Error analysis of a fractional-step method for reactive fluid flows with Arrhenius activation energy

Propagation problems of reaction fronts in viscous fluids are crucial in many industrial and chemical engineering processes. The interactions between the reaction properties and the fluid dynamics yield a complex and nonlinear model of Navier–Stokes equations for the flow with a strong coupling with two reaction–advection–diffusion equations for the temperature and the degree of conversion. To alleviate difficulties related to the numerical approximation of such systems, we propose a fractional-step method to split the problem into several substeps, and based on a viscosity-splitting approach that separates the convective terms from the diffusion terms during the time integration. A first-order scheme is employed for the time integration of each substep of the proposed fractional-step method. The proposed method also preserves the full original boundary conditions for the velocity which eliminates any potential inconsistencies on the pressure, and it allows for nonhomogeneous Dirichlet and Neumann boundary conditions for the temperature and degree of conversion that are physically more appealing. In the present work, we perform an error analysis and provide error estimates for all involved solutions in their relevant norms. A rigorous stability analysis is also carried out in this study and the proposed method is demonstrated to be consistent and stable with no restrictions on the time step. Numerical results obtained for a problem with known analytical solutions are presented to verify the theoretical analysis and to assess the performance of the proposed method. The method is also implemented for solving a two-dimensional flame-like propagation problem in viscous fluids. The obtained computational results for both examples support the theoretical expectations for a stable and accurate numerical solver for reactive fluids with Arrhenius activation energy.