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Lecture 15 The Turbulent Burning Velocity

1

The turbulent burning velocity is defined as the average rate of propagationof the flame through the turbulent premixed gas mixture.

But this may only apply to the wrinkled flamelet and reaction sheet regimes; itis less obvious, however, that it applies in the distributed-reaction regime wherethe basic structure of a flame may no longer exist.

2

In the laminar case, solutions of the governing equations of the form f(x−SLt)implies that the whole structure propagates to the left with a speed SL - thelaminar flame speed.

In the turbulent case it is not clear that a turbulent speed is indeed a well-defined notion. The assumption that it exists could be supported by the obser-vations that turbulent flames propagate a well-defined distance, and if in steadyturbulent flows they possess a measurable inclination angle.

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Kobayashi & Kawazoe, 2000

v�/SL = 0 P = 1 MPav�/SL = 0.58 P = 0.1 MPa v�/SL = 2.45 P = 1 MPa

Images of CH4-air Bunsen-type flames

3

CH4-air inverted conical flame in a low-intensity turbulent flow

unburned

burned

Sattler et al. PCI, 2002

The figure displays the image of a wrinkled methane-air V-flame in a turbulent flow of low intensity.The image is bright at the reactants (as a result of Mie scattering of laser light off the micrometer sizedsilicone oil droplets that were added to the fresh mixture and consumed in the reaction zone),and dark in the products.

4

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Flame stretch turbulent burning

a 20-

@0

KLe= 0.1

0

10-

0 S

z * S

0 a a 0

4

*0-

2 4 6 0 Uk /U

Figure 1. Experimental values of ut/u, against /utl for 0.08 <KLe < 0.12. ?, Abdel-Gayed et al. (1984); e, Abdel-Gayed et al. (1985); o, Abdel-Gayed et al. (1988); e, Andrews et al. (1975); -, Ballal (1979); g, Ballal & Lefebvre (1975); v, Bollinger & Williams (1949); 0, Dandekar & Gouldin (1982); A, Grover et al. (1963); B, Hamamoto et al. (1984); k, Karlovitz (1954); K, Karlovitz et al. (1951); A, Karpov et al. (1959); Q, Khramtsov (1959); o, Kido et al. (1983); *, Petrov & Talantov (1959); q, Singh (1975); o, Smith & Gouldin (1978); 0, Sokolik et al. (1967); V, Vinckier & Van Tiggelen (1968); c, Wagner (1955); A, Williams et al. (1949); x, Wohl & Shore (1955); Z, Zotin & Talantov (1966). Figure 2. Experimental values of ut/u, against us/u, for 0.75 < KLe < 1.25. (For key see figure 1.)

6.

4

ao

2

t~

0

0 x x

C] x 0

0

0

000 x

0 0 x x

0 0 Li0 0 X

x ~ ~ O

00

10-3 10-1 KLe or K

10

Figure 3. Mean deviations of experimental points from the correlation curve for different ranges of KLe and K. x, KLe correlation; o, K correlation (Le < 1.3); n, K correlation (Le > 1.3).

Phil. Trans. . Soc. Lond. A (1992)

Figure 1

12-

Figure 2

365

8-

ZN

E S

05 .b e a

a 0

a 0

Em j

0 10 20 Ut I

30

x

Bradley et al., Proc. Royal Soc. 1992

ST /SL

v�/SL v�/SL

The experimental data collected from various studies exhibit a wide scatter(here K is the Karlovitz number).

5

Damkohler’s ConjectureZ. Elektrochem., 1940.

Damkohler identiifed two distinct limiting regimes:(i) a small scale turbulence regime where small eddies interact with the trans-port mechanisms within the flame.(ii) a large scale turbulence, where the flame is thin compared to the smallestturbulence scale, turbulence-flame interaction is purely kinematic

These two regimes correspond, in the present terminology to the thin reactionzone and corrugated flamelet regimes, respectively.

The expressions for the turbulent burning velocity ST proposed by Damkohler,and the modifications due to Shelkin (NACA TM 1110, 1947) are discussednext.

6

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ST

SL=

�DT

Dth

Since the turbulent diffusivity DT ∼ v�� and Dth ∼ SLlf , we have

ST

SL=

�v�

SL

�

lf

In the small scale turbulence (reaction sheet) regime, the turbulent eddies mod-

ify the transport processes in the preheat zone.

By analogy to the scaling relation for the laminar burning velocity SL ∼�

Dth/tfhe proposed that ST ∼

�DT/tf where DT is the turbulent diffusivity. Hence

7

Af

AST

∆Af

SL

The turbulent speed ST , in the one-dimensional configuration shown in thefigure, is the incoming mean flow velocity. Then

m = ρuAST

Since all the reactants pass through the wrinkled flame, the mass flow ratecan be also calculated from the total contributions of the mass flowing throughthe segments ∆Af comprising the wrinkled flame, assuming that each segmentpropagates normal to itself at the laminar flame speed SL. Thus

m = ρuAfSL

AST = AfSL

In the large scale turbulence (corrugated flamelet) regime, Damkohler resortedto a geometrical argument, with analogy to a Bunsen flame.

8

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ST

SLθ

θ

∆Af

∆A

∆Af

∆A=

1

cos θ=

�1 + tan2 θ

tan θ =v�

SL

ST = SL

�

1 +

�v�

SL

�2

The interaction between the wrinkled flame front and the turbulent flow field isassumed purely kinematic. Using the geometrical analogy with a Bunsen flame,the velocity component normal to the surface ∆Af is the laminar flame speedSL, while that tangential to the surface is due to the turbulent eddy and isproportional to the turbulent intensity v�. Then

ST ∼ SL

�1 +

1

2

�v�

SL

�2�

for v� � SL

ST ∼ v� for v� � SL

9

More generally, let F (x, t) = 0 represents the flame sheet

Af

A=

�|∇F |

|∇F · e|

�

where e is a unit vector in the direction of propagation

and an suitable average is taken.∆Af

∆A

∆Af

e

the area of a surface element onthe flame sheet ∆Af = |∇F |

the projection in the directionof propagation is ∆A = |∇F · e|

x = f(y, z, t)

Af

A=

�1 + |∇f |2

If the flame surface does not fold back on itself 1,it may be represented explicitly in the form

1 For more general surfaces, see the discussion in Williams (1985) and Peters (2000)

with x in the direction of propagation; along e

⇒

ST = SL

�1 + |∇f |2

10

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This expression, for the turbulent flame speed, can be also obtained from thehydrodynamic theory, which applies to laminar as well as turbulent flow.

We consider the flame to be in statistical steady state within an isotropic ho-mogeneous turbulent incident field

By definition, the flame speed is given by Sf = v · n − Vf where velocitiesare evaluated just ahead of the flame. Then

f = 0choose

v = (ST + u�, u�2, u

�3)

u� = 0

ST + u�1 − u�

2fy − u�3fz − ft = Sf

�1 + f2

y + f2z )

ST

x

y

x = f(y, z, t)u1 = ST + u�1

11

upon averaging along the transverse directions y, z and in time,

ST + u�1 − u�

2fy − u�3fz − ft = Sf

�1 + f2

y + f2z )

ST = Sf

�1 + f2

y + f2z

ft = 0 : the flame is in statistical steady state

u�2fy = u�

3fz = 0 : fy and −fy are statistically identical (similarly for fz)

This equation highlights two main contributions to the turbulent propagationspeed, namely a contribution due to flame area increase (as in Damkohler con-jecture) and one due to Sf , the displacement speed of the front relative to theincoming flow.

12

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as in Damkohler’s proposition.

It remains to relate the average area ratio to the properties of the incomingturbulence.

ST = SL

�1 + f2

y + f2z

Neglecting the effect of flame stretch, Sf = SL and

1. Effect of mean area increase on the turbulent burning velocity

• Approximate “asymptotic” relations

• Numerical calculations employing the hydrodynamic theory(two-dimensional “turbulence”)

13

ST = SL

�

1 +

�∂

∂y

�u�1dt

�2+

�∂

∂z

�u�1dt

�2

The leading order solution of the flame structure (in the asymptotic hydrody-namic model), for weak transverse gradients (fy � 1, fz � 1), shows thatft = u�

1/SL i.e., the flame is advected upstream by the velocity perturbations.

If Taylor hypothesis1 is invoked, and isotropy assumed, the statistics of thetransverse gradients could be related to the statistics of the non dimensionallongitudinal fluctuation u�

1/ST , so that

ST = SL

�

1 + 2

�u�1

ST

�2

1 Taylor hypothesis provides a relation between temporal and spatial correla-tions, in the limit of weak turbulence and when there is a predominantly meanflow in one direction.

which may be evaluated if the pdf of u�1/ST is known, or with further simplifi-

cations.

14

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15

Some additional details of slide 14 are given in the next three slides

ST = SL

�

1 +

�∂

∂y

�u�1dt

�2+

�∂

∂z

�u�1dt

�2

The leading order solution of the flame structure (in the asymptotic hydrody-namic model), for weak transverse gradients (fy � 1, fz � 1), shows thatft = u�

1/SL i.e., the flame is advected upstream by the velocity perturbations.

ST = SL

�1 + a2y + a2z)

If the streamwise Eulerian displacement of the fluid elements by turbulence isa =

�u�1dt, we obtain

note that within the current approximation a determines the extent of longi-tudinal motion of the flame sheet, and the gradients ay and az the increase inarea of the flame sheet.

16

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If Taylor hypothesis1 is invoked,

∂

∂y

�u�1dt =

1

ST

∂

∂y

�u�1dx =

1

ST

�∂u�

1

∂ydx

1 Taylor hypothesis provides a relation between temporal and spatial correla-tions, in the limit of weak turbulence and when there is a predominantly meanflow in one direction. Specifically, If the mean flow is in the x-direction, say,and u�/u � 1, then

∂

∂t= u

∂

∂x

and if the turbulence is isotropic, then

1

ST

�∂u�

1

∂ydx =

1

ST

�∂u�

1

∂xdx =

u�1

ST

ST = SL

�

1 + 2

�u�1

ST

�2

17

ST = SL

�

1 +2

3C

�v�

ST

�2

Additional simplifications (Williams, 1985) yields an expression of the form

where C is a correction factor, and v�2 = 3u�12 with v� the turbulent intensity

(assuming isotropic turbulence).

Squaring the LHS and solving the quadratic equation for ST yields

18

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Additional simplifications (such as bringing the average inside the square root,invoking the condition of isotropic turbulence, etc..) suggest expressions theform

and for low turbulent intensity

ST

SL= 1 + C

�v�

SL

�2

ST

SL= 1 + C

�v�

SL

�n

Additional results based on a simpler model, i.e., the Michelson-Sivashinskyequation will be also presented illustrating in particular the effect of hydrody-namic instability on the turbulent flame speed.

Creta & Matalon, JFM 2011

Creta et al., CTM 2011

A similar scaling was recently proposed based on numerical calculations employ-ing the hydrodynamic theory (the flame treated as a sheet propagating with aspeed SF that depends on the local stretch rate), where C exhibits a dependenceon thermal expansion and turbulent scale.

19

Statistically stationary flame profiles

κ = 0v�/SL = 0.4

0.5

• The variance of both distributions increases with turbulence intensity,indicating thicker flame brushes and higher local curvatures.

• The mean curvature is zero; the planar (on the average) flames are likelyto be convex as they are to be concave.

20

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The turbulent propagation speed as a function of turbulence intensity, parametrizedfor different values of the expansion ratio σ. The solid lines are quadratic fitsof the kind

ST /SL = 1 + c (u�0/SL)

2

with c = c(σ, �) a constant that depends in general on the expansion ratio σand, as we shall see, on turbulence scale �.

ST/S

L

v�/SL

MS equation

21

v�/SL

Assuming the quadratic law, the figure displaying the turbulent propagationspeed as a function of σ, visualizes the function c = c(σ), for a given �, whicheffectively collapse on a unique curve.

The effect of an increase of thermal expansion enhances the turbulent flamespeed, but plateau towards a certain value when σ increases.

22

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ST

SL∼ 1 + a c(σ)χ(�)

�v�

SL

�2

The dependence of the turbulent propagation speed on the integral; scale � showsthat a particular intermediate scale exists at which ST experiences a maximum.For a given turbulence intensity there exists, therefore, a preferred eddy sizethat most effectively perturbs the flame front.

The proposed expression for the turbulent propagation speed

highlights the quadratic dependence from turbulence intensity being modulatedby two coefficients depending respectively on expansion ratio and on integralscale, and a is a proportionality constant.

23

ST = Sf

�1 + f2

y + f2z

Sf = SL − LK

Beyond the mere geometric corrugation of the flame’s surface due to turbu-lence influencing the burning rate, an important role in turbulent propagationis played by the mean stretching of the flame which can influence the local flamespeed.

2. Effect of the mean flame stretching on the turbulent burning velocity

So far, effect of stretch was neglected by setting K = 0 and Sf = SL. With

where, we recall that the flame stretch rate consists of the effects of curvatureand strain; i.e., K = SLκ+KS .

ST = SL (1 + f2x)

1/2

� �� �−L K(1 + f2x)

1/2

� �� � .

area increase stretching

hydrodynamic strain

24

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Having established that the mean curvature (for a planar flame brush) is zero,

the mean flame speed

Sf = SL − LSLκ− LKS = SL − LKS .

The overall statistical properties of hydrodynamic strain during the flame prop-agation indicate that KS has a positive mean (KS > 0), which reveals a netexpanding effect of hydrodynamic strain during turbulent propagation.

The mean hydrodynamic strain rate was also found to increase with turbulenceintensity.

Since the mean flame speed Sf is linearly correlated to the mean strain rate, withthe correlation having a negative slope for positive Markstein lengths (L > 0),the mean flame speed will decrease with turbulent intensity in a similar way asthe mean strain rate increases.

25

The decrease of Sf with turbulent result in a decrease in turbulent propagationspeed ST /SL with respect to a flame for which stretch effects are neglected.

As is clearly visible in the figure, the turbulent propagation speed is observed todrop even below the laminar speed, at least for moderate intensities. At greaterintensities, the increase in flame surface area dominates over the decrease inmean flame speed Sf , resulting in the turbulent propagation speed increasingabove the laminar speed.

Rescaling the turbulent speed with themean flame speed, however, is seen toeliminate the effect due to the decreaseof Sf and recovers the quadratic scalingwhich represents the geometric effectdue to area increase alone.

ST =�1 + a χ(�) c(σ) (u�

0/SL)2�[SL − LKS ],

26