Closure modeling for the conditional pressure gradient in turbulent premixed combustion

In turbulent premixed combustion, turbulence is affected by both the influence of combustion heat release and turbulent shear effects. At low Karlovitz number, heat release effects are more dominant than turbulent shear effects, but, at high Karlovitz number, turbulent shear effects dominate the turbulence dynamics. Closure models for the velocity and scalar fields that only account for one of those effects are incapable of generally describing combustion-turbulence interactions at any finite Karlovitz number. A manifold-based turbulence model that solves the conditional mean momentum transport equation conditioned on a progress variable has the potential to provide such a general description, but closure models for the equation are required. Among the unclosed terms, the conditional mean pressure gradient is significant at both low and high Karlovitz numbers but has fundamentally different behaviors in each regime. In this work, a general closure model is developed to capture the conditional pressure gradient in both Karlovitz number regimes by considering heat release and turbulent shear effects. The deviation of the conditional pressure gradient from the unconditional pressure gradient is modeled with two components. One represents the pressure decrease across the flame due to combustion heat release, and the other represents the hydrodynamic pressure fluctuations due to turbulent shear. The heat release component depends on the heat release rate and the flame-normal vector, and the turbulent shear component depends on the conditional Reynolds stresses and the gradient of the progress variable probability density function. The model is validated against Direct Numerical Simulations of spatially-evolving turbulent premixed hydrogen/air planar jet flames at low and high Karlovitz numbers. The proposed model well captures the conditional pressure gradient at both Karlovitz numbers and shows a natural transition of its dominant term: the heat release component at low Karlovitz number and the turbulent shear component at high Karlovitz number.