Lecture 12 The Level Set Approach for Turbulent Premixed Combus=on

Transcription

1 Lecture 12 The Level Set Approach for Turbulent Premixed Combus=on

2 A model for premixed turbulent combus7on, based on the non- reac7ng scalar G rather than on progress variable, has been developed in recent years It avoids complica7ons associated with counter- gradient diffusion and, since G is 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 flow field into two regions where G > G 0 is the region of burnt gas and G < G 0 is that of the unburnt mixture The choice of G 0 is arbitrary and typically taken as G 0 =

3 Recapitula=on Flame front normal vector in direc7on of the unburnt gas Flame displacement speed dx f /dt A field equa7on can now be derived by differen7a7ng with respect to 7me This leads to DG Dt =

4 Introducing and one obtains the field equa7on This equa7on was introduced by Williams (1985) It is known as the G- equa7on

5 G- equa7on is applicable to thin flame structures propaga7ng with a well- defined burning velocity It therefore is well- suited for the descrip7on of premixed turbulent combus7on in the corrugated flamelets regime, where it is assumed that the laminar flame thickness is smaller than the smallest turbulent length scale, the Kolmogorov scale Therefore, the en7re flame structure is embedded within a locally quasi- laminar flow field and the laminar burning velocity remains well- defined

6 The G- equa7on has no diffusion term G is a scalar quan7ty which is defined at the flame surface only, while the surrounding G- field is not yet uniquely defined This originates from the fact that the kinema7c balance is valid only for the flame surface and defini7on of remaining G- field is (within some constraints) arbitrary G- field typically defined to be the closest distance from the flame

7 The burning velocity s L appearing in may be modified to account for the effect of flame stretch Performing two- scale asympto7c analyses of corrugated premixed flames, Pelce and Clavin (1982), Matalon and Matkowsky (1982) derived first order correc7on terms for small curvature and strain Expression for the modified burning velocity becomes

8 The expression for the modified burning velocity becomes Here s L 0 is the burning velocity of the unstretched flame, κ is the curvature and S is the strain rate Flame curvature is defined in terms of the G- field as where has been used Flame curvature is posi7ve if flame is convex with respect to unburnt mixture

9 The strain rate imposed on the flame by velocity gradients is defined as The Markstein length is of the same order of magnitude and propor7onal to the laminar flame thickness The ra7o is called the Markstein number

10 For the case of a one- step reac7on with a large ac7va7on energy, constant transport proper7es and a constant heat capacity, the Markstein length with respect to the unburnt mixture reads, for example (Clavin &Williams, 1982 and Matalon & Matkowsky, 1982) Here is the Zeldovich number, where E is the ac7va7on energy, and Le is the Lewis number of the deficient reactant

11 The Lewis number is approximately unity for methane flames and larger than unity for fuel- rich hydrogen and all fuel- lean hydrocarbon flames other than methane Therefore, since the first 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

12 Whenever the Markstein length is nega7ve, as in lean hydrogen- air mixtures, diffusional- thermal instabili7es tend to increase the flame 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 of diffusional- thermal instabili7es and instabili7es induced by gas expansion could lead to strong flame accelera7ons

13 Effects of curvature and strain on laminar burning velocity Curvature Effect on Laminar Burning Velocity from Experiments and Theory Strain Effect on Laminar Burning Velocity from Numerical Simula7ons Laminar premixed stoichiometric methane/air counterflow flames Laminar premixed stoichiometric methane/air spherically expanding flames φ = 0.8 Note: s L,u s L,b /7 φ = a

14 Introducing into the G- equa7on, it may be wrifen as Here is defined as the Markstein diffusivity The curvature term adds a second order deriva7ve to the G- equa7on

15 This avoids the forma7on of cusps that would result from for a constant value of s L 0 While the solu7on of the G- equa7on with a constant s L 0 is solely determined by specifying the ini7al condi7ons, the parabolic character of requires that the boundary condi7ons for each iso- surface G must be specified

16 The G- equa7on model in the corrugated flamelets 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 à Mul7- scale model G- field defined as distance func7on Buschmann 1996

17 The Level Set Approach for the Thin Reac=on Zones Regime The equa7on is suitable for thin flame structures in the corrugated flamelets regime, En7re flame structure quasi- steady Laminar burning velocity well defined However, not suitable for thin reac7on zones regime Deriva7on of level set equa7on valid in the thin reac7on zones regime

18 Since the inner layer shown previously is responsible for maintaining the reac7on process alive, we define the thin reac7on zone as the inner layer The loca7on of the inner layer will be determined by the iso- scalar surface of the temperature seing T(x,t)=T 0, where T 0 is the inner layer temperature

19 The temperature equa7on reads where D is the thermal diffusivity ω T the chemical source term Similar to for the scalar G, the iso- temperature surface T(x,t)=T 0 sa7sfies the condi7on

20 Gibson (1968) has derived an expression for the displacement speed s d of an iso- surface of non- reac7ng diffusive scalars Extending this result to the reac7ve scalar T this leads to where the displacement speed s d is given by Here the index 0 defines condi7ons immediately ahead of the thin reac7on zone

21 The normal vector on the iso- temperature surface is defined as We now want to formulate a G- equa7on describing the loca7on of the thin reac7on zones such that the iso- surface T(x,t)=T 0 coincides with the iso- surface defined by G(x,t)=G 0 Then, the normal vector defined by is equal to that defined by and also points towards the unburnt mixture

22 Using and together with leads the G- equa7on

23 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 another for diffusion normal to the iso- surface This is consistent with the defini7on of the curvature if the iso- surface G(x,t)=G 0 is replaced by the iso- surface T(x,t)=T 0 and if ρd is assumed constant

24 Introducing into one obtains Here is to be expressed by in terms of the G- field

25 The quan77es s n and s r are contribu7ons due to normal diffusion and reac7on to the displacement speed of the thin reac7on zone and are defined as In a steady unstretched planar laminar flame 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 thin reac7on zone will influence the local displacement speed

26 Then the sum of is a fluctua7ng quan7ty that couples the G- equa7on to the solu7on of the balance equa7ons of the reac7ve scalars There is reason to expect, however, that s L,s is of the same order of magnitude as the laminar burning velocity The evalua7on of DNS- data by Peters et al. (1998) confirms this es7mate

27 In that paper it was also found that the mean values of s n and s r slightly depend on curvature This leads to a modifica7on of the diffusion coefficient which partly takes Markstein effects into account We will ignore these modifica7ons here and consider the following level set equa7on for flame structures of finite thickness This equa7on is defined at the thin reac7on zone v, s L,s, and D are values at that posi7on

28 The G- equa7on in the thin reac7on zones regime is very similar to that for the corrugated flamelets regime Important is the difference between and D and the disappearance of the strain term The lafer is implicitly contained in the burning velocity s L,s

29 In an analy7cal study of the response of one- dimensional constant density flames 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 with the assump7on of a single thin reac7on zone

30 It suggests that could also have been derived from for the limit of high frequency perturba7ons of the flame structure This strongly supports it as level set equa7on for flame structures of finite thickness and shows that unsteadiness of that structure is an important feature in the thin reac7on zones regime

31 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 flame surface

32 It should be noted again that the level set equa7ons and are only defined at the flame surface, while is valid in the en7re field

33 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 Level set describing thin reac7on zone Broadened preheat zone Burned gas Unburned gas DNS of Premixed Combus7on, Knudsen 2010

34 A Common Level Set Equa=on for Both Regimes The G- equa7on applies to different regimes in premixed turbulent combus7on: Corrugated flamelets regime Thin reac7on zones regime In order to show this, we will analyze the order of magnitude of the different terms in the second equa7on

35 This can be done by normalizing the independent variables and the curvature in this equa7on with respect to Kolmogorov length, 7me and velocity scales Using η 2 /t η = ν one obtains In flames, D/ν is of order unity Since Kolmogorov eddies can perturb the flow as well as the G- field, all deriva7ves, the curvature and the velocity v * are typically of order unity

36 However, since s L,s is of the same order of magnitude as s L, the defini7on shows that the ra7o s L,s /v η 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 flamelets regime, s L,s /v η becomes large and propaga7on dominates

37 We want to base the following analysis on an equa7on which contains only the leading order terms in both regimes Therefore we take the propaga7on term with a constant laminar burning velocity s L 0 from the corrugated flamelets 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

38 The leading order equa7on valid in both regimes then reads For consistency with other field equa7ons that will be used as a star7ng point for turbulence modeling, we have mul7plied all terms in this equa7on with ρ. This will allow to apply Favre averaging to all equa7ons Furthermore, we have set ρ s L 0 = ρ u s L,u constant and denoted this by paranthesis

39 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 is convenient to ignore at first its non- uniqueness outside the surface G(x, t) = G 0. Then the G- equa7on would have similar proper7es as other field equa7ons used in fluid dynamics and scalar mixing This would allow to define, at point x and 7me t in the flow field, a probability density func7on P(G;x,t) for the scalar G

40 Then, the probability density of finding the flame surface G(x,t)=G 0 at x and t is given by This quan7ty can be measured, for instance, by coun7ng the number of flame crossings in a small volume ΔV located at x over a small 7me difference Δt

41 Experimental data for the pdf (Wirth et al., 1992, 1993) from a transparent spark- igni7on engine Smoke par7cles, which burnt out immediately in the flame front, were added to the unburnt mixture Thereby, the front could be visualized by a laser sheet as the borderline of the region where Mie scafering of par7cles could be detected The pdf represents the pdf of fluctua7ons around the mean flame contour of several instantaneous images

42 By comparing the measured pdf with a Gaussian distribu7on, it is seen to be slightly skewed to the unburnt gas side This is due to the non- symmetric influence of the laminar burning velocity on the shape of the flame 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 other experimental pdfs Buschmann

43 Without loss of generality, we now want to consider a one- dimensional steady turbulent flame propaga7ng in x- direc7on We will analyze its structure by introducing the flame- normal coordinate x, such that all turbulent quan77es are a func7on of this coordinate only Then, the pdf of finding the flame surface at a par7cular loca7on x within the flame brush simplifies to P(G 0 ; x), which we write as P(x) We normalize P(x) by and define the mean flame posi7on x f as

44 The turbulent flame brush thickness can also be defined using P(x) With the defini7on of the variance a plausible defini7on is We note that from P(x) two important proper7es of a premixed turbulent flame, namely the mean flame posi7on and the flame brush thickness can be calculated

45 Peters (1992) considered turbulent modeling of the G- equa7on in the corrugated flamelets regime and derived Reynolds- averaged equa7ons for the mean and the variance of G The main sink term in the variance equa7on was defined as This quan7ty was called kinema7c restora7on in order to emphasize the effect of local laminar flame propaga7on in restoring the G- field and thereby the flame surface Corruga7ons produced by turbulence, which would exponen7ally increase the flame surface area with 7me of a non- diffusive iso- scalar surface are restored by this kinema7c effect

46 From that analysis resulted a closure assump7on which relates the main sink term to the variance of G and the integral 7me scale where c ω =1.62 is a constant of order unity This expression shows that kinema7c restora7on plays a similar role in reducing fluctua7ons of the flame front as scalar dissipa7on does in reducing fluctua7ons of diffusive scalars It was also shown by Peters (1992) that kinema7c restora7on is ac7ve at the Gibson scale, since the cut- off of the iner7al range in the scalar spectrum func7on occurs at that scale

47 Equa=ons for the Mean and the Variance of G In order to obtain a formula7on that is consistent with the well- established use of Favre averages in turbulent combus7on, we split G and the velocity vector v into Favre means and fluctua7ons Here, the Favre means are at first viewed as uncondi7onal averages At the end, however, only the respec7ve condi7onal averages are of interest

48 The turbulent burning velocity s T 0 is obtained by averaging over By seing one finds for the sta7onary unstretched flame front Using a number of addi7onal closure assump7ons described in Peters (2000), one finally obtains the following equa7ons for the Favre mean and variance of G:

49 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 mean of G that is introduced

50 In order to solve a model for the turbulent burning velocity s 0 T must be provided. A first step would be to use empirical correla7ons from the literature Alterna7vely, a modeled balance equa7on for the mean gradient will be derived next According to Kerstein (1988), this quan7ty represents the flame surface area ra7o, which is propor7onal to the turbulent burning velocity