Subgrid Spark Ignition

During spark ignition, a high-voltage electrical current creates a plasma arc between electrodes which provides enough energy to ignite a fuel/air mixture and creates a hot gas-ball (flame kernel). The flame propagates outwards from the site of ignition.

There are two main difficulties with spark modeling. The first is the complex physics of the initial arc creation, laminar flame propagation, and the transition to turbulence. The second issue is that the initial spark kernel is very small, and commonly not well resolved on the CFD grid. The Subgrid Spark Ignition model attempts to mitigate these difficulties by solving a 0D equation for the spark radius until the spark is resolved on the CFD mesh.

In Simcenter STAR-CCM+, a modeled ODE equation for the burnt gas mass radius $r_{b}$ is solved. The burnt volume from the 0D solution is then averaged onto a fixed volume sphere (final flame kernel sphere) within the CFD mesh.

The reaction progress within the flame kernel sphere increases uniformly from unburnt to burnt as the 0D spark radius increases to the final flame kernel sphere radius. After which, the flame is resolved on the CFD grid.

Arc Breakdown and Initial Flame Kernel: The arc breakdown creates a small kernel of initial temperature $T_{i}$ and spark radius $r_{b} (0)$ —which is logged in the Output window in metres, for example: Kernel Radius for spark #1 = 0.00058087; The initial spark radius (burnt gas mass radius) $r_{b} (0)$ is given by [802]:
Figure 1. EQUATION_DISPLAY

$r_{b} (0) = \sqrt{\frac{γ - 1}{γ} \frac{E_{b d}}{p d_{g a p} π (1 - \frac{T_{u}}{T_{b} (0)})}}$

(4009); $γ$ is the ratio of specific heats, $p$ is the gas pressure in bar, $d_{g a p}$ is the inter-electrodes distance (1mm), $T_{u}$ is the unburnt gas temperature, $T_{b} (0)$ is the initial temperature, and $E_{b d}$ is the breakdown energy, given by [800], [801]:
Figure 2. EQUATION_DISPLAY

$E_{b d} = \frac{{(4.3 + \frac{136 p}{T_{u}} + \frac{324 p}{T_{u} d_{g a p}})}^{2}}{C^{2} d_{g a p}}$

(4010); in which $C = 245 k V / {(J m m)}^{1 / 2}$ .; The initial temperature $T_{b} (0)$ is given by [802]:
Figure 3. EQUATION_DISPLAY

$T_{b} (0) = T_{u} [\frac{1}{γ} (\frac{T_{b d}}{T_{u}} - 1) + 1]$
(4011); where $T_{b d} = 60000 K$ is the breakdown temperature.

Ignition Flame Kernel Expansion: As the flame propagates outwards from the flame kernel, the burnt gas mass $m_{b}$ evolves according to:
Figure 4. EQUATION_DISPLAY

$\frac{d m_{b}}{d t} = ρ_{u} S_{l} A_{k} Ξ$
(4012); where $ρ_{u}$ is the unburnt gas density, $S_{l}$ is the laminar flame speed, $A_{k}$ is the flame kernel surface area, and $Ξ$ is the turbulence wrinkling factor [803]:; Figure 5. EQUATION_DISPLAY

$Ξ ≔ \frac{S_{t}}{S_{l}} = 1 + A {(\frac{v'}{S_{l}})}^{5 / 6}$

(4013); where $S_{t}$ is the turbulent flame speed, $v'$ is the velocity fluctuation, and $A$ is the turbulent flame speed coefficient.; Therefore, the ODE equation for the burnt radius is:; Figure 6. EQUATION_DISPLAY

$\frac{d r_{b}}{d t} = \frac{ρ_{u}}{ρ_{b}} S_{l} Ξ = \frac{T_{b}}{T_{u}} S_{l} Ξ = (τ + 1) S_{t}$
(4014); Where $T_{b}$ is the temperature in the ignition kernel—which is assumed to be equal to the equilibrium temperature. $τ$ is the expansion ratio:
Figure 7. EQUATION_DISPLAY

$τ ≔ \frac{T_{b}}{T_{u}} - 1$
(4015)

Progress Variable Distribution: The fixed/final radius of the flame kernel sphere $r_{b} (f i n a l)$ is defined as:
Figure 8. EQUATION_DISPLAY

$r_{b} (f i n a l) = \max (5 r_{b} (0), 5 Δ x, \min (10 Δ x, 0.1 L_{t})$

(4016); where $r_{b} (0)$ is the initial spark radius in meters, $Δ x$ is the average linear cell size, and $L_{t}$ is defined as:
Figure 9. EQUATION_DISPLAY

$L_{t} = \frac{k^{3 / 2}}{ε}$

(4017); The progress variable for ignition is:
Figure 10. EQUATION_DISPLAY

$c (r_{b}) = {(\frac{r_{b}}{r_{b} (f i n a l)})}^{3}$
(4018)