BackgroundComputational combustion and thermoacoustics have become key issues in the design of motors and turbines. The goal of our research has been to develop, analyze and apply accurate numerical methods for the solution of the compressible Euler and Navier-Stokes equations with source terms in combustion and thermoacoustics. Since the source term is considered in the flux discretization, our new approach is much more accurate than conventional Riemann solvers. For more accurate time integration of stiff combustion problems, high order implicit-explicit Runge-Kutta methods have been developed.
Rankine-Hugoniot-Riemann SolverSound waves in combustion are amplified according to Rayleigh's criterion, if pressure and heat release fluctuate in phase. For the calculation of acoustic instabilities in flames, my former Ph.D. candidate Patrick Jenny developed a semi-implicit finite volume method for 1D and 2D reacting flows. The source terms including the diffusive fluxes and the cross-flow fluxes are taken into account in the inviscid flux discretization by the new flux discretization called Rankine-Hugoniot-Riemann solver. Thereby, large errors in pressure and mass flow of conventional Riemann solvers can be avoided. The superior accuracy of the new approach was tested for 1D and 2D premixed laminar flames both for steady combustion and for thermoacoustics.
Convergence AccelerationIn low Mach number flow, the Mach number can be considerably increased by artificially lowering the pressure and thereby decreasing the speed of sound. Since the time step of an explicit method in slow flow is proportional to the Mach number, the time step can be chosen much larger for the modified flow with higher Mach number and the steady state can be reached much faster. With that Mach transformation, Patrick Jenny could considerably reduce the long computing times for premixed laminar flames. If the algorithm is not implemented as a predictor-corrector method but in one step, I could show that the present convergence acceleration for steady state constitutes an efficient preconditioning of the time derivatives in low Mach number flow.
Conservative Methods for Gas MixturesPatrick Jenny, Hans Thomann and I developed, analyzed and tested a physically motivated simple correction of conservative Euler solvers for gas mixtures. With the correction algorithm, considerable errors in pressure and velocity near moving contact discontinuities can be avoided. In a conventional method, these errors are caused by numerical mixing, if the temperature or the ratio of specific heats differ at the moving contact discontinuity.
Implicit-Explicit Runge-Kutta MethodsErik Lindblad and I developed, analyzed and tested second and fourth order implicit-explicit Runge-Kutta methods for separable stiff ordinary differential equations, where the non-stiff part is treated explicitly and the stiff part implicitly. In collaboration with Damir M. Valiev, Jarmo Rantakokko, Per Lötstedt, Michael A. Liberman and me, Erik Lindblad started to apply these methods to the time integration of the compressible Navier-Stokes equations for stiff combustion problems.
Department of Energy and Process Engineering,
NTNU, Kolbjørn Hejes vei 2, NO-7491 Trondheim, Norway