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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.

Electrokinetic flow instabilities occur under high electric fields in the presence of electrical conductivity gradients. Such instabilities are a key factor limiting the robust performance of complex electrokinetic bio-analytical systems, but can also be exploited for rapid mixing and flow control for microscale devices. This paper reports a representative flow instability phenomenon studied using a microfluidic T-junction with a cross-section of 11 µm by 155 µm. In this system, aqueous electrolytes of 10:1 conductivity ratio were electrokinetically driven into a common mixing channel by a steady electric field. Convectively unstable waves were observed with a nominal threshold field of 0.5 kV cm −1 , and upstream propagating waves were observed at 1.5 kV cm −1 . A physical model has been developed for this instability which captures the coupling between electric and flow fields. A linear stability analysis was performed on the governing equations in the thin-layer limit, and Briggs-Bers criteria were applied to select physically unstable modes and determine the nature of instability. The model predicts both qualitative trends and quantitative features that agree very well with experimental data, and shows that conductivity gradients and their associated bulk charge accumulation are crucial for such instabilities. Comparison between theory and experiments suggests the convective role of electro-osmotic flow. Scaling analysis and numerical results show that the instability is governed by two key controlling parameters: the ratio of dynamic to dissipative forces which governs the onset of instability, and the ratio of electroviscous to electro-osmotic velocities which governs the convective versus absolute nature of instability.

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