Numerical simulations are used to probe Rayleigh–Darcy convection in fluid-saturated porous media towards the ultimate regime. The present three-dimensional dataset, up to Rayleigh–Darcy number $\textit {Ra}=8\times 10^4$, suggests that the appropriate scaling of the Nusselt number is $\textit {Nu}=0.0081\textit {Ra}+0.067\textit {Ra}^{0.61}$, fitting the computed data for $\textit {Ra}\gtrsim 10^3$. Extrapolation of current predictions to the ultimate linear regime yields the asymptotic law $\textit {Nu}=0.0081 \textit {Ra}$, about $16\,\%$ less than indicated in previous studies. Upon examination of the flow structures near the boundaries, we confirm previous indications of small flow cells hierarchically nesting into supercells, and we show evidence that the supercells at the boundary are the footprints of the megaplumes that dominate the interior part of the flow. The present findings pave the way for more accurate modelling of geophysical systems, with special reference to geological $\textrm {CO}_2$ sequestration.

The effect of a small background constant horizontal mean flow on the parametric subharmonic instability (PSI) of locally confined internal wave beams is discussed for the case where the beam frequency is close to twice the inertial frequency due to background rotation. Under this condition, PSI is particularly potent because of the vanishing of the group velocity at the inertial frequency, which prolongs contact of near-inertial subharmonic perturbations with the primary wave. The mean flow generally stabilizes the very short-scale limit of such perturbations. By contrast, the stability of longer-scale perturbations hinges on the strength and the direction of the mean flow; particularly, a negative mean flow (antiparallel to the horizontal projection of the beam group velocity) can extend the sub-inertial range of PSI. However, a large enough mean flow of either sign ultimately weakens PSI.

Regulation of fluid flow by deformations of the surrounding elastic structure is observed in many natural and artificial system, such as in the cardiovascular system. As the first step to study the regulation of oscillating flows, we consider synchronization of vortex shedding past a cylinder within an elastic structure with a sinusoidal external forcing. We use phase-reduction theory to evaluate the synchronization characteristics of the oscillating fluid–structure coupled dynamics. We find that the phase-sensitivity function, which characterizes the phase-response of the oscillation, is significantly affected by the Cauchy number and slightly affected by the fluid-to-structure density ratio and Poisson's ratio of the structure material, for fixed model configuration and Reynolds number. The predicted synchronization characteristics are in close agreement with results from direct numerical simulations. The synchronization region is maximized when the sinusoidal perturbation is applied near the downstream end of the cylinder. These findings open further possibility for the utilization of phase-reduction theory to characterize synchronization in other practical problems exhibiting fluid–structure coupled dynamics, such as in biological systems and the control of microfluidics.

In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves.

In the asymptotic theory of interaction noise from contra-rotating propellers with many blades, previous work in the frequency domain has shown that, for each combination mode/tone, the critical points that dominate the acoustic radiation can be linked to two criteria (sonic and normal-edge) and to a radial event line that spins around the annulus at the same speed as the mode. Thus the speed is different for each combination tone. In real time, however, the interactions precess around the annulus at a single speed, governed by the ‘firing order’ of the interactions, with that speed being (in general) considerably different from that of either the front or rear blade rows. The precession of the interaction is described here in detail and then demonstrated by application to three relevant architectures, including a rotor–stator configuration. The paper then considers how the blade leading-edge design affects the radial variation in the interaction location and analyses the interactions from the viewpoint of a far-field observer. The paper also connects previous time- and frequency-domain results by showing that the sonic and normal-edge criteria can be derived in the time domain using the precession speed. The case of equal blade numbers is also included, for which the precession speed becomes infinite but for which, nonetheless, the sonic and normal-edge criteria still apply.

Laboratory experiments are conducted to investigate the mechanism controlling the formation of stable and unstable acoustic fountains at the free surface of a quiescent body of water. Fountains are induced by focused ultrasonic, a new modality that allows for better spatiotemporal control of water flow. Particle image velocimetry was used to characterize the induced flow field in the vicinity of the ultrasonic focal spot. We used two types of ultrasonic transducers with distinct wave frequencies. We examined three fountain formation regimes by varying the pressure level of the transducers, namely weak, intermediate (stable) and highly forced fountains (explosive). Between different regimes, the fountain height underwent a step-change in response to the increase in acoustic pressure. A force estimation obtained from the flow field shows that the magnitude of axial momentum flux is orders of magnitude lower than that of gravity and surface tension, indicating that the dominant driving force for the fountain generation is the acoustic radiation force (Nightingale et al., Ultrasound Med. Biol., vol. 28, 2002, pp. 227–235). We propose a simple model to estimate the shape of a stable fountain; it accounts for the applied acoustic pressure, gravity, surface tension and axial momentum. The model neglects viscous force, which precludes capturing the intermediate fountain surface curvature. However, the model successfully predicts the geometry of the weak and intermediate fountains.

We report for the first time on triadic resonances in a rotating convection system. Using direct numerical simulations, we find that convective modes in a rotating spherical fluid can excite a pair of inertial modes whose frequencies and wavenumbers match the triadic resonance conditions. Depending on the structures of the convective modes, triadic resonances can lead to the growth of either a pair of modes with lower frequencies and wavenumbers, or a pair of modes with higher frequencies and wavenumbers, providing a possible mechanism for the bi-directional energy cascade. Increased thermal forcing leads to fully developed turbulence, which also exhibits wave-like motions, and is reminiscent of the energy spectrum of inertial wave turbulence. Our results suggest that the interaction of inertial waves plays an important role in rotating convection, which is of great importance in understanding the dynamics of planetary and stellar interiors.

Stability analyses based on the rates of change of perturbations were performed to study the growth mechanisms of second-mode instability in hypersonic boundary layers. The results show that the streamwise velocity perturbation is strengthened by the concurrence of the momentum transfer due to the wall-normal velocity fluctuation and the streamwise gradient of the pressure perturbation near the wall, while the wall-normal velocity perturbation is dominated by the wall-normal gradient of the pressure perturbation. Meanwhile, the change of fluctuating internal energy is sustained by the advection of perturbed thermal energy in the vicinity of the critical layer and by the dilatation fluctuation near the wall. The energy transport by the wall-normal velocity fluctuation accounts for the growth of second-mode instability, and the growth rate depends on the relative phase of the energy transport by the wall-normal velocity fluctuation to the total time rate of change of fluctuating internal energy in the vicinity of the critical layer. Moreover, this relative phase is associated with the mutual interaction between the critical-layer fluctuation and the near-wall fluctuation. Porous walls recast this mutual interaction by delaying the phase of the wall-normal energy transport near the wall, resulting in the stabilization of the second mode.

The celebrated wall-scaling works for many statistical averages in turbulent flows near smooth walls, but the streamwise velocity fluctuation, $u^{\prime }$, is thought to be among the few exceptions. In particular, the near-wall mean-square peak, $\overline {u'u'}^+_p$ – where the superscript $+$ indicates normalization by the friction velocity $u_\tau$, the subscript $p$ indicates the peak value and the overbar indicates time averaging – is known to increase with increasing Reynolds number. The existing explanations suggest a logarithmic growth with respect to $Re$, where $Re$ is the Reynolds number based on $u_\tau$ and the thickness of the wall flow. We show that this boundless growth calls into question the veracity of wall-scaling and so cannot be sustained, and we establish an alternative formula for the peak magnitude that approaches a finite limit $\overline {u'u'}^+_\infty$ owing to the natural constraint of boundedness on the dissipation rate at the wall. This new formula agrees well with the existing data and, in contrast to the logarithmic growth, supports the classical wall-scaling for turbulent intensity at asymptotically high Reynolds numbers.

Equilibrium solutions for hollow vortices in straining flow in a corner are obtained by solving a free-boundary problem. Conformal maps from a canonical doubly connected annular domain to the physical plane combining the Schottky–Klein prime function with an appropriate algebraic map lead to a problem similar to Pocklington's propagating hollow dipole. The result is a two-parameter family of solutions depending on the corner angle and on the non-dimensional ratio of strain to circulation.

Elevated wall temperatures and impinging shock interactions are prevalent features in hypersonic flight. Currently, there is a lack of literature regarding experimental studies examining both features in a flight-representative environment. This work details hot-wall, hypersonic, impinging shock/boundary-layer interaction experiments performed in the T4 Stalker Tube. The model configuration was a two-dimensional heated flat plate and a shock generator. The surface of the graphite flat plate was resistively heated to a mean temperature from $T_w=298\ \textrm {K}$ to $T_w\approx 675\ \textrm {K}$ during an experimental run. An oblique shock, generated by a plate that was inclined at $10^{\circ }$ or $12^{\circ }$ to the free stream, was impinged on the heated flat plate to induce boundary layer separation. The primary flow condition produced Mach 7 flight-equivalent nozzle-supply enthalpy with a unit Reynolds number of $4.93\times 10^6\ \textrm {m}^{-1}$. More flow conditions with lower unit Reynolds numbers and flow enthalpies were considered to examine flow separation characteristics. Schlieren and infrared thermography captured the flow field and the wall temperature distribution, respectively. The results showed that the size of the flow separation grew with a higher $T_w$ and a lower unit Reynolds number. Moreover, the scaled separation of the present data showed a high discrepancy with existing separation correlations developed from a supersonic impinging shock and a hypersonic compression ramp, mainly due to the higher shock strength. Instead, the present data followed a scaling law that includes the pressure ratio across the impinging shock with a slight dependence on the wall temperature ratio.

Laminar flows through pipes driven at steady, pulsatile or oscillatory rates undergo a subcritical transition to turbulence. We carry out an extensive linear non-modal stability analysis of these flows and show that for sufficiently high pulsation amplitudes the stream-wise vortices of the classic lift-up mechanism are outperformed by helical disturbances exhibiting an Orr-like mechanism. In oscillatory flow, the energy amplification depends solely on the Reynolds number based on the Stokes-layer thickness, and for sufficiently high oscillation frequency and Reynolds number, axisymmetric disturbances dominate. In the high-frequency limit, these axisymmetric disturbances are exactly similar to those recently identified by Biau (J. Fluid Mech., vol. 794, 2016, R4) for oscillatory flow over a flat plate. In all regimes of pulsatile and oscillatory pipe flow, the optimal helical and axisymmetric disturbances are triggered in the deceleration phase and reach their peaks in typically less than a period. Their maximum energy gain scales exponentially with Reynolds number of the oscillatory flow component. Our numerical computations unveil a plausible mechanism for the turbulence observed experimentally in pulsatile and oscillatory pipe flow.

This paper presents a quantitative evaluation of the Reynolds analogy factor $s=2c_h/c_f$ for self-similar turbulent boundary layers with pressure gradients in the streamwise direction via direct numerical simulation. Both sub- and supersonic cases are considered at nearly adiabatic wall conditions. The Reynolds analogy factor is found to be greatly increased for adverse-pressure-gradient cases and decreased for favourable-pressure-gradient cases. Although the boundary layers considered cover a comparatively large Reynolds-number range from small to moderate Reynolds numbers, no Reynolds-number dependency of $s$ is found in the parameter range investigated. Mach-number influences on $s$ are found to be small; $s$ decreases slightly with increasing Mach number. The influence of pressure gradients on $s$ turns out to be well approximated by the analytical relation derived by So (Intl J. Heat Mass Transfer, vol. 37, 1994, pp. 27–41) for incompressible flow if a fixed, calibrated Reynolds number is used. Moreover, the effects of non-self-similarity prior to the self-similar region are assessed.

High-resolution simulations of temporally evolving mixing layers, for convective Mach numbers ranging from $M_c=0.2$ to $M_c=2.0$ with density ratios $s=1$ and $s=7$, are analysed to characterize compressibility effects on the structure and evolution of turbulence in this compressible flow. Published experimental results are used to validate simulation results. Examination of the turbulence scales in the present data suggests an internal regulation mechanism. Correlated eddying motions were found to be in support of a ‘sonic eddy hypothesis’. Eddy scales in all spatial directions are found to be a progressively smaller fraction of the overall mixing-layer thickness with increasing $M_c$, forming independent layers of eddying motions at high $M_c$. These reduced spatial scales serve to reduce the effective velocity scale for turbulent motions, suppressed Reynolds stresses, turbulent kinetic energy (TKE) production and dissipation, and the mixing-layer thickness growth rate.

Flapping-wing-based propulsion is ubiquitous in Nature, and it is free in all directions. In this work, the hydrodynamic behaviour of an unconstrained flapping foil, which can self-propel in both longitudinal and lateral directions, is numerically studied. It is found that the flapping foil can keep self-propelling in a straight line along the longitudinal direction, together with a passive oscillation in the lateral direction. Moreover, the effects of multiple parameters on the performance of the flapping swimmer are investigated, including the flapping frequency and amplitude, the mass ratio between foil and fluid, and the thickness–chord ratio of the foil. It is shown that the propulsive speed, the power consumption and the lateral oscillating motion obey some simple scaling laws. The results obtained here may provide some light on understanding biological flapping-wing-based propulsion.

We present here the first study on the stability of plane Poiseuille flow when the fluid is stratified in density perpendicularly to the plane of horizontal shear. Using laboratory experiments, linear stability analyses and direct numerical simulations, we describe the appearance of an instability that results from a resonance of internal gravity waves and Tollmien–Schlichting waves carried by the flow. This instability takes the form of long meanders confined in thin horizontal layers stacked along the vertical axis.

With the growing interest in scale-resolving simulations of spatially evolving boundary layers, synthetic turbulence generation (STG) has become a valuable tool for providing unsteady turbulent boundary conditions through a sum over a finite number of spatio-temporal Fourier modes with amplitude, direction and phase determined by a random number set. Recent developments of STG methods are designed to match target profiles for anisotropic and inhomogeneous Reynolds stresses. In this paper, it is shown that, for practical values of the number of modes, a given set of random numbers may produce Reynolds stress profiles that are 30 % off their target. To remedy this situation, the error in the STG stress prediction is decomposed into a steady-state bias and a purely unsteady part affecting the time convergence. Direct relationships between the random number vectors and both types of error are developed, allowing large collections of random number sets to be rapidly scanned and the best performers selected for a much improved agreement with the target. The process is verified for the inflow to a direct numerical simulation of a flat plate at $Re_\theta = 1000$. This paper demonstrates sufficient time convergence over a few flow-through times as well as a correction of the method's biases.

We investigate experimentally the behaviour of an elongated flexible cylinder settling at moderate Reynolds number under the effect of buoyancy in a fluid otherwise at rest. The experiments uncover the development of large-amplitude periodic deformations of the cylinder (of the order of its diameter) in specific parameter ranges. Bending oscillations are observed to occur for two base flow situations, involving either a steady or an unsteady wake. In both cases, the sequence of oscillatory deformations emerging when the cylinder length is increased involves the bending modes of an unsupported cylinder with free ends. Comparison of the deformation frequency measured for the falling cylinder with the vortex shedding frequency expected for a non-deformable cylinder at the same Reynolds number indicates that the deformation is coupled to the wake unsteadiness. It also suggests that the cylinder degrees of freedom in deformability allow wake instability to be triggered at Reynolds numbers that would be subcritical for fixed rigid cylinders.

The production of broadband frequency spectra from narrowband wave forcing in geophysical flows remains an open problem. Here we consider a related theoretical problem that points to the role of time-dependent vortical flow in producing this effect. Specifically, we apply multi-scale analysis to the transport equation of wave action density in a homogeneous stationary random background flow under the Wentzel–Kramers–Brillouin approximation. We find that, when some time dependence in the mean flow is retained, wave action density diffuses both along and across surfaces of constant frequency in wavenumber–frequency space; this stands in contrast to previous results showing that diffusion occurs only along constant-frequency surfaces when the mean flow is steady. A self-similar random background velocity field is used to show that the magnitude of this frequency diffusion depends non-monotonically on the time scale of variation of the velocity field. Numerical solutions of the ray-tracing equations for rotating shallow water illustrate and confirm our theoretical predictions. Notably, the mean intrinsic wave frequency increases in time, which by wave action conservation implies a concomitant increase of wave energy at the expense of the energy of the background flow.

Steady two-dimensional Rayleigh–Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios ${\rm \pi} /5\leqslant \varGamma \leqslant 4{\rm \pi}$, where $\varGamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\leqslant Ra\leqslant 10^{11}$, and four orders of magnitude in the Prandtl number, $10^{-2}\leqslant Pr\leqslant 10^2$. At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$. The prefactor of this scaling depends on $\varGamma$ and is largest at $\varGamma \approx 1.9$. The Reynolds number $Re$ for large-$Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\varGamma \approx 4.5$. All of these large-$Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox (Phys. Fluids, vol. 21, 2009, 083603). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\varGamma$ is smaller.