The time evolution of localized disturbances in an elliptical flow confined in an elliptical cylinder is studied by direct numerical simulation (DNS). The base flow is subject to the elliptic instability. The unstable growth of localized disturbances predicted by the short-wavelength stability analysis is captured. The time evolution can be divided into four stages: linear, weakly nonlinear, nonlinear and turbulent. In the linear stage a single wavepacket grows exponentially without changing its shape. The exponential growth is accompanied by large oscillations which have time period half that of the fluid particles in the elliptical flow. An averaged wavepacket, which is a train of bending waves that has a finite spatial extent, also grows exponentially, while the oscillations of the growth rate are small. The averaged growth rate increases as the kinematic viscosity decreases; the inviscid limit is close to the value predicted by the short-wavelength stability analysis. In the weakly nonlinear stage the energy stops growing. The vortical structure of the initial disturbances is deformed into wavy patterns. The energy spectrum loses the peak at the initial wavenumber, developing a broad spectrum, and the flow goes into the next stage. In the nonlinear stage weak vorticity is scattered in the whole domain although strong vorticity is still localized. The probability density functions (p.d.f.) of a velocity component and its longitudinal derivative are similar to those of isotropic turbulence; however, the energy spectrum does not have an inertial range showing the Kolmogorov spectrum. Finally in the turbulent stage fine-scale structures appear in the vorticity field. The p.d.f. of the longitudinal derivative of velocity shows the strong intermittency known for isotropic turbulence. The energy spectrum attains an inertial range showing the Kolmogorov spectrum. The turbulence is not symmetric because of rotation and strain; the component of vorticity in the compressing direction is smaller than the other two components. The energy of the mean flow as well as the total energy decreases. The ratio of the lost energy to the initial energy of the mean flow is large in the core region.

Forcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier–Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifth- and sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. Fluid Mech., vol. 289, 1995, pp. 179–198) and Malenica & Molin (J. Fluid Mech., vol. 302, 1995, pp. 203–229) was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.

The paper describes a numerical investigation of linear and nonlinear instability in high-speed boundary layers. Both a frozen gas and a finite-rate chemically reacting gas are considered. The weakly nonlinear instability in the presence of a large-amplitude two-dimensional wave is investigated for the case of fundamental resonance. Depending on the amplitude of this two-dimensional primary wave, strong growth of oblique secondary perturbations occurs for favourable relative phase differences between the two. For essentially the same primary amplitude, secondary amplification is almost identical for a reacting and a frozen gas. Therefore, chemical reactions do not directly affect the growth of secondary perturbations, but only indirectly through the change of linear instability and hence amplitude of the primary wave. When the secondary disturbances reach a sufficiently large amplitude, strongly nonlinear effects stabilize both primary and secondary perturbations.

The two-dimensional inviscid incompressible steady flow past an inclined flat plate is considered. A locus of asymmetric equilibrium configurations for vortex pairs is detected. It is shown that the flat geometry has peculiar properties compared to other geometries: (i) in order to satisfy the Kutta condition at both edges, which ensures flow regularity, the total circulation and the force acting on the plate must be zero; and (ii) the Kutta condition and the free vortex equilibrium conditions are not independent of each other. The non-existence of symmetric equilibrium configurations for an orthogonal plate is extended to more general asymmetric flows.

The receptivity of a laminar swept-wing boundary layer to a spanwise array of circular roughness elements is investigated by means of direct numerical simulations (DNS). The initial amplitude of a steady crossflow mode generated by the shallow roughness elements does not vary strictly linearly with the roughness height, as often assumed. Rather, a fundamental, superlinear dependence of the receptivity amplitude on the roughness height is found. In order to account for shape effects, the roughness geometry is Fourier decomposed to its spanwise spectral content, and elements with a reduced spectrum are investigated. If only modes are present that synthesise a regular structure of alternating bumps and dimples of equal shape and size, the receptivity amplitude is strictly linear for each mode and nominal roughness heights up to at least 15 % of the local displacement thickness.

The flow structure on a rotating wing (flat plate) is characterized over a range of Rossby number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ro} = r_g/C$, in which $r_g$ and $C$ are the radius of gyration and chord of the wing, as well as travel distance $\mathit{Ro} = r_g \Phi /C$, where $\Phi $ is the angle of rotation. Stereoscopic particle image velocimetry (SPIV) is employed to determine the flow patterns on defined planes, and by means of reconstruction, throughout entire volumes. Images of the $Q$-criterion and spanwise vorticity, velocity and vorticity flux are employed to represent the flow structure. At low Rossby number, the leading-edge, tip and root vortices are highly coherent with large dimensionless values of $Q$ in the interior regions of all vortices and large downwash between these components of the vortex system. For increasing Rossby number, however, the vortex system rapidly degrades, accompanied by loss of large $Q$ within its interior and downstream displacement of the region of large downwash. These trends are accompanied by increased deflection of the leading-edge vorticity layer away from the surface of the wing, and decreased spanwise velocity and vorticity flux in the trailing region of the wing, which are associated with the degree of deflection of the tip vortex across the wake region. Combinations of large Rossby number $\mathit{Ro} =r_g/C$ and travel distance $r_g \Phi /C$ lead to separated flow patterns similar to those observed on rectilinear translating wings at high angle of attack $\alpha $. In the extreme case where the wing travels a distance corresponding to a number of revolutions, the highly coherent flow structure is generally preserved if the Rossby number is small; it degrades substantially, however, at larger Rossby number.

We consider the periodic injection and extraction of fluid from a line well in a horizontal saturated aquifer of finite thickness as part of an aquifer thermal energy storage system. We focus on the case in which the injected fluid is dense relative to the original fluid in the aquifer and we explore the competition between the driving pressure and buoyancy force in controlling the dispersal of the injected fluid through the aquifer. We show that, after each cycle, a progressively larger fraction of the injected fluid is extracted, while the remainder of the injected fluid gradually migrates away from the well such that, after time $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}t$, the position of the leading edge of the injected fluid, $x_{nose}(t)$, scales as $ x_{nose}(t)\sim x_{nose}(\tau )\sqrt{t/\tau }$, where $\tau $ is the period of injection. If the fluid is extracted from the base of the layer, then, near the well, the thickness of the injected fluid at the end of the extraction cycle tends to a constant value, which decreases with injection rate. We also show that there is a class of self-similar exchange-flow solutions that develop when a saturated porous layer of thickness $H$ is in contact with a stratified fluid reservoir, filled to thickness $F_0H<H$ with relatively dense fluid, and with original reservoir fluid above this level. We show that these solutions coincide exactly with the far-field flow produced by the injection–extraction cycles. We successfully test the models with a series of analogue experiments of both the injection–extraction flow and the exchange flow using a Hele-Shaw cell. In the case that the fluid is injected and extracted from the top of the aquifer, the value $F_0$ tends to unity in all cases, although the convergence time depends on the rate and period of injection, the buoyancy speed and the vertical extent and the porosity of the aquifer. We use the model to explore how the concentration of reservoir fluid in the produced fluid varies as the system evolves from cycle to cycle, and we also examine the time required to transport a localised but distant contaminant to the production well through the far-field exchange flow. Finally, we consider the analogous axisymmetric injection–extraction flow problem, and show, through both numerical solution of the governing equations and experiment, that, although there is no simple class of similarity solutions, the fraction of injected fluid that is extracted progressively increases in each cycle.

Material pitting from cavitation bubble collapse is investigated numerically including two-way fluid–structure interaction (FSI). A hybrid numerical approach which links an incompressible boundary element method (BEM) solver and a compressible finite difference flow solver is applied to capture non-spherical bubble dynamics efficiently and accurately. The flow codes solve the fluid dynamics while intimately coupling the solution with a finite element structure code to enable simulation of the full FSI. During bubble collapse high impulsive pressures result from the impact of the bubble re-entrant jet on the material surface and from the collapse of the remaining bubble ring. A pit forms on the material surface when the impulsive pressure is large enough to result in high equivalent stresses exceeding the material yield stress. The results depend on bubble dynamics parameters such as the size of the bubble at its maximum volume, the bubble standoff distance from the material wall, and the pressure driving the bubble collapse. The effects of these parameters on the re-entrant jet, the following bubble ring collapse pressure, and the generated material pit characteristics are investigated.

A general mathematical framework is presented for modelling the pulling of optical glass fibres in a draw tower. The only modelling assumption is that the fibres are slender; cross-sections along the fibre can have general shape, including the possibility of multiple holes or channels. A key result is to demonstrate how a so-called reduced time variable $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\tau $ serves as a natural parameter in describing how an axial-stretching problem interacts with the evolution of a general surface-tension-driven transverse flow via a single important function of $\tau $, herein denoted by $H(\tau )$, derived from the total rescaled cross-plane perimeter. For any given preform geometry, this function $H(\tau )$ may be used to calculate the tension required to produce a given fibre geometry, assuming only that the surface tension is known. Of principal practical interest in applications is the ‘inverse problem’ of determining the initial cross-sectional geometry, and experimental draw parameters, necessary to draw a desired final cross-section. Two case studies involving annular tubes are presented in detail: one involves a cross-section comprising an annular concatenation of sintering near-circular discs, the cross-section of the other is a concentric annulus. These two examples allow us to exemplify and explore two features of the general inverse problem. One is the question of the uniqueness of solutions for a given set of experimental parameters, the other concerns the inherent ill-posedness of the inverse problem. Based on these examples we also give an experimental validation of the general model and discuss some experimental matters, such as buckling and stability. The ramifications for modelling the drawing of fibres with more complicated geometries, and multiple channels, are discussed.

The application of turbulent plume theory in describing the dynamics of emptying filling boxes, control volumes connected to an infinite exterior through a series of openings along the upper and lower boundaries, has yielded novel strategies for the natural ventilation of buildings. Making the plume laminar and having it fall through a porous medium yields a problem of fundamental significance in its own right, insights from which may be applied, for example, in minimizing the contamination of drinking water by geologically sequestered $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\mathrm{CO}}_2$ or the chemicals leached from waste piles. After reviewing the theory appropriate to rectilinear and axisymmetric plumes in porous media, we demonstrate how the model equations may be adapted to the case of an emptying filling box. In this circumstance, the long-time solution consists of two ambient layers, each of which has a uniform density. The lower and upper layers comprise fluid that is respectively discharged by the plume and advected into the box through the upper opening. Our theory provides an estimate for both the height and thickness of the associated interface in terms of, for example, the source volume and buoyancy fluxes, the outlet area and permeability, and the depth-average solute dispersion coefficient, which is itself a function of the far-field horizontal flow speed. Complementary laboratory experiments are provided for the case of a line source plume and show very good agreement with model predictions. Our measurements also indicate that the permeability, $k_f$, of the lower opening (or fissure) decreases with the density of the fluid being discharged, a fact that has been overlooked in some previous studies, wherein $k_f$ is assumed to depend only on the fissure geometry.

The problem of near-trapping of linear water waves in the time domain for rigid bodies or variations in bathymetry is considered. The singularity expansion method (SEM) is used to give an approximation of the solution as a projection onto a basis of modes. This requires a modification of the method so that the modes, which grow towards infinity, can be correctly normalized. A time-dependent solution, which allows for possible trapped modes, is introduced through the generalized eigenfunction method. The expression for the trapped mode and the expression for the near-trapped mode given by the SEM are shown to be closely connected. A numerical method that allows the SEM to be implemented is also presented. This method combines the boundary element method with an eigenfunction expansion, which allows the solution to be extended analytically to complex frequencies. The technique is illustrated by numerical simulations for geometries that support near-trapping.

Lock–release laboratory experiments are performed to examine saline and particle-laden flows down a slope into both constant-density and linearly stratified ambients. Both hypopycnal (surface-propagating) currents and hyperpycnal (turbidity) currents are examined, with the focus being upon the influence of ambient stratification on turbidity currents. Measurements are made of the along-slope front speed and the depth at which the turbidity current separates from the slope and intrudes into the ambient. These results are compared to the predictions of a theory that characterizes the flow evolution and separation depth in terms of the slope $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}s$, the entrainment parameter $E$ (the ratio of entrainment to flow speed), the relative stratification parameter $S$ (the ratio of the ambient density difference to the relative current density) and a new parameter $\gamma $ defined to be the ratio of the particle settling to entrainment speed. The implicit prediction for the separation depth, $H_s$, is made explicit by considering limits of small and large separation depth. In the former case of a ‘weak’ turbidity current, entrainment and particle settling are unimportant and separation occurs where the density of the ambient fluid equals the density of the fluid in the lock. In the latter case of a ‘strong’ turbidity current, entrainment and particle settling crucially affect the separation depth. Consistent with theory, we find that the separation depth indeed depends on $\gamma $ if the particle size (and hence settling rate) is sufficiently large and if the current propagates many lock lengths before separating from the slope. A composite prediction that combines the explicit formulae for the separation depth for weak and strong turbidity currents agrees well with experimental measurements over a wide parameter range.

Existing experimental and theoretical studies are discussed which lead to the clear hypothesis of a hitherto unidentified convective instability mode that dominates within the boundary-layer flow over slender rotating cones. The mode manifests as Görtler-type counter-rotating spiral vortices, indicative of a centrifugal mechanism. Although a formulation consistent with the classic rotating-disk problem has been successful in predicting the stability characteristics over broad cones, it is unable to identify such a centrifugal mode as the half-angle is reduced. An alternative formulation is developed and the governing equations solved using both short-wavelength asymptotic and numerical approaches to independently identify the centrifugal mode.

First, we review analytical and observational studies on third-order structure functions including velocity and buoyancy increments in rotating and stratified turbulence and discuss how these functions can be used in order to estimate the flux of energy through different scales in a turbulent cascade. In particular, we suggest that the negative third-order velocity–temperature–temperature structure function that was measured by Lindborg & Cho (Phys. Rev. Lett., vol. 85, 2000, p. 5663) using stratospheric aircraft data may be used in order to estimate the downscale flux of available potential energy (APE) through the mesoscales. Then, we calculate third-order structure functions from idealized simulations of forced stratified and rotating turbulence and compare with mesoscale results from the lower stratosphere. In the range of scales with a downscale energy cascade of kinetic energy (KE) and APE we find that the third-order structure functions display a negative linear dependence on separation distance $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}} r $, in agreement with observation and supporting the interpretation of the stratospheric data as evidence of a downscale energy cascade. The spectral flux of APE can be estimated from the relevant third-order structure function. However, while the sign of the spectral flux of KE is correctly predicted by using the longitudinal third-order structure functions, its magnitude is overestimated by a factor of two. We also evaluate the third-order velocity structure functions that are not parity invariant and therefore display a cyclonic–anticyclonic asymmetry. In agreement with the results from the stratosphere, we find that these functions have an approximate $ r^{2} $-dependence, with strong dominance of cyclonic motions.

Sensitivity analysis has successfully located the most efficient regions in which to apply passive control in many globally unstable flows. As is shown here and in previous studies, the standard sensitivity analysis, which is linear (first order) with respect to the actuation amplitude, predicts that steady spanwise wavy alternating actuation/modification has no effect on the stability of planar flows, because the eigenvalue change integrates to zero in the spanwise direction. In experiments, however, spanwise wavy modification has been shown to stabilize the flow behind a cylinder quite efficiently. In this paper, we generalize sensitivity analysis by examining the eigenvalue drift (including stabilization/destabilization) up to second order in the perturbation, and show how the second-order eigenvalue changes can be computed numerically by overlapping the adjoint eigenfunction with the first-order global eigenmode correction, shown here for the first time. We confirm the prediction against a direct computation, showing that the eigenvalue drift due to a spanwise wavy base flow modification is of second order. Further analysis reveals that the second-order change in the eigenvalue arises through a resonance of the original (2-D) eigenmode with other unperturbed eigenmodes that have the same spanwise wavelength as the base flow modification. The eigenvalue drift due to each mode interaction is inversely proportional to the distance between the eigenvalues of the modes (which is similar to resonance), but also depends on mutual overlap of direct and adjoint eigenfunctions (which is similar to pseudoresonance). By this argument, and by calculating the most sensitive regions identified by our analysis, we explain why an in-phase actuation/modification is better than an out-of-phase actuation for control of wake flows by spanwise wavy suction and blowing. We also explain why wavelengths several times longer than the wake thickness are more efficient than short wavelengths.

Homboe (Geophys. Publ., vol. 24, 1962, pp. 67–112) postulated that resonant interaction between two or more progressive, linear interfacial waves produces exponentially growing instabilities in idealized (broken-line profiles), homogeneous or density-stratified, inviscid shear layers. Here we have generalized Holmboe’s mechanistic picture of linear shear instabilities by (i) not initially specifying the wave type, and (ii) providing the option for non-normal growth. We have demonstrated the mechanism behind linear shear instabilities by proposing a purely kinematic model consisting of two linear, Doppler-shifted, progressive interfacial waves moving in opposite directions. Moreover, we have found a necessary and sufficient (N&S) condition for the existence of exponentially growing instabilities in idealized shear flows. The two interfacial waves, starting from arbitrary initial conditions, eventually phase-lock and resonate (grow exponentially), provided the N&S condition is satisfied. The theoretical underpinning of our wave interaction model is analogous to that of synchronization between two coupled harmonic oscillators. We have re-framed our model into a nonlinear autonomous dynamical system, the steady-state configuration of which corresponds to the resonant configuration of the wave interaction model. When interpreted in terms of the canonical normal-mode theory, the steady-state/resonant configuration corresponds to the growing normal mode of the discrete spectrum. The instability mechanism occurring prior to reaching steady state is non-modal, favouring rapid transient growth. Depending on the wavenumber and initial phase-shift, non-modal gain can exceed the corresponding modal gain by many orders of magnitude. Instability is also observed in the parameter space which is deemed stable by the normal-mode theory. Using our model we have derived the discrete spectrum non-modal stability equations for three classical examples of shear instabilities: Rayleigh/Kelvin–Helmholtz, Holmboe and Taylor–Caulfield. We have shown that the N&S condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the normal-mode theory.

By characterising the hydrodynamic stresses generated by statistically homogeneous and isotropic turbulence in rigid aggregates, we estimate theoretically the rate of turbulent breakup of colloidal aggregates and the size distribution of the formed fragments. The adopted method combines direct numerical simulation of the turbulent field with a discrete element method based on Stokesian dynamics. In this way, not only is the mechanics of the aggregate modelled in detail, but the internal stresses are evaluated while the aggregate is moving in the turbulent flow. We examine doublets and cluster–cluster isostatic aggregates, where the failure of a single contact leads to the rupture of the aggregate and breakup occurs when the tensile force at a contact exceeds the cohesive strength of the bond. Owing to the different role of the internal stresses, the functional relationship between breakup frequency and turbulence dissipation rate is very different in the two cases. In the limit of very small and very large values, the frequency of breakup scales exponentially with the turbulence dissipation rate for doublets, while it follows a power law for cluster–cluster aggregates. For the case of large isostatic aggregates, it is confirmed that the proper scaling length for maximum stress and breakup is the radius of gyration. The cumulative fragment distribution function is nearly independent of the mean turbulence dissipation rate and can be approximated by the sum of a small erosive component and a term that is quadratic with respect to fragment size.

The linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.

We investigate the shock-induced turbulent mixing between a light and a heavy gas, where a Richtmyer–Meshkov instability (RMI) is initiated by a shock wave with Mach number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ma}= 1.5$. The prescribed initial conditions define a deterministic multimode interface perturbation between the gases, which can be imposed exactly for different simulation codes and resolutions to allow for quantitative comparison. Well-resolved large-eddy simulations are performed using two different and independently developed numerical methods with the objective of assessing turbulence structures, prediction uncertainties and convergence behaviour. The two numerical methods differ fundamentally with respect to the employed subgrid-scale regularisation, each representing state-of-the-art approaches to RMI. Unlike previous studies, the focus of the present investigation is to quantify the uncertainties introduced by the numerical method, as there is strong evidence that subgrid-scale regularisation and truncation errors may have a significant effect on the linear and nonlinear stages of the RMI evolution. Fourier diagnostics reveal that the larger energy-containing scales converge rapidly with increasing mesh resolution and thus are in excellent agreement for the two numerical methods. Spectra of gradient-dependent quantities, such as enstrophy and scalar dissipation rate, show stronger dependences on the small-scale flow field structures as a consequence of truncation error effects, which for one numerical method are dominantly dissipative and for the other dominantly dispersive. Additionally, the study reveals details of various stages of RMI, as the flow transitions from large-scale nonlinear entrainment to fully developed turbulent mixing. The growth rates of the mixing zone widths as obtained by the two numerical methods are ${\sim } t^{7/12}$ before re-shock and ${\sim } (t-t_0)^{2/7}$ long after re-shock. The decay rate of turbulence kinetic energy is consistently ${\sim } (t-t_0)^{-10/7}$ at late times, where the molecular mixing fraction approaches an asymptotic limit $\varTheta \approx 0.85$. The anisotropy measure $\langle a \rangle _{xyz}$ approaches an asymptotic limit of ${\approx }0.04$, implying that no full recovery of isotropy within the mixing zone is obtained, even after re-shock. Spectra of density, turbulence kinetic energy, scalar dissipation rate and enstrophy are presented and show excellent agreement for the resolved scales. The probability density function of the heavy-gas mass fraction and vorticity reveal that the light–heavy gas composition within the mixing zone is accurately predicted, whereas it is more difficult to capture the long-term behaviour of the vorticity.

In the companion paper (Chen & Yeh, J. Fluid Mech., vol. 749, 2014, pp. 577–596), collisions of counter-propagating solitary waves were studied experimentally by analysing the measured water-surface variations. Here we study the flow fields associated with the collisions. With the resolved velocity data obtained in the laboratory, the flow fields are analysed in terms of acceleration, vorticity, and velocity-gradient tensors in addition to the velocity field. The data show that flow acceleration becomes maximum slightly before and after the collision peak, not in accord with the linear theory which predicts the maximum acceleration at the collision peak. Visualized velocity-gradient-tensor fields show that fluid parcels are stretched vertically prior to reaching the state of maximum wave amplitude. After the collision peak, fluid parcels are stretched in the horizontal direction. The boundary-layer evolution based on the vorticity generation and diffusion processes are discussed. It is shown that flow separation occurs at the bed during the collision. The collision creates small dispersive trailing waves. The formation of the trailing waves is captured by observing the transition behaviour of the velocity-gradient-tensor field: the direction of stretching of fluid parcels alternates during the generation of the trailing waves.