The reactive Navier–Stokes equations with adaptive mesh refinement and a detailed chemical reactive mechanism (11 species, 27 steps) were adopted to investigate a detonation engine considering the injection and supersonic mixing processes. Flame acceleration and deflagration-to-detonation transition (DDT) in a premixed/inhomogeneous supersonic hydrogen–air mixture with and without transverse jet obstacles were addressed. Results demonstrate the difficulty in undergoing DDT in the premixed/inhomogeneous supersonic mixture within a smooth chamber. By contrast, multiple transverse jets injected into the chamber aid detonation transition by introducing perturbed vortices, shock waves and a suitable blockage ratio. Increasing distance between the leading shock and the flame tip impedes detonation transition due to an insufficient blockage ratio. The extremely perturbed distributions of fuel-lean and fuel-rich mixtures lead to more complicated flame structures. Also, a larger flame thickness appears in the inhomogeneous mixture compared with the premixed mixture, resulting in a lower combustion temperature. The key findings are that the DDT, detonation quenching and reinitiation are generated in the inhomogeneous supersonic mixture, but both DDT mechanisms are ascribed to a strong Mach stem with the Zel'dovich gradient mechanism. Additionally, the obtained results demonstrate that an intensely fuel-lean mixture (equivalence ratio = 0.15) results in a partially decoupled flame front. However, detonation reinitiation and subsequent self-sustained detonation occur when a fierce shock wave propagates through a highly sensitive mixture, even within a smaller and elongated area. Moreover, the inhomogeneous mixture also augments the propagation speed and detonation cell structure instabilities and delays the sonic point resulting from the extending non-equilibrium reaction.

Direct numerical simulations are performed to explore the evolution behaviour of the turbulent/non-turbulent interface (TNTI) in a temporally evolving turbulent plane jet, using the evolution equation for the TNTI surface area. A novel algorithm is used to calculate the surface area of the TNTI and entrainment flux. It is shown that the surface area remains relatively constant, which leads to the mean entrainment velocity being inversely proportional to the square root of time. On average, the effects of the stretching and curvature/viscous terms on the TNTI area roughly counterbalance each other, while the curvature/inviscid term associated with vortex stretching is virtually zero. More specifically, the stretching term contributes to the production of the surface area, while the curvature/viscous term is associated with a destruction in the surface area. The local effect of the curvature/viscous term exhibits high spatial intermittency with small-scale extreme/intense events, whereas the effect of the large-scale stretching term is more continuous. To shed light on the contribution of curvature/viscous term to the evolution of the surface area, we decompose it into three components. The effect of the curvature/normal diffusion term (the curvature/viscous dissipation term) in the bulging regions (the valley regions) mainly contributes to the production of the area. The continuous decrease of the average mean curvature is associated with the production of the bulging regions and the destruction of the valley regions. Finally, although the entrainment velocity is mainly dominated by the normal diffusion effect, all three components related to the viscous effect are indispensable to the production and destruction of the TNTI area. This numerical study contributes to a better understanding of the evolution of the TNTI area.

Prandtl's secondary flows of the second kind generated by laterally varying roughness are studied using the linearised Reynolds-averaged Navier–Stokes approach proposed by Zampino et al. (J. Fluid Mech., vol. 944, 2022, p. A4). The momentum equations are coupled to the Spalart–Allmaras model while the roughness is captured by adapting established strategies for homogeneous roughness to heterogeneous surfaces. Linearisation of the governing equations yields a framework that enables a rapid exploration of the parameter space associated with heterogeneous surfaces, in the limiting case of small spanwise variations of the roughness properties. Channel flow is considered, with longitudinal high- and low-roughness strips arranged symmetrically. By varying the strip width, it is found that linear mechanisms play a dominant role in determining the size and intensity of secondary flows. In this setting, secondary flows may be interpreted as the time-averaged output response of the turbulent mean flow subjected to a steady forcing produced by the wall heterogeneity. In fact, the linear model predicts that secondary flows are most intense when the strip width is about 0.7 times the half-channel height, in excellent agreement with available data. Furthermore, a unified framework to analyse combinations of heterogeneous roughness properties and laterally varying topographies, common in applications, is discussed. Noting that the framework assumes small spanwise variations of the surface properties, two separate secondary-flow-inducing source mechanisms are identified, i.e. the lateral variation of the virtual origin from which the turbulent structure develops and the lateral variation of the streamwise velocity slip, capturing the acceleration/deceleration perceived by the bulk flow over troughs and crests of non-planar topographies.