a b s t r a c tFlows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor-Green vortex, Shu-Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

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In diesel engines, combustion is initiated by a two-staged autoignition that includes both low-and high-temperature chemistry. The location and timing of both stages of autoignition are important parameters that influence the development and stabilisation of the flame. In this study, a twodimensional direct numerical simulation (DNS) is conducted to provide a fully resolved description of ignition at diesel engine-relevant conditions. The DNS is performed at a pressure of 40 atmospheres and at an ambient temperature of 900 K using dimethyl ether (DME) as the fuel, with a 30 species reduced chemical mechanism. At these conditions, similar to diesel fuel, DME exhibits two-stage ignition. The focus of this study is on the behaviour of the low-temperature chemistry (LTC) and the way in which it influences the high-temperature ignition. The results show that the LTC develops as a "spotty" first-stage autoignition in lean regions which transitions to a diffusively supported cool-flame and then propagates up the local mixture fraction gradient towards richer regions. The cool-flame speed is much faster than can be attributed to spatial gradients in firststage ignition delay time in homogeneous reactors. The cool-flame causes a shortening of the second-stage ignition delay times compared to a homogeneous reactor and the shortening becomes

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