Numerical Stability and Reduced Order Chemistry Modeling in Detonation Simulations

The coupling between shocks and chemistry in detonations poses a challenge for simulations. In this thesis, a simulation framework is developed to address key components of detonation modeling: numerical stability of shocks and discontinuities, and computational efficiency in chemistry modeling. To ensure numerical stability in the vicinity of shocks, a variety of methods have been used, including shock-capturing schemes such as weighted essentially non-oscillatory (WENO) schemes, as well as the addition of artificial diffusivities to the governing equations. In this work, all necessary viscous/diffusion terms are derived from first principles, and the performance of these analytical terms is demonstrated within a centered differencing framework. The physical Euler equations are spatially-filtered with a Gaussian-like filter. Sub-filter scale (SFS) terms arise in the momentum and energy equations. Analytical closure is provided for each of them by leveraging the jump conditions for a shock. No SFS terms are present in the continuity or species equations. For contact discontinuities, the analytical SFS terms are identically zero. However, numerically, the transport of a contact discontinuity may result in artificial oscillations due to dispersive errors. To treat contact discontinuities, a WENO-like correction term is applied to the enthalpy transport. Implemented within a centered difference code, this filtered framework performs well for a range of shock-dominated flows without introducing excessive diffusion. In addition to providing new insight into the placement and form of required diffusion terms in the governing equations, this framework is general and may be used with any numerical scheme. Chemistry modeling in detonations typically relies on two broad approaches: simplified models with one- or two-step chemistry, and detailed chemistry. These approaches require choosing between computational efficiency or physical accuracy. In detailed chemistry simulations, there are physical constraints that must be met when transporting species mass fractions; nonlinear transport schemes such as WENO do not satisfy these constraints automatically. A new method is presented to ensure that the sum of mass fractions equals 1, without penalizing inert species. The approach is better able to capture the physical instability expected for detonations. To reduce the cost of chemistry while maintaining accurate physics, tabulated chemistry has been used extensively for flames/deflagrations in the low Mach number framework. In the simplest tabulated chemistry model for premixed flames, a progress variable, describing the progress of reactions in the system, is transported in the simulation. This progress variable is then used to look up all other species, transport properties, and thermodynamic variables from a pre-computed table. Unfortunately, there is no existing tabulation approach designed specifically for detonations. As such, this work extends the tabulated chemistry method to detonations. To describe the enthalpy and specific heat capacity, the temperature is selected as a second table coordinate. The two table coordinates are able to capture virtually all variations in the progress variable source term. The Zel'dovich-von Neumann-Döring (ZND) model is found to be the most appropriate one-dimensional problem for generation of the table. The ZND tabulation approach is validated for both one-dimensional stable and pulsating and two-dimensional regular and irregular detonations in various hydrogen-oxygen mixtures. The tabulated chemistry simulations are able to reproduce the detailed chemistry results in terms of propagation speed, cellular structures, and source term statistics at a reduced computational cost, demonstrating the benefits of this approach for predictive modeling of detonations.