Turbulent Flames

Turbulent flames can be divided into different flame regimes depending on velocity and length scale ratios as reviewed in the following section in brief.

The influence of turbulence on burning velocity will be discussed in relation to the defined flame regimes.

2.3.3.1 Flame Regimes

The interaction between turbulence and combustion is an important factor to define the burning velocity. Depending on the structure of this interaction, i.e.

velocity and length scale ratios, different flame regimes can be distinguished as proposed by Borghi [110] and Peters [64, 94, 111]. Figure 2.42 shows the flame regime diagram in terms of velocity ratiou⁰/S_l,0 over length scale ratio Λ/δFas introduced by Borghi [110] and extended by Peters [64]. Peters defines a turbulent Reynolds number

Ret= u⁰Λ S_l,0δF

(2.72)

based on turbulent velocity fluctuations u⁰ and turbulent macro scale Λ as-suming a laminar flame thickness of δF = ν/S_l,0 (cf. Eq (2.47)). In the flame regime diagram Re_t = 1 separates laminar flames (Re_t < 1) from turbulent flames (Re_t>1). As laminar flames have been discussed previously, this sec-tion will focus on the turbulent flame regimes with Re_t>1.

In order to distinguish turbulent flame regimes, Peters [64] defines two Karlovitz numbers, Ka and Ka_IL. The turbulent Karlovitz number

Ka= t_F t_η =δ²_F

η² (2.73)

represents the ratio of chemical time scale t_F = δF/S_l,0 to the Kolmogorov timescale of the smallest turbulent eddiest_η=(ν/²)^1/2. WithδF=ν/S_l,0, Ka can be expressed by flame thicknessδF and Kolmogorov length scaleη=(ν³/²)^1/4 (cf. Eq. (2.13)). With ² = u⁰³/Λ (cf. Eq (2.14)), the Karlovitz number can be rewritten in terms ofu⁰/Sl,0andΛ/δF:

Ka= µ u⁰

S_l,0

¶³₂µΛ δF

¶₋¹₂

. (2.74)

In the flame regime diagram Ka = 1 separates the flamelet regime from the thin reaction zone regime.

The second Karlovitz number

KaIL= δ²_IL

η² (2.75)

is based on the inner layer thicknessδ^IL. With Eq. (2.33), it relates to Ka via

Ka_IL=C_ILKa . (2.76)

The line Ka_IL=1 separates the thin reaction zone regime from the broken re-action zone regime.

In the wrinkled and corrugated flamelet regime the flame thickness δF is smaller than the Kolmogorov scaleη. This means that the smallest turbulent eddies do not perturb the flame structure. Transport of reactants into the pre-heat and reaction zone is only based on molecular diffusion. The wrinkled

2.3 Premixed Combustion

˙

m S_t

S_l

A_F A

Figure 2.43:Schematic of a premixed turbulent flame in a duct (adapted from [94]).

flamelet regime represents flames with turbulent velocity fluctuations smaller than the laminar burning velocity (u⁰/S_l,0<1).

The thin reaction zone regime is characterized by the idea that the smallest turbulent eddies can penetrate into the flame structure as η < δF. However, withη>δIL, the eddies are not able to enter the inner layer. The interaction of small eddies with the preheat zone increases scalar mixing and, therefore, the reaction rate and the turbulent burning velocity.

In the broken reaction zone regime turbulent eddies are small enough to pen-etrate the reaction zone. As cold reactants can be transported into the reaction zone, local quenching of the reaction can occur. This poses an upper limit to the increase of burning velocity due to turbulence.

2.3.3.2 Turbulent Burning Velocity

In literature, numerous concepts and correlations can be found to represent the influence of turbulence on burning velocity. An overview is for example given by Driscoll [112]. In 1940, Damköhler [113] was the first to develop a theoretical expression for the turbulent burning velocityS_t. For a comprehen-sive summary of Damköhler’s approach the reader is referred to Peters [64, 94].

The basic principle of Damköhler’s theory is presented in Fig. 2.43. Assuming a turbulent flame at a fixed position in a turbulent flow, the mass flow rate ˙m

is equal to the mass flow rate consumed by the turbulent flame:

˙

m=ρuS_lA_F=ρuS_tA. (2.77) The mass flow rate consumed by the flame can be expressed either with the turbulent burning velocity S_t and the flow’s cross-sectional area A (dashed line) or the local laminar burning velocityS_l perpendicular to the flame front and the turbulent flame surface area A_F. Based on Eq. (2.77) the ratio of lami-nar and turbulent burning velocity

S_t S_l = A_F

A (2.78)

is defined by the ratio of flame surface area and flow cross-sectional area.

Damköhler [113] identified two flame regimes, the large scale and the small scale turbulence regime which are comparable to the corrugated flamelet and the thin reaction zone regime introduced in the previous section. He devel-oped expressions for the area ratio AF/A for both cases.

In the large scale turbulence or corrugated flamelet regime, turbulent eddies do not penetrate the flame front. Consequently, the area increase due to com-bustion is proportional to the ratio of turbulent velocity fluctuations and lam-inar burning velocity:

A_F A ∝ u⁰

S_l . (2.79)

This relation combined with Eq. (2.78) leads to

S_t∝u⁰. (2.80)

If the turbulent eddies are able to perturb the preheat zone in the small scale turbulence or thin reaction zone regime, diffusion of reactants into the pre-heat zone is increased and the molecular diffusion coefficientD has to be re-placed by the turbulent valueD_t. According to Damköhler [113], the turbulent burning velocity can be expressed by

S_t∝

2.3 Premixed Combustion

where the laminar burning velocity is related to molecular diffusion into the preheat zone divided by the chemical timescale at which reactants are con-sumed by the combustion reaction. From Eq. (2.81) and (2.82) the proportion-ality related to turbulence properties and laminar flame characteristics:

S_t

A correlation related to Damköhler’s approach has been widely used to repre-sent experimental turbulent burning velocity data [94]:

S_t

According to Damköhler [113] the model parameterC_Sshould depend on the length scale ratioΛ/δF and the exponentc₃ should be set to 0.5. He used the unstretched valueS_l,0for the laminar burning velocityS_l.

As the conditions analyzed in this work typically lie around the boundary be-tween corrugated flamelet and thin reaction zone regime, Eq. (2.85) will be used in Chap. 5 and 6 to compute turbulent burning velocities. In order to in-clude the effect of flame stretch, the laminar burning velocityS_l will be set to the stretched valueSl,s introduced in Sec. 2.3.2.3.