The closure developed by Zimont et al. [38] [40] [41] is used for the turbulent burning velocity:

(7–60)

The leading factor, , is a modeling coefficient that has the universal value (default), with the exception of H2/Air flames where is recommended [41].

The stretching factor, , accounts for reduction of the flame velocity due to large strain rate (large dissipation rate of the turbulent kinetic energy). This effect is modeled in terms of the probability for turbulence eddy dissipation, , being larger than a critical value . For , flamelet extinction takes place, while for , the stretching effect is ignored completely. Assuming a lognormal distribution for , the stretching factor is given by:

(7–61)

where denotes the complimentary error function and is the standard deviation of the distribution of , with being an empirical model coefficient (default ).

is the thermal diffusivity of the unburned mixture. The turbulent flame speed closure model is completed with the following models for integral velocity fluctuations level:

(7–62)

integral turbulent length scale:

(7–63)

and Kolmogorov length scale.

(7–64)

The critical dissipation rate, , is computed from a specified critical velocity gradient, , and the kinematic viscosity of the fluid, , according to:

(7–65)

For steady laminar flow the critical velocity gradient for quenching, , can be obtained numerically. However, for turbulent flows, the critical value must be larger than in laminar cases because the smallest turbulent eddies, which are responsible for the largest strain rates, do not persist long enough to quench a flame front locally. Furthermore, different model problems may result in significant variation of the critical values obtained. For these reasons, the quenching critical velocity gradient has to be tuned for industrial simulations. In fact, it is the only significant parameter for tuning the model.

Theory or numerical modeling can suggest a range of physically plausible values of . For example, the inverse of the chemical time scale of the reaction, , scaled by a factor in the range 0.1 to 1.0 is a reasonable starting point. For gas turbine combustion chambers (burning a lean methane/air mixture) values in the range:

to

depending on the configuration, have been used successfully [40] [41] It should be noted that these recommended values are for atmospheric temperature and pressure.