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Lecture 12

The Level Set Approach for Turbulent PremixedCombus=on

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A model for premixed turbulent combus7on, based on the non-reac7ng scalar Grather than on progress variable, has been developed in recent years

It avoids complica7ons associated with counter-gradient diffusion and, since Gis non-reac7ng, there is no need for a source term closure

An equa7on for G can be derived

by considering an iso-scalar surface

This surface divides the ow eld intotwo regions where

G > G0 is the region of burnt gas

andG < G0 is that of the unburnt mixture

The choice of G0 is arbitrary and typically taken as G0 = 0

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Recapitula=on Flame front normal vector in direc7on of the unburnt gas

Flame displacement speed d x f /dt

A eld equa7on can now be derivedby differen7a7ng

with respect to 7me

This leads to

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DG

Dt= 0

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Introducing

and

one obtains the eld equa7on

This equa7on was introduced byWilliams (1985)

It is known as the G-equa7on

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G-equa7on is applicable to thin ame structures propaga7ng with a well-dened burning velocity

It therefore is well-suited for the descrip7on of premixed turbulentcombus7on in the corrugated amelets regime, where it is assumed that

the laminar ame thickness is smaller than the smallest turbulent length

scale, the Kolmogorov scale

Therefore, the en7re ame structure is embedded within a locally quasi-

laminar ow eld and the laminar burning velocity remains well-dened

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The G-equa7on

has no diffusion term

G is a scalar quan7ty which is dened at the ame surface only , while thesurrounding G-eld is not yet uniquely dened

This originates from the fact that the kinema7c balance

is valid only for the ame surface and deni7on of remaining G-eld is (within

some constraints) arbitrary

G-eld typically dened to be the closest distance from the ame

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The burning velocity sL appearing in

may be modied to account for the effect of ame stretch

Performing two-scale asympto7c analyses of corrugated premixed ames,Pelce and Clavin (1982), Matalon and Matkowsky (1982) derived rst order

correc7on terms for small curvature and strain

Expression for the modied burning velocity becomes

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The expression for the modied burning velocity becomes

Here sL0 is the burning velocity of the unstretched ame, is the curvatureand S is the strain rate

Flame curvature is dened in terms of the G-eld as

where

has been used Flame curvature is posi7ve if ame is convex with respect to unburnt mixture

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The strain rate imposed on the ame by velocity gradients is dened as

The Markstein length is of the same order of magnitude and propor7onalto the laminar ame thickness

The ra7o is called the Markstein number

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For the case of a one-step reac7on with a large ac7va7on energy , constanttransport proper7es and a constant heat capacity, the Markstein length with

respect to the unburnt mixture reads, for example (Clavin &Williams, 1982and Matalon & Matkowsky, 1982)

Here

is the Zeldovich number, where E is the ac7va7on energy, and Le is the

Lewis number of the decient reactant

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The Lewis number is approximately unity for methane ames and largerthan unity for fuel-rich hydrogen and all fuel-lean hydrocarbon ames

other than methane

Therefore, since the rst term on the r.h.s. of

is always posi7ve, the Markstein length is posi7ve for most prac7cal

applica7ons of premixed hydrocarbon combus7on, occurring typically

under stoichiometric or fuel-lean condi7ons

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Whenever the Markstein length is nega7ve , as in lean hydrogen-air mixtures,diffusional-thermal instabili7es tend to increase the ame surface area

This is believed to be an important factor in gas cloud explosions of hydrogen-

air mixtures

Although turbulence tends to dominate such local effects the combus7on ofdiffusional-thermal instabili7es and instabili7es induced by gas expansion

could lead to strong ame accelera7ons

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Strain Effect on Laminar Burning Velocityfrom Numerical Simula7ons

Laminar premixed stoichiometricmethane/air counterow ames

Curvature Effect on Laminar BurningVelocity from Experiments and Theory

Effects of curvature and strain on laminar burning velocity

Laminar premixed stoichiometricmethane/air spherically expanding

ames

Note: sL,u sL,b/

= 0.8

= 1

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Introducing

into the G-equa7on, it may be wri en as

Here

is dened as the Markstein diffusivity

The curvature term adds a second order deriva7ve to the G-equa7on

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This avoids the forma7on of cusps that would result from

for a constant value of sL0

While the solu7on of the G-equa7on with a constant sL0 is solely determinedby specifying the ini7al condi7ons, the parabolic character of

requires that the boundary condi7ons for each iso-surface G must be

specied

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The G-equa7on model in the corrugated amelets regime Flame thin compared to small turbulent scales Level set describes interface between unburned and burned Flame structure embedded in the interface, but varia7ons on much

smaller scale Mul7-scale model

G-eld dened as distance func7on

Buschmann 1996

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The Level Set Approach for the Thin Reac=on Zones Regime

The equa7on

is suitable for thin ame structures in the corrugated amelets regime, En7re ame structure quasi-steady Laminar burning velocity well dened

However, not suitable for thin reac7on zones regime

Deriva7on of level set equa7on valid in the thin reac7on zones

regime

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Since the inner layer shown previously is responsible for maintaining the

reac7on process alive, we dene the thin reac7on zone as the inner layer

The loca7on of the inner layerwill be determined by the

iso-scalar surface

of the temperature se ng

T ( x ,t )=T 0, where T 0 is the

inner layer temperature

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The temperature equa7on reads

where D is the thermal diffusivity T the chemical source term

Similar to

for the scalar G, the iso-temperature surface T ( x ,t )=T 0 sa7ses the condi7on

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Gibson (1968) has derived an expression for the displacement speed sd of aniso-surface of non-reac7ng diffusive scalars

Extending this result to the reac7ve scalar T this leads to

where the displacement speed sd is given by

Here the index 0 denes condi7ons immediately ahead of the thin reac7onzone

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The normal vector on the iso-temperature surface is dened as

We now want to formulate a G-equa7on describing the loca7on of the thinreac7on zones such that the iso-surface T( x ,t )=T0 coincides with the iso-

surface dened by G( x ,t )=G0

Then, the normal vector dened by

is equal to that dened by

and also points towards the unburnt mixture

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Using and

together with

leads the G-equa7on

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Peters et al. (1998) show that the diffusive term appearing in the brackets in

this equa7on may be split into one term accoun7ng for curvature and anotherfor diffusion normal to the iso-surface

This is consistent with the deni7on of the curvature

if the iso-surface G( x ,t )=G0 is replaced by the iso-surface T ( x ,t )=T 0 and

if D is assumed constant

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Introducing

into

one obtains

Here

is to be expressed by in terms of the G-eld

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The quan77es sn and sr are contribu7ons due to normal diffusion and reac7onto the displacement speed of the thin reac7on zone and are dened as

In a steady unstretched planar laminar ame we would have

In the thin reac7on zones regime, however, the unsteady mixing and diffusion

of chemical species and the temperature in the regions ahead of the thinreac7on zone will inuence the local displacement speed

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Then the sum of

is a uctua7ng quan7ty that couples the G-equa7on to the solu7on of the

balance equa7ons of the reac7ve scalars

There is reason to expect, however, that sL,s is of the same order of magnitudeas the laminar burning velocity

The evalua7on of DNS-data by Peters et al. (1998) conrms this es7mate

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In that paper it was also found that the mean values of sn and sr slightlydepend on curvature

This leads to a modica7on of the diffusion coefficient which partly takesMarkstein effects into account

We will ignore these modica7ons here and consider the following level setequa7on for ame structures of nite thickness

This equa7on is dened at the thin reac7on zonev , sL,s, and D are values at that posi7on

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The G-equa7on in the thin reac7on zones regime

is very similar to that for the corrugated amelets regime

Important is the difference between and D and the disappearance of thestrain term

The la er is implicitly contained in the burning velocity sL,s

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In an analy7cal study of the response of one-dimensional constant densityames to 7me-dependent strain and curvature, Joulin (1994) has shown that

in the limit of high frequency perturba7ons, the effect of strain disappears

en7rely and Lewis-number effects also disappear in the curvature term such

that

This analysis was based on one-step large ac7va7on energy asympto7cs withthe assump7on of a single thin reac7on zone

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The important difference between the level set formula7on

and the equa7on for the reac7ve scalar

is the appearance of a burning velocity which replaces normal diffusion and

reac7on at the ame surface

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It should be noted again that the level set equa7ons

and

are only dened at the ame surface,while

is valid in the en7re eld

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The G-equa7on model in the thin reac7on zones regime Flame broadened by turbulence, but Reac7on zone s7ll thin compared to turbulent scales and appears as

interface Level set describes loca7on of reac7on zone

DNS of Premixed Combus7on, Knudsen 2010

Level set describing thinreac7on zone

Burned gasBroadened preheat zone

Unburned gas

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A Common Level Set Equa=on for Both Regimes

The G-equa7on applies to different regimes in premixed turbulentcombus7on:

Corrugated amelets regime

Thin reac7on zones regime

In order to show this, we will analyze the order of magnitude of the differentterms in the second equa7on

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This can be done by normalizing the independent variables and the curvaturein this equa7on with respect to Kolmogorov length, 7me and velocity scales

Using 2/ t = one obtains

In ames, D/ is of order unity

Since Kolmogorov eddies can perturb the ow as well as the G-eld, allderiva7ves, the curvature and the velocity v * are typically of order unity

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However, since sL,s is of the same order of magnitude as sL, the deni7on

shows that the ra7o sL,s/ v is propor7onal to Ka -1/2 .

Since Ka > 1 in the thin reac7on zones regime, it follows for this regime that

In the thin reac7on zones regime , the propaga7on term in the equa7on

is therefore small and the curvature term will be dominant

In the corrugated amelets regime , sL,s/ v becomes large and propaga7ondominates

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We want to base the following analysis on an equa7on which contains onlythe leading order terms in both regimes

Therefore we take the propaga7on term with a constant laminar burningvelocity sL0 from the corrugated amelets regime and the curvature term

mul7plied with the diffusivity D from the thin reac7on zones regime

The strain term will be neglected in both regimes

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The leading order equa7on valid in both regimes then reads

For consistency with other eld equa7ons that will be used as a star7ng pointfor turbulence modeling, we have mul7plied all terms in this equa7on with .

This will allow to apply Favre averaging to all equa7ons

Furthermore, we have set sL0 = u sL,u constant and denoted this byparanthesis

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Modeling Premixed Turbulent Combus=on Based on the Level Set Approach

If the G-equa7on is to be used as a basis for turbulence modeling, it isconvenient to ignore at rst its non-uniqueness outside the surface G( x , t ) =

G0.

Then the G-equa7on would have similar proper7es as other eld equa7onsused in uid dynamics and scalar mixing

This would allow to dene, at point x and 7me t in the ow eld, a probabilitydensity func7on P(G; x ,t ) for the scalar G

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Then, the probability density of nding the ame surface G( x ,t )=G0 at x and t is given by

This quan7ty can be measured, for instance, by coun7ng the number of amecrossings in a small volume V located at x over a small 7me difference t

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Experimental data for the pdf (Wirth et al., 1992, 1993) from a transparentspark-igni7on engine

Smoke par7cles, which burntout immediately in the ame

front, were added to the

unburnt mixture

Thereby, the front could bevisualized by a laser sheet as

the borderline of the region

where Mie sca ering of

par7cles could be detected The pdf represents the pdf of

uctua7ons around the mean ame contour of several instantaneous images

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By comparing the measured pdf with a Gaussiandistribu7on, it is seen to be slightly skewed to the

unburnt gas side

This is due to the non-symmetric inuence of thelaminar burning velocity on the shape of the ame

front

There are rounded leading edges towards

the unburnt mixture, but sharp and narrow

troughs towards the burnt gas

This non-symmetry is also found in otherexperimental pdfs

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Buschmann 1996

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Without loss of generality, we now want to consider a one-dimensionalsteady turbulent ame propaga7ng in x -direc7on

We will analyze its structure by introducing the ame-normal coordinate x ,such that all turbulent quan77es are a func7on of this coordinate only

Then, the pdf of nding the ame surface at a par7cular loca7on x within the

ame brush simplies to P(G0; x ), which we write as P( x )

We normalize P( x ) by

and dene the mean ame posi7on x f as

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The turbulent ame brush thickness can also be dened using P( x )

With the deni7on of the variance

a plausible deni7on is

We note that from P( x ) two important proper7es of a premixed turbulentame, namely the mean ame posi7on and the ame brush thickness can be

calculated

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Peters (1992) considered turbulent modeling of the G-equa7on in thecorrugated amelets regime and derived Reynolds-averaged equa7ons for the

mean and the variance of G

The main sink term in the variance equa7on was dened as

This quan7ty was called kinema7c restora7on in order to emphasize theeffect of local laminar ame propaga7on in restoring the G-eld and thereby

the ame surface

Corruga7ons produced by turbulence, which would exponen7ally increase theame surface area with 7me of a non-diffusive iso-scalar surface are restored

by this kinema7c effect

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From that analysis resulted a closure assump7on which relates the main sinkterm to the variance of G and the integral 7me scale

where c =1.62 is a constant of order unity

This expression shows that kinema7c restora7on plays a similar role inreducing uctua7ons of the ame front as scalar dissipa7on does in reducing

uctua7ons of diffusive scalars

It was also shown by Peters (1992) that kinema7c restora7on is ac7veat the Gibson scale, since the cut-off of the iner7al range in the scalar

spectrum func7on occurs at that scale

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Equa=ons for the Mean and the Variance of G

In order to obtain a formula7on that is consistent with the well-establisheduse of Favre averages in turbulent combus7on, we split G and the velocity

vector v into Favre means and uctua7ons

Here, the Favre means are at rst viewed as uncondi7onal averages

At the end, however, only the respec7ve condi7onal averages are of interest

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The turbulent burning velocity sT 0 is obtained by averaging over

By se ng one nds for the sta7onary unstretched ame front

Using a number of addi7onal closure assump7ons described in Peters (2000),one nally obtains the following equa7ons for the Favre mean and variance of

G:

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It is easily seen that

has the same form as

and therefore shares its mathema7cal proper7es

It also is valid at only

The solu7on outside of that surface depends on the ansatz for the Favre meanof G that is introduced

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In order to solve

a model for the turbulent burning velocity s0T must be provided.

A rst step would be to use empirical correla7ons from the literature

Alterna7vely, a modeled balance equa7on for the mean gradient will bederived next

According to Kerstein (1988), this quan7ty represents the ame surface areara7o , which is propor7onal to the turbulent burning velocity

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