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19.1.2 Spark Model Theory

The spark model in FLUENT is based on the work done by Lipatnikov [ 216] and extended to other combustion models. The derivation of the model can be done in the context of the Zimont premixed combustion model.



Zimont Premixed Flame Model


The transport equation for the mean reaction progress variable, $c$, is given by Equation  19.1-1


 \frac{\partial {\rho c}}{\partial t} + \nabla \cdot ( \rho {... ...\left( D_t \nabla c \right) + \rho_u U_t \vert \nabla c \vert (19.1-1)

where $D_t$ is the turbulent diffusivity, $\rho_u$ is the density of the unburned mixture and $U_t$ is the turbulent flame speed. Since the spark is often very small compared to the grid size of the model and is often laminar in nature, the Zimont model is modified such that


 \frac{\partial {\rho c}}{\partial t} + \nabla \cdot ( \rho {... ... D_{tt}) \nabla c \right) + \rho_u U_{t} \vert \nabla c \vert (19.1-2)

where $\kappa$ is the laminar thermal diffusivity and the effective diffusivity $D_{tt}$ is given by


 D_{tt} = \left\{ \begin{array}{ll} D_t \left( 1-\exp\left... ...0$\ } \\ D_t & \mbox{if $t_{td} < 0$\ } \end{array} \right. (19.1-3)

where is $t_{td} = t-t_{ig}$ and $t_{ig}$ denotes the time at which the spark is initiated. Additionally, $\tau'$ is an effective diffusion time, set by the user.

Only turbulent scales that are smaller than the spark radius can contribute to turbulent spark diffusion, so the expression for the effective turbulent diffusivity, $D_{tt}$, is ramped up as the spark grows. This creates higher temperatures at the location of the spark and can cause convergence difficulties. In addition to convergence difficulties, small changes in the diffusion time can change the result significantly. Because of these issues, the diffusion time can be controlled by the user, and has a default value of 1e-5 seconds.



Other Combustion Models


The spark model is compatible with all combustion models in FLUENT. However, the premixed and partially premixed models differ in that the progress variable inside the spark region is set equal to 1, a burned state, for the duration of the spark event. Other combustion models have the energy input into the cell. If the temperature exceeds 2500 K or the spark duration is exceeded, no energy from the spark model will be added to the spark cells.

The spark model can be used in models other than the premixed and partially premixed combustion models, however, the user must balance energy input and diffusivity to produce a high enough temperature to initiate combustion, which can be a nontrivial undertaking. The model's use has been extended to be compatible with the other models, however, in some cases it simply creates a high temperature region and does not guarantee the initiation of combustion.


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