Convective boundary layers are governed by an interplay of vertical turbulent convection and shear-driven turbulence. Here, we investigate vertical velocity and buoyancy fields in convective boundary layers for varying atmospheric conditions by combining probability density function methods and direct numerical simulations. The evolution equations for the probability density functions of vertical velocity and buoyancy contain unclosed terms in the form of conditional averages. We estimate these terms from our direct numerical simulations data, and discuss their physical interpretation. Furthermore, using the method of characteristics, we investigate how these unclosed terms jointly determine the average evolution of a fluid element in a convective boundary layer, and how it relates to the evolution of the probability density functions of vertical velocity and buoyancy as a function of height. Thereby, our work establishes a connection between the turbulent dynamics of convective boundary layers and the resulting statistics.

The instability characteristics and laminar–turbulent transition of a series of laminar separation bubbles (LSBs) formed due to a single sinusoidal surface waviness are investigated in the absence of external disturbances or forcing. A scaling based on the geometrical parameters of the waviness and flow Reynolds number is found that enables the prediction of flow separation on the wall leeward side. The analysis of three-dimensional instabilities of two-dimensional base flows reveals a relation between the number of changes in the curvature sign of the recirculating streamlines and the number of unstable centrifugal modes that coexist for the same flow. When multiple curvature changes occur, in addition to the usual steady mode reported for two-dimensional recirculation bubbles, a new self-excited mode with a higher growth rate emerges, localised near the highest streamline curvature, close to the reattachment point. A detailed analysis of the mode growth and saturation using DNS reveals that the localised mode only disturbs the LSB locally, while the usual one leads to a global distortion of the bubble in the spanwise direction; this has a distinctive impact on the self-excited secondary instabilities. Then, the complete transition scenario is studied for two selected LSB cases. The first one only presents an unstable eigenmode, namely the usual centrifugal mode in recirculating flows. The second case presents three unstable eigenmodes: two centrifugal eigenmodes (the usual and the localised ones) and a two-dimensional eigenmode associated with the self-sustained Kelvin–Helmholtz waves. These results show how completely different transition scenarios can emerge from subtle changes in the LSB characteristics.

We perform direct numerical simulations of centrifugal convection with an oscillating rotational velocity of small amplitude to study the effects of oscillatory boundary motion. The oscillation period is the main control parameter, with its range corresponding to a Womersley number in the range $1\lt Wo\lt 300$. Oscillating boundaries generate a circumferential shear flow, which significantly inhibits heat transfer, with maximum suppression $87\,\%$ observed in the present parameter space. Through analysis of the background flow, we find that as the oscillation period increases, the increasing penetration depth of the oscillation and weakening local shear strength result in non-monotonic changes in heat transfer. Under high-frequency oscillation, the characteristic length scale of the viscous layer induced by the oscillation is smaller than the convective length scale, and shear manifests primarily as a continuous suppression of the boundary layer. In contrast, under low-frequency oscillation, the shear flow covers the entire region but with weak strength. The suppression effect of such shear flow exhibits periodicity, leading to alternating phases of convection inhibition and convection generation. The present findings explore the physical mechanisms behind the suppression of convective heat transfer by oscillation, and offer a new strategy for controlling convection systems, with potential implications for both fundamental research and industrial applications.

A generalised multiparameter model for linear modal stability and sensitivity analysis is developed. The stability and sensitivity equations are derived from a generalised vector-form governing equation comprised of multiple dimensionless parameters that represent different physical forces affecting the system’s stability. By introducing adjoint variables and constructing the Lagrangian identity, a differential relationship between the eigenvalue of the perturbation mode and dimensionless parameters is determined and defined as the global sensitivity gradient. It provides the constraint that must be satisfied for changes in different dimensionless parameters along the isoeigenvalue curve, which aids in the fast computation of the neutral curve. Moreover, the global sensitivity gradient can directly and intuitively evaluate the competitive relationship among the influences of various parameters on system instability. Based on the global sensitivity gradient, an optimal stability control strategy for transitioning from an unstable state to a stable state is discussed. Additionally, the relative sensitivity function is also introduced to investigate the influence of relative parameter variations on instability. To demonstrate the effectiveness of this method, three applications are presented: two-dimensional flow around a circular cylinder with a single dimensionless parameter Re; three-dimensional axisymmetric magnetohydrodynamic (MHD) flow around a sphere with two parameters Re and $N$; and two-dimensional MHD mixed convection with three parameters Re, ${\textit{Gr}}$ and $\textit{Ha}$.

Power minimisation in branched fluidic networks has gained significant attention in biology and engineering. The optimal network is defined by channel radii that minimise the sum of viscous dissipation and the volumetric energetic cost of the fluid. For limit cases including laminar flows, high-Reynolds-number turbulence or smooth-channel approximations, optimal solutions are known. However, current methods do not allow optimisation for a large intermediate part of the parameter space which is typically encountered in realistic fluidic networks that exhibit turbulent flow. Here, we present a unifying optimisation approach based on the Darcy friction factor, which has been determined for a wide range of flow regimes and fluid models and is applicable to the entire parameter space: (i) laminar and turbulent flows, including networks that exhibit both flow types, (ii) non-Newtonian fluids (powerlaw, Bingham and Herschel–Bulkley) and (iii) networks with arbitrary wall roughness, including non-uniform relative roughness. The optimal channel radii are presented analytically and graphically. All existing limit cases are recovered, and a concise framework is presented for systematic optimisation of fluidic networks. Finally, the parameter $x$ in the optimisation relationship $Q\propto R^{x}$, with $Q$ the flow rate and $R$ the channel radius, was approximated as a function of the Reynolds number, revealing in which case the entire network can be optimised based on one optimal channel radius, and in which case all radii must be optimised individually. Our approach can be extended to a wide range of fluidic networks for which the friction factor is known, such as different channel curvatures, bubbly flows or specific wall slip conditions.

The spatiotemporal dynamics of a turbulent boundary layer subjected to an unsteady pressure gradient are studied. A dynamic sequence of favourable to adverse pressure gradients (FAPGs) is imposed by deforming a section of the wind tunnel ceiling, transitioning the pressure gradient from zero to a strong FAPG within 0.07 s. At the end of the transient, the acceleration parameter is $K$ = $6 \times 10^{-6}$ in the favourable pressure gradient (FPG) region and $K$ = $-4.8 \times 10^{-6}$ in the adverse pressure gradient (APG) region. The resulting unsteady response of the boundary layer is compared with equivalent steady pressure gradient cases in terms of turbulent statistics and coherent structures. While the steady FAPG effects, as shown by Parthasarathy & Saxton-Fox (2023), caused upstream stabilisation in the FPG, a milder APG response downstream, and the formation of an internal layer, the unsteady case presented in this paper shows a reduced stabilisation in the FPG region, a stronger APG response and a weaker internal layer. This altered response is hypothesised to stem from the different spatiotemporal pressure gradient histories experienced by turbulent structures when the pressure gradient changes at a time scale comparable to their convection.