A numerical method for the multidimensional hydrodynamic model of flames propagating in closed vessels

Abstract

A numerical methodology has been developed to simulate the evolution of multidimensional premixed flames in closed vessels. The mathematical formulation is based on a hydrodynamic theory, wherein the flame is a surface of discontinuity that separates burned products from unburned gas. The flame front propagates into a mixture which is continuously compressed, causing a rise in pressure and temperature that affects the flow field and modifies the local burning rate. The latter depends on the voluminal stretch rate, which combines the effects of local variations in flame front curvature, flame thickness and hydrodynamic strain, and on the rate of the overall pressure rise. The burning rate is modulated by an effective pressure-dependent Markstein length, that exhibits a dependence on the extent of heat release and diffusion properties of the reactants, and decreases continuously as the pressure rises and the flame becomes thinner. A hybrid embedded-manifold/Navier–Stokes methodology is proposed to numerically solve this free-boundary hydrodynamic problem. It consists of two modules; the first involves solving the fluid dynamic equations and the second employs a level-set approach to advance the flame front in time. The two modules are coupled through a mass conservation constraint and a high-order geometrical closest point method used to evaluate fluid dynamical and geometrical quantities defined strictly on the flame surface. An immersed boundary method is utilized to implement boundary conditions at the walls of vessels of arbitrary shape. The numerical approach is validated against exact analytical solutions of planar and cylindrical flames, and is shown to describe highly corrugated flame conformations resulting from intrinsic combustion instabilities, in rectangular and circular domains. The methodology is adept at developing a comprehensive understanding of the effects of instabilities and low-intensity turbulence on the propagation of premixed flames in closed vessels.

Novelty and significance statement

This paper is based on a novel hydrodynamic theory that describes the propagation of premixed flames in closed vessels of arbitrary shapes, wherein the flame is treated as a surface of discontinuity. The nontrivial free-boundary problem is solved by an advanced numerical methodology that consists of two coupled modules: a variable-density Navier–Stokes solver for determining the flow field, and a level-set approach for tracking the evolution of the flame. It is the first such methodology that accounts for the continuous rise in pressure and temperature due to adiabatic compression, and for the dependence of the flame speed on a pressure-dependent Markstein length. The methodology is sufficiently robust and adept at handling multidimensional flames. It has been validated by comparing numerical results of planar and cylindrical flames against exact analytical solutions and has been shown to describe the nonlinear development of the Darrieus–Landau instability, highlighting the effects of pressure rise.