence (Lecture 1) Premixed-gas flames Premixed-gas flames Turbulent burning velocity (Lecture 1) Turbulent burning velocity (Lecture 1) Regimes of turbulent combustion (Lecture 1) Regimes of turbulent combustion (Lecture 1) Flamelet models (Lecture 1) Flamelet models (Lecture 1) Non-flamelet models (Lecture 1) Non-flamelet models (Lecture 1) Flame quenching via turbulence (Lecture 1) Flame quenching via turbulence (Lecture 1) Case study I: Liquid flames (Lecture 2) Case study I: Liquid flames (Lecture 2) (turbulence without thermal expansion) Case study II: Flames in Hele-Shaw cells (Lecture 2) Case study II: Flames in Hele-Shaw cells (Lecture 2) (thermal expansion without turbulence) Nonpremixed gas flames (Lecture 3) Nonpremixed gas flames (Lecture 3) Edge flames (Lecture 3) Edge flames (Lecture 3)

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2 AME Spring Lecture 7 Motivation Almost all flames used in practical combustion devices are turbulent because turbulent mixing increases burning rates, allowing more power/volume Almost all flames used in practical combustion devices are turbulent because turbulent mixing increases burning rates, allowing more power/volume Even without forced turbulence, if the Grashof number gd 3 / 2 is larger than about 10 6 (g = 10 3 cm/s 2, 1 cm 2 /s d > 10 cm), turbulent flow will exist due to buoyancy Even without forced turbulence, if the Grashof number gd 3 / 2 is larger than about 10 6 (g = 10 3 cm/s 2, 1 cm 2 /s d > 10 cm), turbulent flow will exist due to buoyancy Examples Examples Premixed turbulent flames Premixed turbulent flames »Gasoline-type (spark ignition, premixed-charge) internal combustion engines »Stationary gas turbines (used for power generation, not propulsion) Nonpremixed flames Nonpremixed flames »Diesel-type (compression ignition, nonpremixed-charge) internal combustion engines »Gas turbines »Most industrial boilers and furnaces

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3 AME Spring Lecture 7 Basics of turbulence Good reference: Tennekes: A First Course in Turbulence Good reference: Tennekes: A First Course in Turbulence Job 1: need a measure of the strength of turbulence Job 1: need a measure of the strength of turbulence Define turbulence intensity (u) as rms fluctuation of instantaneous velocity u(t) about mean velocity ( ) Define turbulence intensity (u) as rms fluctuation of instantaneous velocity u(t) about mean velocity ( ) Kinetic energy of turbulence = mass*u 2 /2; KE per unit mass (total in all 3 coordinate directions x, y, z) = 3u 2 /2 Kinetic energy of turbulence = mass*u 2 /2; KE per unit mass (total in all 3 coordinate directions x, y, z) = 3u 2 /2

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4 AME Spring Lecture 7 Basics of turbulence Job 2: need a measure of the length scale of turbulence Job 2: need a measure of the length scale of turbulence Define integral length scale (L I ) as Define integral length scale (L I ) as A measure of size of largest eddies A measure of size of largest eddies Largest scale over which velocities are correlated Largest scale over which velocities are correlated Typically related to size of system (tube or jet diameter, grid spacing, …) Typically related to size of system (tube or jet diameter, grid spacing, …) Here the overbars denote spatial (not temporal) averages A(r) is the autocorrelation function at some time t A(r) is the autocorrelation function at some time t Note A(0) = 1 (fluctuations around the mean are perfectly correlated at a point) Note A(0) = 1 (fluctuations around the mean are perfectly correlated at a point) Note A() = 0 (fluctuations around the mean are perfectly uncorrelated if the two points are very distant) Note A() = 0 (fluctuations around the mean are perfectly uncorrelated if the two points are very distant) For truly random process, A(r) is an exponentially decaying function A(r) = exp(-r/L I ) For truly random process, A(r) is an exponentially decaying function A(r) = exp(-r/L I )

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5 AME Spring Lecture 7 Basics of turbulence In real experiments, generally know u(t) not u(x) - can define time autocorrelation function A(x, ) and integral time scale I at a point x In real experiments, generally know u(t) not u(x) - can define time autocorrelation function A(x, ) and integral time scale I at a point x Here the overbars denote temporal (not spatial) averages With suitable assumptions L I = (8/π) 1/2 u I With suitable assumptions L I = (8/π) 1/2 u I Define integral scale Reynolds number Re L uL I / ( = kinematic viscosity) Define integral scale Reynolds number Re L uL I / ( = kinematic viscosity) Note generally Re L Re flow = Ud/ ; typically u 0.1U, L I 0.5d, thus Re L 0.05 Re flow Note generally Re L Re flow = Ud/ ; typically u 0.1U, L I 0.5d, thus Re L 0.05 Re flow

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6 AME Spring Lecture 7 Basics idea of turbulence Large scales Large scales Viscosity is unimportant (high Re L ) - inertial range of turbulence Viscosity is unimportant (high Re L ) - inertial range of turbulence Through nonlinear interactions of large-scale features (e.g. eddies), turbulent kinetic energy cascades down to smaller scales Through nonlinear interactions of large-scale features (e.g. eddies), turbulent kinetic energy cascades down to smaller scales Interaction is NOT like linear waves: Interaction is NOT like linear waves: (e.g. waves on a string) that can pass through each other without changing or interacting in an irreversible way Interaction is described by inviscid Navier-Stokes equation + continuity equation Interaction is described by inviscid Navier-Stokes equation + continuity equation Smaller scales: viscosity dissipates the KE generated at larger scales Smaller scales: viscosity dissipates the KE generated at larger scales Steady-state: rate of energy generation at large scales = rate of dissipation at small scales Steady-state: rate of energy generation at large scales = rate of dissipation at small scales Example: stir water in cup at large scale, but look later and see small scales - where did they come from??? Example: stir water in cup at large scale, but look later and see small scales - where did they come from??? nonlinear term

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7 AME Spring Lecture 7 Energy dissipation Why is energy dissipated at small scales? Why is energy dissipated at small scales? Viscous term in NS equation ( 2 u) scales as u/( x) 2, convective term (u )u as (u) 2 / x, ratio convective / viscous ~ u x/ = Re x - viscosity more important as Re (thus x) decreases Viscous term in NS equation ( 2 u) scales as u/( x) 2, convective term (u )u as (u) 2 / x, ratio convective / viscous ~ u x/ = Re x - viscosity more important as Re (thus x) decreases Define energy dissipation rate ( ) as energy dissipated per unit mass (u) 2 per unit time ( I ) by the small scales Define energy dissipation rate ( ) as energy dissipated per unit mass (u) 2 per unit time ( I ) by the small scales ~ (u) 2 / I ~ (u) 2 /(L I /u) = C(u) 3 /L I (units Watts/kg = m 2 /s 3 ) ~ (u) 2 / I ~ (u) 2 /(L I /u) = C(u) 3 /L I (units Watts/kg = m 2 /s 3 ) With suitable assumptions C = 3.1 With suitable assumptions C = 3.1 Note to obtain u/S L = 10 for stoich. HC-air (S L = 40 cm/s) with typical L I = 5 cm, 4000 W/kg Note to obtain u/S L = 10 for stoich. HC-air (S L = 40 cm/s) with typical L I = 5 cm, 4000 W/kg Is this a lot of power? Is this a lot of power? If flame propagates at speed S T u across a distance L I, heat release per unit mass is Y f Q R and the time is L I /S T, thus heat release per unit mass per unit time is Y f Q R S T /L I (0.068)(4.5 x 10 7 J/kg)(4 m/s)/(0.05 m) 2.4 x 10 8 W/kg If flame propagates at speed S T u across a distance L I, heat release per unit mass is Y f Q R and the time is L I /S T, thus heat release per unit mass per unit time is Y f Q R S T /L I (0.068)(4.5 x 10 7 J/kg)(4 m/s)/(0.05 m) 2.4 x 10 8 W/kg Ratio of power required to sustain turbulence (4000 W/kg) to attainable heat release rate 1.6 x Ratio of power required to sustain turbulence (4000 W/kg) to attainable heat release rate 1.6 x Also: turbulent kinetic energy (not power) / heat release (not rate) = (3/2 u 2 )/Y f Q R 8 x ! Also: turbulent kinetic energy (not power) / heat release (not rate) = (3/2 u 2 )/Y f Q R 8 x ! Answer: NO, very little power (or energy) Answer: NO, very little power (or energy)

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8 AME Spring Lecture 7 Kolmogorov universality hypothesis (1941) In inertial range of scales, the kinetic energy (E) per unit mass per wavenumber (k) (k = 2π/, = wavelength) depends only on k and In inertial range of scales, the kinetic energy (E) per unit mass per wavenumber (k) (k = 2π/, = wavelength) depends only on k and Use dimensional analysis, E = (m/s) 2 /(1/m) = m 3 /s 2, k = m -1 Use dimensional analysis, E = (m/s) 2 /(1/m) = m 3 /s 2, k = m -1 E(k) ~ a k b or m 3 /s 2 ~ ( m 2 /s 3 ) a (1/m) b a = 2/3, b = -5/3 E(k) ~ a k b or m 3 /s 2 ~ ( m 2 /s 3 ) a (1/m) b a = 2/3, b = -5/3 E(k) = 1.6 2/3 k -5/3 (constant from experiments or numerical simulations) (e.g. C = 1.606, Yakhot and Orzag, 1986) E(k) = 1.6 2/3 k -5/3 (constant from experiments or numerical simulations) (e.g. C = 1.606, Yakhot and Orzag, 1986) Total kinetic energy (K) over a wave number range (k 1 < k < k 2 ) Total kinetic energy (K) over a wave number range (k 1 < k < k 2 ) Characteristic intensity at scale L = u(k) = (2K/3) 1/2 = /3 L 1/3 Characteristic intensity at scale L = u(k) = (2K/3) 1/2 = /3 L 1/3 Cascade from large to small scales (small to large k) ends at scale where dissipation is important Cascade from large to small scales (small to large k) ends at scale where dissipation is important Smallest scale in the turbulent flow Smallest scale in the turbulent flow This scale (Kolmogorov scale) can depend only on and This scale (Kolmogorov scale) can depend only on and Dimensional analysis: wave number k Kolmogorov ~ ( / 3 ) 1/4 or length scale L K = C( 3 / ) 1/4, C 11 Dimensional analysis: wave number k Kolmogorov ~ ( / 3 ) 1/4 or length scale L K = C( 3 / ) 1/4, C 11 L K /L I = C( 3 / ) 1/4 /L I = C( 3 /(3.1u 3 /L I )) 1/4 /L I = 14 (uL I / ) -3/4 = 14 Re L -3/4 L K /L I = C( 3 / ) 1/4 /L I = C( 3 /(3.1u 3 /L I )) 1/4 /L I = 14 (uL I / ) -3/4 = 14 Re L -3/4 Time scale t K ~ L K /u(L K ) ~ ( / ) 1/2 Time scale t K ~ L K /u(L K ) ~ ( / ) 1/2

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9 AME Spring Lecture 7 Kolmogorov universality hypothesis (1941)

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10 AME Spring Lecture 7 There are three types of turbulent eddies K. Nomura, UCSD S. Tieszen, Sandia large entrainment eddies inertial-range eddies dissipation-scale eddies (Diagram courtesy A. Kerstein, Sandia National Labs)

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11 AME Spring Lecture 7 Application of Kolmogorov scaling Taylor scale (L T ): scale at which average strain rate occurs Taylor scale (L T ): scale at which average strain rate occurs L T u/(mean strain) L T u/(mean strain) L T ~ u/(t K ) -1 ~ u/(( / ) 1/2 ) -1 ~ u 1/2 /(u 3 /L I ) 1/2 ~ L I Re L -1/2 L T ~ u/(t K ) -1 ~ u/(( / ) 1/2 ) -1 ~ u 1/2 /(u 3 /L I ) 1/2 ~ L I Re L -1/2 With suitable assumptions, mean strain 0.147( / ) 1/2 With suitable assumptions, mean strain 0.147( / ) 1/2 Turbulent viscosity T Turbulent viscosity T Molecular gas dynamics: ~ (velocity of particles)(length particles travel before changing direction) Molecular gas dynamics: ~ (velocity of particles)(length particles travel before changing direction) By analogy, at scale k, T (k) ~ u(k)(1/k) By analogy, at scale k, T (k) ~ u(k)(1/k) Total effect of turbulence on viscosity: at scale L I, T ~ uL I Total effect of turbulence on viscosity: at scale L I, T ~ uL I Usually written in the form T = CK 2 /, C = Usually written in the form T = CK 2 /, C = Or T / = Re L Or T / = Re L Turbulent thermal diffusivity T = T /Pr T Turbulent thermal diffusivity T = T /Pr T Pr goes from molecular value (gases 0.7, liquids ,000; liquid metals 0.05) at Re L = 0 to Pr T 0.72 as Re L (Yakhot & Orzag, 1986) Pr goes from molecular value (gases 0.7, liquids ,000; liquid metals 0.05) at Re L = 0 to Pr T 0.72 as Re L (Yakhot & Orzag, 1986) T / = Re L as Re L T / = Re L as Re L

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12 AME Spring Lecture 7 Turbulent premixed combustion - intro Study of premixed turbulent combustion is important because Study of premixed turbulent combustion is important because Turbulence increases mean flame propagation rate (S T ) and thus mass burning rate (= S T A projected ) Turbulence increases mean flame propagation rate (S T ) and thus mass burning rate (= S T A projected ) If this trend increased ad infinitum, arbitrarily lean mixtures (low S L ) could be burned arbitrarily fast by using sufficiently high u If this trend increased ad infinitum, arbitrarily lean mixtures (low S L ) could be burned arbitrarily fast by using sufficiently high u...but too high u' leads to extinction - nixes that idea...but too high u' leads to extinction - nixes that idea Current theories don't agree with experiments Current theories don't agree with experiments...and don't agree with each other...and don't agree with each other Numerical simulations difficult at high u'/S L (>10) Numerical simulations difficult at high u'/S L (>10) Applications to Applications to Auto engines Auto engines Gas turbine designs - possible improvements in emission characteristics with lean premixed combustion compared to non-premixed combustion (remember why?) Gas turbine designs - possible improvements in emission characteristics with lean premixed combustion compared to non-premixed combustion (remember why?)

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13 AME Spring Lecture 7 = S T /S L Bradley et al. (1992) Compilation of data from many sources Compilation of data from many sources = u/S L

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14 AME Spring Lecture 7 Turbulent burning velocity Experimental results shown in Bradley et al. (1992) smoothed data from many sources, e.g. fan-stirred bomb Experimental results shown in Bradley et al. (1992) smoothed data from many sources, e.g. fan-stirred bomb

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15 AME Spring Lecture 7 Bradley et al. (1992) … and we are talking MAJOR smoothing!

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16 AME Spring Lecture 7 Turbulent burning velocity Models of premixed turbulent combustion dont agree with experiments nor each other! Models of premixed turbulent combustion dont agree with experiments nor each other!

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17 AME Spring Lecture 7 Modeling of premixed turbulent flames Models often dont agree well with experiments Models often dont agree well with experiments Most model employ assumptions not satisfied by real flames, e.g. Most model employ assumptions not satisfied by real flames, e.g. Adiabatic (sometimes ok) Adiabatic (sometimes ok) Homogeneous, isotropic turbulence over many L I (never ok) Homogeneous, isotropic turbulence over many L I (never ok) Low Ka or high Da (thin fronts) (sometimes ok) Low Ka or high Da (thin fronts) (sometimes ok) Lewis number = 1 (sometimes ok, e.g. CH 4 -air) Lewis number = 1 (sometimes ok, e.g. CH 4 -air) Constant transport properties (never ok, 25x increase in and across front!) Constant transport properties (never ok, 25x increase in and across front!) u doesnt change across front (never ok, thermal expansion across flame generates turbulence) (but viscosity increases across front, decreases turbulence, sometimes almost cancels out) u doesnt change across front (never ok, thermal expansion across flame generates turbulence) (but viscosity increases across front, decreases turbulence, sometimes almost cancels out) Constant density (never ok!) Constant density (never ok!)

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18 AME Spring Lecture 7 Characteristics of turbulent flames Most important property: turbulent flame speed (S T ) Most important property: turbulent flame speed (S T ) Most models based on physical models of Damköhler (1940) Most models based on physical models of Damköhler (1940) Behavior depends on Karlovitz number (Ka) Behavior depends on Karlovitz number (Ka) Kolmogorov turbulence: u = u; L = L T = Kolmogorov turbulence: u = u; L = L T = Alternatively, Ka = ( /L K ) 2 ~ (S L / )/(Re L -3/4 L I ) 2 ~ Re L -1/2 (u/S L ) 2 Alternatively, Ka = ( /L K ) 2 ~ (S L / )/(Re L -3/4 L I ) 2 ~ Re L -1/2 (u/S L ) 2 - same scaling results based on length scales Sometimes Damköhler number (Da) is reported instead Sometimes Damköhler number (Da) is reported instead Da 1 = I / chem ~ (L I /u)/( /S L 2 ) ~ Re L (u/S L ) -2 Da 1 = I / chem ~ (L I /u)/( /S L 2 ) ~ Re L (u/S L ) -2 Da 2 = Taylor / chem ~ (Re L -1/2 L I /u)/( /S L 2 ) ~ Re L 1/2 (u/S L ) -2 ~ 1/Ka Da 2 = Taylor / chem ~ (Re L -1/2 L I /u)/( /S L 2 ) ~ Re L 1/2 (u/S L ) -2 ~ 1/Ka Low Ka: Huygens propagation, thin fronts that are wrinkled by turbulence but internal structure is unchanged Low Ka: Huygens propagation, thin fronts that are wrinkled by turbulence but internal structure is unchanged High Ka: Distributed reaction zones, broad fronts High Ka: Distributed reaction zones, broad fronts Defined using cold- gas viscosity

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19 AME Spring Lecture 7 Characteristics of turbulent flames

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20 AME Spring Lecture 7 = u/S L = L I / = L I (S L / ) = Re L /(u/S L ) Laminar flames Broadened flamesBroadened flames (Ronney & Yakhot, 1992) Distributed reaction zones Ka = Re 1/2 Turbulent combustion regimes (Peters, 1986)

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21 AME Spring Lecture 7 Turbulent combustion regimes Comparison of flamelet and distributed combustion (Yoshida, 1988) Comparison of flamelet and distributed combustion (Yoshida, 1988) Flamelet: temperature is either T or T ad, never between, and probability of product increases through the flame Distributed: significant probability of temperatures between T or T ad, probability of intermediate T peaks in middle of flame

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22 AME Spring Lecture 7 Flamelet modeling Damköhler (1940): in Huygens propagation regime, flame front is wrinkled by turbulence but internal structure and S L are unchanged Damköhler (1940): in Huygens propagation regime, flame front is wrinkled by turbulence but internal structure and S L are unchanged Propagation rate S T due only to area increase via wrinkling: Propagation rate S T due only to area increase via wrinkling: S T /S L = A T /A L S T /S L = A T /A L

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23 AME Spring Lecture 7 Flamelet modeling Low u/S L : weakly wrinkled flames Low u/S L : weakly wrinkled flames S T /S L = 1 + (u/S L ) 2 (Clavin & Williams, 1979) - standard for many years S T /S L = 1 + (u/S L ) 2 (Clavin & Williams, 1979) - standard for many years Actually Kerstein and Ashurst (1994) showed this is valid only for periodic flows - for random flows S T /S L - 1 ~ (u/S L ) 4/3 Actually Kerstein and Ashurst (1994) showed this is valid only for periodic flows - for random flows S T /S L - 1 ~ (u/S L ) 4/3 Higher u/S L : strongly wrinkled flames Higher u/S L : strongly wrinkled flames Schelkin (1947) - A T /A L estimated from ratio of cone surface area to base area; height of cone ~ u/S L ; result Schelkin (1947) - A T /A L estimated from ratio of cone surface area to base area; height of cone ~ u/S L ; result Other models based on fractals, probability-density functions, etc., but mostly predict S T /S L ~ u/S L at high u/S L with the possibility of bending or quenching at sufficiently high Ka ~ (u/S L ) 2 Other models based on fractals, probability-density functions, etc., but mostly predict S T /S L ~ u/S L at high u/S L with the possibility of bending or quenching at sufficiently high Ka ~ (u/S L ) 2

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24 AME Spring Lecture 7 Flamelet modeling Many models based on G-equation (Kerstein et al. 1987) Many models based on G-equation (Kerstein et al. 1987) Any curve of G = C is a flame front (G C burned) Any curve of G = C is a flame front (G C burned) Flame advances by self-propagation and advances or retreats by convection Flame advances by self-propagation and advances or retreats by convection

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25 AME Spring Lecture 7 Flamelet modeling G-equation formulation, renormalization group (Yakhot, 1988) (successive scale elimination); invalid (but instructive) derivation: G-equation formulation, renormalization group (Yakhot, 1988) (successive scale elimination); invalid (but instructive) derivation:

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26 AME Spring Lecture 7 Fractal models of flames What is the length (L) is the coastline of Great Britain? Depends on measurement scale: L(10 miles) < L(1 mile) What is the length (L) is the coastline of Great Britain? Depends on measurement scale: L(10 miles) < L(1 mile) Plot Log(L) vs. Log(measurement scale); if its a straight line, its a fractal with dimension (D) = 1 - slope Plot Log(L) vs. Log(measurement scale); if its a straight line, its a fractal with dimension (D) = 1 - slope If object is very smooth (e.g. circle), D = 1 If object is very smooth (e.g. circle), D = 1 If object has self-similar wrinkling on many scales, D > 1 - expected for flames since Kolmogorov turbulence exhibits scale-invariant properties If object has self-similar wrinkling on many scales, D > 1 - expected for flames since Kolmogorov turbulence exhibits scale-invariant properties If object is very rough, D 2 If object is very rough, D 2 If object 3D, everything is same except length area, D D + 1 (space- filling surface: D 3) If object 3D, everything is same except length area, D D + 1 (space- filling surface: D 3) Generic 2D object 3D flame surface

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27 AME Spring Lecture 7 Fractal models of flames Application to flames (Gouldin, 1987) Application to flames (Gouldin, 1987) Inner (small-scale) and outer (large-scale) limits (cutoff scales) Inner (small-scale) and outer (large-scale) limits (cutoff scales) Outer: L I ; inner: L K or L G (no clear winner…) Outer: L I ; inner: L K or L G (no clear winner…) L G = Gibson scale, where u(L G ) = S L L G = Gibson scale, where u(L G ) = S L Recall u(L) ~ 1/3 L 1/3, thus L G ~ S L 3 / Recall u(L) ~ 1/3 L 1/3, thus L G ~ S L 3 / Since ~ u 3 /L I thus L G ~ L I (S L /u) 3 Since ~ u 3 /L I thus L G ~ L I (S L /u) 3 S T /S L = A T /A L = A(L inner )/A(L outer ) S T /S L = A T /A L = A(L inner )/A(L outer ) For 3-D object, A(L inner )/A(L outer ) = (L outer /L inner ) 2-D For 3-D object, A(L inner )/A(L outer ) = (L outer /L inner ) 2-D What is D? Kerstein (1988) says it should be 7/3 for a flame in Kolmogorov turbulence (consistent with experiments), but Haslam & Ronney (1994) show its 7/3 even in non-Kolmogorov turbulence! What is D? Kerstein (1988) says it should be 7/3 for a flame in Kolmogorov turbulence (consistent with experiments), but Haslam & Ronney (1994) show its 7/3 even in non-Kolmogorov turbulence! With D = 7/3, L inner = L K, S T /S L = (L K /L I ) 2-7/3 ~ (Re L -3/4 ) -1/3 ~ Re L 1/4 With D = 7/3, L inner = L K, S T /S L = (L K /L I ) 2-7/3 ~ (Re L -3/4 ) -1/3 ~ Re L 1/4 With D = 7/3, L inner = L G, S T /S L = (L G /L I ) 2-7/3 ~ (u/S L ) -3 ) -1/3 ~ u/S L With D = 7/3, L inner = L G, S T /S L = (L G /L I ) 2-7/3 ~ (u/S L ) -3 ) -1/3 ~ u/S L Consistent with some experiments (like all models…) Consistent with some experiments (like all models…)

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28 AME Spring Lecture 7 Effects of thermal expansion Excellent review of flame instabilities: Matalon, 2007 Excellent review of flame instabilities: Matalon, 2007 Thermal expansion across flame known to affect turbulence properties, thus flame properties Thermal expansion across flame known to affect turbulence properties, thus flame properties Exists even when no forced turbulence - Darrieus-Landau instability - causes infinitesimal wrinkle in flame to grow Exists even when no forced turbulence - Darrieus-Landau instability - causes infinitesimal wrinkle in flame to grow Some models (e.g. Pope & Anand, 1987) predict thermal expansion decreases S T /S L for fixed u/S L but most predict an increase Some models (e.g. Pope & Anand, 1987) predict thermal expansion decreases S T /S L for fixed u/S L but most predict an increase Williams, 1985 A flame, u, P A flame, u, P Bernoulli: P+1/2 u 2 = const FLOW FLOW

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29 AME Spring Lecture 7 Effects of thermal expansion Byckov (2000): Byckov (2000): Same as Yakhot (1988) if no thermal expansion ( = 1) Same as Yakhot (1988) if no thermal expansion ( = 1) Also says for any, if u/S L = 0 then S T /S L = 1; probably not true Also says for any, if u/S L = 0 then S T /S L = 1; probably not true

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30 AME Spring Lecture 7 Effects of thermal expansion Cambray and Joulin (1992): one-scale (not really turbulent) flow with varying thermal expansion = 1-1/ Cambray and Joulin (1992): one-scale (not really turbulent) flow with varying thermal expansion = 1-1/ S T /S L increases as increases (more thermal expansion) S T /S L increases as increases (more thermal expansion) If 0, then S T /S L does not go to 1 as u/S L goes to zero - inherent thermal expansion induced instability and wrinkling causes S T /S L > 1 even at u/S L = 0 If 0, then S T /S L does not go to 1 as u/S L goes to zero - inherent thermal expansion induced instability and wrinkling causes S T /S L > 1 even at u/S L = 0 Effect of expansion-induced wrinkling decreases as u/S L Effect of expansion-induced wrinkling decreases as u/S L

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31 AME Spring Lecture 7 Effects of thermal expansion Time-dependent computations in decaying turbulence suggest Time-dependent computations in decaying turbulence suggest Early times, before turbulence decays, turbulence dominates Early times, before turbulence decays, turbulence dominates Late times, after turbulence decays, buoyancy dominates Late times, after turbulence decays, buoyancy dominates Next lecture: thermal expansion effects on S T with no turbulence, and turbulence effects on S T with no thermal expansion Next lecture: thermal expansion effects on S T with no turbulence, and turbulence effects on S T with no thermal expansion Boughanem and Trouve, 1998 Initial u'/S L = 4.0 (decaying turbulence); integral-scale Re = 18 Total reaction rate / reaction rate of flat steady flame More or less = S T /S L Time / turbulence integral time scale Well after initial turbulence decays - S T /S L depends totally on density ratio Before turbulence decays - S T /S L is independent of density ratio = / f - 1 = / f - 1

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32 AME Spring Lecture 7 Upward propagation: buoyantly unstable, should increase S T compared to no buoyancy effects Upward propagation: buoyantly unstable, should increase S T compared to no buoyancy effects Buoyancy affects largest scales, thus scaling of buoyancy effects should depend on integral time scale / buoyant time scale = (L I /u)/(L I /g) 1/2 = (gL I /u 2 ) 1/2 = (S L /u)(gL I /S L 2 ) 1/2 Buoyancy affects largest scales, thus scaling of buoyancy effects should depend on integral time scale / buoyant time scale = (L I /u)/(L I /g) 1/2 = (gL I /u 2 ) 1/2 = (S L /u)(gL I /S L 2 ) 1/2 Libby (1989) (g > 0 for upward propagation) Libby (1989) (g > 0 for upward propagation) No systematic experimental test to date No systematic experimental test to date Buoyancy effects Lewis (1970): experiments on flames in tubes in a centrifugal force field (50 g o < g < 850 g o ), no forced turbulence: S T ~ g 0.387, almost independent of pressure - may be same as S ~ g 1/2 Lewis (1970): experiments on flames in tubes in a centrifugal force field (50 g o < g < 850 g o ), no forced turbulence: S T ~ g 0.387, almost independent of pressure - may be same as S ~ g 1/2

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33 AME Spring Lecture 7 Time-dependent computations in decaying turbulence suggest Time-dependent computations in decaying turbulence suggest Early times, before turbulence decays, turbulence dominates Early times, before turbulence decays, turbulence dominates Late times, after turbulence decays, buoyancy dominates - upward propagating: higher S T /S L ; downward: almost no wrinkling, S T /S L 1 Late times, after turbulence decays, buoyancy dominates - upward propagating: higher S T /S L ; downward: almost no wrinkling, S T /S L 1 Boughanem and Trouve, 1998 Initial u'/S L = 4.0 (decaying turbulence); integral-scale Re = 18 Effects of buoyancy Total reaction rate / reaction rate of flat steady flame More or less = S T /S L Time / turbulence integral time scale Well after initial turbulence decays - S T /S L depends totally on buoyancy Before turbulence decays - S T /S L is independent of buoyancy Buoyancy parameter: Ri = g / S L 3

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34 AME Spring Lecture 7 Lewis number effects Turbulence causes flame stretch & curvature Turbulence causes flame stretch & curvature Since both + and - curved flamelets exist, effects nearly cancel (Rutland & Trouve, 1993) Since both + and - curved flamelets exist, effects nearly cancel (Rutland & Trouve, 1993) Mean flame stretch is biased towards + stretch - benefits flames in low-Le mixtures Mean flame stretch is biased towards + stretch - benefits flames in low-Le mixtures H 2 -air: extreme case, lean mixtures have cellular structures similar to flame balls -Pac-Man combustion H 2 -air: extreme case, lean mixtures have cellular structures similar to flame balls -Pac-Man combustion Causes change in behavior at 8% - 10% H 2, corresponding to 0.25 (Al-Khishali et al., 1983) - same as transition from flame ball to continuous flames in non-turbulent mixtures Causes change in behavior at 8% - 10% H 2, corresponding to 0.25 (Al-Khishali et al., 1983) - same as transition from flame ball to continuous flames in non-turbulent mixtures

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35 AME Spring Lecture 7 Distributed combustion modeling Much less studied than flamelet combustion Much less studied than flamelet combustion Damköhler (1940): Damköhler (1940): A 0.25 (gas); A 6.5 (liquid) Assumption T L probably not valid for high ; recall Assumption T L probably not valid for high ; recall …but probably ok for small …but probably ok for small Example: 2 equal volumes of combustible gas with E = 40 kcal/mole, 1 volume at 1900K, another at 2100K Example: 2 equal volumes of combustible gas with E = 40 kcal/mole, 1 volume at 1900K, another at 2100K (1900) ~ exp(-40000/(1.987*1900)) = 3.73 x 10 4 (1900) ~ exp(-40000/(1.987*1900)) = 3.73 x 10 4 (2100) ~ exp(-40000/(1.987*2100)) = 1.34 x 10 4 (2100) ~ exp(-40000/(1.987*2100)) = 1.34 x 10 4 Average = 2.55 x 10 4, whereas (2000) = 2.2 x 10 4 (16% difference)! Averaging over ±5% T range gives 16% error! Averaging over ±5% T range gives 16% error!

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36 AME Spring Lecture 7 Distributed combustion modeling Pope & Anand (1984) Pope & Anand (1984) Constant density, 1-step chemistry, 12.3: S T /S L 0.3 (u/S L )[ log 10 (Re L ) - 2 log 10 (u/S L )] - Close to Damköhler (1940) (?!) for relevant Re L

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37 AME Spring Lecture 7 Distributed combustion modeling When applicable? T L I T 6 T /S T (Ronney & Yakhot, 1992), S T /S L 0.25 Re L 1/2 Ka Re L 1/2 (based on cold-gas viscosity) Experiments (Abdel-Gayed et al., 1989) » »Ka 0.3: significant flame quenching within the reaction zone - independent of Re L Applicable range uncertain, but probably only close to quenching Experiment - Bradley (1992) S T /S L 5.5 Re close to models over limited applicable range Implications for bending Flamelet models generally S T ~ u Distributed combustion models generally S T ~ u Transition from flamelet to distributed combustion - possible mechanism for bending

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38 AME Spring Lecture 7 Broadened flamesBroadened flames Ronney & Yakhot, 1992: If Re 1/2 > Ka > 1, some but not all scales (i.e. smaller scales) fit inside flame front and broaden it since Ronney & Yakhot, 1992: If Re 1/2 > Ka > 1, some but not all scales (i.e. smaller scales) fit inside flame front and broaden it since T > T > Combine flamelet (Yakhot) model for large (wrinkling) scales with distributed (Damköhler) model for small (broadening) scales Combine flamelet (Yakhot) model for large (wrinkling) scales with distributed (Damköhler) model for small (broadening) scales Results show S T /S L bending and even decrease in S T /S L at high u/S L as in experiments Results show S T /S L bending and even decrease in S T /S L at high u/S L as in experiments Still no consideration of thermal expansion, etc. Still no consideration of thermal expansion, etc.

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39 AME Spring Lecture 7 Quenching by turbulence Low Da / high Ka: thin-flame models fail Distributed combustion Quenching at still higher Ka Quenching attributed to mass-extinguishment of flamelets by zero-mean turbulent strain...but can flamelet quenching cause flame quenching? Hypothetical system: flammable mixture in adiabatic channel with arbitrary zero-mean flow disturbance 1st law: No energy loss from system adiabatic flame temperature is eventually reached 2nd law: Heat will be transferred to unburned gas Chemical kinetics: (T>T H ) >> (T L ) Propagating front will always exist (???)

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40 AME Spring Lecture 7 Quenching by turbulence

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41 AME Spring Lecture 7 Turbulent flame quenching in gases "Liquid flame" experiments suggest flamelet quenching does not cause flame quenching "Liquid flame" experiments suggest flamelet quenching does not cause flame quenching Poinsot et al. (1990): Flame-vortex interactions Poinsot et al. (1990): Flame-vortex interactions Adiabatic: vortices cause temporary quenching Adiabatic: vortices cause temporary quenching Non-adiabatic: permanent flame quenching possible Non-adiabatic: permanent flame quenching possible Giovangigli and Smooke (1992) no intrinsic flammability limit for planar, steady, laminar flames Giovangigli and Smooke (1992) no intrinsic flammability limit for planar, steady, laminar flames So what causes quenching in standard experiments? So what causes quenching in standard experiments? Radiative heat losses? Radiative heat losses? Ignition limits? Ignition limits?

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42 AME Spring Lecture 7 Turbulent flame quenching in gases High Ka (distributed combustion): High Ka (distributed combustion): Turbulent flame thickness ( T ) >> L Turbulent flame thickness ( T ) >> L Larger T more heat loss (~ T ), S T increases less Larger T more heat loss (~ T ), S T increases less heat loss more important at high Ka heat loss more important at high Ka Ka q Re L 0.76 for typical L I = 0.05 m Ka q Re L 0.76 for typical L I = 0.05 m Also: large T large ignition energy required Also: large T large ignition energy required Ka q 2.3 Re L Ka q 2.3 Re L limit regimes qualitatively consistent with experiments 2 limit regimes qualitatively consistent with experiments

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43 AME Spring Lecture 7 Turbulent flame quenching in gases

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44 AME Spring Lecture 7 References Abdel-Gayed, R. G. and Bradley, D. (1985) Combust. Flame 62, 61. Abdel-Gayed, R. G., Bradley, D. and Lung, F. K.-K. (1989) Combust. Flame 76, 213. Al-Khishali, K. J., Bradley, D., Hall, S. F. (1983). "Turbulent combustion of near-limit hydrogen-air mixtures," Combust. Flame Vol. 54, pp Anand, M. S. and Pope, S. B. (1987). Combust. Flame 67, 127. H. Boughanem and A. Trouve (1998). Proc. Combust. Inst. 27, 971. Bradley, D., Lau, A. K. C., Lawes, M., Phil. Trans. Roy. Soc. London A, (1992) Bray, K. N. C. (1990). Proc. Roy. Soc.(London) A431, 315. Bychov, V. (2000). Phys. Rev. Lett. 84, Cambray, P. and Joulin, G. (1992). Twenty-Fourth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, p. 61. Clavin, P. and Williams, F. A. (1979). J. Fluid Mech. 90, 589. Damköhler, G. (1940). Z. Elektrochem. angew. phys. Chem 46, 601. Gouldin, F. C. (1987). Combust. Flame 68, 249. Kerstein, A. R. (1988). Combust. Sci. Tech. 60, 163. Kerstein, A. R. and Ashurst, W. T. (1992). Phys. Rev. Lett. 68, 934. Kerstein, A. R., Ashurst, W. T. and Williams, F. A. (1988). Phys. Rev. A., 37, Lewis, G. D. (1970). Combustion in a centrifugal-force field. Proc. Combust. Inst. 13,

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45 AME Spring Lecture 7 References Libby, P. A. (1989). Theoretical analysis of the effect of gravity on premixed turbulent flames, Combust. Sci. Tech. 68, Matalon, M. (2007). Intrinsic flame instabilities in premixed and nonpremixed combustion. Ann. Rev. Fluid. Mech. 39, Poinsot, T., Veyante, D. and Candel, S. (1990). Twenty-Third Symposium (International) on Combustion, Combustion Institute, Pittsburgh, p Pope, S. B. and Anand, M. S. (1984). Twentieth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, p Peters, N. (1986). Twenty-First Symposium (International) on Combustion, Combustion Institute, Pittsburgh, p Ronney, P. D. and Yakhot, V. (1992). Combust. Sci. Tech. 86, 31. Rutland, C. J., Ferziger, J. H. and El Tahry, S. H. (1990). Twenty-Third Symposium (International) on Combustion, Combustion Institute, Pittsburgh, p Rutland, C. J. and Trouve, A. (1993). Combust. Flame 94, 41. Sivashinsky, G. I. (1990), in: Dissipative Structures in Transport Processes and Combustion (D. Meinköhn, ed.), Springer Series in Synergetics, Vol. 48, Springer-Verlag, Berlin, p. 30. Yakhot, V. (1988). Combust. Sci. Tech. 60, 191. Yakhot, V. and Orzag, S. A. (1986). Phys. Rev. Lett. 57, Yoshida, A. (1988). Proc. Combust. Inst. 22,

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