Linear modal instabilities of flow over untapered wings with aspect ratios $AR=4$ and 8, based on the NACA 0015 profile, have been investigated numerically over a range of angles of attack, $\alpha$, and angles of sweep, $\varLambda$, at chord Reynolds numbers $100\le Re\le 400$. Laminar base flows have been generated using direct numerical simulation and selective frequency damping, as appropriate. Several families of unstable three-dimensional linear global (TriGlobal) eigenmodes have been identified and their dependence on geometric parameters has been examined in detail at $Re=400$. The leading global mode A is associated with the peak recirculation in the three-dimensional laminar separation bubble formed on the wing and becomes unstable when recirculation reaches $\textit {O}(10\,\%)$. On unswept wings, this mode peaks in the midspan region of the wake and moves towards the wing tip with increasing $\varLambda$, following the displacement of peak recirculation; its linear amplification leads to wake unsteadiness. Additional amplified modes exist at nearly the same and higher frequencies compared to mode A. The critical $Re$ has been identified and it is shown that amplification increases with increasing sweep, up to $\varLambda \approx 10^\circ$. At higher $\varLambda$, all global modes become less amplified and are ultimately stable at $\varLambda =30^\circ$. An increase in amplification of the leading mode with sweep was not observed over the $AR=4$ wing, where tip vortex effects were shown to dominate.

Modal decomposition techniques have become important mathematical methodologies in analysing complex flow physics. The data-driven methods, such as proper orthogonal decomposition and dynamic mode decomposition are used to extract coherent structures in the form of spatial modes. These methods can be applied to both numerical and experimental data to identify the characteristic dynamics of the system. However, the classical data-driven modal decomposition methods are only applicable to problems with a fixed shape and are not suitable for systems with fluid–structure interaction or systems with shape-changing geometries. In this paper we propose a novel method utilizing a conformal mapping technique to solve this issue. Through different examples with deforming geometry, the capability of the method to accurately capture the flow features is demonstrated. The proposed method is found to be suitable for a wide range of applications and a possible candidate for reduced-order flow modelling of complex shape-changing systems.

This article presents a data-driven model based on modal decomposition, applied to approximate the low-order statistics of the spatially averaged wall-shear stress in a turbulent channel flow over a porous wall with two anisotropic permeabilities, producing drag increase or reduction when compared with the case of an isotropic porous wall. The model is comparable to a neural network architecture using a linear map to a classification. To create this model, we use high-order dynamic mode decomposition (DMD) to identify the structures describing the main flow dynamics, and then test different linear combinations of these modes to estimate the time evolution of the stress at the porous interface. The coefficients of the model are obtained by training the model against the results of direct numerical simulations over different time intervals. Depending on the number and the way of combining the DMD modes, the reduced-order models presented can reconstruct the wall-shear stress with relative error smaller than 0.01 % and reproduce its statistical variations for at least 1500 time units with relative error in the standard deviation or the mean smaller than 5 %. The model has also been tested to approximate the statistics of the wall-shear stress over the whole wall, showing that the regeneration of the flow structures can be reproduced by the nonlinear interaction of modes. Finally, considering the DMD modes as communities in a neural network, we examine the influence of the mode-to-mode interaction on the nonlinear flow dynamics, which explains the performance of the different models.

Vortex-induced vibration of a curved flexible cylinder placed in the test section of a recirculating water tunnel and fixed at both ends is studied experimentally. Both the concave and the convex orientations (with respect to the incoming flow direction) are considered. The cylinder was hung by its own weight with a dimensionless radius of curvature of $R/D=66$, and a low mass ratio of $m^{*} = 3.6$. A high-speed imaging technique was employed to record the oscillations of the cylinder in the cross-flow direction for a reduced velocity range of $U^{*} = 3.7 - 48.4$, corresponding to a Reynolds number range of $Re= 165 - 2146$. Mono- and multi-frequency responses as well as transition from low-mode-number to high-mode-number oscillations were observed. Regardless of the type of curvature, both odd and even mode shapes were excited in the cross-flow directions. However, the response of the system, in terms of the excited modes, amplitudes and frequencies of the oscillations, was observed to be sensitive to the direction of the curvature (i.e. concave vs convex), in particular at higher reduced velocities, where mode transition occurred. Hydrogen bubble flow visualization exhibited highly three-dimensional vortex shedding patterns in the wake of the cylinder, where there existed spatial and temporal evolution of the vortex shedding modes along the length of the cylinder. The time-varying intermittent vortex shedding in the wake of the cylinder was linked to the spanwise travelling wave behaviour of the vortex-induced vibration response. The observed spatially altering wake corresponded to the multi-modal excitation and mode transition along the length of the cylinder.

The near wake of a hemisphere immersed in a laminar boundary layer is studied utilizing time-resolved tomographic particle image velocimetry (TPIV). Focus is placed on the three-dimensional vortical structures and the formation details of hairpin vortices before the onset of transition. The three-dimensional instantaneous pressure field of the hemisphere wake is reconstructed for better understanding the flow mechanism. Experiments are carried out with Reynolds number $Re_{r}=1370$, based on the hemisphere radius $R$. Features of periodicity of the near wake are analysed using proper orthogonal decomposition and Fourier transformation. The velocity fluctuation in the wall-normal direction is shown to be crucial to the formation of hairpin vortices in the near wake. By investigating the transport of mass and vorticity, and the correlation between pressure and hairpin vortex strength, the formation mechanism is revealed clearly. Specifically, the main hairpin vortices (MHVs) are formed within the reaction of outer high-speed flow and near-wall flow. The formation of the head portion is followed by the leg portion formation. The shedding of the MHVs is highly correlated with the pressure, as well as the pressure gradient in the wall-normal direction. For the side hairpin vortices (SHVs), the leg portion is formed first, followed by the generation of the head portion thanks to induction of the re-oriented standing vortices. The generation of the SHVs can be regarded as the downstream bridging of the standing vortices, similar to the generation of hairpin vortices due to the connection of streamwise vortices in turbulent boundary layers.

Pulsatile channel and pipe flows constitute a fundamental flow configuration with significant bearing on many applications in the engineering and medical sciences. Rotating machinery, hydraulic pumps or cardiovascular systems are dominated by time-periodic flows, and their stability characteristics play an important role in their efficient and proper operation. While previous work has mainly concentrated on the modal, harmonic response to an oscillatory or pulsatile base flow, this study employs a direct–adjoint optimisation technique to assess short-term instabilities, identify transient energy-amplification mechanisms and determine their prevalence within a wide parameter space. At low pulsation amplitudes, the transient dynamics is found to be similar to that resulting from the equivalent steady parabolic flow profile, and the oscillating flow component appears to have only a weak effect. After a critical pulsation amplitude is surpassed, linear transient growth is shown to increase exponentially with the pulsation amplitude and to occur mainly during the slow part of the pulsation cycle. In this latter regime, a detailed analysis of the energy transfer mechanisms demonstrates that the huge linear transient growth factors are the result of an optimal combination of Orr mechanism and intracyclic normal-mode growth during half a pulsation cycle. Two-dimensional sinuous perturbations are favoured in channel flow, while pipe flow is dominated by helical perturbations. An extensive parameter study is presented that tracks these flow features across variations in the pulsation amplitude, Reynolds and Womersley numbers, perturbation wavenumbers and imposed time horizon.

We carry out direct numerical simulations (DNS) of flow in a turbulent square duct by focusing on heat transfer effects, considering the case of unit Prandtl number. Reynolds numbers up to ${{Re}}_{\tau } \approx 2000$ are considered that are much higher than in previous studies, and that yield clear scale separation between inner- and outer-layer dynamics. Close similarity between the behaviour of the temperature and the streamwise velocity fields is confirmed as in previous studies related to plane channels and pipes. Just like the mean velocity, the mean temperature is found to exhibit logarithmic layers as a function of the nearest wall, but with a different slope. The most important practical implication is the validity of the traditional hydraulic diameter as the correct reference length for reporting heat transfer data, as we show rigorously here. Temperature and velocity fluctuations also have similar behaviour, but apparently logarithmic growth of their inner-scaled peak variances is not observed here, unlike in canonical wall-bounded flows. Analysis of the split contributions to the heat transfer coefficient shows that mean cross-stream convection associated with secondary motions is responsible for about $5\,\%$ of the total. Finally, we use the DNS database to highlight shortcomings of traditional linear closures for the turbulent heat flux, and show that substantial modelling improvement in principle may be obtained by retaining at least the three terms in the vector polynomial integrity basis expansion.

The mean vertical convective heat transport $\langle wT \rangle$ between isothermal plates driven by uniform internal heating is investigated by means of rigorous bounds. These are obtained as a function of the Rayleigh number R by constructing feasible solutions to a convex variational problem, derived using a formulation of the classical background method in terms of a quadratic auxiliary function. When the fluid's temperature relative to the boundaries is allowed to be positive or negative, numerical solution of the variational problem shows that best previous bound $\langle wT \rangle \leqslant 1/2$ (Goluskin & Spiegel, Phys. Lett. A, vol. 377, issue 1–2, 2012, pp. 83–92) can only be improved up to finite R. Indeed, we demonstrate analytically that $\langle wT \rangle \leqslant 2^{-21/5} {\textit {R}}^{1/5}$ and therefore prove that $\langle wT\rangle < 1/2$ for ${\textit {R}} < 65\,536$. However, if the minimum principle for temperature is invoked, which asserts that internal temperature is at least as large as the temperature of the isothermal boundaries, then numerically optimised bounds are strictly smaller than $1/2$ until at least ${\textit {R}}=3.4\times 10^{5}$. While the computational results suggest that the best bound on $\langle wT\rangle$ approaches $1/2$ asymptotically from below as ${\textit {R}}\rightarrow \infty$, we prove that typical analytical constructions cannot be used to prove this conjecture.

The modelling of natural convection in porous media is receiving increased interest due to its significance in environmental and engineering problems. State-of-the-art simulations are based on the classic macroscopic Darcy–Oberbeck–Boussinesq (DOB) equations, which are widely accepted to capture the underlying physics of convection in porous media provided the Darcy number, $Da$, is small. In this paper we analyse and extend the recent pore-resolved direct numerical simulations (DNS) of Gasow et al. (J. Fluid Mech, vol. 891, 2020, p. A25) and show that the macroscopic diffusion, which is neglected in DOB, is of the same order (with respect to $Da$) as the buoyancy force and the Darcy drag. Consequently, the macroscopic diffusion must be modelled even if the value of $Da$ is small. We propose a ‘two-length-scale diffusion’ model, in which the effect of the pore scale on the momentum transport is approximated with a macroscopic diffusion term. This term is determined by both the macroscopic length scale and the pore scale. It includes a transport coefficient that solely depends on the pore-scale geometry. Simulations of our model render a more accurate Sherwood number, root mean square (r.m.s.) of the mass concentration and r.m.s. of the velocity than simulations that employ the DOB equations. In particular, we find that the Sherwood number $Sh$ increases with decreasing porosity and with increasing Schmidt number $(Sc)$. In addition, for high values of $Ra$ and high porosities, $Sh$ scales nonlinearly. These trends agree with the DNS, but are not captured in the DOB simulations.

To investigate the transport of heavy small particles (inertial particles) in high-Reynolds- number wall turbulence, we conduct direct numerical simulations of inertial particles in turbulent channel flow at the friction Reynolds number $Re_\tau =1000$. In the statistically steady state, particles distribute inhomogeneously; particles with different relaxation times form voids with different sizes in the bulk of the flow, whereas they form streak-like clusters with different widths along the streamwise direction. To explore the origin of the multiscale voids and clusters of particles, we identify objectively the axes of tubular vortices and the spines of streaks with different sizes. These identifications enable us to show quantitatively that (i) vortices sweep out the particles of the relaxation time comparable with their turnover time, irrespective of their size and existing height, and that, among these particles, (ii) those swept out by wall-detached tubular vortices form clusters isotropically around them, whereas (iii) those swept out by wall-attached vortices are attracted by a nearby low-speed streak. These explain the reason why the multiscale clusterings are well described in terms of the local Stokes number defined by the turnover time of multiscale vortices. Furthermore, these descriptions of the particle transport give us a clear view to understand velocity statistics and wall-deposition mechanism of inertial particles in high-Reynolds-number wall turbulence.

The aim of this paper is to provide a complete description of the bifurcation scenario of a uniform flow past a rotating circular cylinder up to $Re = 200$. Linear stability theory is used to depict the neutral curves and analyse the arising unstable global modes. Three codimension-two bifurcation points are identified, namely a Takens–Bogdanov, a cusp and generalised Hopf, which are closely related to qualitative changes in orbit dynamics. The occurrence of the cusp and Takens–Bogdanov bifurcations for very close parameters (corresponding to an imperfect codimension-three bifurcation) is shown to be responsible for the existence of multiple steady states, as already observed in previous studies. Two bistability regions are identified, the first with two stable fixed points and the second with a fixed point and a cycle. The presence of homoclinic and heteroclinic orbits, which are classical in the presence of Takens–Bogdanov bifurcations, is confirmed by direct numerical simulations. Finally, a weakly nonlinear analysis is performed in the neighbourhood of the generalised Hopf, showing that above this point the Hopf bifurcation is subcritical, leading to a third range of bistability characterised by both a stable fixed point and a stable cycle.