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The local burning velocity and the flame displacement speed are the dominant properties in the mechanism of turbulent premixed combustion. The flame displacement speed and the local burning velocity have been investigated separately, because the flame displacement speed can be used for the discussion of flame-turbulence interactions and the local burning velocity can be used for the discussion of the inner structure of turbulent premixed flames. In this study, to establish the basis for the discussion on the effects of turbulence on the inner structure of turbulent premixed flames, the indirect relationship between the flame displacement speed and the local burning velocity was investigated by the flame stretch, the flame curvature, and the tangential strain rate using DNS database with different density ratios. It was found that for the local tangential strain rate and the local flame curvature, the local burning velocity and the flame displacement speed had the opposite correlations in each density ratio case. Therefore, it is considered that the local burning velocity and the flame displacement speed have a negative correlation.

Since the late 20th century, the possibilities of the exhaustion of fossil fuels and the alternative fuels have been discussed. In the beginning of the 21st century, however, the fossil fuels still remain the main source of energy and it can be predicted easily that the current situation will be running for at least several decades. Therefore, further enhancement of the efficiency and reduction of the environmental load are required for the combustion technology. In many combustors, turbulent combustion is mainly utilized. The local burning velocity and the flame displacement speed are the dominant properties in the mechanism of turbulent premixed combustion. The local burning velocity is the instantaneous local quantity based on the local consumption rate of an unburned mixture, while the flame displacement speed is the local quantity based on the flame normal speed, in which the flame surface, defined as the isosurface of temperature or mass fraction of an unburned mixture, moves relatively to a local flow [

The flame displacement speed and the local burning velocity have been investigated separately, because the flame displacement speed can be used for the discussion of flame-turbulence interactions and the local burning velocity can be used for the discussion of the inner structure of turbulent premixed flames. In this study, to establish the basis for the discussion on the effects of turbulence on the inner structure of turbulent premixed flames, the indirect relationship between the flame displacement speed and the local burning velocity was investigated by the flame stretch, the flame curvature, and the tangential strain rate using DNS database with different density ratios.

DNS database with different density ratios ρ_{u}/ρ_{b} was used for the analysis of both the local burning velocity and the flame displacement speed. These database were ρ_{u}/ρ_{b} = 2.50, termed case Lm; ρ_{u}/ρ_{b} = 5.00, termed case Mm; and ρ_{u}/ρ_{b} = 7.53, termed case Hm. The simulations were carried out on the VPP700 installed at RIKEN [

case | Lm | Mm | Hm |
---|---|---|---|

ρ_{u}/ρ_{b} | 2.50 | 5.00 | 7.53 |

Le | 1.0 | 1.0 | 1.0 |

0.416 | 0.523 | 0.600 | |

0.157 | 0.191 | 0.216 | |

ū_{in} (m/s) | 0.786 | 0.992 | 1.146 |

0.88 | 1.01 | 1.26 | |

13.0 | 10.7 | 9.44 | |

15.9 | 18.0 | 21.8 | |

56.7 | 56.7 | 56.7 | |

95.5 | 95.5 | 95.5 |

collocation scheme in the directions perpendicular to the mean flow for spatial discretisation. A third-order three-step Runge-Kutta method was used for the time evolution and an overall single-step irreversible reaction, which was the Arrhenius type, was used to describe the chemical kinetics. The inflow and outflow boundaries were described on the basis of Navier-Stokes characteristic boundary conditions (NSCBC) [

Initially, a laminar premixed flame was formed, which grew to form a turbulent premixed flame. The inflow velocity of the unburned mixture was adjusted while monitoring the turbulent burning velocity until the turbulent premixed flame became fully developed and stabilised. The instantaneous turbulent burning velocity varied temporally; however, the time-averaged turbulent burning velocity, which can be measured experimentally, was steady. The database was constructed without changing the inflow velocity. Each case in the database consisted of almost 200 sampled data points at 51.68 μs intervals (which was longer than the DNS time step). The conditions described in the database correspond to the boundary between wrinkled flamelets and corrugated flamelets in the turbulent combustion regime diagram [

A turbulent flame surface at each sampling time was identified as the isosurface of the prescribed reaction progress variable, c_{T}, with which the reaction rate in the planar flame takes a maximum for each case in the database, i.e., c_{T} = 0.89 for case Lm and c_{T} = 0.90 for the other cases. The reaction progress variable is defined as:

where T is the temperature, T_{a} is the adiabatic flame temperature, and T_{u} is the temperature of the unburned mixture (300 K). Typical shapes of the turbulent flame surface are shown in

The local burning velocity, u_{c}, was evaluated at the flame surface as follows:

Substituting a normal to the local flame surface, n, for the infinitesimal volume, dV, Equation (2) can be rewritten as

It follows that the local burning velocity can be evaluated by integrating the local reaction rate along the normal

to the local flame surface. In the evaluation of the local burning velocity at a given point at the flame surface, the normal to the flame surface at the point may cross the flame surface at another point. To avoid this problem, we find the minimum distance of the normal between the point on the flame surface and the other point crossing the flame surface, and then obtain half of the distance between these points, as shown in

Using the balance equation of the reaction progress variable,

the flame displacement speed, u_{d}, at each intersection on the flame surface is defined as:

where n is the unit normal vector to the flame surface towards unburned mixture, u is the flow velocity vector.

In order to evaluate the correlation between the local burning velocity and the flame displacement speed using the joint probability density function (joint pdf) appropriately, the both scales of the local burning velocity and the flame displacement speed must be the same. The local burning velocity is based on the consumption rate of the unburned mixture, and the flame displacement speed is based on the flame normal speed of the isosurface of the prescribed reaction progress variable defined as the flame surface. Thus, an equation of continuity:

yields

where ρ is the function of the reaction progress variable. In fact, the flame displacement speed is divided by T/T_{u} for the same scale of both the local burning velocity and the displacement speed. It is convenient, however, to substitute T_{b} for T because T_{b} can be measured easily in experiments when the DNS analysis is compared with experiments.

The local flame stretch rate, κ, which involves the local tangential strain rate, a_{t}, and local flame curvature, ∇×n, are defined as [

Joint pdfs of the local flame stretch rate with both the local burning velocity and flame displacement speed, which were reduced by

Joint pdfs of the local tangential strain rate with both the local burning velocity and flame displacement speed, which were reduced by

From the observations above, the trend of the correlation between the local burning velocity and the flame stretch rate is caused by the facts that the local burning velocity has a very weakly negative correlation with the

tangential strain rate and has a weakly positive correlation with the flame curvature for all density ratios. As for the correlation between the flame displacement speed and the flame stretch rate. For case Lm, the flame displacement speed had almost no correlation with the tangential strain rate and had a negative correlation with the flame curvature, as a result, these trends made the correlation between the flame displacement speed and the flame stretch rate negative. For case Mm and Hm, the flame displacement speed had weakly positive correlations with the tangential strain rate and had negative correlations with the flame curvature; as a result, these correlations lead to the insensitive trends of the flame displacement speed to the flmae stretch rate.

From the discussion above, for the tangential strain rate, the local burning velocity had very weakly negative correlations, while the flame displacement speed had almost positive correlations. For the flame curvature, the local burning velocity had weakly positive correlations, while the flame displacement speed had negative correlations. Therefore, it is considered that the local burning velocity and the flame displacement speed have a negative correlation under the conditions of this study.

The indirect relationship between the local burning velocity and the flame displacement speed was investigated using the DNS data of turbulent premixed flames with different density ratios. It was found that for the local tangential strain rate and the local flame curvature, the local burning velocity and the flame displacement speed had the opposite correlations in each density ratio case. Under the conditions of this study, it is considered that the local burning velocity and the flame displacement speed have a negative correlation.

A_{t}: Turbulent flame area

a_{t}: Local tangential strain rate

c_{T}: Reaction progress variable

Le: Lewis number

l_{t}: Integral length scale

n: Normal to a flame surface

n: Unit normal vector to a flame surface towards unburned mixture

T: Temperature

u: Flow velocity

u: Flow velocity vector

u’: Turbulence intensity

u_{c}: Local burning velocity

u_{d}: Flame displacement speed

ū_{in}: Mean inflow velocity

u_{L}: Laminar burning velocity

u_{T}: Turbulent burning velocity

V: Total volume of the computational domain

Y: Mass fraction

δ_{f}: Flame thickness

κ: Local flame stretch rate

λ: Taylor microscale

ρ: Density

0: Without flame stretch

a: In adiabatic condition

b: In burned product

u: In unburned mixture