Axisymmetric vortex method for low-Mach number, diffusion-controlled combustion

A grid-free, Lagrangian method for the accurate simulation of low-Mach number, variable-density, diffusion-controlled reacting flow is presented. A fast-chemistry model in which the conversion rate of reactants to products is limited by the local mixing rate is assumed in order to reduce the combustion problem to the solution of a convection–diffusion–generation equation with volumetric expansion and vorticity generation at the reaction fronts. The solutions of the continuity and vorticity equations, and the equations governing the transport of species and energy, are obtained using a formulation in which particles transport conserved quantities by convection and diffusion. The dynamic impact of exothermic combustion is captured through accurate integration of source terms in the vorticity transport equations at the location of the particles, and the extra velocity field associated with volumetric expansion at low Mach number computed to enforced mass conservation. The formulation is obtained for an axisymmetric geometry where the impact of radial symmetry is imposed partly through the introduction of an “adaptive” core function, used in the discretization of the vorticity field, whose shape depend on the location of the computational element with respect to the axis of symmetry, and partly through the implementation of the Green’s functions of the Poisson’s equation and the diffusion equation. The core function is used to compute the velocity field, and in the simulation of diffusion, where the formulation allows different computational elements to have different core sizes, using an extension of the vorticity redistribution approach. A fractional-step method is applied to decouple the convection and diffusion operators in the equations in the nonreacting flow. In the reacting flow simulations, operator splitting is applied to decouple the convection, generation and diffusion operators, while a second-order predictor–corrector integration is used in the convection and generation steps. Numerical tests are used to examine the convergence rates of the algorithm using a number of generic examples.