Intermittency effects are numerically studied in turbulent bubbling Rayleigh–Bénard (RB) flow and compared to the standard RB case. The vapour bubbles are modelled with a Euler–Lagrangian scheme and are two-way coupled to the flow and temperature fields, both mechanically and thermally. To quantify the degree of intermittency we use probability density functions, structure functions, extended self-similarity (ESS) and generalized extended self-similarity (GESS) for both temperature and velocity differences. For the standard RB case we reproduce scaling very close to the Obukhov–Corrsin values common for a passive scalar and the corresponding relatively strong intermittency for the temperature fluctuations, which are known to originate from sharp temperature fronts. These sharp fronts are smoothed by the vapour bubbles owing to their heat capacity, leading to much less intermittency in the temperature but also in the velocity field in bubbling thermal convection.

We present several new spanwise-localized equilibrium and travelling-wave solutions of plane Couette and channel flows. The solutions exhibit concentrated regions of vorticity that are centred over low-speed streaks and flanked on either side by high-speed streaks. For several travelling-wave solutions of channel flow, the vortex structures are concentrated near the walls and form particularly isolated and elemental versions of coherent structures in the near-wall region of shear flows. One travelling wave appears to be the invariant solution corresponding to a near-wall coherent structure educed from simulation data by Jeong et al. (J. Fluid Mech., vol. 332, 1997, pp. 185–214) and analysed in terms of transient growth modes of streaky flow by Schoppa & Hussain (J. Fluid Mech., vol. 453, 2002, pp. 57–108). The solutions are constructed by a variety of methods: application of windowing functions to previously known spatially periodic solutions, continuation from plane Couette to channel flow conditions, and from initial guesses obtained from turbulent simulation data. We show how the symmetries of localized solutions derive from the symmetries of their periodic counterparts, analyse the exponential decay of their tails, examine the scale separation and scaling of their streamwise Fourier modes, and show that they develop critical layers for large Reynolds numbers.

The combustion stability of a liquid-propellant rocket engine experiencing a random, finite perturbation from steady-state conditions is examined. The probability is estimated for a nonlinear resonant limit-cycle oscillation to be triggered by a random disturbance. Transverse pressure waves are considered by using a previously published two-dimensional nonlinear pressure wave equation coupled with Euler equations governing the velocity components. The cylindrical combustion chamber is a complex system containing multiple co-axial methane–oxygen injectors; each co-axial jet is analysed for mixing and burning on its own local grid scheme, with the energy release rate coupled to the wave oscillation on the more global grid. Two types of stochastic forcing for the random disturbance are explored: a travelling Gaussian pressure pulse and an oscillating pressure dipole source. The random variables describing the pulse are magnitude, location, duration and orientation of the disturbances. The polynomial chaos expansion (PCE) method is used to determine the long-time behaviour and infer the asymptote of the solution to the governing partial differential equations. Depending on the random disturbance, the asymptote could be the steady-state solution or a limit-cycle oscillation, e.g. a first tangential travelling wave mode. The asymptotic outcome is cast as a stochastic variable which is determined as a function of input random variables. The accuracy of the PCE application is compared with a Monte Carlo calculation and is shown to be significantly less costly for similar accuracy.

The effects of spanwise rotation on the channel flow across a symmetric sudden expansion are investigated using direct numerical simulation. Four rotation regimes are considered with the same Reynolds number $\mathit{Re}=5000$ and expansion ratio $\mathit{Er}=3/2$. Upstream from the expansion, inflow turbulent conditions are generated realistically for each rotation rate through a very simple and efficient technique of recycling without the need for any precursor calculation. As the rotation is increased, the flow becomes progressively asymmetric with stabilization (destabilization) effects on the cyclonic (anticyclonic) side, respectively. These rotation effects, already present in the upstream channel, lead further downstream to an increase (reduction) of the separation size behind the cyclonic (anticyclonic) step. In the cyclonic separation, the free-shear layer created behind the step corner leads to the formation of large-scale spanwise vortices that become increasingly two-dimensional as the rotation is increased. Conversely, in the anticyclonic region, the turbulent structures in the separated layer are more elongated in the streamwise direction and also more active in promoting reattachment. For the highest rotation rate, a secondary separation is observed further downstream in the anticyclonic region, leading to the establishment of an elongated recirculation bubble that deflects the main flow towards the cyclonic wall. The highest level of turbulent kinetic energy is obtained at high rotation near the cyclonic reattachment in a region where stabilization effects are expected. The phenomenological model of absolute vortex stretching is found to be useful in understanding how the rotation influences the dynamics in the various regions of the flow.

This paper describes a detailed experimental study using hot-wire anemometry of the laminar–turbulent transition region of a rotating-disk boundary-layer flow without any imposed excitation of the boundary layer. The measured data are separated into stationary and unsteady disturbance fields in order to elaborate on the roles that the stationary and the travelling modes have in the transition process. We show the onset of nonlinearity consistently at Reynolds numbers, $R$, of $\sim $510, i.e. at the onset of Lingwood’s (J. Fluid Mech., vol. 299, 1995, pp. 17–33) local absolute instability, and the growth of stationary vortices saturates at a Reynolds number of $\sim $550. The nonlinear saturation and subsequent turbulent breakdown of individual stationary vortices independently of their amplitudes, which vary azimuthally, seem to be determined by well-defined Reynolds numbers. We identify unstable travelling disturbances in our power spectra, which continue to grow, saturating at around $R=585$, whereupon turbulent breakdown of the boundary layer ensues. The nonlinear saturation amplitude of the total disturbance field is approximately constant for all considered cases, i.e. different rotation rates and edge Reynolds numbers. We also identify a travelling secondary instability. Our results suggest that it is the travelling disturbances that are fundamentally important to the transition to turbulence for a clean disk, rather than the stationary vortices. Here, the results appear to show a primary nonlinear steep-fronted (travelling) global mode at the boundary between the local convectively and absolutely unstable regions, which develops nonlinearly interacting with the stationary vortices and which saturates and is unstable to a secondary instability. This leads to a rapid transition to turbulence outward of the primary front from approximately $R=565$ to 590 and to a fully turbulent boundary layer above 650.

We investigate how the domain depth affects the turbulent behaviour in spatially developing mixing layers by means of large-eddy simulations based on a spectral vanishing viscosity technique. Analyses of spectra of the vertical velocity, of Lumley’s diagrams, of the turbulent kinetic energy and of the vortex stretching show that a two-dimensional behaviour of the turbulence is promoted in spatial mixing layers by constricting the fluid motion in one direction. This finding is in agreement with previous works on turbulent systems constrained by a geometric anisotropy, pioneered by Smith, Chasnov & Waleffe (Phys. Rev. Lett., vol. 77, 1996, pp. 2467–2470). We observe that the growth of the momentum thickness along the streamwise direction is damped in a confined domain. An almost fully two-dimensional turbulent behaviour is observed when the momentum thickness is of the same order of magnitude as the confining scale.

The formation and evolution of flow structures of a finite-span synthetic jet issuing into a quiescent flow were investigated experimentally using stereoscopic particle image velocimetry (SPIV). The effect of two geometrical parameters, the orifice aspect ratio and the neck length, were explored at a Strouhal number of 0.115 and a Reynolds number of 615. Normalized orifice neck lengths of 2, 4 and 6 and aspect ratios of 6, 12, and 18 were examined. It was found that the effect of the aspect ratio is much larger than the effect of the neck length, and as the aspect ratio increases the size of the edge vortices decreases and the presence of secondary structures is more evident. Moreover, axis switching was observed and its streamwise location increases as the aspect ratio increases. The effect of the neck length on the flow structures and the evolution of the synthetic jet was found to be secondary, where the effect was only in the very near field (i.e. close to the jet’s orifice).

Steady two-dimensional nonlinear flexural–gravity hydraulic falls past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal and is covered above by a thin ice plate. Cosserat theory is used to model the sheet of ice as a thin elastic shell, and boundary integral equation techniques are then employed to find critical flow solutions. By utilising a second obstruction, solutions with a train of waves trapped between two obstructions are investigated.

In a uniformly rotating fluid, inertial waves propagate along rays that are inclined to the rotation axis by an angle that depends on the wave frequency. In closed domains, multiple reflections from the boundaries may cause inertial waves to focus onto particular structures known as wave attractors. These attractors are likely to appear in fluid containers with at least one boundary that is neither parallel nor normal to the rotation axis. A closely related process also applies to internal gravity waves in a stably stratified fluid. Such structures have previously been studied from a theoretical point of view, in laboratory experiments, in linear numerical calculations and in some recent numerical simulations. In the present paper, two-dimensional direct numerical simulations of an inertial wave attractor are presented. By varying the amplitude at which the system is forced periodically, we are able to describe the transition between the linear and nonlinear regimes as well as the characteristic properties of the two situations. In the linear regime, we first recover the results of the linear calculations and asymptotic theory of Ogilvie (J. Fluid Mech., vol. 543, 2005, pp. 19–44) who considered a prototypical problem involving the focusing of linear internal waves into a narrow beam centred on a wave attractor in a steady state. The velocity profile of the beam and its scalings with the Ekman number, as well as the asymptotic value of the dissipation rate, are found to be in agreement with the linear theory. We also find that, as the beam builds up around the wave attractor, the power input by the applied force reaches its limiting value more rapidly than the dissipation rate, which saturates only when the beam has reached its final thickness. In the nonlinear regime, the beam is strongly affected and becomes unstable to a subharmonic instability. This instability transfers energy to secondary waves possessing shorter wavelengths and lower frequencies. The onset of the instability of a narrow inertial wave beam is investigated by means of a separate linear analysis and the results, such as the onset of the instability, are found to be consistent with the global simulations of the wave attractor. The excitation of such secondary waves described theoretically in this work has also been seen in recent laboratory experiments on internal gravity waves.

The linear stability of transient diffusive boundary layers in porous media has been studied extensively for its applications to carbon dioxide sequestration. The onset of nonlinear convection, however, remains understudied because the transient base state invalidates the traditional stability methods that are used for autonomous systems. We demonstrate that the onset time of nonlinear convection, $t=t_{\mathit{on}}$, can be determined from an expansion that is two orders of magnitude faster than a direct numerical simulation. Using the expansion, we explore the sensitivity of $t_{\mathit{on}}$ to the initial perturbation magnitude and wavelength, as well as the initial time at which a perturbation is initiated. We find that there is an optimal initial time and wavelength that minimize $t_{\mathit{on}}$, and we obtain analytical relationships for these parameters in terms of aquifer properties and initial perturbation magnitude. This importance of the initial perturbation time and magnitude is often overlooked in previous studies. To investigate perturbation evolution at late-times, $t>t_{\mathit{on}}$, we perform direct numerical simulations that reveal two unique features of transient diffusive boundary layers. First, when a boundary layer is perturbed with a single horizontal Fourier mode, nonlinear mechanisms generate a zero-wavenumber response whose magnitude eventually surpasses that of the fundamental mode. Second, when a boundary layer is simultaneously perturbed with many Fourier modes, the late-time perturbation magnitude is concentrated in the zero-wavenumber mode, and there is no clearly dominant, non-zero, wavenumber. These unique results are further interpreted by comparison with direct numerical simulations of Rayleigh–Bénard convection.

This study investigates the collision statistics of inertial particles in inverse-cascading two-dimensional (2D) homogeneous isotropic turbulence by means of a direct numerical simulation (DNS). A collision kernel model for particles with small Stokes number ($\mathit{St}$) in 2D flows is proposed based on the model of Saffman & Turner (J. Fluid Mech., vol. 1, 1956, pp. 16–30) (ST56 model). The DNS results agree with this 2D version of the ST56 model for $\mathit{St}\lesssim 0.1$. It is then confirmed that our DNS results satisfy the 2D version of the spherical formulation of the collision kernel. The fact that the flatness factor stays around 3 in our 2D flow confirms that the present 2D turbulent flow is nearly intermittency-free. Collision statistics for $\mathit{St}= 0.1$, 0.4 and 0.6, i.e. for $\mathit{St}<1$, are obtained from the present 2D DNS and compared with those obtained from the three-dimensional (3D) DNS of Onishi et al. (J. Comput. Phys., vol. 242, 2013, pp. 809–827). We have observed that the 3D radial distribution function at contact ($g(R)$, the so-called clustering effect) decreases for $\mathit{St}= 0.4$ and 0.6 with increasing Reynolds number, while the 2D $g(R)$ does not show a significant dependence on Reynolds number. This observation supports the view that the Reynolds-number dependence of $g(R)$ observed in three dimensions is due to internal intermittency of the 3D turbulence. We have further investigated the local $\mathit{St}$, which is a function of the local flow strain rates, and proposed a plausible mechanism that can explain the Reynolds-number dependence of $g(R)$. Meanwhile, 2D stochastic simulations based on the Smoluchowski equations for $\mathit{St}\ll 1$ show that the collision growth can be predicted by the 2D ST56 model and that rare but strong events do not play a significant role in such a small-$\mathit{St}$ particle system. However, the probability density function of local $\mathit{St}$ at the sites of colliding particle pairs supports the view that powerful rare events can be important for particle growth even in the absence of internal intermittency when $\mathit{St}$ is not much smaller than unity.

We propose and analyse a new strategy of shear flow turbulence control that can be realized by the following steps: (i) imposing specially designed seed velocity perturbations, which are non-symmetric in the spanwise direction, at the walls of a flow; (ii) the configuration of the latter ensures a gain of shear flow energy and the breaking of turbulence spanwise reflection symmetry: this leads to the generation of spanwise mean flow; (iii) that changes the self-sustained dynamics of turbulence and results in a considerable reduction of the turbulence level and the production of turbulent kinetic energy. In fact, by this strategy the shear flow transient growth mechanism is activated and the formed spanwise mean flow is an intrinsic, nonlinear composition of the controlled turbulence and not directly introduced in the system. In the present paper, a weak near-wall volume forcing is designed to impose the velocity perturbations with required characteristics in the flow. The efficiency of the proposed scheme has been demonstrated by direct numerical simulation using plane Couette flow as a representative example. A promising result was obtained: after a careful parameter selection, the forcing reduces the turbulence kinetic energy and its production by up to one-third. The strategy can be naturally applied to other wall-bounded flows, e.g. channel and boundary-layer flows. Of course, the considered volume force is theoretical and hypothetical. Nevertheless, it helps to gain knowledge concerning the design of the seed velocity field that is necessary to be imposed in the flow to achieve a significant reduction of the turbulent kinetic energy. This is convincing with regard to a new control strategy, which could be based on specially constructed blowing/suction or riblets, by employing the insight gained by the comprehension of the results obtained using the investigated methodology in this paper.

The dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion is often described as a diffusive process, with an effective diffusivity that is enhanced compared with the molecular value. However, this description fails to capture the tails of the scalar concentration distribution in initial-value problems. To remedy this, we develop a large-deviation theory of scalar dispersion that provides an approximation to the scalar concentration valid at much larger distances away from the centre of mass, specifically distances that are $O(t)$ rather than $O(t^{1/2})$, where $t \gg 1$ is the time from the scalar release. The theory centres on the calculation of a rate function characterizing the large-time form of the scalar concentration. This function is deduced from the solution of a one-parameter family of eigenvalue problems which we derive using two alternative approaches, one asymptotic, the other probabilistic. We emphasize the connection between the large-deviation theory and the homogenization theory that is often used to compute effective diffusivities: a perturbative solution of the eigenvalue problems in the appropriate limit reduces at leading order to the cell problem of homogenization theory. We consider two classes of flows in some detail: shear flows and periodic flows with closed streamlines (cellular flows). In both cases, large deviation generalizes classical results on effective diffusivity and captures new phenomena relevant to the tails of the scalar distribution. These include approximately finite dispersion speeds arising at large Péclet number $\mathit{Pe}$ (corresponding to small molecular diffusivity) and, for two-dimensional cellular flows, anisotropic dispersion. Explicit asymptotic results are obtained for shear flows in the limit of large $\mathit{Pe}$. (A companion paper, Part 2, is devoted to the large-$\mathit{Pe}$ asymptotic treatment of cellular flows.) The predictions of large-deviation theory are compared with Monte Carlo simulations that estimate the tails of concentration accurately using importance sampling.

A standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor $\mathit{Pe}^{1/2}$ larger than molecular diffusivity when the Péclet number $\mathit{Pe}$ is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time $t$ by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances $|\boldsymbol {x}|$ from the scalar release point that satisfy $|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the $O(\mathit{Pe}^{1/2})$ effective diffusivity, are recovered from our analysis in the limit $|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when $|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$ and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances $|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.

We reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

Similarity solutions based on velocity potential theory are found to be possible in the case of an expanding paraboloid entering water when gravity is ignored. Numerical solutions are obtained based on the boundary element method. Iteration is used for the nonlinear boundary conditions on the unknown free surface, together with regular remeshing. Results are obtained for paraboloids with different slenderness (or bluntness). Flow features and pressure distributions are discussed along with the physical implications. It is also concluded that similarity solutions may be possible in more general cases.

Direct numerical simulations are employed to investigate the interactions of bidisperse turbidity currents with three-dimensional seafloor topography in the form of Gaussian bumps. Results for two different bump heights are compared against currents propagating over a flat surface. The bump heights are chosen such that the current largely flows over the smaller bump, while it primarily flows around the taller bump. Furthermore, the effects of the settling velocity are investigated by comparing turbidity currents with corresponding compositional gravity currents. The influence of the bottom topography on the front velocity of turbidity currents is seen to be much weaker than the influence of the particle settling velocity. Consistent with earlier work on gravity currents propagating over flat boundaries, the influence of the Reynolds number on the front velocity of currents interacting with three-dimensional bottom topography is found to be small, as long as $\mathit{Re}\geq O(1000)$. The lobe-and-cleft structures, on the other hand, exhibit a stronger influence of the Reynolds number. The current/bump interaction deforms the bottom boundary-layer vorticity into traditional horseshoe vortices, with a downwash region in the centre of the wake. At the same time, the vorticity originating in the mixing layer between the current and the ambient interacts with the bump in such a way as to form ‘inverted horseshoe vortices’, with an upwash region in the wake centre. Additional streamwise vortical structures form as a result of baroclinic vorticity generation. The dependence of the sedimentation rate and streamwise vorticity generation on the height of the bump are discussed, and detailed analyses are presented of the energy budget and bottom wall-shear stress. It is shown that for typical laboratory-scale experiments, the range of parameters explored in the present investigation will not give rise to bedload transport or sediment resuspension. Based on balance arguments for the kinetic and potential energy components, a scaling law is obtained for the maximum bump height over which gravity currents can travel. This scaling law is validated by simulation results, and it provides a criterion for distinguishing between ‘short’ and ‘tall’ topographical features. For turbidity currents, this scaling result represents an upper limit. An interesting non-monotonic influence of the bump height is observed on the long-term propagation velocity of the current. On the one hand, the lateral deflection of the current by the bump leads to an effective increase in the current height and its front velocity in the region away from the bump. At the same time, taller bumps result in a more vigorous three-dimensional evolution of the current, accompanied by increased levels of dissipation, which slows the current down. For small bumps, the former mechanism dominates, so that on average the current front propagates faster than its flat bottom counterpart. For currents interacting with larger bumps, however, the increased dissipation becomes dominant, so that they exhibit a reduced front velocity as compared to currents propagating over flat surfaces.

In this paper, the stability of falling films with different flow conditions at the inlet is studied. This is done with an algorithm for the numerical investigation of stability of steady-state solutions to dynamical systems, which is based on an Arnoldi-type iteration. It is shown how this algorithm can be applied to free boundary problems in hydrodynamics. A volume-of-fluid solver is employed to predict the time evolution of perturbations to the steady state. The method is validated by comparison to data from temporal and spatial stability theory, and to experimental results. The algorithm is used to analyse the flow fields of falling films with inlet and outlet, taking the inhomogeneity caused by different inlet conditions into account. In particular, steady states with a curved interface are analysed. A variety of reasonable inlet conditions is investigated. The instability of the film is convective and perturbations at the inlet could be of importance since they are exponentially amplified as they are transported downstream. However, the employed algorithm shows that there is no significant effect of the inlet condition. It is concluded that the flow characteristics of falling films are stable with respect to the considered time-independent inlet conditions.

In this paper, we investigate computationally the effects of membrane hardness on the dynamics of strain-hardening capsules in planar extensional Stokes flows. As the flow rate increases, all capsules reach elongated steady-state configurations but the cross-section of the more strain-hardening capsules preserves its elliptical shape while the less strain-hardening capsules become lamellar. The capsule deformation in strong extensional flows is accompanied with very pointed edges, i.e. large edge curvatures and thus small local edge length scales, which makes the current investigation a multi-length interfacial dynamics problem. Our computational results for elongated strain-hardening capsules are accompanied with a scaling analysis which provides physical insight on the extensional capsule dynamics. The two distinct capsule conformations we found, i.e. the slender spindle and lamellar capsules, are shown to represent two different types of steady-state extensional dynamics. The former are stabilized mainly via the membrane’s shearing resistance and the latter via its area-dilatation resistance, associated with the elongation tension normal forces and thus both types differ from bubbles which are stabilized mainly via the lateral surface-tension normal forces. Our steady-state deformation results can be used to identify the elastic properties of a real capsule, i.e. the membrane’s shear and area-dilatation moduli, utilizing a single experimental technique.

Bubble dynamics near a rigid boundary are associated with wide and important applications in cavitation erosion in many industrial systems and medical ultrasonics. This classical problem is revisited with the following two developments. Firstly, computational studies on the problem have commonly been based on an incompressible fluid model, but the compressible effects are essential in this phenomenon. Consequently, a bubble usually undergoes significantly damped oscillation in practice. In this paper this phenomenon will be modelled using weakly compressible theory and a modified boundary integral method for an axisymmetric configuration, which predicts the damped oscillation. Secondly, the computational studies so far have largely been concerned with the first cycle of oscillation. However, a bubble usually oscillates for a few cycles before it breaks into much smaller ones. Cavitation erosion may be associated with the recollapse phase when the bubble is initiated more than the maximum bubble radius away from the boundary. Both the first and second cycles of oscillation will be modelled. The toroidal bubble formed towards the end of the collapse phase is modelled using a vortex ring model. The repeated topological changes of the bubble are traced from a singly connected to a doubly connected form, and vice versa. This model considers the energy loss due to shock waves emitted at minimum bubble volumes during the beginning of the expansion phase and around the end of the collapse phase. It predicts damped oscillations, where both the maximum bubble radius and the oscillation period reduce significantly from the first to second cycles of oscillation. The damping of the bubble oscillation is alleviated by the existence of the rigid boundary and reduces with the standoff distance between them. Our computations correlate well with the experimental data (Philipp & Lauterborn, J. Fluid Mech., vol. 361, 1998, pp. 75–116) for both the first and second cycles of oscillation. We have successively reproduced the bubble ring in direct contact with the rigid boundary at the end of the second collapse phase, a phenomenon that was suggested to be one of the major causes of cavitation erosion by experimental studies.