Typically, the initial spark size is small relative to the cell size, and the spark is under-resolved on the CFD mesh. Modeling the spark by burning a few cells around the spark location shows strong sensitivity to the grid and time-step size, and flame speed and flame brush diffusion can be erroneous due to the insufficient space and time resolution. In addition, for cases where the initial spark is smaller than the cell size, ignition proceeds too quickly.

To mitigate this sensitivity, Ansys Fluent solves a sub-grid equation for the spark evolution. The spark shape is assumed to be perfectly spherical, with an infinitely thin flame front. The spark radius, , grows in time, , according to the ODE,

(10–1)

where is the density of the unburned fluid ahead of the flame front, is the density of the burnt fluid behind of the flame, and is the turbulent flame speed.

The sub-grid spark model is transferred to the CFD grid through a representative volume of CFD cells. This volume is spherical and has a fixed diameter computed as the local turbulent length scale at the spark location. However, in order to ensure that this representative volume is not too large (relative to the size of the combustor) or too small (relative to the cell size), the radius of the representative sphere, , is calculated as,

(10–2)

where is the user-specified initial spark radius, is the cell length scale, and is the turbulent length scale.

Alternatively may be specified as a fixed value via the spark-model text interface.

is taken to define the representative volume because, once the spark diameter reaches this size, the flame speed is affected by all the turbulent scales present. The spark radius will continue to increase until the length scale is reached, even if the simulation time exceeds the user-specified duration. Note, that the specified duration is only used to calculate the rate of spark energy input and the time when this input ceases. At this condition Ansys Fluent automatically switches the spark flame speed model off, so that the flame speed is modeled using the flame speed model that you have selected in the Species dialog box, throughout the domain.

For the G-Equation model, reaction progress in the spark is calculated in the usual way for this model as given by Equation 8–76.

The temperature and species composition (denoted by ) within the representative spherical volume at every time-step are calculated as,

(10–3)

where denotes the equilibrium burnt composition, and is the unburned composition. These compositions are fixed in time and uniform in space.

Since the thermo-chemical state behind the spark flame front is instantaneously equilibrated as the spark propagates, spark energy is not required to ignite the mixture. By default, the spark energy is set to zero for all combustion models, and the burnt temperature is the equilibrium temperature. However, spark energy can be set to a positive value in the user interface, in which case the temperature behind the spark will be higher than the equilibrium temperature.

Ansys Fluent offers the following models for the turbulent flame speed, in Equation 10–1:

Turbulent Curvature

In the Turbulent Curvature model, the turbulent flame speed is calculated as,

(10–4)

where is the laminar flame speed, is the current spark radius, is the laminar diffusivity, and is the turbulent diffusivity. and are evaluated at the spark location, and is the turbulent flame speed evaluated at the turbulent length scale of the spark radius. Since turbulent scales larger than the spark radius convect the spark but do not increase its area and flame speed, only turbulent length scales up to the spark radius can affect the turbulent flame speed of the spark. For the premixed and partially-premixed models, the spark flame speed model is the same flame speed model as selected for the main combustion. For Species Transport cases, by default the Zimont flame speed model is used. and are added as additional material inputs in the interface for the species transport models.

Note that the effect of the flame curvature is to decrease both the laminar and turbulent flames speeds. Since the initial spark radius is a user-input, decreasing slows the spark propagation and increases the burning time.

Turbulent Length

In the Turbulent Length model, the turbulent flame speed is calculated as,

(10–5)

That is, the Turbulent Length model ignores the effects of flame curvature on the flame speed.

Herweg-Maly

The turbulent flame speed is calculated using the model proposed by Herweg and Maly [243]:

(10–6)

where is a function for effect of strain on the laminar burning velocity calculated as,

If the Blint modifier is being used

where
	= 2.0
	= 0.7
	is the unburned density
	is the burned density

Laminar

In the Laminar model, the turbulent flame speed in Equation 10–1 is modeled as the laminar flame speed, . Since the Laminar flame speed can be modeled with a user-defined function (UDF), the Laminar option can be used to define your own function for the turbulent flame speed.