We investigate thermal boundary layer (BL) asymmetry in turbulent Rayleigh–Bénard convection (RBC) under both spherical and annular geometries using different BL theories. Unlike planar RBC, the spherical and annular configurations exhibit asymmetric thermal BLs near the inner and outer boundaries due to boundary curvature and non-uniform radial gravity. We generalise three BL frameworks – the Prandtl–Blasius BL model, the steady free-convective model and the fluctuating BL model – and apply them to both geometries. Direct numerical simulations (DNSs), based on the Oberbeck–Boussinesq equations, are performed in three-dimensional spherical RBC and three-dimensional annular RBC for various radius ratios ($\eta$), gravity profiles and also Prandtl numbers ($ \textit{Pr}$), to compare with the predictions of the extended BL models. We find that the BL asymmetries predicted by both the extended steady free-convective BL and the fluctuating BL agree well with DNS results, with the fluctuating BL model providing the best agreement for the mean temperature profiles. A force-balance analysis further shows that this better performance is consistent with the DNS observation that, in the wall-normal direction within the thermal BL, buoyancy is balanced primarily by the pressure-gradient force. This is consistent with the assumption underlying the steady free-convective and fluctuating BL models. Moreover, the fluctuating BL model explicitly accounts for the contribution of turbulent fluctuations to the heat flux, which further improves its agreement with the DNS mean temperature profiles. We derive analytical expressions for the bulk temperature and the thermal BL thickness ratio as functions of the radius ratio and gravity profile across different Prandtl-number regimes. These expressions are obtained by integrating the similarity thermal equation for both the inner and outer BLs using an approximate similarity streamfunction, and by closing the solutions through a heat-flux matching condition. The resulting leading-order expressions obtained from both the steady free-convective and fluctuating BL models are shown to be the same, and they agree well with DNS data. This analytical result provides a robust and practical tool for quantifying BL asymmetry in curved RBC systems.

We consider free-surface flows driven by turbulence beneath the surface, particularly the strong free-surface turbulence (FST) regime, characterised by large Froude number ${\textit {Fr}}^2_T=\varepsilon /u_{\textit{rms}} {g}\,\gtrsim 0.1$. We study the surface layer, where air and water are highly mixed and turbulence modelling is challenging. We develop a definition of the surface-layer thickness $\delta _s$ based on the vertical derivative of intermittency $\gamma$ at the mean free surface $\bar {\eta }$, which, unlike previous definitions, is independent of the tail behaviour of $\gamma$. From direct numerical simulation (DNS) of statistically stationary, horizontally homogeneous strong FST, we show that scaling by $z^* = (z-\bar {\eta })/\delta _s$ collapses $\gamma$ across a wide range of ${\textit {Fr}}^2_T\in [0.03,0.3]$. The distribution more closely follows logistic rather than Gaussian behaviour. From the near-surface turbulence obtained from DNS, we make two general observations. First, we show that for strong FST there is minimal direct effect of the free surface on the isotropy, turbulence kinetic energy $\tilde {k}$ or dissipation rate $\varepsilon$ beneath the surface layer ($z^*\lt -0.5$). Instead, turbulence is only indirectly affected through the flux of kinetic energy into the surface layer. Second, we show that many relevant metrics within the surface layer ($z^*\in [-0.5, 0.5]$) collapse when appropriately scaled by $u_{\textit{rms}}^2=2\tilde {k}/3$ and $\varepsilon$ measured at $z^*=-0.5$. These observations suggest the possibility of a turbulence closure model which avoids direct modelling of $\tilde {k}$ and $\varepsilon$ in the surface layer. Towards this, we show that, across a wide range of ${\textit {Fr}}_T^2$, surface-layer thickness can be predicted by $\delta _s \approx 11.1 \,u_{\textit{rms}}^2 {g}^{-1}$ and energy flux into the surface layer by $W \approx 0.41 \,\varepsilon \delta _s$.

This work employs structured input–output analysis (SIOA) augmented by an eddy viscosity model (SIOA-e) to investigate turbulent flows over rigid and compliant walls. The SIOA-e framework demonstrates the capability in identifying both streamwise and spanwise dominant characteristic wavelengths for rigid wall turbulence. For compliant walls, the SIOA-e method predicts optimal compliant wall parameters associated with positive damping coefficients when minimizing input–output gain for near-wall cycle and very large-scale motions, respectively. The reduction of input–output gain due to the compliant wall is achieved by wall displacement resembling blowing and suction opposite to the wall-normal velocity of dominant streamwise vortices. However, optimized compliant wall parameters based on specific wavenumber–frequency combinations may amplify flow structures for other wavenumber–frequency pairs, potentially leading to an overall drag increase. For example, compliant wall parameters tuned for suppressing large-scale structures can affect both large- and small-scale structures. We also employ input–output analysis to predict convective velocity of wall displacement and pressure for turbulent flow over the compliant wall, and the predicted convective velocity of wall displacement is 0.53 times centreline velocity, which aligns well with recent experimental measurements.

We investigate the incompressible flow inside a two-dimensional square cavity, driven by the sliding motion of its four lids, all at the same speed and with facing lids moving in opposite directions. The problem has three symmetries: two mirror symmetries with respect to the diagonals and a $\pi$ rotation invariance about the centre of the cavity. The base flow, a steady state that has all three symmetries, is the unique solution at sufficiently low values of the Reynolds number ($ \textit{Re}$) and acts as a global attractor. At higher $ \textit{Re}$, it has become unstable and shares the phase space with a globally attracting space–time symmetric periodic orbit that, in addition to the rotational invariance, is also invariant under evolution over half a period followed by reflection about either of the diagonals. In between, a wealth of solution branches and intervening bifurcations mediate the transition process. In particular, a pair of steady states that break the mirror symmetries but are mirror-symmetry images of each other regulate the appearance and disappearance of a second space–time symmetric periodic orbit and a pair of asymmetric periodic orbits that are also mirror images of each other. The catalogue of instabilities includes both local (two pitchfork, two Hopf, a saddle-node and a cyclic fold) and global (two heteroclinic and one homoclinic) bifurcations. The sequence of transitions is explained in terms of a one-dimensional path through the parameter space of a codimension-four bifurcation: the double zero bifurcation with Z$_2$ symmetry and degeneracy of the third order terms.

This study investigates necklace-vortex systems forming when a laminar shear-wake, generated by two streams merging at the trailing edge of a splitter plate, interacts with a circular cylinder placed downstream in the wake. Hydrogen-bubble flow visualisations were employed in a water channel capable of producing laminar shear-wake flows. In the absence of the cylinder, oppositely signed vorticity in the shear-wake undergoes mutual annihilation. The introduction of the cylinder interrupts this evolution, promoting off-wall flow separation upstream of the cylinder and vortex roll-up. The study primarily focuses on two non-dimensional parameters, the Reynolds number $ \textit{Re}_m$ and the shear ratio $ \textit{SR}$, and presents a mapping of the observed vortex regimes. Increasing $ \textit{Re}_m$ promotes either the formation of additional vortices or unsteadiness. Increasing $ \textit{SR}$ generally suppresses vortex formation or attenuates unsteadiness, except near $ \textit{SR}\approx 0$ at low to moderate $ \textit{Re}_m$, where the two-vortex system is unstable to additional vortex generation. Observed configurations range from no-vortex states to one- or two-vortex systems at low Reynolds numbers, and to three-, four- and five-vortex systems at larger Reynolds numbers, with unsteadiness becoming prominent beyond the three-vortex regime and predominant in four- and five-vortex systems. Beyond regime mapping, we delve into the structure of a steady two- and three-vortex system at low to moderate $ \textit{Re}_m$. This provides insights into the emergence and evolution of the vortex system, which is analysed in the context of the vorticity-transport equations.