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The burning rate in a spherically turbulent premixed flame is explored using direct numerical simulations, and a model of ordinary differential equations is proposed. The numerical dataset, from a previous work, is obtained from direct numerical simulations of confined spherical flames in isotropic turbulence over a range of Reynolds numbers. We begin the derivation of the model with an equation for the burning rate for the domain under consideration, and using a thin flame assumption and a two-fluid approach, we find the normalized turbulent burning rate to be controlled by the increase in flame surface area due to turbulent wrinkling, and correction factor which is observed to be consistently less than unity. A Reynolds scaling hypothesis for the flame turbulent wrinkling from a previous work using the same numerical dataset is used to model the term controlling the increase in flame surface area. The correction factor is hypothesized to reflect flame stretch effects, and hence this factor is modeled using Markstein theory applied to global averaged quantities. The ordinary differential equations are rewritten to reflect easily observable quantities such as the chamber pressure and mean flame radius, and then expressed in dimensionless form to assess dependence on various dimensionless parameters. The model predictions are found to be in good agreement with the numerical data within expected variances. Additionally, Markstein theory is found to be sufficient in describing the effects of flame stretch in the turbulent premixed flames under consideration. 

In the flamelet regime of turbulent premixed combustion the enhancement in the burning rates originates primarily from surface wrinkling. In this work we investigate the Reynolds number dependence of burning rates of spherical turbulent premixed methane/air flames in decaying isotropic turbulence with direct numerical simulations. Several simulations are performed by varying the Reynolds number, while keeping the Karlovitz number the same, and the temporal evolution of the flame surface is compared across cases by combining the probability density function of the radial distance of the flame surface from the origin with the surface density function formalism. Because the mean area of the wrinkled flame surface normalized by the area of a sphere with radius equal to the mean flame radius is proportional to the product of the turbulent flame brush thickness and peak surface density within the brush, the temporal evolution of the brush and peak surface density are investigated separately. The brush thickness is shown to scale with the integral scale of the flow, evolving due to decaying velocity fluctuations and stretch. When normalized by the integral scale, the wrinkling scale defined as the inverse of the peak surface density is shown to scale with Reynolds number across simulations and as turbulence decays. As a result, the area ratio and the burning rate are found to increase as ${Re}_{\lambda }^{1.13}$ , in agreement with recent experiments on spherical turbulent premixed flames. We observe that the area ratio does not vary with turbulent intensity when holding the Reynolds number constant.