7. Turbulent Premixed Flames • Background:

- Structure of turbulent premixed flames;

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- Instantaneous flame fronts (left) and turbulent flame brush envelope (right).

• Definitions:

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- Laminar flame thickness:

δL ∼ /Sα L = D/SL = /Sν L (1)

- Above equality implies that we assumed, Schmidt Number: Sc = /D ν = 1

Lewis Number: Le = /D α = 1 Prandtl Number: Pr = / ν α = 1 - Turbulent Reynolds number

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Re (2)

where Λ is the integral length scale of turbulence.- Turbulent Damköhler number: ratio of

characteristic flow time, τflow, to the characteristic chemical time, τc.

Da = (3)

- Characteristic flow time: τflow = Λ/uI - Characteristic chemical time: τc = δL/SL - Then Damköhler number is:

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Da = (3a)

- Turbulence length scales: λ: Taylor microscale η: Kolmogorov length scale

- Karlovitz number:

Ka = (4)

- Turbulent Reynolds number based on λ:

Re λ= uI /λ ν (5)

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- Turbulent Reynolds number based on η:

Re η= uI /η ν (6)

ReΛ≈ Re (7)

• Turbulent Burning Velocity:

- One of the most important unresolved problems in premixed turbulent combustion is the determination of the turbulent burning velocity.

- Above statement assumes that turbulent burning velocity is a well-defined quantity that only depends on local mean properties.

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- However, there is no consensus in literature whether the turbulent burning velocity is a characteristic quantity that can be defined unambigously for different geometries.

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Turbulent Reynolds Number, ReΛ

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- Turbulent premixed flame propagation was first investigated by Damköhler (1940).

- He identified two limiting cases based on the magnitude of the scale of turbulence as compared to the thickness of the laminar premixed flame.

- For large scale turbulence, Damköhler assumed that the interaction between a turbulent premixed flame (wrinkled flame) front and the turbulent flame front is purely kinematic.

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Laminar flame structure. - Damköhler equated the mass flux m˙ through the instantaneous turbulent flame surface area AT with the mass

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flux through the cross-sectional area

Ao. He used SL for mass flux through AT, and ST for mass flux through A.

m˙ = ρuSLAT = ρuSTAo (8)

(9)

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- Using geometric approximations, Damköhler proposed that (for large-scale, small-intensity turbulence),

(10)

In view of Eq.2,

(11)

- uI, turbulent fluctuating velocity in the unburned gas.

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LSIu

=1+LSTS

- Using similar geometric arguments, Schelkin showed that:

(12)

- Relationship proposed by Klimov:

(13)

- Clavin &Williams:

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2/1�2W

LSIu2w

1+^

=LSTS

- Gülder:

- For small-scale and high-intensity turbulence, Damköhler argued that turbulence only modifies the transport between the reaction zone and the unburned gas.

ST/SL (∼ DT/D)1/2 (16)

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Since DT ∝ uI Λ and D ∝ SLδL

- Then we have,

(17)

- For small-scale high-intensity turbulence conditions (usually called as distributed reaction regime), there are not many formulations available. In this regime, turbulent mixing is rapid as compared to the chemistry.

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- For the distributed reaction regime the following semi-empirical model has been proposed, Gülder (1990):

(18)

State-of-the-art:

- Definition of turbulent burning velocity is not uniform/universal.

- Experimental data scatter is significant between different experimental rigs.

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4/3WIuLSw

4.=6LSTS

- Numerical simulation:- Flamelet model/assumption- Turbulent burning closure- Direct numerical simulation

Experimental Measurement Methods:

- Conical stationary flames on cylindrical nozzles.- Swirling flames.- Constant volume vessels.- Stagnation point flames.- Laser-based diagnostics to study flame structure.

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- Statistical approaches to estimate the flame front surface area.

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