Eddy Break-Up

The Eddy Break-Up model is intended for modeling reacting flow with fast chemistry where the reaction rate is determined by the rate at which turbulence can mix the reactants and heat.

Reaction System

EBU models characterize the reacting flow system by a specified number of species and chemical reactions. The standard EBU model assumes that the reaction rate is determined solely by the turbulent mixing time scale. However, the hybrid kinetics and combined time-scale versions of the EBU model assume that the rate of reaction is also affected by the kinetic reaction rate from finite rate kinetics.

Source Term Definition

The kinetic reaction source term for each species is obtained from multiplying the rate from Eqn. (3362) by the molecular weight ${M_{w}}_{i}$ . The species source term $S_{i}$ is modeled as a function of chemical reaction rates and characteristic time scales. The specific formulation of the source term and time scales depend on the EBU model selection.

Standard EBU: The Standard EBU model assumes that when species are mixed, they are immediately burnt. The chemical source term is calculated from the mixing time scale.; The reaction rate is modeled through an expression that accounts for the turbulent micromixing process. For a reaction of the form Eqn. (3353), the rate of fuel depletion is assumed to be:
Figure 1. EQUATION_DISPLAY

$r_{F} = - \frac{ρ}{{M_{w}}_{F}} (\frac{1}{τ_{t u r b}}) A_{e b u} min [Y_{F}, \frac{Y_{O}}{s_{O}}] moles / (m^{3} s)$

(3454)
or, when the EBU coefficients option, Use Products for Rate, is activated:
Figure 2. EQUATION_DISPLAY

$r_{F} = r_{F, m i x} = - \frac{ρ}{{M_{w}}_{F}} (\frac{1}{τ_{t u r b}}) A_{e b u} min [Y_{F}, \frac{Y_{O}}{s_{O}}, B_{e b u} (\frac{Y_{P 1}}{s_{P 1}} + \frac{Y_{P 2}}{s_{P 2}} + \dots + \frac{Y_{P j}}{s_{P j}})]$

(3455)
In the above equation,
Figure 3. EQUATION_DISPLAY

$s_{O} \equiv υ_{O} m_{O} / υ_{F} {M_{w}}_{F}$

(3456)

Figure 4. EQUATION_DISPLAY

$s_{P i} \equiv | υ_{P i} | m_{P i} / υ_{F} {M_{w}}_{F}$

(3457)
when using the K-Epsilon model:
Figure 5. EQUATION_DISPLAY

$τ_{t u r b} = \frac{k}{ε}$

(3458)
where $k$ is the turbulent kinetic energy, and $ε$ is its dissipation rate. ${M_{w}}_{F}$ is the molecular weight of fuel in a cell. The $m i n$ operator on the right-hand side of Eqn. (3454) indicates that the concentration of the limiting reactant is used to determine a mass fraction scale when calculating the reactant consumption rate. Eqn. (3454) essentially states that the integrated micromixing rate is proportional to the mean (macroscopic) concentration of the limiting reactant, divided by the time scale of the large eddies. Eqn. (3455) is an optional modification to Eqn. (3454) for premixed flames in which fuel and oxidizer are already mixed at the molecular scale. The reaction rate is determined by the rate at which the products are mixed with reactants.
Hybrid EBU: The Hybrid EBU model assumes that the minimum value of mixing and chemical kinetic time scale is rate-limiting and calculates the source term using Eqn. (3362) multiplied by the molecular weight ${M_{w}}_{i}$ of species $i$ .; This model is expressed as:
Figure 6. EQUATION_DISPLAY

$r_{i} = - \min (| r_{i, k i n} (ρ, Y_{1}, Y_{2,} Y_{3}, ..., Y_{N}, T) |, | r_{i, m i x} |)$
(3459)
Combined Time-Scale EBU: The Combined Time-Scale EBU model combines the mixing time scale and chemical time scale to calculate the source term using Eqn. (3412). The turbulent mixing time scale $τ_{m i x}$ used in the standard EBU model, decreases with decreasing distance from solid surfaces. This leads to over-prediction of reaction rates in near-wall regions. To alleviate this problem, the mixing time scale is augmented by a time scale that is derived from the chemical reaction rate from finite-rate kinetics, Eqn. (3362), defined as:
Figure 7. EQUATION_DISPLAY

$τ_{k i n} = \frac{ρ Y_{F}}{m_{F}} (\frac{1}{| r_{i, k i n} (\bar{ρ}, {\bar{Y}}_{1}, {\bar{Y}}_{2,} {\bar{Y}}_{3}, ..., {\bar{Y}}_{N}, \bar{T}) |})$
(3460)
As a result, the time scale, $τ$ , that is used to calculate reaction rate (Eqn. (3454) or Eqn. (3455)) is calculated by:
Figure 8. EQUATION_DISPLAY

$τ = τ_{m i x} + τ_{k i n}$
(3461)
Therefore, the reaction rate that is predicted by the combined time-scale variant is smaller than that from the standard EBU model.
Kinetics Only EBU: The Kinetics Only EBU model assumes that the reaction rate is dictated by finite-rate chemical kinetics. Therefore, the source term is calculated by Eqn. (3412) using the reaction rate that is calculated by Eqn. (3362).

Solution Procedure

The Eddy Break up model uses the source term in Eqn. (3412) directly in the transport equations, with no subcycling. Therefore, this model is suitable for a mechanism with a limited number of reactions that are not stiff.

For transient (unsteady) simulations, the chemistry is calculated over the duration of the time-step. For steady state simulations, the chemistry is calculated over the cell residence time (chemistry time step) $τ_{r e s}$ in Eqn. (3328).

If source term limiting is activated, the reaction rate

r_{F}

in Eqn. (3454) or Eqn. (3455) is limited to make sure that the amount of reactants in a cell is less than or equal to the amount of reactants that were in the cell before the reaction started:

Figure 9. EQUATION_DISPLAY

r_{F, \lim} = \min (r_{F}, \frac{m Y_{F}}{τ_{r e s}})

(3462)

Where $m$ is the total mass in the cell.

Limiting source terms prevents the reaction rate from driving the fuel mass fraction $Y_{F}$ and oxidizer mass fraction $Y_{O}$ to a negative value, which is not physically possible.