The linear and nonlinear dynamics of centrifugal instability in Taylor–Couette flow are investigated when fluids are stably stratified and highly diffusive. One-dimensional local linear stability analysis (LSA) of cylindrical Couette flow confirms that the stabilising role of stratification in centrifugal instability is suppressed by strong thermal diffusion (i.e. low Prandtl number $Pr$). For $Pr\ll 1$, it is verified that the instability dependence on thermal diffusion and stratification with the non-dimensional Brunt–Väisälä frequency $N$ can be prescribed by a single rescaled parameter $P_{N}=N^{2}Pr$. From direct numerical simulation (DNS), various nonlinear features such as axisymmetric Taylor vortices at saturation, secondary instability leading to non-axisymmetric patterns or transition to chaotic states are investigated for various values of $Pr\leqslant 1$ and Reynolds number $Re_{i}$. Two-dimensional bi-global LSA of axisymmetric Taylor vortices, which appear as primary centrifugal instability saturates nonlinearly, is also performed to find the secondary critical Reynolds number $Re_{i,2}$ at which the Taylor vortices become unstable by non-axisymmetric perturbation. The bi-global LSA reveals that $Re_{i,2}$ increases (i.e. the onset of secondary instability is delayed) in the range $10^{-3}\lt Pr\lt 1$ at $N=1$ or as $N$ increases at $Pr=0.01$. Secondary instability leading to highly non-axisymmetric or irregular chaotic patterns is further investigated by three-dimensional DNS. The Nusselt number $Nu$ is also computed from the torque at the inner cylinder for various $Pr$ and $Re_{i}$ at $N=1$ to describe how the angular momentum transfer increases with $Re_{i}$ and how $Nu$ varies differently for saturated and chaotic states.

Neural network models have been employed to predict the instantaneous flow close to the wall in a viscoelastic turbulent channel flow. Numerical simulation data at the wall are used to predict the instantaneous velocity fluctuations and polymeric-stress fluctuations at three different wall-normal positions in the buffer region. Such an ability of non-intrusive predictions has not been previously investigated in non-Newtonian turbulence. Our comparative analysis with reference simulation data shows that velocity fluctuations are predicted reasonably well from wall measurements in viscoelastic turbulence. The network models exhibit relatively improved accuracy in predicting quantities of interest during the hibernation intervals, facilitating a deeper understanding of the underlying physics during low-drag events. This method could be used in flow control or when only wall information is available from experiments (for example, in opaque fluids). More importantly, only velocity and pressure information can be measured experimentally, while polymeric elongation and orientation cannot be directly measured despite their importance for turbulent dynamics. We therefore study the possibility to reconstruct the polymeric-stress fields from velocity or pressure measurements in viscoelastic turbulent flows. The neural network models demonstrate a reasonably good accuracy in predicting polymeric shear stress and the trace of the polymeric stress at a given wall-normal location. The results are promising, but also underline that a lack of small scales in the input velocity fields can alter the rate of energy transfer from flow to polymers, affecting the prediction of the polymeric-stress fluctuations.

Understanding the vertical coherence of the pressure structure and its interaction with velocity fields is critical for elucidating the mechanisms of acoustic generation and radiation in hypersonic turbulent boundary layers. This study employs linear coherence analysis to examine the self-similar coherent structures in the velocity and pressure fields within a Mach 6 hypersonic boundary layer, considering a range of wall-to-recovery temperature ratios. The influence of wall cooling on the geometric characteristics of these structures, such as inclination angles and three-dimensional aspect ratios, is evaluated. Specifically, the streamwise velocity exhibits self-similar coherent structures with the streamwise/wall-normal aspect ratio ranging from 16.5 to 38.7, showing a linear increases with decreasing wall temperatures. Similar linear dependence between the streamwise/wall-normal aspect ratio and the wall temperatures are observed for the Helmholtz-decomposed streamwise velocity and the pressure field. In terms of velocity–pressure coupling, the solenoidal component exhibits stronger interactions with the pressure fields in the near-wall region, while the dilatational component has stronger interactions with the pressure field at large scales with the increase of height. Such coupling generally follows the distance-from-the-wall scaling of the pressure field, except in cooled wall cases. Using the linear stochastic estimation, the pressure field across the boundary layer is predicted by inputting the near-wall pressure/velocity signal along with the transfer kernel. The result demonstrates that near-wall pressure signals provide the most accurate description of the pressure field in higher regions of the boundary layer. As wall-mounted sensors can measure near-wall pressure fluctuations, this study presents a potential approach to predict the off-wall pressure field correlated with the near-wall structures based on wall-pressure measurements.

We present direct numerical simulations of a supersonic, zero-pressure-gradient, adiabatic turbulent boundary layer at a free-stream Mach number of $M_\infty =2$, over cubical roughness elements. The simulations are complemented by a subsonic rough-wall boundary layer over the same geometry, alongside reference smooth-wall data, allowing us to elucidate compressibility effects. The simulations feature turbulent flow transitioning from a smooth to a rough surface with an extended computational domain to facilitate recovery. At the smooth-to-rough transition, we compare the development of an internal boundary layer between the subsonic and supersonic cases, introducing a novel definition of its height that is less sensitive to local compressibility effects. We demonstrate that, although the internal boundary-layer growth is similar to the subsonic case, a delayed equilibrium is expected for the supersonic case due to the sudden growth of the external boundary-layer thickness at the onset of roughness. Turbulence statistics are then evaluated far from the surface transition, where various compressibility transformations reveal outer-layer similarity for the mean velocity. We find that the classical van Driest II transformation can also be applied to rough walls, at least in the adiabatic case. Analysis of thermal statistics for the supersonic case confirms the significant influence that roughness has on both mean and fluctuating temperature fields, which, unlike velocity fields, do not display outer-layer similarity. Nonetheless, we find that the temperature–velocity relation established for smooth walls is also valid over rough surfaces, implying that the mean temperature field can be predicted solely based on the mean velocity.

The acoustic receptivity of Tollmien–Schlichting (TS) waves due to two-dimensional sharp-edged rectangular bumps and gaps in a compressible boundary layer was investigated by direct numerical simulations. The conclusions were based on a new procedure proposed for obtaining the receptivity amplitudes which appeared to be more robust and accurate than previous ones. The procedure is particularly important for the correct evaluation of the receptivity of gaps. The receptivity amplitudes for gaps were smaller than those for bumps, except for the nominally zero height/depth roughness element, where, consistent with a linear behaviour, they had the same absolute value. The procedure also revealed in detail the behaviour in the region downstream of the roughness element where the TS wave is formed (the formation region). This region extends for approximately $50\delta ^*_{b}$, regardless of bump height or gap depth. For bumps, the receptivity scaled superlinearly with bump height while for the gaps it scaled sublinear with depth. This behaviour is associated with the different velocity profiles caused by bumps and gaps in the formation region. We also discussed issues regarding comparison with experiments. Investigation of the effect of compressibility confirmed that, in the subsonic regime, the receptivity reduces with Mach number. Finally, we addressed the receptivity scaling with the acoustic wave amplitude. It was found that the receptivity scales linearly with the acoustic wave amplitude in a range for which experiments available in the literature indicated a superlinearly scaling. Reasons for these conflicting results are discussed.

It is known that the complex eigenfrequencies of one-dimensional systems of large but finite extent are concentrated near the asymptotic curve determined by the dispersion relation of an infinite system. The global instability caused by uppermost pieces of this curve was studied in various problems, including hydrodynamic stability and fluid–structure interaction problems. In this study, we generalise the equation for the asymptotic curve to arbitrary frequencies. We analyse stable local topology of the curve and prove that it can be a regular point, branching point or dead-end point of the curve. We give a classification of unstable local tolopogies, and show how they break up due to small changes of the problem parameters. The results are demonstrated on three examples: supersonic panel flutter, flutter of soft fluid-conveying pipe, and the instability of rotating flow in a pipe. We show how the elongation of the system yields the attraction of the eigenfrequencies to the asymptotic curve, and how each locally stable curve topology is reflected on the interaction of eigenfrequencies.

The motion and deformation of a neutrally buoyant drop in a rectangular channel experiencing a pressure-driven flow at a low Reynolds number has been investigated both experimentally and numerically. A moving-frame boundary-integral algorithm was used to simulate the drop dynamics, with a focus on steady-state drop velocity and deformation. Results are presented for drops of varying undeformed diameters relative to channel height ($D/H$), drop-to-bulk viscosity ratio ($\lambda$), capillary number ($Ca$, ratio of deforming viscous forces to shape-preserving interfacial tension) and initial position in the channel in a parameter space larger than considered previously. The general trend shows that the drop steady-state velocity decreases with increasing drop diameter and viscosity ratio but increases with increasing $Ca$. An opposite trend is seen for drops with small viscosity ratio, however, where the steady-state velocity increases with increasing $D/H$ and can exceed the maximum background flow velocity. Experimental results verify theoretical predictions. A deformable drop with a size comparable to the channel height when placed off centre migrates towards the centreline and attains a steady state there. In general, a drop with a low viscosity ratio and high capillary number experiences faster cross-stream migration. With increasing aspect ratio, there is a competition between the effect of reduced wall interactions and lower maximum channel centreline velocity at fixed average velocity, with the former helping drops attain higher steady-state velocities at low aspect ratios, but the latter takes over at aspect ratios above approximately 1.5.

Experimental and numerical observations in turbulent shear flows point to the persistence of the anisotropy imprinted by the large-scale velocity gradient down to the smallest scales of turbulence. This is reminiscent of the strong anisotropy induced by a mean passive scalar gradient, which manifests itself by the ‘ramp–cliff’ structures. In the shear flow problem, the anisotropy can be characterised by the odd-order moments of $\partial _y u$, where $u$ is the fluctuating streamwise velocity component, and $y$ is the direction of mean shear. Here, we extend the approach proposed by Buaria et al. (Phys. Rev. Lett., 126, 034504, 2021) for the passive scalar fields, and postulate that fronts of width $\delta \sim \eta Re_\lambda ^{1/4}$, where $\eta$ is the Kolmogorov length scale, and $Re_\lambda$ is the Taylor-based Reynolds number, explain the observed small-scale anisotropy for shear flows. This model is supported by the collapse of the positive tails of the probability density functions (PDFs) of $(\partial _y u)/(u^{\prime }/\delta )$ in turbulent homogeneous shear flows (THSF) when the PDFs are normalised by $\delta /L$, where $u^{\prime }$ is the root-mean-square of $u$ and $L$ is the integral length scale. The predictions of this model for the odd-order moments of $\partial _y u$ in THSF agree well with direct numerical simulation (DNS) and experimental results. Moreover, the extension of our analysis to the log-layer of turbulent channel flows (TCF) leads to the prediction that the odd-order moments of order $p (p \gt 1)$ of $\partial _y u$ have power-law dependencies on the wall distance $y^{+}$: $\langle (\partial _y u)^p \rangle /\langle (\partial _y u)^2 \rangle ^{p/2} \sim (y^{+})^{(p-5)/8}$, which is consistent with DNS results.

We define ‘surface layer’ (SL) as an inertia-dominated turbulence region outside a viscous or roughness surface-adjacent sub-layer (SAS) that is characterised by linear scaling of specific coherence length scales on wall-normal distance, $z$. We generalise the mechanisms that underlie the formation of the classical inertial SL in the shear-dominated turbulent boundary layer (TBL) to wall-bounded turbulent flows with zero mean shear. Using particle image velocimetry data from two wind tunnel facilities, we contrast the classical TBL SL with a non-classical shear-free SL generated within grid turbulence advected over an impermeable plate using two grids with different turbulence length scales. Integral-scale variations with $z$ and other statistics are quantified. In both shear-dominated and shear-free SLs we observe well-defined linear increases in $z$ of the streamwise integral scale of vertical velocity fluctuations. In grid turbulence the shear-free SL initiates just above the SAS that confines friction-generated motions. By contrast, the TBL SL forms with non-zero mean shear rate that extends streamwise coherence lengths of streamwise fluctuations. In both flow classes only the integral scales of vertical fluctuating velocity increase linearly with $z$, indicating that the SL is generated by the blockage of vertical fluctuations in the vertical. Whereas the SAS in the TBL is much thinner than in the grid-turbulence flows, the generation of a shear-free SL by the interaction of turbulence eddies and a surface depends on the relative thinness of the SAS. We conclude that the common generalisable SL mechanism is direct blockage of vertical fluctuations by the impermeable surface.

Significant progress has been made in understanding planetary core dynamics using numerical models of rotating convection (RC) in spherical shell geometry. However, the behaviour of forces in these models within various dynamic regimes of RC remains largely unknown. Directional anisotropy, scale dependence and the role of dynamically irrelevant gradient contributions in incompressible flows complicate the representation of dynamical balances in spherical shell RC. In this study, we systematically compare integrated and scale-dependent representations of mean and fluctuation forces and curled forces (which contain no gradient contributions) separately for the three components ($\hat {r},\hat {\theta },\hat {\phi }$). The analysis is performed with simulations in a range of convective supercriticality $Ra_T/Ra_T^{c}=1.2{-}297$ where $Ra_T$ and $Ra^{c}_T$ are the Rayleigh and critical Rayleigh numbers, respectively and Ekman number $E=10^{-3}{-}10^{-6}$, with fixed Prandtl number $Pr=1$, along with no-slip and fixed flux boundaries. We have excluded regions from each boundary of the spherical shell, with a thickness equivalent to ten velocity boundary layers, which provides a consistent representation of the bulk dynamics between the volume-averaged force and curled force balance in the parameter space studied. Radial, azimuthal and co-latitudinal components exhibit distinct force and curled force balances. The total magnitudes of the mean forces and mean curled forces exhibit a primary thermal wind balance; the corresponding fluctuating forces are in a quasi-geostrophic primary balance, while the fluctuating curled forces transition from a Viscous–Archimedean–Coriolis balance to an Inertia–Viscous–Archimedean–Coriolis balance with increasing $Ra_T/Ra_T^{c}$. The curled force balances are more weakly scale-dependent compared to the forces, and do not show clear cross-over length scales. The fluctuating force and curled force balances are broadly consistent with three regimes of RC (weakly nonlinear, rapidly rotating and weakly rotating), but do not exhibit sharp changes with $Ra_{T}/Ra_{T}^{c}$, which inhibits the identification of precise regime boundaries from these balances.

This study investigates the energy exchange between coherent structures in flows over four low-aspect-ratio (low-) plates using the tomographic particle image velocimetry dataset originally obtained by Zhu et al. (2024. J. Fluid Mech. 983, A35). The chord-based Reynolds number is $5400$, with fixed angle of attack $6 ^\circ$. In this study, multiscale proper orthogonal decomposition is applied to extract the coherent structures, including those associated with the vortex-shedding frequency $St_1$ and its subharmonic counterpart. Subsequently, the coherent kinetic energy budget is analysed with a focus on inter-scale energy transfer. This study demonstrates that the energy transfer between the scales centred at $St_1$ and $0.5\,St_1$ can exhibit a reverse or forward direction, depending on the transformation pattern of the leading-edge vortices (LEVs). Specifically, different triadic interactions are excited during the LEV transformation, and manifest themselves during the formation of hairpin vortices downstream. Understanding this nonlinear energy transfer is essential for elucidating mechanisms underlying the development of turbulence in three-dimensional flows over low- plates.

Crowdy et al. (2023 Phys. Rev. Fluids, vol. 8, 094201), recently showed that liquid suspended in the Cassie state over an asymmetrically spaced periodic array of alternating cold and hot ridges such that the menisci spanning the ridges are of unequal length will be pumped in the direction of the thermocapillary stress along the longer menisci. Their solution, applicable in the Stokes flow limit for a vanishingly small thermal Péclet number, provides the steady-state temperature and velocity fields in a semi-infinite domain above the superhydrophobic surface, including the uniform far-field velocity, i.e. pumping speed, the key engineering parameter. Here, a related problem in a finite domain is considered where, opposing the superhydrophobic surface, a flow of liquid through a microchannel is bounded by a horizontally mobile smooth wall of finite mass subjected to an external load. A key assumption underlying the analysis is that, on a unit area basis, the mass of the liquid is small compared with that of the wall. Thus, as shown, rather than the heat equation and the transient Stokes equations governing the temperature and flow fields, respectively, they are quasi-steady and, as a result, governed by the Laplace and Stokes equations, respectively. Under the further assumption that the ridge period is small compared with the height of the microchannel, these equations are resolved using matched asymptotic expansions which yield solutions with exponentially small asymptotic errors. Consequently, the transient problem of determining the velocity of the smooth wall is reduced to an ordinary differential equation. This approach is used to provide a theoretical demonstration of the conversion of thermal energy to mechanical work via the thermocapillary stresses along the menisci.

We derive a depth-averaged model consistent with the $\mu (I)$ rheology for an incompressible granular flow down an inclined plane. The first two variables of the model are the depth and the depth-averaged velocity. The shear is also taken into account via a third variable called enstrophy. The obtained system is a hyperbolic system of conservation laws, with an additional equation for the energy. The system is derived from an asymptotic expansion of the flow variables in powers of the shallow-water parameter. This method ensures that the model is fully consistent with the rheology. The velocity profile is a Bagnold profile at leading order and the first-order correction to this profile can be calculated for flows that are not steady uniform. The first-order correction to the classical granular friction law is also consistently written. As a consequence, the instability threshold of the steady uniform flow is the same for the depth-averaged model and for the governing equations. In addition, a higher-order version that contains diffusive terms is also presented. The spatial growth rate, the phase velocity and the cutoff frequency of the version with diffusion are in good agreement with the experimental data and with the theoretical predictions for the rheology. The mathematical structure of the equations enables us to use well-known and stable numerical solvers. Numerical simulations of granular roll waves are presented. The model has the same limitations as the $\mu (I)$ rheology, in particular for the solid/ liquid and liquid/gas transitions, and needs therefore a regularisation for these transitions.

The formation of Kelvin–Helmholtz-like rollers (referred to as K–H rollers) over riblet surfaces has been linked to the drag-increasing behaviour seen in certain riblet geometries, such as sawtooth and blade riblets, when the riblet size reaches sufficiently large viscous scales (Endrikat et al. (2021a), J. Fluid Mech. 913, A37). In this study, we focus on the sawtooth geometry of fixed physical size, and experimentally examine the response of these K–H rollers to further increases in viscous scaled riblet sizes, by adopting the conventional approach of increasing freestream speeds (and consequently, the friction Reynolds number). Rather than continual strengthening, the present study shows a gradual weakening of these K–H rollers with increasing sawtooth riblet size. This is achieved by an analysis of the roller geometric characteristics using both direct numerical simulations and hot-wire anemometry databases at matched viscous scaled riblet spacings, with the former used to develop a novel methodology for detecting these rollers via streamwise velocity signatures (e.g. as acquired by hot wires). Spectral analysis of the streamwise velocity time series, acquired within riblet grooves, reveals that the frequencies (and the streamwise wavelengths) of the K–H rollers increase with increasing riblet size. Cross-correlation spectra, estimated from unique two-point hot-wire measurements in the cross-plane, show a weakening of the K–H rollers and a reduction in their wall-normal coherence with increasing riblet size. Besides contributing to our understanding of the riblet drag-increasing mechanisms, the present findings also have implications for the heat transfer enhancing capabilities of sawtooth riblets, which have been associated previously with the formation of K–H rollers. The present study also suggests conducting future investigations by decoupling the effects of viscous scaled riblet spacing and friction Reynolds numbers, to characterise their influence on the K–H rollers independently.

We conduct direct numerical simulations to investigate the synchronisation of Kolmogorov flows in a periodic box, with a focus on the mechanisms underlying the asymptotic evolution of infinitesimal velocity perturbations, also known as conditional leading Lyapunov vectors. This study advances previous work with a spectral analysis of the perturbation, which clarifies the behaviours of the production and dissipation spectra at different coupling wavenumbers. We show that, in simulations with moderate Reynolds numbers, the conditional leading Lyapunov exponent can be smaller than a lower bound proposed previously based on a viscous estimate. A quantitative analysis of the self-similar evolution of the perturbation energy spectrum is presented, extending the existing qualitative discussion. The prerequisites for obtaining self-similar solutions are established, which include an interesting relationship between the integral length scale of the perturbation velocity and the local Lyapunov exponent. By examining the governing equation for the dissipation rate of the velocity perturbation, we reveal the previously neglected roles of the strain rate and vorticity perturbations, and uncover their unique geometrical characteristics.