Extension of the Spectral Difference Method to Premixed Laminar and Turbulent Combustion

A Spectral Difference (SD) algorithm on tensor-product elements which solves the reacting compressible Navier-Stokes equations is presented. The classical SD algorithm is shown to be unstable for test cases involving a multi-species gas whose thermodynamic properties depend on temperature and species mass fractions. It is observed that the extrapolation of conservative variables from the solution points to the flux points is the origin of the instability produced by the classic SD algorithm. An alternative scheme, where primitive variables are computed at the solution points and then extrapolated to the flux points, is shown to make the SD method stable for such cases. In addition, a new, more stable methodology for computing the diffusive fluxes at an interface is detailed. To allow the simulation of combustion, characteristic and wall boundary conditions for multi-species flows in the SD framework are introduced, and the thickened flame turbulent combustion model is adapted to the SD context. Validation test cases from one-dimensional and two-dimensional laminar flames to a three-dimensional turbulent flame are performed showing excellent agreement with each of the reference solutions. To the authors' knowledge, this is the first time that a 3D turbulent flame is simulated using the SD method. The interest of employing a high-order method, such as the SD, in combustion is demonstrated: high values of the polynomial degree improve the quality of the results which advocates for the use of p-refinement when simulating reacting flows.