Numerical Simulation of Unsteady Combustion Using the Transport Element Method

The transport element method is described and implemented in the simulation of the non-premixed reacting shear layer. The method, a natural extention of the vortex element method, resolves the low Mach number variable density ow and the exothermic reacting eld. The eeect of combustion on the ow is accommodated by incorporating a volumetric expansion velocity component and by modifying the integration of the vorticity equation to include expansion-related and baroclinic terms. The reacting eld equations describing a single step, irreversible, chemical reaction, are simpliied by the introduction of Schvab-Zeldovich (SZ) conserved scalars whose transport is suucient to compute the evolution of combustion in the case of innnite reaction rate. In the case of nite rate chemistry the evolution of one primitive scalar, the product mass-fraction, is also computed. The vorticity, conserved scalar gradient and product mass fraction are discretized amongst elds of transport elements. Their time evolution is implemented by advecting the elements at the local velocity while simultaneously integrating their transport equations along particle trajectories. The integration of the vorticity and the conserved scalar gradient equations is simpliied using ideas from kinematics. A novel core expansion scheme that avoids the problems associated with the conventional implementation is used to simulate diiusion. Field quantities are obtained using convolutions over the elements. Results indicate that the method is able to accurately reproduce the essential features of the ow. Convergence of the solution in time is approximately linear. Moreover, the nite reaction rate solution at low Karlovitz number bear strong similarities to that of the innnite reaction rate model. This similarity is exploited in validating the part of the numerical methodology related to the integration of the product mass-fraction equation.