The flow past a $6:1$ prolate spheroid at a moderate pitch angle $\alpha =10^\circ$ is investigated with a focus on the turbulent wake in a high-fidelity large eddy simulation (LES) study. Two length-based Reynolds numbers, ${\textit{Re}}_L=3\times 10^4$ and $9\times 10^4$, and four Froude numbers, ${\textit{Fr}} = \infty \text{(unstratified)}, 6, 1.9 \text{ and }1$, are selected for the parametric study. Spectral proper orthogonal decomposition (SPOD) analysis of the flow reveals the leading coherent modes in the unsteady separated flow at the tail of the body. At the higher ${\textit{Re}}_L=9\times 10^4$, a high-frequency spanwise flapping of shear layers on either side of the body is observed in the separated boundary layer for all cases. The flapping does not perturb the lateral symmetry of the wake. At ${\textit{Fr}}=\infty$, a low-frequency oscillating laterally asymmetric mode, which is found in addition to the shear-layer mode, leads to a sidewise unsteady lateral load. All temporally averaged wakes at ${\textit{Re}}=9\times 10^4$ are found to be spanwise symmetric in the mean as opposed to the lower ${\textit{Re}}=3\times 10^4$, at which the ${\textit{Fr}}=\infty \text{ and }6$ wakes exhibit asymmetry. The turbulent kinetic energy (TKE) budget is compared among cases. Here, ${\textit{Fr}}=\infty$ exhibits higher production and dissipation compared with ${\textit{Fr}}=6 \text{ and }1.9$. The streamwise vortex pair in the wake induces a significant mean vertical velocity ($U_z$). Therefore, in contrast to straight-on flow, the terms involving gradients of $U_z$ matter to TKE production. Buoyancy reduces $U_z$ and also the Reynolds shear stresses involving $u^{\prime}_z$. Through this indirect mechanism, buoyancy exerts control on the wake TKE budget, albeit being small relative to production and dissipation. Buoyancy, through the baroclinic torque, is found to qualitatively affect the streamwise vorticity. In particular, the primary vortex pair is extinguished in the intermediate wake and two new vortex pairs form with opposite-sense circulation relative to the primary.

This paper investigates the transient characteristics of uniform momentum zones (UMZs) in a rapidly accelerating turbulent pipe flow using direct numerical simulation datasets starting from an initial friction Reynolds number ($Re_{\tau 0}) = 500$ up to a final friction Reynolds number ($Re_{\tau 1}) = 670$. Instantaneous UMZs are identified following the identification methodology proposed by Adrian et al. (2000 J. Fluid Mech. vol. 422, pp. 1–54). The present results reveal that, as the flow rapidly accelerates, the average number of UMZs drops. However, as the flow recovers, it is regained. This result is complemented by the temporal evolution of the average number of internal shear layers. The temporal evolution of UMZs reveals that UMZs sustain their hierarchical flow arrangement with slower zones near the wall and faster zones away from the wall throughout the rapid turbulent flow acceleration. The results show that UMZs speed up during the inertial and pre-transition phases, and progressively slow down during the transition and core-relaxation stages. It is also revealed that UMZs near the wall respond first to flow instability and show earlier signs of recovery based on UMZ kinematic results. Finally, the dominant quadrant behaviour of Reynolds shear stress within UMZs has been investigated. It is found that, prior to the flow excursion, the UMZs nearest to the wall are always $Q2$ dominated, while the rest of the UMZs are always $Q4$ dominated. This behaviour is detected to not change during and after the flow excursion, suggesting that this is a characteristic behaviour of UMZs in accelerating turbulent wall-bounded flows.

For smectic C* (SmC*) liquid crystals, configured in a bookshelf-type geometry between two horizontal parallel plates, with the bottom plate fixed and the top plate free to move, it is known from experiment that pumping can occur when an electric field is applied, i.e. an upward movement of the top plate through mechanical vibrations when the electric field is suddenly reversed. In this paper we revisit an earlier mathematical model for fast electric field reversal by removing an assumption made there on the velocity field; instead, we arrive at a time-dependent, two-dimensional squeeze-film model, which can ultimately be formulated in terms of a highly nonlinear integro-differential equation. Subsequent analysis leads to an unexpected solvability condition involving the five SmC* viscosity coefficients regarding the existence and uniqueness of solutions. Furthermore, we find that, when solutions do exist, they imply that the plate can move down as well as up, with the final resting position turning out to be dependent on the initial conditions; this is in stark contrast to the results of the earlier model.

Permanent gravity waves propagating in deep water, spanning amplitudes from infinitesimal to their theoretical limiting values, remain a classical yet challenging problem due to its inherent nonlinear complexities. Traditional analytical and numerical methods encounter substantial difficulties near the limiting wave condition due to singularities at sharp wave crests. In this study, we propose a novel hybrid framework combining the homotopy analysis method (HAM) with machine learning (ML) to efficiently compute convergent series solutions of Stokes waves in deep water for arbitrary wave amplitudes from small to theoretical limiting values, which show excellent agreement with established benchmarks. We introduce a neural network trained using only 20 representative cases whose series solution are given by means of HAM, which can rapidly predict series solutions across arbitrary steepness levels, substantially improving computational efficiency. Additionally, we develop a neural network to gain the inverse mapping from the conformal coordinates $(\theta , r)$ to the physical coordinates $(x,y)$, facilitating explicit and intuitive representations of series solutions in physical plane. This HAM–ML hybrid framework represents a powerful and efficient approach to compute convergent series in a whole range of physical parameters for water waves with arbitrary wave height including even limiting waves. In this way we establish a new paradigm to quickly obtain convergent series solutions of complex nonlinear systems for a whole range of physical parameters, thereby significantly broadening the scope of series solutions that can be easily gained by means of HAM even for highly nonlinear problems in science and engineering.

Magneto-gravity-precessional instability, which results from the excitation of resonant magneto-inertia-gravity (MIG) waves by a background shear generic to precessional flows, is addressed here. Two simple background precession flows, that of Kerswell (1993 Geophys. Astrophys. Fluid Dyn. vol 72, no. 1–4, pp. 107–144), and that of Mahalov (1993 Phys. Fluids A: Fluid Dyn. vol. 5, no. 4, pp. 891–900), are considered. We analytically perform an asymptotic analysis to order ${ O}(\varepsilon ),$ where $\varepsilon$ denotes the Poincaré number, i.e. the precession parameter, and determine the maximum growth rate of the destabilizing subharmonic resonances of MIG waves: that between two fast modes, that between two slow modes and that between a fast mode and a slow mode (mixed modes). The domains of the $(K_0 B_0/\varOmega _0, N/\varOmega _0)\hbox{-}$plane for which this instability operates are identified, where $1/K_0$ denotes a characteristic length scale, $B_0$ is the unperturbed Alfvén velocity, $\varOmega _0$ is the rotation rate and $N$ denotes the Brunt–Väisälä frequency. We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). At large $K_0B_0/\varOmega _0,$ stable stratification acts to suppress the destabilizing resonance between two fast modes as well as that between two slow modes, whereas it revives the destabilizing resonance between a fast mode and a slow mode provided $N\lt \varOmega _0,$ because, without stratification, the maximal growth rate of this instability approaches zero as $K_0B_0/\varOmega _0\rightarrow +\infty .$ This would be relevant for the generation of the mean electromotive force, and hence, the $\alpha \hbox{-}$effect in helical magnetized precessional flows under weak stable stratification. Diffusive effects on the instability is considered in the simple case where the magnetic and thermal Prandtl numbers are both equal to one.

We examine the linear stability of a shear flow driven by wind stress at the free surface and rotation at the lower boundary, mimicking oceanic flows influenced by surface winds and the Earth’s rotation. The linearised eigenvalue problem is solved using the Chebyshev spectral collocation method and a long-wave asymptotic analysis. Our results reveal new long-wave instability modes that emerge for non-zero rotational Reynolds numbers. It is observed that the most unstable mode, characterised by the lowest critical parameters, corresponds to long-wave spanwise disturbances with vanishing streamwise wavenumber. The asymptotic analysis, which shows excellent agreement with numerical results, analytically confirms the existence of this instability. Thus, the present study demonstrates the hitherto unreported combined influence of wind stress and the Earth’s rotation on ocean dynamics.

Two-way diffusion equations arising in kinetic problems relating to electron scattering and in Brownian particle dynamics present singularities absent from conventional diffusion equations. Although calculations by Stein & Bernstein, and Fisch & Kruskal have revealed the formation of entry and exit slope discontinuities at the critical points where the velocity changes sign, the analytical structure of these discontinuities remains unclear. Here we fill this gap via a local similarity variable analysis, illustrated through the two-way diffusion equation $y \partial n/\partial x=\partial ^2 n/\partial y^2$ in $-1 \leq y \leq 1$; $0 \leq x \leq L$, with $n(x,\pm 1)=0$ with various entry conditions $n(0,y)_{y\gt 0}$, and the exit condition $n(L,y)_{y\lt 0}=0$. The similarity variable $\eta =y/x^{1/3}$ permits the analytical characterization of the entry discontinuity, except for constants determined by matching with numerical solutions obtained with two numerical schemes: separation of variables following the construction of Beals, or finite-difference discretization of the transient partial differential equation, which converges in time to a solution almost identical to the separation of variables solution. Although the slope discontinuity depends markedly on the initial condition $n(0,y)_{y\gt 0}$, a simple general similarity solution structure emerges empirically, always involving a spontaneous singular contribution $C |y|^{1/2}$ at $x=0,y\lt 0$. Slow convergence of both numerical solutions near $\{x,y\}=\{0,0\}$ is attributed to the poor eigenfunction representation of the ever-present singular solution component $|y|^{1/2}$. The similarity approach applies equally to other two-way diffusion equations when the coefficient of $\partial n/\partial x$ changes sign linearly with $y$. It can also be extended to situations where this coefficient is discontinuous at the critical points.

The convection velocity in high-Reynolds-number pipe flow was investigated using two-point correlations obtained from two laser Doppler velocimetry systems. The Reynolds number ranged from ${\textit{Re}}_{{\tau}}=3000$ to 20 800, and profiles were obtained from $y/R=0.002$ up to the pipe centre, where $R$ is the pipe radius. This study examines the scaling behaviour of convection velocity profiles derived from raw velocity signals, and the convection characteristics of very large-scale motions (VLSMs) and large-scale motions extracted via scale-separated or time-resolved velocity signals. The profiles show that convection velocities from raw signals exceed the local mean velocity near the wall and gradually approach it toward the centre. These profiles can be scaled using inner variables, namely $y^+$ and $\Delta x^+$, where $\Delta x^+$ represents the measurement distance. Scale-separated convection velocities for VLSM-scale structures – defined as those larger than $5R$ – were higher than the unfiltered values and remained nearly constant up to $y^+ \leq 2000$ at ${\textit{Re}}_{{\tau}} \approx 20\,000$. In this constant region, the convection velocity of VLSMs scaled well with the bulk velocity $U_{\textit{b}}$, taking values of approximately $0.85U_{\textit{b}}$. Furthermore, analysis of the time-resolved data highlights that, when applying Taylor’s frozen turbulence hypothesis, it is essential to consider both the scale dependence and the temporal fluctuations of the convection velocity, which reflect the underlying spatio-temporal dynamics of the flow structures. The present study provides valuable data for discussions on converting frequency-domain measurements into wavenumber space using Taylor’s hypothesis.

Surface roughness is often present in flight systems travelling at high speeds, but its interaction with compressible turbulence is not well understood. Using direct numerical simulations, we study how prism-shaped roughness influences supersonic turbulent boundary layers at a free-stream Mach number $M_\infty =2$. The dataset includes four simulations featuring cubic- and diamond-shaped elements in aligned and staggered configurations. All cases have an initial smooth region where a fully turbulent boundary layer transitions to a rough wall with positively skewed roughness elements relative to the smooth-wall zero plane. This causes a sudden boundary layer growth at the smooth-to-rough transition, generating an oblique shock wave. Individual roughness elements downstream do not generate shock or expansion waves, as they do not protrude into the supersonic region. For cubical elements, the staggered arrangement increases drag and produces more pronounced boundary layer growth than the aligned case. Rotating the cubes along their vertical axis further enhances these effects, yielding the highest drag. Interestingly, diamond-shaped elements in a staggered arrangement exhibit a dynamics similar to aligned cubes, producing lower drag than other cases. We explain the relative drag induced by each roughness shape by examining viscous and pressure drag components separately. The analysis reveals that, for staggered diamonds, the flow skims more easily over roughness, drastically reducing recirculation in troughs and gaps. In other cases, wake interactions are more prominent, causing spikes of highly positive and negative skin friction, a feature often neglected in reduced-order model formulations.

Droplet vaporisation can exhibit distinct shrinkage kinetic laws depending on the experimental set-up and ambient conditions. In this work, we present a unified approach that combines experiment and theory to identify true shrinkage kinetics across a broad range of droplet vaporisation processes extending beyond the classical D2-law – particularly under realistic conditions involving support fibres or/and inevitable convective effects. Experimentally, we assume a power law $D^n= D_0^n- \textit{Kt}$, where K is the vaporisation rate constant, and re-express it as $(D/D_0)^n = 1 - t/t_{\textit{life}}$ in terms of the normalised droplet diameter $D/D_0$ and time t$ / $tlife relative to the droplet’s initial diameter D0 and lifetime tlife. Taking D as the diameter of a volume-equivalent sphere, the exponent n can be reliably extracted from the slope of the log–log plot of $( 1 - t/t_{\textit{life}})$ against $D/D_0$. The robustness of this method is demonstrated by re-confirming the D2-law for pure fuel droplet evaporation and validating the $D^{3/2} $-law for droplet evaporation under forced convection. We further apply this method to droplet combustion, revealing a significant departure from the D2-law with n $=$ 2.56 ± 0.20–2.65 ± 0.17 across various liquid fuels, unaffected by the presence of support fibres. An even more pronounced departure, with n approaching 3, is observed in droplet combustion within a continuous flame sustained by an auxiliary burner. Theoretically, we develop a more general theory to describe these droplet combustion processes, showing that the observed positive departures mainly result from flame-driven buoyant convection with 2.33 < n < 3, capturing well the experimental data. The same theoretical framework can also account for the negative departures in convection-driven vaporisation processes without flame, thereby providing a unified interpretation for the fundamental distinctions between flame-driven and non-flame-driven droplet vaporisation processes. The present study not only identifies distinct shrinkage power laws that emerge from complexities in these processes, but also reveals the central role of an inherent length scale – arising from underlying convective mechanisms – in shaping the true shrinkage kinetics that lead to violations of the D2-law.

We examine the dynamic interactions between the large-scale coherent motion and the small-scale turbulence in the passive scalar field of a circular cylinder wake, where the coherent motion exhibits strong periodicity. A combination of four X-wires and four cold wires was used to simultaneously measure the three velocity and temperature fluctuations at nominally the same location. Measurements were taken at $x/d=10$, 20 and 40 in the mean shear plane at Reynolds number 2500, based on the cylinder diameter $d$ and the free-stream velocity. The phase-averaging technique is used to distinguish the large-scale coherent motion from the stochastic motion, enabling the construction of phase-averaged structure functions of the passive scalar in the scale phase plane. The maximum of the coherent scalar $\tilde {\theta }$ closely aligns with the minima of the phase-averaged strain $\langle S \rangle$ and the vortex centre, suggesting that heat is contained within the interior of the vortex. The scale-by-scale distributions of the scalar variance and the streamwise velocity variance exhibit a similar phase dependence associated with the coherent motion. This dependence is perceptible even at the smallest scales. However, as the distance from the cylinder increases, the perceivable scale range decreases and eventually disappears. An expression is formulated to describe the time-averaged second-order structure function of coherent scalar and the time-averaged second-order mixed structure function between the coherent scalar and coherent streamwise velocity at $x/d= 10$ and 20, where the coherent motion is prominent. Furthermore, the scale-by-scale contribution of the coherent scalar variance to the total scalar variance is evaluated. Also, we derive the scale-by-scale scalar variance transport equations that account for the coherent motion in both general and isotropic formulations. Assuming local isotropy, it is found that the equation agrees approximately with the experimental data across all scales at $x/d= 40$. Finally, the differences between the scale-by-scale transport equation for the stochastic scalar variance and that for the stochastic turbulent kinetic energy are discussed.

Integral modelling of turbulent buoyant plumes is crucial for rapid predictions of plume characteristics. While the governing equations are typically derived using self-similarity and a Boussinesq approximation, these assumptions may not hold for plumes originating from finite-area sources with large density ratios. This work evaluates the accuracy of integral-scale models for non-Boussinesq lazy plumes using high-fidelity numerical simulations of turbulent helium plumes. We analyse the plume kinematics by computing vertical fluxes, plume radius and radial profiles, establishing some disparities between common practice and physical accuracy. We identify how the definition of the plume radius changes the perception of the plume structure when the flow is not self-similar and derive a relationship between the flux-based and threshold-based definitions without requiring self-similarity. We then examine the plume dynamics by evaluating the source terms from the governing plume equations. Our results support neglecting diffusive and viscous effects but emphasise the importance of the mean pressure gradient, even in the self-similar regime. Two coefficients need to be modelled: the well-known entrainment coefficient and the lesser-known momentum correction coefficient, which is a correction required for the momentum equation to account for self-similar and slender approximations. The momentum correction coefficient is found to be approximately constant and slightly greater than the assumed value of 1. The standard entrainment coefficient models perform well up to a local Richardson number three times the asymptotic value but overpredict entrainment for larger Richardson numbers. We propose a correction using the known finite limit of entrainment at infinite Richardson number.

The nonlinear Tollmien–Schlichting waves mechanism of subcritical transitional flow in quasi-two-dimensional flow and two-dimensional (2-D) plane Poiseuille flow have been investigated (Camobreco et al. 2023 J. Fluid Mech., vol. 963, p. R2; Huang et al. 2024 J. Fluid Mech., vol. 994, p. A6). However, the subcritical transitional flow threshold has remained unsolved for 2-D shear flows since the problem was proposed in Trefethen et al. (1993 Science vol. 261, no. 5121, pp. 578–584). In this study, we proposed a theoretical analysis based on the nonlinear non-modal analysis and asymptotic analysis to quantify the scaling law for subcritical transitional flow of 2-D plane Poiseuille flow. The subcritical transitional flow induced by the critical disturbance experiences the nonlinear edge state with invariant disturbance kinetic energy (Huang et al. 2024 J. Fluid Mech. vol. 994, p. A6). Consequently, the required magnitude along with the edge state is predicted by asymptotic analysis, and the a priori threshold is achieved theoretically. All stages are validated by the numerical minimal seeds of different channels. The proposed theory predicts that the scaling laws are $O(Re^{-11/3})$ and $O(\textit{Re}^{-7/3})$ for the critical disturbances and their edge state, respectively. While the numerical thresholds of the subcritical transitional flow are $ \textit{Re}^{-11/3 \pm 0.06}$ and $ \textit{Re}^{-7/3 \pm 0.05}$, respectively.

A burning droplet in normal gravity inevitably encounters buoyant convection set up by the flame, which can significantly impact its shrinkage kinetics traditionally described by the D2-law. However, the detailed mechanism governing droplet vapourisation under such self-generated flame-driven buoyant convection remains elusive. Here, we present both experimental and theoretical evidence highlighting the critical role of buoyant convection in droplet combustion. Experimentally, we precisely measure the values of the shrinkage exponent n for various liquid fuels, revealing a significant departure from the D2-law. While the measured n values consistently fall within the narrow range 2.6–2.7, they exhibit a slight increase with the fuel’s boiling point. A more general and in-depth theory is also developed to explain such small but systematic variations, revealing that differences in flow and thermal boundary layer structures – arising from varying combustion intensities – may account for the observed trends. Our theory predicts three distinct values of n, namely 2.6, 8/3 ≈ 2.67 and 35/13 ≈ 2.69, successfully capturing slight differences in n among various fuels. This is the first study demonstrating that the shrinkage kinetics in droplet vapourisation driven by flame-induced buoyant convection is nearly universal, determined solely by the underlying transport mechanisms, although these can be significantly altered due to their high susceptibility to detailed fuel chemistry and combustion kinetics. The present theoretical framework not only enables accurate prediction and control of burning droplet behaviour, but also is extendable to analyse more complex combustion processes involving a broader range of fuel types and flow conditions.

Particles in compressible shear flows experience lifting effects due to asymmetric pressure and viscous forces across the particle surface, rotation induced by asymmetric viscous forces (Magnus effect), and asymmetric compression and viscous effects if near a wall (wall effect). This work focuses on the lifting force on a solid spherical particle due to asymmetric pressure and shear stress distributions driven by density and velocity gradients. We show via direct numerical simulation and verify using scaling arguments that the lifting force in unbounded laminar compressible shear flows is a function of dynamic pressure gradient. We show that steady flow regimes demonstrate predictable lifting forces. Unsteady flow regimes demonstrate asymmetric vortex shedding which creates lift in directions not readily predictable. Thus, predicting lift requires the ability to predict wake structure. We develop approximate delineations between wake types at Reynolds numbers up to 20 000. We use the non-dimensional dynamic pressure gradient, Mach number, Reynolds number and predicted wake structure to develop a shear-induced lift model. The proposed model can be used in conjunction with a drag model to simulate particle motion in compressible shear flow.

The present study focuses on the influence of gas swirl on the spray behaviour from a two-fluid coaxial atomiser with high gas-to-liquid dynamic pressure ratios $M$ by varying both the liquid Reynolds number ${\textit{Re}}_l$ and the gas Weber number ${\textit{We}}_g$. The investigations identify the deviations of the carrier phase velocity fields, droplet distribution, and dispersion when swirl is introduced to the gas phase compared with the non-swirling conditions. The changes in the axial, radial and tangential velocities of the continuous phase due to the introduction of swirl are highlighted while retaining a self-similar behaviour. The slip velocity of the large droplets in swirling sprays is negative, unlike the known positive value for non-swirling sprays. The shape of the radial profiles of the mean drop size is investigated along ${\textit{We}}_g$, notably revealing an inflection point for swirling sprays at high-${\textit{We}}_g$ values. A global assessment of the drop size uncovered that swirl leads to its increase for low $M$ while assisting spray formation at high $M$. Additionally, the radial profiles of axial fluxes for swirling sprays have a wider bell-shaped curve compared with non-swirling sprays at high $M$, unlike the off-centre maxima found for low $M$. However, the mentioned dependencies of drop sizes and fluxes cannot be determined by $M$ solely for intermediate gas-to-liquid momentum ratios ($23\lt M\lt 46$), and vary with ${\textit{Re}}_l$ and ${\textit{We}}_g$. In addition, the response of at least the mean droplets at the edge of the spray to the large gas eddies shows a linear relation with swirl intensity.

Unsteady aerodynamic forces in flapping wings arise from complex, nonlinear flow structures that challenge predictive modelling. In this work, we introduce a data-driven framework that links experimentally observed flow structures to sectional pressure loads on physical grounds. The methodology combines proper orthogonal decomposition and quadratic stochastic estimation (QSE) to model and interpret these forces using phase-resolved velocity fields from particle image velocimetry measurements. The velocity data are decomposed in a wing-fixed frame to isolate dominant flow features, and pressure fields are reconstructed by solving the Poisson equation for incompressible flows. The relationship between velocity and pressure modes is captured through QSE, which accounts for nonlinear interactions and higher-order dynamics. We introduce an uncertainty-based convergence criterion to ensure model robustness. Applied to a flapping airfoil, the method predicts normal and axial forces with less than 6 % average error using only two velocity modes. The resulting model reveals an interpretable underlying mechanism: linear terms in the QSE model the circulatory force linked to the formation of vortices on the wing, while quadratic terms capture the nonlinear component due to added-mass effects and flow–vorticity interactions. This data-driven yet physically grounded approach offers a compact tool for modelling the unsteady aerodynamics in flapping systems with potential to generalise to other problems.