Studies of Kelvin–Helmholtz (KH) instability have typically modelled the initial flow as an isolated shear layer. In geophysical cases, however, the instability often occurs near boundaries and may therefore be influenced by boundary proximity effects. Ensembles of direct numerical simulations are conducted to understand the effect of boundary proximity on the evolution of the instability and the resulting turbulence. Ensemble averages are used to reduce sensitivity to small variations in initial conditions. Both the transition to turbulence and the resulting turbulent mixing are modified when the shear layer is near a boundary: the time scales for the onset of instability and turbulence are longer, and the height of the KH billow is reduced. Subharmonic instability is suppressed by the boundary because phase lock is prevented due to the diverging phase speeds of the KH and subharmonic modes. In addition, the disruptive influence of three-dimensional secondary instabilities on pairing is more profound as the two events coincide more closely. When the shear layer is far from the boundary, the shear-aligned convective instability is dominant; however, secondary central-core instability takes over when the shear layer is close to the boundary, providing an alternate route for the transition to turbulence. Both the efficiency of the resulting mixing and the turbulent diffusivity are dramatically reduced by boundary proximity effects.

We derive a mathematical model of an electromagnetic (EM) field generated by tsunamigenic seabed deformation over an ocean of constant depth. We solve the governing Maxwell equations for the EM field, coupled with a potential flow model of Cauchy–Poisson type for the transient fluid motion forced by seabed deformation. Our new model advances previous studies, where simplified formulae without direct forcing were assumed for the wave field. Using complex integration and large-time asymptotics, we obtain a novel analytical solution for the magnetic field propagating at large distance from the seabed deformation in two dimensions. We show that this magnetic field is made of two terms, one proportional to an Airy function, and thus propagating similarly to the surface gravity wave, and one proportional to a Scorer function, which exhibits a phase lag with respect to the surface gravity wave. Such a phase lag explains the time difference between the arrival of the EM field and the surface gravity wave generated by seabed deformation, which were observed in recent measurements and numerical results. Finally, we discuss the opportunity to detect EM fields as precursors of surface gravity waves in tsunami early warning systems. We introduce a novel non-dimensional parameter to identify the propagation regime of the magnetic field, i.e. self-induction versus diffusion-dominated. We show that tsunami early warning via EM field is possible for diffusion-dominated regimes when the water depth is less than 2 km. Our findings provide a rigorous analytical explanation of existing observations and numerical results.

We interpret the Taylor–Green cellular vortex model in terms of the Kolmogorov length and velocity scales, in order to study the balance between aggregation and breakup of cohesive sediment in fine-scale turbulence. One-way coupled numerical simulations, which capture the effects of cohesive, lubrication and direct contact forces on the flocculation process, reproduce the non-monotonic relationship between the equilibrium floc size and shear rate observed in previous experiments. The one-way coupled results are confirmed by select two-way coupled simulations. Intermediate shear gives rise to the largest flocs, as it promotes preferential concentration of the primary particles without generating sufficiently strong turbulent stresses to break up the emerging aggregates. We find that the optimal intermediate shear rate increases for stronger cohesion and smaller particle-to-fluid density ratios, and we propose a simple model for the equilibrium floc size that agrees well with experimental data reported in the literature.

The $n$th-order velocity structure function $S_n$ in homogeneous isotropic turbulence is usually represented by $S_n \sim r^{\zeta _n}$, where the spatial separation $r$ lies within the inertial range. The first prediction for $\zeta _n$ (i.e. $\zeta _3=n/3$) was proposed by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941) using a dimensional argument. Subsequently, starting with Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85), models for the intermittency of the turbulent energy dissipation have predicted values of $\zeta _n$ that, except for $n=3$, differ from $n/3$. In order to assess differences between predictions of $\zeta _n$, we use the Hölder inequality to derive exact relations, denoted plausibility constraints. We first derive the constraint $(p_3-p_1)\zeta _{2p_2} = (p_3 -p_2)\zeta _{2p_1} +(p_2-p_1)\zeta _{2p_3}$ between the exponents $\zeta _{2p}$, where $p_1 \leq p_2 \leq p_3$ are any three positive numbers. It is further shown that this relation leads to $\zeta _{2p} = p \zeta _2$. It is also shown that the relation $\zeta _n=n/3$, which complies with $\zeta _{2p} = p \zeta _2$, can be derived from constraints imposed on $\zeta _n$ using the Cauchy–Schwarz inequality, a special case of the Hölder inequality. These results show that while the intermittency of $\epsilon$, which is not ignored in the present analysis, is not incompatible with the plausible relation $\zeta _n=n/3$, the prediction $\zeta _n=n/3 +\alpha _n$ is not plausible, unless $\alpha _n =0$.

The paper presents a Fluid Structure Interaction (FSI) method for hydroelastic water entry. The method assumes that the momentum exchange between the fluid and solid body can be used for the calculation of pressure, deformation and stresses arising during impact. The flexible fluid–structure interactions of flat plates entering water are solved using a computational code that employs the finite volume method to discretise both fluid and solid equations. This provides a better matching of momentum on the fluid–solid interface. The momentum arising in the solid body that emerges after the impact is defined as the momentum exchange, and is shown to increase linearly under the increase of non-dimensional impact speed. The ratio of the maximum pressure arising in an elastic body entering water to that of a rigid body is termed relative pressure and is shown to decrease linearly as a function of momentum exchange. The latter verifies the main hypothesis of this paper, namely that ‘the pressure acting on an elastic body can be predicted using an unsophisticated equation that uses the momentum exchange.’ The deformation and stresses arising in elastic plates entering water are demonstrated to be functions of momentum exchange and can be found using simple equations formulated via parametrisation of data. It is concluded that subject to further validation, the method could be extended for the prediction of hydroelastic response of other sections/bodies entering water.

Viscous fingering or Saffman–Taylor instability shows fingering interfacial patterns when a more mobile fluid displaces a less mobile one in porous media. The effective interfacial tension (EIT) is like capillary force, acting at the miscible interface on time scales shorter than interface relaxation. It has been numerically reported so far that the fingers are wider with EIT compared with those without EIT. A recent experiment observed finger widening with increasing flow rate in a miscible system with EIT, which is not observed in classical immiscible and miscible systems. In this study, we have numerically investigated the effect of $Pe$ (which corresponds to the injection flow rates in the experiment) on miscible viscous fingering with/without EIT. We have shown that the fingers are monotonically thinner with an increase in $Pe$ in the system without EIT, while finger widening with increasing $Pe$ is observed in the system with EIT. Furthermore, we have also examined a one-dimensional underlying concentration profile and EIT profile by using a one-dimensional diffusion–convection model because EIT is proportional to the squared concentration gradient. We have then shown that the concentration gradient is steeper and, thus, the EIT is larger as $Pe$ is larger. Therefore, this is the first numerical study that can theoretically verify finger widening with increasing flow rate, which occurs only in a miscible system with EIT to the extent of our targeting EIT values, and explain the mechanism by one-dimensional analysis.

Flow patterns of causal significance to three-dimensional isotropic turbulence are identified through the recently introduced algorithm of Jiménez (J. Fluid Mech., vol. 854, 2018, R1). Localised perturbations are introduced at arbitrary regions of a triple-periodic decaying flow at ${Re}_\lambda \approx 190$, and their evolution is used as a marker of the significance of said regions to the flow. Their dimensions are found to be an important parameter, with sizes of the order of the integral scale being controlled by the kinetic energy content, and sizes within the dissipative range, by the enstrophy and dissipation. The three quantities are found to be important at intermediate (inertial) scales. Prominent differences emerge between regions of high and low significance. The former typically contain strong gradients and/or kinetic energy and the latter are weak. An analysis of the structure of significant and insignificant flow patterns reveals that strain is more efficient than vorticity at propagating the contents of the perturbation to other regions of the flow. Moreover, the flow patterns of significant regions are found to be more complex, typically containing vortex clusters, while simpler vortex sheets are found in insignificant regions. The present results suggest that strategies aiming to manipulate the flow should focus on strain-dominated vortex clusters, avoiding enstrophy-dominated vortex sheets. This is confirmed through an assimilation experiment, in which greater synchronisation between two simulations is achieved when simulations share significant regions rather than insignificant ones. These conclusions have implications for both the control of turbulent flows and the making of predictions based on limited or noisy measurements.

Modelling the noise emitted by turbulent jets is made difficult by their acoustic inefficiency: only a tiny fraction of the near-field turbulent kinetic energy is propagated to the far field as acoustic waves. As a result, jet-noise models must accurately capture this small, acoustically efficient component hidden among comparatively inefficient fluctuations. In this paper, we identify this acoustically efficient near-field source from large-eddy simulation data and use it to inform a predictive model. Our approach uses the resolvent framework, in which the source takes the form of nonlinear fluctuation terms that act as a forcing on the linearised Navier–Stokes equations. First, we identify the forcing that, when acted on by the resolvent operator, produces the leading spectral proper orthogonal decomposition modes in the acoustic field for a Mach 0.4 jet. Second, the radiating components of this forcing are isolated by retaining only portions with a supersonic phase speed. This component makes up less than 0.05 % of the total forcing energy but generates most of the acoustic response, especially at peak (downstream) radiation angles. Finally, we propose an empirical model for the identified acoustically efficient forcing components. The model is tested at other Mach numbers and flight-stream conditions and predicts noise within 2 dB accuracy for a range of frequencies, downstream angles and flight conditions.

The wall dependence of length scales used to describe large- and small-scale structures of turbulence is examined using highly resolved experiments in zero-pressure-gradient turbulent boundary layers and pipe flows spanning the range $2000< Re_\tau <37\ 700$. Of particular interest is the influence of external intermittency on the scaling of these length scales. It is found that when suitable scaling parameters are selected and external intermittency is accounted for, the dissipative motions follow inner scaling even into the outer-scaled regions of the flow, and that certain large-scale descriptions follow outer scaling even in the inner-scaled regions of the flow. The wall dependence is the same for both internal pipe and external boundary layer flows, and the different length scales can be related to recognizable features in the longitudinal wavenumber spectrum.

The perturbations existing on a breaking wavefront can be a potential explanation for the slamming pressure variability in wave impacts. Here, we investigate the effect of these perturbations by forced vertical slamming of a two-dimensional circular cylinder with constant downward velocity on standing waves. Through experimental modelling and numerical simulation, the slamming force is measured for several standing wave amplitudes and wavelengths. The standing wave phase is tuned such that the impact occurs symmetrically at the instant of maximum crest or trough. Our observations show that slamming coefficients vary with the standing wave amplitude when the wavelength is kept constant and vice versa. The trough impact slamming coefficient can be more than two times the flat impact, and up to four times the crest impact. The experimental results are reproduced by numerical simulations and they agree reasonably well in general. Two analytical approaches based on the von Kármán (NACA, vol. 321, 1929, pp. 1–8) and Wagner (Z. Angew. Math. Mech., vol. 12, 1932, pp. 913–215) methods, which consider the effect of water surface curvature, are introduced. The slamming coefficient calculated from these methods can provide a bound in which the slamming coefficient can be found for each standing wave amplitude and wavelength. Further insight is achieved by numerical simulations of impact on the shorter wavelength to diameter ratio of $0.05<\lambda /D<0.4$. As the wavelength to diameter ratio becomes smaller, the cylinder impacts the water surface at several locations. As a result, multiple peaks occur, and the trapped air at different locations between the cylinder and the water surface yields oscillations with different frequencies on the slamming coefficient time history.

Spatial linear stability analysis is used to study the axisymmetric screech tones generated by twin converging round nozzles at low supersonic Mach numbers. Vortex-sheet and finite-thickness models allow for identification of the different waves supported by the flow at different conditions. Regions of the frequency–wavenumber domain for which the upstream-propagating guided jet modes are observed to be neutrally stable are observed to vary as a function of solution symmetry, jet separation, $S$, and the velocity profile used. Screech-frequency predictions performed using wavenumbers obtained from both models agree well with experimental data. Predictions obtained from the finite-thickness model better align with the screech tones measured experimentally and so are seen to be an improvement on predictions made with the vortex sheet. Additionally, results from the finite-thickness model predict both symmetric and antisymmetric screech tones for low $S$ that are found in the vortex-sheet model only at greater $S$. The present results indicate that the feedback loop generating these screech tones is similar to that observed for single-jet resonance, with equivalent upstream and downstream modes.

Indirect noise is a significant contributor to aircraft engine noise, which needs to be minimized in the design of aircraft engines. Indirect noise is caused by the acceleration of flow inhomogeneities through a nozzle. High-fidelity simulations showed that some flow inhomogeneities can be chemically reacting when they leave the combustor and enter the nozzle (Giusti et al., Trans. ASME J. Engng Gas Turbines Power, vol. 141, issue 1, 2019). The state-of-the-art models, however, are limited to chemically non-reacting (frozen) flows. In this work, first, we propose a low-order model to predict indirect noise in nozzle flows with reacting inhomogeneities. Second, we identify the physical sources of sound, which generate indirect noise via two physical mechanisms: (i) chemical reaction generates compositional perturbations, thereby adding to compositional noise; and (ii) exothermic reaction generates entropy perturbations. Third, we numerically compute the nozzle transfer functions for different frequency ranges (Helmholtz numbers) and reaction rates (Damköhler numbers) in subsonic flows with hydrogen and methane inhomogeneities. Fourth, we extend the model to supersonic flows. We find that hydrogen inhomogeneities have a larger impact on indirect noise than methane inhomogeneities. Both the Damköhler number and the Helmholtz number markedly influence the phase and magnitude of the transmitted and reflected waves, which affect sound generation and thermoacoustic stability. This work provides a physics-based low-order model which can open new opportunities for predicting noise emissions and instabilities in aeronautical gas turbines with multi-physics flows.

We present pore-resolved compressible direct numerical simulations of turbulent flows grazing over perforated plates, that closely resemble the acoustic liners found on aircraft engines. Our direct numerical simulations explore a large parameter space including the effects of porosity, thickness and viscous-scaled diameter of the perforated plates, at friction Reynolds numbers $\textit {Re}_\tau = 500\unicode{x2013}2000$, which allows us to develop a robust theory for estimating the added drag induced by acoustic liners. We find that acoustic liners can be regarded as porous surfaces with a wall-normal permeability and that the relevant length scale characterizing their added drag is the inverse of the wall-normal Forchheimer coefficient. Unlike other types of porous surfaces featuring Darcian velocities inside the pores, the flow inside the orifices of acoustic liners is fully turbulent, with a magnitude of the wall-normal velocity fluctuations comparable to the peak in the near-wall cycle. We provide clear evidence of a fully rough regime for acoustic liners, also confirmed by the increasing relevance of pressure drag. Once the fully rough asymptote is reached, canonical acoustic liners provide an added drag comparable to that of sand-grain roughness with viscous-scaled height matching the inverse of the viscous-scaled Forchheimer permeability of the plate.

We study the statistics of passive scalars (either temperature or concentration of a diffusing substance) at friction Reynolds number ${Re}_{\tau }=1140$, for turbulent flow within a smooth straight pipe of circular cross-section, in the range of Prandtl numbers from ${Pr}=0.00625$, to ${Pr}=16$, using direct numerical simulations (DNS) of the Navier–Stokes equations. Whereas the organization of passive scalars is similar to the axial velocity field at ${Pr} = O(1)$, similarity is impaired at low Prandtl number, at which the buffer-layer dynamics is filtered out, and at high Prandtl number, at which the passive scalar fluctuations become confined to the near-wall layer. The mean scalar profiles at ${Pr} \gtrsim 0.0125$ are found to exhibit logarithmic overlap layers, and universal parabolic distributions in the core part of the flow. Near-universality of the eddy diffusivity is exploited to derive accurate predictive formulas for the mean scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling formulas are derived for the thickness of the conductive (diffusive) layer, for the peak scalar variance, and its production rate. The DNS data are leveraged to synthesize a modified form of the classical predictive formula of Kader & Yaglom (Intl J. Heat Mass Transfer, vol. 15, 1972, pp. 2329–2351), which is capable of accounting accurately for the dependence on both Reynolds and Prandtl numbers, for ${Pr} \gtrsim 0.25$.

Cilia are micro-scale hair-like organelles. They can exhibit self-sustained oscillations which play crucial roles in flow transport or locomotion. Recent studies have shown that these oscillations can spontaneously emerge from dynamic instability triggered by internal stresses via a Hopf bifurcation. However, the flow transport induced by an instability-driven cilium still remains unclear, especially when the fluid is non-Newtonian. This study aims at bridging these gaps. Specifically, the cilium is modelled as an elastic filament, and its internal actuation is represented by a constant follower force imposed at its tip. Three generalized Newtonian behaviours are considered, i.e. the shear-thinning, Newtonian and shear-thickening behaviours. Effects of four key factors, including the filament zero-stress shape, Reynolds number ($Re$), follower-force magnitude and fluid rheology, on the filament dynamics, fluid dynamics and flow transport are explored through direct numerical simulation at $Re$ of 0.04 to 5 and through a scaling analysis at $Re \approx 0$. The results reveal that even though it is expected that inertia vanishes at $Re \ll 1$, inertial forces do alter the filament dynamics and deteriorate the flow transport at $Re\ge 0.04$. Regardless of $Re$, the flow transport can be improved when the flow is shear thinning or when the follower force increases. Furthermore, a linear stability analysis is performed, and the variation of the filament beating frequency, which is closely correlated with the filament dynamics and flow transport, can be predicted.

The objective of this paper is to understand the effects of waves and vaporous cavitation upon the hydrodynamic and hydroelastic responses of a flexible surface-piercing hydrofoil, adding to the subcavitating results presented in Part 1. In general, the presence of a sufficiently large vaporous cavitation bubble facilitated the formation of a ventilated cavity, substantially reducing the angle of attack or speed required to induce fully ventilated flow, relative to subcavitating conditions. A new co-analysis procedure was used to isolate synchronous hydrodynamic modes and structural operating deflection shapes. Significant dynamic load amplification occurred when the resonant frequencies of the first twisting and second bending modes coalesced in both fully wetted and partially cavitating flows. The presence of waves did not diminish the effect of frequency coalescence, but did encourage intermittent lock-in with both leading edge cavity shedding and trailing edge vortex shedding at certain speeds. Partial cavity shedding typically had a negligible impact on the power spectral densities of structural motions because of incoherent cavity shedding. However, lock-in between the cavity shedding frequency and modal coalescence frequency led to shifting of the primary frequency peak, as well as amplified harmonics and interactions between the cavity shedding frequency and vortex shedding frequency. The transition from partially cavitating to fully ventilated flow caused sudden and large drops in the mean hydrodynamic loads and deformations, as well as substantial reductions in the intensity of the fluctuations.

The helix approach is a new individual pitch control method to mitigate wake effects of wind turbines. Its name is derived from the helical shape of the wake caused by a rotating radial force exerted by the turbine. While its potential to increase power production has been shown in previous studies, the physics of the helical wake are not well understood to date. Open questions include whether the increased momentum in the wake stems from an enhanced wake mixing or from the wake deflection. Furthermore, its application to a row of more than two turbines has not been examined before. We study this approach in depth from both an analytical and numerical perspective. We examine large-eddy simulations (LES) of the wake of a single turbine and find that the helix approach exhibits both higher entrainment and notable deflection. As for the application to a row of turbines, we show that the phase difference between two helical wakes is independent of ambient turbulence. Examination of LES of a row of three turbines shows that power gains greatly depend on the phase difference between the helices. We find a maximum increase in the total power of approximately 10 % at a phase difference of $270^\circ$. However, we do not optimise the phase difference any further. In summary, we provide a set of analytical tools for the examination of helical wakes, show why the helix approach is able to increase power production, and provide a method to extend it to a wind farm.

We study the influence of side walls on the stability of falling liquid films. We combine a temporal biglobal stability analysis based on the linearised Navier–Stokes equations with experiments measuring the spatial growth rate of sinusoidal waves flowing downstream an inclined channel. Very good agreement was found when comparing the theoretical and experimental results. Strong lateral confinement of the channel stabilises the flow. In the wavenumber-Reynolds number space, the instability region experiences a fragmentation due to selective damping of moderate wavenumbers. For this range of parameters, the three-dimensional confined problem shows several prominent stability modes which are classified into two categories, the well-known Kapitza hydrodynamic instability mode (H-mode) and a new unstable mode, we refer to it as wall-mode (W-mode). The two mode types are stabilised differently, where the H-modes are stabilised at small wavenumbers, while the W-modes experience stabilisation at high wavenumbers, and at sufficiently small channel widths, only the W-mode is observed. The reason behind the unique H-modes stabilisation is that they become analogous to waveguide modes, which can not propagate below a certain cut-off wavenumber. The spatial structure of the eigenmodes experiences significant restructuring at wavenumbers smaller than the most damped wavenumber. The mode switching preserves the spatial symmetry of the unstable mode.

We investigate the streamwise vorticity generation mechanism and distribution pattern in an unbounded steady inertial flow past a fixed Platonic polyhedron. Three angular positions are selected: an edge facing the flow (E), a face facing the flow (F) and a vertex facing the flow (V). We provide compelling evidence that the generation of the streamwise vorticity is primarily caused by the tilting of the transverse vorticity that originates from the particle front surface. Each inclined face on the front surface generates a pair of opposite-signed streamwise vortices. They are advected to the particle wake and form a chiral vorticity pattern which preserves the symmetry of the particle front surface. Two particles at dual angular positions exhibit highly similar vorticity patterns. Our study reveals a striking similarity between the vorticity patterns and the far-field optics diffraction pattern of a light beam past a polygonal aperture. We discover the deterministic vorticity generation mechanism to predict the streamwise vorticity patterns based on the distribution of edges and inclined faces on the particle front surface. Conversely, the vorticity patterns themselves can serve as a diagnostic tool to infer the geometry of the opaque particle front surface. Additionally, the vorticity patterns can be used to predict the stable angular position of a freely settling angular particle, which tends to be such that the number of streamwise vorticity pairs in the wake is maximized.

Ice-infiltrated sediment, or frozen fringe, is responsible for phenomena such as frost heave, ice lenses and metres of debris-rich ice under glaciers. Understanding frozen fringes is important as frost heave is responsible for damaging infrastructure at high latitudes and sediment freeze-on at the base of glaciers can modulate subglacial friction, influencing the rate of global sea level rise. Here we describe the thermomechanics of liquid water flow and freezing in ice-saturated sediments, focusing on the conditions relevant for subglacial environments. The force balance that governs the frozen fringe thickness depends on the weight of the overlying material, the thermomolecular force between ice and sediments across liquid premelted films and the water pressure required by Darcy flow. We combine this mechanical model with an enthalpy method that conserves energy across phase change interfaces on a fixed computational grid. The force balance and enthalpy model together determine the evolution of the frozen fringe thickness and our simulations predict frost heave rates and ice lens spacing. Our model accounts for premelting at ice–sediment contacts, partial ice saturation of the pore space, water flow through the fringe, the thermodynamics of the ice–water–sediment interface and vertical force balance. We explicitly account for the formation of ice lenses, regions of pure ice that cleave the fringe at the depth where the interparticle force vanishes. Our model results allow us to predict the thickness of a frozen fringe and the spacing of ice lenses in subaerial and subglacial sediments.