Numerical investigations of body-wake interactions were carried out by simulating the flow over a rod–airfoil configuration using high-order implicit large eddy simulation (HILES) for the incoming velocity $U_{\infty }=72~\text{m}~\text{s}^{-1}$ and a Reynolds number based on the airfoil chord $4.8\times 10^{5}$. The flow over five different rod–airfoil configurations with different distances of $L/d=2$, 4, 6, 8 and 10, respectively, were calculated for the analysis of body-wake interaction phenomena. Various fundamental mechanisms dictating the intricate flow phenomena including force varying regulation, flow structures and flow patterns in the interaction region, turbulent fluctuations and their suppression, noise radiation and fluid resonant oscillation, have been studied systematically. Due to the airfoil downstream, a relatively higher base pressure is exerted on the surface of the cylinder upstream, and the pressure fluctuation on the surface of the rod–airfoil configuration with $L/d=2$ is significantly suppressed, resulting in a reduction of the fluctuating lift. Following the distance between the cylinder and airfoil strongly decreases, Kármán-street shedding is suppressed due to the blocking effect. The flow in this interaction region has two opposite tendencies: the influence of the airfoil on the steady flow is to accelerate it and the counter-rotating vortices connecting with the leading edge of the airfoil tend to slow the flow down. There may be two flow patterns associated with the interference region, i.e. the Kármán-street suppressing mode and the Kármán-street shedding mode. The primary vortex shedding behind the cylinder upstream, and the shedding wake impingement onto the airfoil downstream, play a dominant role in the production of turbulent fluctuations. When primary vortex shedding is suppressed, the intensity of impingement is weakened, resulting in a significant suppression of the turbulent fluctuations. Due to these factors, a special broadband noise without a manifestly distinguishable peak is radiated by the rod–airfoil configuration with $L/d=2$. The fluid resonant oscillation within the flow interaction between the turbulent wake and the bodies was further investigated by adopting a feedback model, which confirmed that the effect of fluid resonant oscillation becomes stronger when $L/d=6$ and 10. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to the body-wake interaction.

Convection in a horizontal duct aligned with a uniform magnetic field is analysed computationally. The motivation of the study is the concept of a liquid metal blanket for a tokamak fusion reactor, in which ducts are oriented toroidally, i.e. parallel to the main component of the magnetic field. Computations of two-dimensional (streamwise-uniform) flows appearing at very strong magnetic fields and of three-dimensional flows in long domains are conducted. Non-uniform volumetric internal heating is applied, while the walls are maintained at a constant temperature. Two-dimensional or nearly two-dimensional turbulent convection is found at high Grashof and Hartmann numbers typical for fusion reactor conditions. The turbulence results in stronger mixing and more uniform distribution of wall heat flux, indicating promising potential of this concept of the blanket.

Transient scaling and integral analyses were performed to predict trends in alloy solidification in a rectangular cavity cooled by an isothermal sidewall. The natural convection fluid flow was approximated by a scaling analysis for a laminar boundary layer at the solidification front, and was coupled to scaling and integral analyses of the energy equation to predict the solidification behaviour of the system. These analyses predicted several relevant aspects of the solidification process, including the time required to extinguish the initial superheat and the maximum local solidification time as a function of the system parameters and material properties. These results were verified by comparison to numerical simulations for an Al–4.5 wt% Cu alloy for various initial and boundary conditions and cavity aspect ratios. The analysis was compared to previous attempts to analyse similar fluid flow and solidification processes, and the limitations of the assumptions used for this analysis were discussed.

Drops impacting at low velocities onto a pool surface can stretch out thin hemispherical sheets of air between the drop and the pool. These air sheets can remain intact until they reach submicron thicknesses, at which point they rupture to form a myriad of microbubbles. By impacting a higher-viscosity drop onto a lower-viscosity pool, we have explored new geometries of such air films. In this way we are able to maintain stable air layers which can wrap around the entire drop to form repeatable antibubbles, i.e. spherical air layers bounded by inner and outer liquid masses. Furthermore, for the most viscous drops they enter the pool trailing a viscous thread reaching all the way to the pinch-off nozzle. The air sheet can also wrap around this thread and remain stable over an extended period of time to form a cylindrical air sheet. We study the parameter regime where these structures appear and their subsequent breakup. The stability of these thin cylindrical air sheets is inconsistent with inviscid stability theory, suggesting stabilization by lubrication forces within the submicron air layer. We use interferometry to measure the air-layer thickness versus depth along the cylindrical air sheet and around the drop. The air film is thickest above the equator of the drop, but thinner below the drop and up along the air cylinder. Based on microbubble volumes, the thickness of the cylindrical air layer becomes less than 100 nm before it ruptures.

In pursuit of a universal law for the rate of entrainment across a density interface driven by the impingement of a localised turbulent flow, the role of the confinement, wherein the environment is within the confines of a box, has to date been overlooked. Seeking to unravel the effects of confinement, we develop a phenomenological model describing the quasi-steady rate at which buoyant fluid is turbulently entrained across a density interface separating two uniform layers within the confines of a box. The upper layer is maintained by a turbulent plume, and the localised impingement of a turbulent fountain with the interface drives entrainment of fluid from the upper layer into the lower layer. The plume and fountain rise from sources at the base of the box and are non-interacting. Guided by previous observations, our model characterises the dynamics of fountain–interface interaction and the steady secondary flow in the environment that is induced by the perpetual cycle of vertical excursions of the interface. We reveal that the dimensionless entrainment flux across the interface $E_{i}$ is governed not only by an interfacial Froude number $\mathit{Fr}_{i}$ but also by a ‘confinement’ parameter ${\it\lambda}_{i}$, which characterises the length scale of interfacial turbulence relative to the depth of the upper layer. By deducing the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’, we shed new light on the effects of confinement on interfacial entrainment. We establish that for small ${\it\lambda}_{i}$, a weak secondary flow has little influence on $E_{i}$, which follows a quadratic power law $E_{i}\propto \mathit{Fr}_{i}^{2}$. For large ${\it\lambda}_{i}$, a strong secondary flow significantly influences $E_{i}$, which then follows a cubic power law $E_{i}\propto \mathit{Fr}_{i}^{3}$. Drawing on these results, and showing that for previous experimental studies ${\it\lambda}_{i}$ exhibits wide variation, we highlight underlying physical reasons for the significant scatter in the existing measurements of the rate of interfacial entrainment. Finally, we explore the implications of our results for guiding appropriate choices of box geometry for experimentally and numerically examining interfacial entrainment.

We examine the interactions between two three-dimensional quasi-geostrophic hetons. The hetons are initially translating towards one another. We address the effect of the vertical distance between the two poles (vortices) constituting each heton on the interaction. We also examine the influence of the horizontal separation between the poles within each heton. In this investigation, the two hetons are facing each other. Two configurations are possible depending on the respective locations of the like-signed poles of the hetons. When they lie at the same depth, we refer to the configuration as symmetric; the antisymmetric configuration corresponds to opposite-signed poles at the same depth. The first step in the investigation uses point vortices to represent the poles of the hetons. This approach allows us to rapidly browse the parameter space and to estimate the possible heton trajectories. For a symmetric pair, the hetons either reverse their trajectory or recombine and escape perpendicularly depending of their horizontal and vertical offsets. On the other hand, antisymmetric hetons recombine and escape perpendicularly as same-depth dipoles. In a second part, we focus on finite core hetons (with finite volume poles). These hetons can deform and may be sensitive to horizontal-shear-induced deformations, or to baroclinic instability. These destabilisations depend on the vertical and horizontal offsets between the various poles, as well as on their width-to-height aspect ratios. They can modify the volume of the poles via vortex merger, breaking and/or shearing out; they compete with the advective evolution observed for singular (point) vortices. Importantly, hetons can break down or reconfigure before they can drift away as expected from a point vortex approach. Thus, a large variety of behaviours is observed in the parameter space. Finally, we briefly illustrate the behaviour of tall hetons which can be unstable to an azimuthal mode $l=1$ when many vertical modes of deformation are present on the heton.

We consider the motion of a fluid-immersed negatively buoyant particle in the vicinity of a thin compressible elastic wall, a situation that arises in a variety of technological and natural settings. We use scaling arguments to establish different regimes of sliding, and complement these estimates using thin-film lubrication dynamics to determine an asymptotic theory for the sedimentation, sliding and spinning motions of a cylinder. The resulting theory takes the form of three coupled nonlinear singular-differential equations. Numerical integration of the resulting equations confirms our scaling relations and further yields a range of unexpected behaviours. Despite the low-Reynolds-number feature of the flow, we demonstrate that the particle can spontaneously oscillate when sliding, can generate lift via a Magnus-like effect, can undergo a spin-induced reversal effect and also shows an unusual sedimentation singularity. Our description also allows us to address a sedimentation–sliding transition that can lead to the particle coasting over very long distances, similar to certain geophysical phenomena. Finally, we show that a small modification of our theory allows us to generalize the results to account for additional effects such as wall poroelasticity.

The rheology of highly concentrated monodisperse emulsions is studied by rigorous multidrop numerical simulations for three types of steady macroscopic flow, (i) simple shear ($\dot{{\it\gamma}}x_{2}$, 0 0), (ii) planar extension (PE) ($\dot{{\it\Gamma}}x_{1},-\dot{{\it\Gamma}}x_{2},0$) and (iii) mixed ($\dot{{\it\gamma}}x_{2}$, $\dot{{\it\gamma}}{\it\chi}x_{1}$, 0), where $\dot{{\it\gamma}}$ and $\dot{{\it\Gamma}}$ are the deformation rates, and ${\it\chi}\in (-1,1)$ is the flow parameter, in order to construct and validate a general constitutive model for emulsion flows with arbitrary kinematics. The algorithm is a development of the multipole-accelerated boundary-integral (BI) code of Zinchenko & Davis (J. Fluid Mech., vol. 455, 2002, pp. 21–62). It additionally incorporates periodic boundary conditions for (ii) and (iii) (based on the reproducible lattice dynamics of Kraynik–Reinelt for PE), control of surface overlapping, much more robust controllable surface triangulations for long-time simulations, and more efficient acceleration. The emulsion steady-state viscometric functions (shear viscosity and normal stress differences) for (i) and extensiometric functions (extensional viscosity and stress cross-difference) for (ii) are studied in the range of drop volume fractions $c=0.45{-}0.55$, drop-to-medium viscosity ratios ${\it\lambda}=0.25{-}10$ and various capillary numbers $\mathit{Ca}$, with 100–400 drops in a periodic cell and 2000–4000 boundary elements per drop. High surface resolution is important for all three flows at small $\mathit{Ca}$. Large system size and strains $\dot{{\it\gamma}}t$ of up to several thousand are imperative in some shear-flow simulations to identify the onset of phase transition to a partially ordered state, and evaluate (although still not precisely) the viscometric functions in this state. Below the phase transition point, the shear viscosity versus $\mathit{Ca}$ shows a kinked behaviour, with the local minimum most pronounced at ${\it\lambda}=1$ and $c=0.55$. The ${\it\lambda}=0.25$ emulsions flow in a partially ordered manner in a wide range of $\mathit{Ca}$ even when $c=0.45$. Increase of ${\it\lambda}$ to 3–10 shifts the onset of ordering to much smaller $\mathit{Ca}$, often outside the simulation range. In contrast to simple shear, phase transition is never observed in PE or mixed flow. A generalized five-parameter Oldroyd model with variable coefficients is fitted to our extensiometric and viscometric functions at arbitrary flow intensities (but outside the phase transition range). The model predictions compare very well with precise simulation results for strong mixed flows, ${\it\chi}=0.25$. Time-dependent PE flow is also considered. Ways to overcome the phase transition and drop breakup limitations on constitutive modelling are discussed.

Direct numerical simulation was used to study laminar and turbulent flows in circular pipes with smoothly corrugated walls. The corrugation wavelength was kept constant at $0.419D$, where $D$ is the mean diameter of the wavy-wall pipe and the corrugation height was varied from zero to $0.08D$. Flow rates were varied in steps between low values that generate laminar flow and higher values where the flow is in the post-transitional turbulent regime. Simulations in the turbulent regime were also carried out at a constant Reynolds number, $\mathit{Re}_{{\it\tau}}=314$, for all corrugation heights. It was found that even in the laminar regime, larger-amplitude corrugations produce flow separation. This leads to the proportion of pressure drop attributable to pressure drag being approximately 50 %, and rising to approximately 85 % in transitional rough-wall flow. The near-wall structure of turbulent flow is seen to be heavily influenced by the effects of flow separation and reattachment. Farther from the wall, the statistical profiles examined exhibit behaviours characteristic of smooth-wall flows or distributed roughness rough-wall flows. These observations support Townsend’s wall-similarity hypothesis. The organized nature of the present roughness allows the mean pressure drop to be written as a function of the corrugation height. When this is exploited in an analysis of the mean dynamical equation, the scaling problem is explicitly revealed to result from the combined influences of roughness and Reynolds number. The present results support the recent analysis and observations of Mehdi et al. (J. Fluid Mech., vol. 731, 2013, pp. 682–712), indicating that the length scale given by the distance from the wall at which the mean viscous force loses leading order is important to describing these combined influences, as well as providing a dynamically self-consistent connection to the scaling structure of smooth-wall pipe flow.

A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container $\mathscr{D}$, as the number of vortices $N$ tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydrodynamic eigenvalue problem of the Laplacian in $\mathscr{D}$. The statistics of the projection of the vorticity distribution onto these eigenfunctions (‘vorticity projections’) are then investigated. The statistics are used first to obtain the density-of-states function and caloric curve for the system, generalising previous results to arbitrary (neutral) distributions of vortex circulations. Explicit expressions are then obtained for the microcanonical (i.e. fixed energy) probability density functions of the vorticity projections in a form that can be compared directly with direct numerical simulations of the dynamics. The energy spectra of the resulting flows are predicted analytically. Ensembles of simulations with $N=100$, in several conformal domains, are used to make a comprehensive validation of the theory, with good agreement found across a broad range of energies. The probability density function of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager–Kraichnan condensation in the literature, between low-energy states with no mean flow in the domain and high-energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of $\mathscr{D}$, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite $N$, are explored using a Markov chain phase-space sampling method.

The scaling of the scour depth of equilibrium at the base of a solid cylinder immersed within an erodible granular bed and impinged by a turbulent shear flow is investigated here, for the first time, by means of the phenomenological theory of turbulence. The proposed theory allows the derivation of a predictive formula that (i) includes all the relevant non-dimensional parameters controlling the process, and (ii) contrary to commonly employed empirical formulae, is free from scale issues. Theoretical predictions agree very well with experimental data, shed light on unresolved issues on the physics of the problem, and clarify the effects of various dimensionless parameters controlling the scouring process.

In this paper, we examine the dimensionality of a single electrically driven vortex bounded by two no-slip and perfectly insulating horizontal walls a distance $h$ apart. The study was performed in the weakly inertial limit by means of an asymptotic expansion, which is valid for any Hartmann number. We show that the dimensionality of the leading order can be fully described using the single parameter $l_{z}^{{\it\nu}}/h$, where $l_{z}^{{\it\nu}}$ represents the distance over which the Lorentz force is able to act before being balanced by viscous dissipation. The base flow happens to introduce inertial recirculations in the meridional plane at the first order, which are shown to follow two radically different mechanisms: inverse Ekman pumping driven by a vertical pressure gradient along the axis of the vortex, or direct Ekman pumping driven by a radial pressure gradient in the Hartmann boundary layers. We demonstrate that when the base flow is quasi-2D, the relative importance of direct and inverse pumping is solely determined by the aspect ratio ${\it\eta}/h$, where ${\it\eta}$ refers to the width of the vortex. Of the two mechanisms, only inverse pumping appears to act as a significant source of helicity.

The plane turbulent wall-jet flow without externally imposed stream is considered. It is assumed that the wall jet does not emerge from a second wall perpendicular to the velocity vector of the initial wall jet. The (kinematic) momentum flux $K(x)$ of the wall jet decreases downstream owing to the shear stress at the wall. This investigation is based on the hypothesis that the total friction force on the wall is smaller than the total inflow momentum flux. In other words, the turbulent wall jet tends to a turbulent ‘half-free jet’ with a non-zero momentum flux $K_{\infty }\;(\text{m}^{3}~\text{s}^{-2})$ far downstream. The fact that the turbulent half-free jet is the asymptotic form of a turbulent wall jet is the basis for a singular perturbation method by which the wall-jet flow is determined. It turns out that the ratio between the wall distance $y_{m}$ of the maximum velocity and the wall distance $y_{0.5}$ of half the maximum velocity decreases downstream to zero. Dimensional analysis leads immediately to a universal function of the dimensionless momentum flux $K(\mathit{Re}_{x})/K_{\infty }$ that depends asymptotically only on the local Reynolds number $\mathit{Re}_{x}=\sqrt{(x-x_{0})K_{\infty }}/{\it\nu}$, where $x_{0}$ denotes the coordinate of the virtual origin. When the values $K$ and ${\it\nu}$ at the position $x-x_{0}$ are known, the asymptotic momentum flux $K_{\infty }$ can be determined. Experimental data on all turbulent plane wall jets (except those emerging from a second plane wall) collapse to a single universal curve. Comparisons between available experimental data and the analysis make the hypothesis $K_{\infty }\neq 0$ plausible. A convincing verification, however, will be possible in the future, preferably by direct numerical simulations.

Measurements are presented in zero-pressure-gradient, flat-plate, turbulent boundary layers for Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=2600$ to $\mathit{Re}_{{\it\tau}}=72\,500$ ($\mathit{Re}_{{\it\theta}}=8400{-}235\,000$). The wind tunnel facility uses pressurized air as the working fluid, and in combination with MEMS-based sensors to resolve the small scales of motion allows for a unique investigation of boundary layer flow at very high Reynolds numbers. The data include mean velocities, streamwise turbulence variances, and moments up to 10th order. The results are compared to previously reported high Reynolds number pipe flow data. For $\mathit{Re}_{{\it\tau}}\geqslant 20\,000$, both flows display a logarithmic region in the profiles of the mean velocity and all even moments, suggesting the emergence of a universal behaviour in the statistics at these high Reynolds numbers.

We propose and apply a Fourier-based symmetry-reduction scheme to remove, or quotient, the streamwise translation symmetry of laser-induced-fluorescence measurements of turbulent pipe flows that are viewed as dynamical systems in a high-dimensional state space. We also explain the relation between Taylor’s hypothesis and the comoving frame velocity $U_{d}$ of the turbulent orbit in state space. In particular, in physical space we observe flow structures that deform as they advect downstream at a speed that differs significantly from $U_{d}$. Indeed, the symmetry-reduction analysis of planar dye concentration fields at Reynolds number $Re=3200$ reveals that the speed $u$ at which high-concentration peaks advect is roughly 1.43 times $U_{d}$. In a physically meaningful symmetry-reduced frame, the excess speed $u-U_{d}\approx 0.43U_{d}$ can be explained in terms of the so-called geometric phase velocity $U_{g}$ associated with the orbit in state space. The ‘self-propulsion velocity’ $U_{g}$ is induced by the shape-changing dynamics of passive scalar structures observed in the symmetry-reduced frame, in analogy with that of a swimmer at low Reynolds numbers.

Both experiments and numerical simulations pertinent to the study of self-similarity in shock-induced turbulent mixing often do not cover sufficiently long times for the mixing layer to become developed in a fully turbulent manner. When the Mach number of the flow is sufficiently low, numerical simulations based on the compressible flow equations tend to become less accurate due to inherent numerical cancellation errors. This paper concerns a numerical study of the late-time behaviour of a single-shocked Richtmyer–Meshkov instability (RMI) and the associated compressible turbulent mixing using a new technique that addresses the above limitation. The present approach exploits the fact that the RMI is a compressible flow during the early stages of the simulation and incompressible at late times. Therefore, depending on the compressibility of the flow field, the most suitable model, compressible or incompressible, can be employed. This motivates the development of a hybrid compressible–incompressible solver that removes the low-Mach-number limitations of the compressible solvers, thus allowing numerical simulations of late-time mixing. Simulations have been performed for a multi-mode perturbation at the interface between two fluids of densities corresponding to an Atwood number of 0.5, and results are presented for the development of the instability, mixing parameters and turbulent kinetic energy spectra. The results are discussed in comparison with previous compressible simulations, theory and experiments.

Laboratory experiments in an open channel bend provide insight into the physics of convex bank flow separation occurring in a variety of channel configurations, including confluences and bifurcations. The edge of the zone of flow separation is characterized by a shear layer, enhanced velocity gradients, tke, turbulent shear stresses and reversal of the streamwise vorticity and vertical velocity. The latter result from turbulence-induced secondary flow near the convex bank. When bankline curvature abruptly increases, flow tends to move away from the convex bank along a straight path, as represented by the inertial forces – including the centrifugal force – in the transverse momentum equation written in curvilinear coordinates. Mass accumulation at the opposite bank leads to a transverse tilting of the water surface, and a pressure gradient towards the convex bank that causes the flow to change direction. The pressure gradient force lags spatially behind the inertial forces, which promotes flow separation. Flow separation typically occurs downstream of the location of maximum change in the bankline curvature, because an abrupt increase in bankline curvature also leads to water surface gradients that cause local flow redistribution towards the convex bank that opposes flow separation. The zone of convex bank flow separation is shaped by the secondary flow induced by streamline curvature and turbulence. The latter is conditioned by the production rate of tke, which crucially depends on the accurate description of the Reynolds stresses. Hydrodynamic, geometric and sedimentologic control parameters of convex bank flow separation are identified and discussed.

We compute the drag on a circular and liquid microdomain diffusing in a two-dimensional fluid lipid bilayer membrane surrounded by a fluid above and below. Under the assumptions that the liquids are incompressible and the flow is of low Reynolds number, Stokes’ equations describe the flow in the two-dimensional membrane as well as in the surrounding three-dimensional fluid. The expression for the drag force on the liquid domain involves Fredholm integral equations of the second kind, which we numerically solve using discrete collocation method based on Chebyshev polynomials. We observe that when the domain is more viscous than the surrounding membrane (including the rigid domain case), the drag force is almost independent of the viscosity contrast between the domain and the surrounding membrane, as also observed earlier in experiments by other researchers. The mobility also varies logarithmically with Boussinesq number ${\it\beta}$ for large ${\it\beta}$. On the other hand, for a less viscous domain the dimensionless drag force reduces with increasing viscosity contrast, and a significant change in the drag force, from that when there is no viscosity contrast or when the domain is rigid, has been observed. Further, the logarithmic behaviour of the mobility no longer holds for less viscous domains. Our method of computing the drag force and diffusion coefficient is valid for arbitrary viscosity contrast between the domain and membrane and any domain size (subject to ${\it\beta}\geqslant 5$).

We propose and numerically implement a constitutive framework for granular media that allows the material to traverse through its many common phases during the flow process. When dense, the material is treated as a pressure-sensitive elasto-viscoplastic solid obeying a yield criterion and a plastic flow rule given by the ${\it\mu}(I)$ inertial rheology of granular materials. When the free volume exceeds a critical level, the material is deemed to separate and is treated as disconnected, stress-free media. A material point method (MPM) procedure is written for the simulation of this model and many demonstrations are provided in different geometries, which highlight the ability of the numerical model to handle transitions through dense and disconnected states. By using the MPM framework, extremely large strains and nonlinear deformations, which are common in granular flows, are representable. The method is verified numerically and its physical predictions are validated against many known experimental phenomena, such as Beverloo’s scaling in silo flows, jointed power-law scaling of the run-out distance in granular-column-collapse problems, and various known behaviours in inclined chute flows.

We formulate a general theory of wave adjustment applicable to any physical system (not necessarily a hydrodynamic one), which, being linearized, possesses linear invariants and a complete system of waves harmonically depending on the time $t$. The invariants are determined by the initial conditions and are zero for the waves, which, therefore, do not transport and affect the invariants. The evolution of such a system can be represented naturally as the sum of a stationary component with non-zero invariants and a non-steady wave part with zero invariants. If the linear system is disturbed by a small perturbation (linear or nonlinear), then the state vector of the system is split into slow balanced and fast wave components. Various scenarios of the wave adjustment are demonstrated with fairly simple hydrodynamic models. The simplest scenario, called ‘fast radiation’, takes place when the waves rapidly (their group speed $c_{gr}$ greatly exceeds the slow flow velocity $U$) radiate away from the initial perturbation and do not interact effectively with the slow component. As a result, at large times, after the waves propagate away, the residual flow is slow and described by a balanced model. The scenario is exemplified by the three-dimensional non-rotating barotropic flow with a free surface. A more complicated scenario, called ‘nonlinear trapping’, occurs if oscillations with small group speed $c_{gr}\leqslant U$ are present in the wave spectrum. In this case, after nonlinear wave adjustment, the state vector is a superposition of the slow balanced component and oscillations with small $c_{gr}$ trapped by this component. An example of this situation is the geostrophic adjustment of a three-dimensional rotating barotropic layer with a free surface. In the third scenario, called ‘incomplete splitting’, the wave adjustment is accompanied by non-stationary boundary layers arising near rigid and internal boundaries at large times. The thickness of such a layer tends to zero and cross-gradients of physical parameters in the layer tend to infinity as $t\rightarrow \infty$. The layer is an infinite number of wave modes whose group speed tends to zero as the mode number tends to infinity. In such a system, complete splitting of motion into fast and slow components is impossible even in the linear approximation. The scenario is illustrated by an example of stratified non-rotating flow between two rigid lids. The above scenarios describe, at least, the majority of known cases of wave adjustment.