Ansys Forte tracks the growth of the ignition kernel by using the Discrete Particle Ignition Kernel Flame (DPIK) model by Fan, Tan, and Reitz [[23] , [94] ]. By assuming a spherical-shaped kernel, the flame front position is marked by Lagrangian particles, and the flame surface density is obtained from the number density of these particles in each computational cell. Assuming the temperature inside the kernel to be uniform, the kernel growth rate is:

(7–1)

where is the kernel radius, is the local unburned gas density, is the gas density inside the kernel region, is the plasma velocity, and is the turbulent flame speed.

One factor not considered in Equation 7–1 is the bulk flow convection effect on the kernel particles. To consider this effect, an alternative formula is implemented:

(7–2)

where is the position vector of flame particle k, is the radial expansion direction pointing from the ignition kernel center to this flame particle, and is the local bulk flow velocity on the unburned side. Equation 7–2 is the default option used to compute the convection of flame particles in Ansys Forte. It is also the default method for considering the convection term of the G-equation in the G-equation stage.

The plasma velocity is given as:

(7–3)

where and are the density and enthalpy of the unburned mixture. and are the density and internal energy of the mixture inside the kernel. is the electrical energy discharge rate, is the electrical energy transfer efficiency due to heat loss to the spark plug. and are user-specified inputs (Energy Release Rate and Energy Transfer Efficiency, respectively) in Ansys Forte. A typical value of is 0.3, as suggested by Heywood [35].

The laminar flame speed in Equation 7–18 was multiplied by a stretch factor, , which accounts for strain and curvature effects, and the modified correlation is used as the turbulent flame speed, , in the kernel stage. is the stretch factor input present in the user interface describing the stretching effect of turbulence on flame speed. A larger value will increase the flame strain cause by turbulence and reduce flame speed. Its effect can be large when flame propagation is weak, for example, under high EGR or lean burn conditions. A typical range is 0.5-1.0. follows the suggested expression of Herweg et al. [34]:

(7–4)

Note that curvature effects are also considered in the combustion model by the last term of Equation 7–15.

The chemistry processes in the kernel-growth stage are treated in the same way as in the G-equation combustion model (described in G-equation Model). Although the transport equation of is not solved here, the field is constructed based on the positions of the kernel particles, thus providing the necessary information for the chemical heat release calculations.

The transition from the kernel model to the turbulent G-equation model is controlled by a comparison of the kernel radius with a critical size that is proportional to the locally averaged turbulence integral length scale, viz.,

(7–5)

where is a model constant and l is the turbulent length scale. is provided as a user input (Kernel Flame to G-equation Switch Constant) in Ansys Forte with typical value 2.0.. If high mesh resolution is used in the spark plug region (there should be at least 4 to 5 layers of cells across the span of the flame structure when the transition happens), a preferred practice is to set to 0.0 and use a constant radius threshold as the transition criterion. For typical automotive spark ignition, 0.1 cm is a good default value for the transition radius threshold.