A nonlinear Schrödinger equation for pure capillary waves propagating at the free surface of a vertically sheared current has been used to study the stability and bifurcation of capillary Stokes waves on arbitrary depth. A linear stability analysis of weakly nonlinear capillary Stokes waves on arbitrary depth has shown that (i) the growth rate of modulational instability increases as the vorticity decreases whatever the dispersive parameter $kh$, where $k$ is the carrier wavenumber and $h$ the depth; (ii) the growth rate is significantly amplified for shallow water depths; and (iii) the instability bandwidth widens as the vorticity decreases. Particular attention has been paid to damping due to viscosity and forcing effects on modulational instability. In addition, a linear stability analysis to transverse perturbations in deep water has been carried out, demonstrating that the dominant modulational instability is two-dimensional whatever the vorticity. Near the minimum of linear phase velocity in deep water, we have shown that generalised capillary solitary waves bifurcate from linear capillary Stokes waves when the vorticity is positive. Moreover, we have shown that the envelope of pure capillary waves in deep water is unstable to transverse perturbations. Consequently, deep-water generalised capillary solitary waves are expected to be unstable to transverse perturbations.

Görtler vortices induced by concave curvature in supersonic turbulent flows are investigated using resolvent analysis and large-eddy simulations at Mach 2.95 and Reynolds number $ Re_{\delta }=63\,500$ based on the boundary-layer thickness $ \delta$. Resolvent analysis reveals that the most amplified coherent structures manifest as streamwise counter-rotating vortices with optimal spanwise wavelength $ 2.4\delta$ at cut-off frequency $f\delta /{u}_{\infty } =0.036$, where $ {u}_{\infty }$ is the freestream velocity. The leading spectral proper orthogonal decomposition modes with spanwise wavelength approximately $ 2\delta$ align well with the predicted coherent structures from resolvent analysis at $f\delta /{u}_{\infty } =0.036$. These predicted and extracted coherent structures are identified as Görtler vortices, driven by the Görtler instability. The preferential spanwise scale of the Görtler vortices is further examined under varying geometric and freestream parameters. The optimal spanwise wavelength is insensitive to the total turning angle beyond a critical value, but sensitive to the concave curvature $ K$ at the same turning angle. A limit spanwise wavelength $ 1.96\delta$, corresponding to an infinite concave curvature as $ K\rightarrow \infty$, is identified and validated. Increasing the freestream Mach number or decreasing the ratio of wall temperature to freestream temperature reduces the optimal wavelength normalised by $ \delta$, while variations in freestream Reynolds number have negligible impact. Additionally, a modified definition of the turbulent Görtler number $ G_{T}$ based on the peak eddy viscosity in boundary layers is proposed and employed to assess the occurrence of Görtler instability.

The supersonic wake of a circular cylinder in Mach 3 flow was studied through spectral proper orthogonal decomposition (SPOD) of high-speed focussing schlieren datasets. A wavenumber decomposition of the SPOD eigenvectors was found to be an effective tool for isolating imaging artefacts from the flow features, resulting in a clearer interpretation of the SPOD modes. The cylinder wake consists of both symmetric and antisymmetric instabilities, with the former being the dominant type. The free shear layers that form after the flow separates from the cylinder surface radiate strong Mach waves that interact with the recompression shocks to release significant disturbances into the wake. The wake shows a bimodal vortex shedding behaviour with a purely hydrodynamic instability mode around a Strouhal number of 0.2 and an aeroacoustic instability mode around Strouhal number of 0.42. The hydrodynamic mode, which is presumably the same as the incompressible case, is weaker and decays rapidly as the wake accelerates due to increasing compressibility. The aeroacoustic mode is the dominant shedding mode and persists farther into the wake because of an indirect energy input received through free-stream acoustic waves. A simple aeroacoustic feedback model based on an interaction between downstream propagating shear-layer instabilities and upstream propagating acoustic waves within the recirculation region is shown to accurately predict the shedding frequency. Based on this model, the vortex shedding in supersonic flows over a circular cylinder occurs at a universal Strouhal number (based on approach free-stream velocity and feedback path length) of approximately 0.3.

In this work we focus on expected flow in porous formations with highly conductive isolated fractures, which are of non-negligible length compared with the scales of interest. Accordingly, the definition of a representative elementary volume (REV) for flow and transport predictions may not be possible. Recently, a non-local kernel-based theory for flow in such formations has been proposed. There, fracture properties like their expected pressure are represented as field quantities. Unlike existing models, where fractures are assumed to be small compared with the scale of interest, a non-local kernel function is used to quantify the expected flow transfer between a point in the fracture domain and a potentially distant point in the matrix continuum. The transfer coefficient implied by the kernel is a function of the fracture characteristics that are in turn captured statistically. So far the model has successfully been applied for statistically homogeneous cases. In the present work we demonstrate the applicability for heterogeneous cases with spatially varying fracture statistics. Moreover, a scaling law is presented that relates the transfer coefficient to the fracture characteristics. Test cases involving discontinuously and continuously varying fracture statistics are presented, and the validity of the scaling law is demonstrated.

The recent identification of an outlying cemetery at the Maya ceremonial centre of Ceibal, Guatemala, is providing new insights into the Preceramic to Middle Preclassic transition in the Maya lowlands, c. 1000 BC. Identified within the Amoch Group complex and dating to c. 1100–800 BC, the use of a dedicated area for the dead is not previously documented in this region for this period. Here, the authors argue that the emergence and subsequent disappearance of this practice was likely interwoven with social change, involving the adoption of ceramics, increasingly sedentary lifeways and, ultimately, the creation of monumental ceremonial centres.

Fluids at supercritical pressures exhibit large variations in density near the pseudo-critical line, such that buoyancy plays a crucial role in their fluid dynamics. Here, we experimentally investigate heat transfer and turbulence in horizontal hydrodynamically developed channel flows of carbon dioxide at $88.5$ bar and $32.6\,^{\circ }\rm C$, heated at either the top or bottom surface to induce a strong vertical density gradient. In order to visualise the flow and evaluate its heat transfer, shadowgraphy is used concurrently with surface temperature measurements. With moderate heating, the flow is found to strongly stratify for both heating configurations, with bulk Richardson numbers $Ri$ reaching up to 100. When the carbon dioxide is heated from the bottom upwards, the resulting unstably stratified flow is found to be dominated by the increasingly prevalent secondary motion of thermal plumes, enhancing vertical mixing and progressively improving heat transfer compared with a neutrally buoyant setting. Conversely, stable stratification, induced by heating from the top, suppresses the vertical motion, leading to deteriorated heat transfer that becomes invariant to the Reynolds number. The optical results provide novel insights into the complex dynamics of the directionally dependent heat transfer in the near-pseudo-critical region. These insights contribute to the reliable design of heat exchangers with highly property-variant fluids, which are critical for the decarbonisation of power and industrial heat. However, the results also highlight the need for further progress in the development of experimental techniques to generate reliable reference data for a broader range of non-ideal supercritical conditions.