The propagation of sound waves in high-temperature and plasma flows is subject to attenuation phenomena that alter both the wave amplitude and speed. This finite change in acoustic wave properties causes ambiguity in the definition of sound speed travelling through a chemically reactive medium. This paper proposes a novel computational study to address such a dependence of sound-wave propagation on non-equilibrium mechanisms. The methodology presented shows that the equations governing the space and time evolution of a small disturbance around an equilibrium state can be formulated as a generalised eigenvalue problem. The solution to this problem defines the wave structure of the flow and provides a rigorous definition of the speed of sound for a non-equilibrium flow along with its absorption coefficient. The method is applied to a two-temperature plasma evolving downstream of a shock, modelled using Park’s two-temperature model with 11 species for air. The numerical absorption coefficient at low temperatures shows excellent agreement with classical theory. At high temperatures, the model is validated for nitrogen and argon across wide temperature ranges with experimental values, showing that classical absorption theory is insufficient to characterise high-temperature flows due to the effect of finite-rate chemistry and vibrational relaxation. The speed of sound is verified in the frozen and equilibrium limits and its non-equilibrium profile is presented with and without viscous effects. It is furthermore shown that the variation in the speed of sound is driven by the dominating reaction mechanisms that the flow is subject to at different thermodynamic conditions.

Developing a consistent near-wall turbulence model remains an unsolved problem. The machine learning method has the potential to become the workhorse for turbulence modelling. However, the learned model suffers from limited generalisability, especially for flows without similarity laws (e.g. separated flows). In this work, we propose a knowledge-integrated additive (KIA) learning approach for learning wall models in large-eddy simulations. The proposed approach integrates the knowledge in the simplified thin-boundary-layer equation with a data-driven forcing term for the non-equilibrium effects induced by pressure gradients and flow separations. The capability learned from each flow dataset is encapsulated using basis functions with the corresponding weights approximated using neural networks. The fusion of capabilities learned from various datasets is enabled using a distance function, in a way that the learned capability is preserved and the generalisability to other cases is allowed. The additive learning capability is demonstrated via training the model sequentially using the data of the flow with pressure gradient but no separation, and the separated flow data. The capability of the learned model to preserve previously learned capabilities is tested using turbulent channel flow cases. The periodic hill and the 2-D Gaussian bump cases showcase the generalisability of the model to flows with different surface curvatures and different Reynolds numbers. Good agreements with the references are obtained for all the test cases.

Shallow cumuli are cloud towers that extend a few kilometres above the atmospheric boundary layer without significant precipitation. We present a novel laboratory experiment, boiling stratified flow, as an analogy to study turbulent mixing processes in the boundary layer by shallow cumulus convection. In the experimental beaker, a syrup layer (representing the atmospheric boundary layer) is placed below a freshwater layer (representing the free troposphere) and heated from below. The temperature is analogous to the water vapour mixing ratio in the atmosphere, while the freshwater concentration is analogous to the potential temperature. When the syrup layer starts boiling, bubbles and their accompanying vortex rings stir the two-layer interface and bring colder fresh water into the syrup layer. Two distinct regimes are identified: transient and steady boiling. If the syrup layer is initially sufficiently thin and diluted, then the vortex rings entrain more cold water than needed to quench superheating in the syrup layer, ending the boiling. If the syrup layer is initially deep and concentrated, then the boiling is steady since the entrainment is weak, causing the entrained colder water to continuously prevent superheating. A theory is derived to predict the entrainment rate and the transition between the two regimes, validated by experimental data. Finally, analogies and differences with the atmospheric processes are discussed.

This paper investigates the linear and nonlinear dynamics of two-dimensional penetrative convection subjected to radiative volumetric thermal forcing, focusing on ice-covered freshwater systems. Linear stability analysis reveals how critical wavenumbers $k_c$ and Rayleigh numbers $Ra_c$ are influenced by the attenuation lengths and incoming heat flux. In this configuration, the system easily becomes unstable with a small $Ra_c$, which is two decades smaller than that of the classical Rayleigh–Bénard convection problem, with typically $O(10)$. Weakly nonlinear analysis figures out that this configuration is supercritical, contrasting with the subcritical case by Veronis (Astrophys. J., vol. 137, 1963, 641–663). Numerical bifurcation solutions are performed from the critical points and over several decades, up to $Ra \sim O(10^6)$. This paper found that the system exhibits multiple steady solutions, and under certain specific conditions, a staircase temperature profile emerges. Meanwhile, we further discuss the influence of incoming heat flux and the Prandtl number $Pr$ on the primary bifurcation. Direct numerical simulations are also carried out, showing that heat is transported more efficiently via unsteady convection.

In inviscid, incompressible flows, the evolution of vorticity is exactly equivalent to that of an infinitesimal material line-element and, hence, vorticity can be traced forward or backward in time in a Lagrangian fashion. This elegant and powerful description is not possible in viscous flows due to the action of diffusion. Instead, a stochastic Lagrangian interpretation is required and was recently introduced, where the origin of vorticity at a point is traced back in time as an expectation over the contribution from stochastic trajectories. We herein introduce for the first time an Eulerian, adjoint-based approach to quantify the back-in-time origin of vorticity in viscous, incompressible flows. The adjoint variable encodes the advection, tilting and stretching of the earlier-in-time vorticity that ultimately leads to the target value. Precisely, the adjoint vorticity is the volume-density of the mean Lagrangian deformation of the earlier vorticity. The formulation can also account for the injection of vorticity into the domain at solid boundaries. We demonstrate the mathematical equivalence of the adjoint approach and the stochastic Lagrangian approach. We then provide an example from turbulent channel flow, where we analyse the origin of high streamwise wall-shear-stress events and relate them to Lighthill’s mechanism of stretching of near-wall vorticity.

At constant pressure, a mixture of water parcels with equal density but differing salinity and temperature will be denser than the parent water parcels. This is known as cabbeling and is a consequence of the nonlinear equation of state for seawater density. With a source of turbulent vertical mixing, cabbeling has the potential to trigger and drive convection in gravitationally stable water columns and there is observational evidence that this process shapes the thermohaline structure of high-latitude oceans. However, the evolution and maintenance of turbulent mixing due to cabbeling has not been fully explored. Here, we use turbulence-resolving direct numerical simulations to investigate cabbeling’s impact on vertical mixing and pathways of energy in closed systems. We find that cabbeling can sustain convection in an initially gravitationally stable two-layer configuration where relatively cold/fresh water sits atop warm/salty water. We show the mixture of the cold/fresh and warm/salty water is constrained by a density maximum and that cabbeling enhances mixing rates by four orders of magnitude. Cabbeling’s effect is amplified as the static stability limit is approached, leading to convection being sustained for longer. We find that available potential energy, which is classically thought to only decrease with mixing, can increase with mixing due to cabbeling’s densification of the mixed water. Our direct numerical dimulations support the notion that cabbeling could be a source of enhanced ocean mixing and that conventional definitions of energetic pathways may need to be reconsidered to take into account densification under mixing.