An anisotropic adaptive, Lagrange–Galerkin numerical method for spray combustion

Abstract We present in this paper a novel numerical method for the simulation of time-dependent, diluted spray combustion problems. Spray combustion is a complex numerical problem owing to the diversity of the phenomena (convection, diffusion, heat transfer, vaporization of droplets and chemical reaction) and the disparity of the scales involved. Our model considers a large number of tiny droplets (liquid phase) surrounded by a gas flow (gas phase), making up a special two-phase system where droplet interaction is neglected on account of the much larger inter-droplet distance compared to the droplet radius typically found in burners. The mathematical formulation of the problem follows a hybrid, Eulerian–Lagrangian approach: the continuous, gas phase is ruled by the conservation equations of mass, momentum, species mass fractions and energy, while the liquid phase is described by a set of ordinary differential equations (ODE) that govern the properties of each droplet along its fluid trajectory. For the resolution of the conservation equations of the gas phase, a Lagrange–Galerkin method in an anisotropic adaptive finite element framework is used. In combination with the aforementioned Lagrange–Galerkin methodology we apply a second order Explicit Runge–Kutta–Chebyshev scheme, also useful to solve the ODE systems of equations for the droplets movement. Explicit Runge–Kutta–Chebyshev preserves numerical stability while also facilitating a decoupled calculation of the unknown variables of the gas and liquid phases, a most beneficial feature from a computationally point of view. Finally, we show several examples to highlight the capabilities of our numerical algorithm in canonical and real-world configurations.