Richtmyer–Meshkov instability (RMI) at a light–heavy single-mode interface over a wide range of post-shock Atwood numbers $A_1$ is studied systematically through elaborate experiments. The interface generation and $A_1$ variation are achieved by the soap-film technology and gas-layer scheme, respectively. Qualitatively, the nonlinear interface evolution features, including spike, bubble and roll-up structures, are more significant in RMI with higher $A_1$. Quantitatively, both the impulsive model and an analytical linear model perform well in predicting the linear growth rate under a wide range of $A_1$ conditions. For the weakly nonlinear stage, the significant spike acceleration occurring when $A_1$ is high, which is observed experimentally for the first time, results in the evolution law of RMI with high $A_1$ being different from the counterpart with low or intermediate $A_1$. None of the considered nonlinear models is found to be applicable for RMI under all $A_1$ conditions, and the predictive capabilities of these models are analysed and summarized. Based on the present experimental results, an empirical nonlinear model is proposed for RMI over a wide range of $A_1$. Further, modal analysis shows that in RMI with high (low or intermediate) $A_1$, high-order harmonics evolve rapidly (slowly) and cannot (can) be ignored. Accordingly, for RMI with high (low or intermediate) $A_1$, the modal model proposed by Zhang & Sohn (Phys. Fluids, vol. 9, 1997, pp. 1106–1124) is less (more) accurate than the one proposed by Vandenboomgaerde et al. (Phys. Fluids, vol. 14, 2002, pp. 1111–1122), since the former ignores perturbation solutions higher than fourth order (the latter retains only terms with the highest power in time).

We present the general analytical solution of the Riemann problem (decay of a jump discontinuity) for non-convex relativistic hydrodynamics. In convex dynamics, an elementary nonlinear wave, i.e. a rarefaction or a shock, originates at the discontinuity and travels towards one of the initial states. Between the left and right waves, an equilibrium state appears represented by a contact discontinuity. The exact solution to the Riemann problem in convex relativistic hydrodynamics was first addressed by Martí & Müller (J. Fluid Mech., vol. 258, 1994, pp. 317–333). In non-convex dynamics, two sequences of elementary nonlinear waves move towards the left and right initial states. Solving the Riemann problem involves determining the types of wave developing and the equilibrium state where they coincide. The procedure consists of constructing the wave curves associated with the nonlinear waves in the pressure–velocity phase space, where the intersection of the wave curves indicates the equilibrium state. We describe the relation between the wave curves, the explicit formulas for their calculation, and the outline of the process for a correct derivation and representation of the waves in the spatial domain. We present examples of the exact solution of a Riemann problem that illustrate the complex phenomena of non-convex dynamics by using the phenomenological non-convex equation of state proposed by Ibáñez et al. (Mon. Not. R. Astron. Soc., vol. 476, 2017, pp. 1100–1110).

Using direct numerical simulations, we investigate the heat transport in bulk and boundary flows separately in rotating Rayleigh–Bénard convection in cylindrical cells. In the bulk we observe a steep scaling relationship between the Nusselt number ($Nu$) and the Rayleigh number ($Ra$), which is consistent with the results from simulations using periodic boundary conditions. For the boundary flow, we observe a power law $Nu_{BF}\sim (Ra/Ra_w)^1$ at the leading order, where $Nu_{BF}$ is the local Nusselt number of the boundary flow and $Ra_w$ is the onset Rayleigh number of the wall mode. We develop a model using the boundary layer marginal stability theory to explain this power law, and further show that a more precise description of the data can be obtained if a higher-order correction is introduced. A striking finding of our study is the observation of a sharp transition in flow state, manifested by a sudden drop in $Nu_{BF}$ with a corresponding collapse of the boundary flow coherency. After the transition, the boundary flow breaks into vortices, leading to a reduction in flow coherency and heat transport efficiency. As the physical properties of the vortices should not depend on the aspect ratio, $Nu_{BF}$ for all aspect ratios collapse together after the transition. Moreover, the centrifugal force helps trigger the breakdown of the coherent boundary flow state. For this reason, $Nu_{BF}$ for the cases with non-zero centrifugal force collapse together. We further develop a method that enables us to separate the contributions from the bulk and boundary flows in the global Nusselt number using only the global $Nu$ and it does not require the centrifugal force to be absent.