High‐order implicit‐explicit additive Runge–Kutta schemes for numerical combustion with adaptive mesh refinement

Abstract

The objective of this study is to develop and apply efficient solution techniques for numerical modeling of combustion with stiff chemical kinetics in practical combustors. The new technique combines a fourth-order implicit-explicit (IMEX) additive Runge-Kutta scheme (ARK) with adaptive mesh refinement (AMR). The IMEX component treats the stiff reactions implicitly but integrates convection and diffusion explicitly in time, and thus permits the solution to advance with larger time-step sizes than that of explicit time-marching methods alone. The AMR further adds computational efficiency by effectively placing high spatial resolution meshes in regions with strong gradients, such as flame fronts. The novelty of this study is in the integration of a fourth-order IMEX ARK method with AMR for a high-order finite-volume scheme and the application to solving complex reacting flows governed by the compressible Navier-Stokes equations with very stiff chemistry in a practical combustor geometry. The effectiveness and performance of the adaptive ARK4 is assessed for complex reacting flows by examining properties, such as the presence of shock waves, the time-scale changes in response to AMR levels, and the size and stiffness of reaction mechanisms for various fuels such as H 2 , CH 4 , and C 3 H 8 . The new adaptive ARK4 method is verified and validated using a convection-diffusion-reaction problem and shock-driven combustion, respectively. The validated algorithm is then applied to solve the stiff C 3 H 8 -air combustion in a bluff-body combustor. A significant speedup of three orders of magnitude is achieved in comparison to the standard ERK4 method at the given solution accuracy.

Adaptive clipping‐and‐redistribution algorithms for bounded and conservative high‐order interpolations applied to discontinuous and reactive flows

“…After the initial refinement as seen in the subfigure (c), negative species mass fractions, as indicated by magenta, are large, and positive overshoots beyond the coarse solution are shown in yellow‐orange. While mass fractions of species must be always positive to be physically meaningful, Chord does have a small tolerance for negative mass fractions to cope with the numerical instability in the nonlinear solvers employed during the solution process 10 . In the subfigure (D), the magnitude of the negative mass fractions is reduced by an order of magnitude, which is sufficient to have kept the overall algorithm stable and allow the solution to develop properly.…”

“…Overall, reducing nonphysical overshoots at the source (in this case, from interpolation), vastly mitigates the impact of renormalization on the behavior of the dynamical system. Although not the focus of this work, balancing chemistry is an important discussion, and we refer readers to works by Christopher et al 10 and Owen et al 27 for details of the approach Chord uses.…”

“…The first case (10) examines the effects of delaying regridding, by initializing on the coarse grid and first advancing for one coarse time step. Beginning at the second time step, the solution is either refined using the base interpolation algorithm of Chord or using the HO-ACR method.…”

“…Overall, reducing nonphysical overshoots at the source (in this case, from interpolation), vastly mitigates the impact of renormalization on the behavior of the dynamical system. Although not the focus of this work, balancing chemistry is an important discussion, and we refer readers to works by Christopher et al 10 and Owen et al 27 for details of the approach Chord uses. Although not matching the "ideal" solution, the partially added refinement is seen to begin capturing fine scale features not visible in the coarse solution while using significantly less computational resources.…”

“…In summary, we have devised a new high-order bound-preserving interpolation scheme, which improves the robustness of AMR for high-order FVMs when modeling combusting flows. The new high-order adaptive clipping-and-redistribution (HO-ACR) method has been implemented, verified, and validated with our in-house CFD code Chord 3,1,4,5,6,7,8,9,10 . Chord employs fourth-order finite-volume discretizations in space and fourth-order Runge-Kutta methods (both fully explicit and additive implicit-explicit) for time integration.…”