Stéphane Popinet

Jet formation in bubbles bursting at a free surface

We study numerically bubbles bursting at a free surface and the subsequent jet formation. The Navier–Stokes equations with a free surface and surface tension are solved using a marker-chain approach. Differentiation and boundary conditions near the free surface are satisfied using least-squares methods. Initial conditions involve a bubble connected to the outside atmosphere by a preexisting opening in a thin liquid layer. The evolution of the bubble is studied as a function of bubble radius. A jet forms with or without the formation of a tiny air bubble at the base of the jet. The radius of the droplet formed at the tip of the jet is found to be about one tenth of the initial bubble radius. A series of critical radii exist, for which a transition from a dynamics with or without bubbles exist. For some parameter values, the jet formation is close to a singular flow, with a conical cavity shape and a large curvature or cusp at the bottom. This is compared to similar singularities investigated in other contexts such as Faraday waves.

Growth and collapse of a vapor bubble in a narrow tube

The fluid mechanical aspects of the axisymmetric growth and collapse of a bubble in a narrow tube filled with a viscous liquid are studied numerically. The tube is open at both ends and connects two liquid reservoirs at constant pressure. The bubble is initially a small sphere and growth is triggered by a large internal pressure applied for a short time. After this initial phase, the motion proceeds by inertia. This model simulates the effect of an intense, localized, brief heating of the liquid which leads to the nucleation and growth of a bubble. The dimensionless parameters governing the problem are discussed and their effects illustrated with several examples. It is also shown that, when the bubble is not located at the midpoint of the tube, a net flow develops capable of pumping fluid from one reservoir to the other. The motivation for this work is offered by the possibility to use vapor bubbles as actuators in fluid-handling microdevices.

Viscous nonlinear theory of Richtmyer–Meshkov instability

We propose a quantitative prediction of the effect of viscosity on the weakly nonlinear impulsive Richtmyer–Meshkov instability between two fluids of arbitrary density and viscosity. This theory is based on an asymptotic analysis of the Navier–Stokes equations using singular perturbation techniques. The law obtained for interface deformation does not agree with former theoretical predictions of the effect of viscosity [K. O. Mikaelian, Phys. Rev. E 47, 375 (1993)], but compares very well with direct numerical simulations we performed using a front-tracking code developed in our laboratory. Application of this law to typical experimental parameters gives a formal demonstration of the relevance of inviscid models for the description of typical shock-tube experiments; at the same time, however, it shows that care should be taken with regard to viscosity in the case of impulsive experiments performed with liquids.

Wave transformation and shoreline water level on Funafuti Atoll, Tuvalu

The influence of sea swell (SS) waves, infragravity (IG) waves, and wave setup on maximum runup (Rmax) is investigated across different tidal stages on Fatato Island, Funafuti Atoll, Tuvalu. Field results illustrate that SS waves are tidally modulated at the shoreline, with comparatively greater wave attenuation and setup occurring at low tide versus high tide. A shoreward increase in IG wave height is observed across the 100 m wide reef flat at all tidal elevations, with no tidal modulation of IG wave height at the reef flat or island shoreline. A 1-D shock-capturing Green-Naghdi solver is used to replicate the field deployment and analyze Rmax. Model outputs for SS wave height, IG wave height and setup at the shoreline match field results with model skill >0.96. Model outputs for Rmax are used to identify the temporal window when geomorphic activity can occur on the beach face. During periods of moderate swell energy, waves can impact the beach face at spring low tide, due to a combination of wave setup and strong IG wave activity. Under mean wave conditions, the combined influence of setup, IG waves and SS waves results in interaction with island sediment at midtide. At high tide, SS and IG waves directly impact the beach face. Overall, wave activity is present on the beach face for 71% of the study period, a significantly longer duration than is calculated using mean water level and topographic data.

Droplet migration in a Hele–Shaw cell: Effect of the lubrication film on the droplet dynamics

Droplet migration in a Hele–Shaw cell is a fundamental multiphase flow problem which is crucial for many microfluidics applications. We focus on the regime at low capillary number and three-dimensional direct numerical simulations are performed to investigate the problem. In order to reduce the computational cost, an adaptive mesh is employed and high mesh resolution is only used near the interface. Paramet-ric studies are performed on the droplet horizontal radius and the capillary number. For droplets with an horizontal radius larger than half the channel height the droplet overfills the channel and exhibits a pancake shape. A lubrication film is formed between the droplet and the wall and particular attention is paid to the effect of the lubrication film on the droplet velocity. The computed velocity of the pancake droplet is shown to be lower than the average inflow velocity, which is in agreement with experimental measurements. The numerical results show that both the strong shear induced by the lubrication film and the three-dimensional flow structure contribute to the low mobility of the droplet. In this low-migration-velocity scenario the interfacial flow in the droplet reference frame moves toward the rear on the top and reverses direction moving to the front from the two side edges. The velocity of the pancake droplet and the thickness of the lubrication film are observed to decrease with capillary number. The droplet velocity and its dependence on capillary number cannot be captured by the classic Hele–Shaw equations, since the depth-averaged approximation neglects the effect of the lubrication film.

The effect of viscosity, surface tension and non-linearity on Richtmyer–Meshkov instability

A weakly non-linear theoretical model of the Richtmyer–Meshkov instability between two viscous fluids with surface tension is proposed. The model is based on the application of singular perturbations techniques to the incompressible Navier–Stokes equations written for two superposed immiscible fluids. A simple analytical law of interface deformation is obtained, in which the effects of viscosity, surface tension and non-linearities appear under the form of independent terms. The model gives also access to the velocity and pressure distribution in the fluids, which can be of interest for estimating vorticity diffusion in the fluids. A comparison with accurate direct numerical simulations confirms the validity of the proposed theory. The interface deformation law is then applied to typical experimental configurations in order to estimate the relative influence of surface tension, viscosity and non-linearities on the growth of perturbations for each of the chosen cases.

Initialising Landslide-Generated Tsunamis for Probabilistic Tsunami Hazard Assessment in Cook Strait

The Cook Strait Canyon is a submarine canyon which lies within 10u2009km of Wellington, the capital city of New Zealand. The canyon flanks are scarred with the evidence of past landslides that may have caused large local tsunamis. City planning and civil defence management require information on the magnitude and frequency of these tsunamis to adequately plan for them. Submarine-landslide-generated tsunamis are by nature local features. While they may be catastrophic in the near field, they are generally far smaller scales than co-seismic tsunamis and their energy does not travel very far. Including them within a comprehensive tsunami hazard assessment requires accounting for a large number of potential landslide sources. Unless we only use simple rules of thumb to approximate tsunami height, this requires considerable computing power. This article describes a technique for expanding two-dimensional vertical-slice tsunami generation by landslide modelling into a two-dimensional horizontal surface which can be used for tsunami propagation and inundation modelling. As such, it spans the gap between full three-dimensional modelling of the landslide and simple initialisation.

Future Reef Growth Can Mitigate Physical Impacts of Sea‐Level Rise on Atoll Islands

We present new detail on how future SLR will modify nonlinear wave transformation processes, shoreline wave energy and wave driven flooding on atoll islands. Frequent and destructive wave inundation is a primary climate-change hazard that may render atoll islands uninhabitable in the near future. However, limited research has examined the physical vulnerability of atoll islands to future SLR and sparse information is available to implement process based coastal management on coral reef environments. We utilize a field-verified numerical model capable of resolving all nonlinear wave transformation processes to simulate how future SLR will modify wave dissipation and overtopping on Funafuti Atoll, Tuvalu, accounting for static and accretionary reef adjustment morphologies. Results show that future SLR coupled with a static reef morphology will not only increase shoreline wave energy and overtopping but will fundamental alter the spectral composition of shoreline energy by decreasing the contemporary influence of low frequency infragravity waves. ‘Business-as-usual emissions (RCP 8.5) will result in annual wave overtopping on Funafuti Atoll by 2030, with overtopping at high tide under mean wave conditions occurring from 2090. Comparatively, vertical reef accretion in response to SLR will prevent any significant increase in shoreline wave energy and mitigate wave driven flooding volume by 72%. Our results provide the first quantitative assessment of how effective future reef accretion can be at mitigating SLR associated flooding on atoll islands and endorse active reef conservation and restoration for future coastal protection.

Transition in a numerical model of contact line dynamics and forced dewetting

Abstract We investigate the transition to a Landau–Levich–Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier–Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle θ Δ , called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from 15 ∘ to 110 ∘ and capillary numbers from 0.00085 to 0.2 where the mesh size Δ is varied in the range of 0.0035 to 0.06 of the capillary length l c . To interpret the results, we use Coxs theory which involves a microscopic distance r m and a microscopic angle θ e . In the numerical case, the equivalent of θ e is the angle θ Δ and we find that Coxs theory also applies. We introduce the scaling factor or gauge function ϕ so that r m = Δ / ϕ and estimate this gauge function by comparing our numerics to Coxs theory. The comparison provides a direct assessment of the agreement of the numerics with Coxs theory and reveals a critical feature of the numerical treatment of contact line dynamics: agreement is poor at small angles while it is better at large angles. This scaling factor is shown to depend only on θ Δ and the viscosity ratio q. In the case of small θ e , we use the prediction by Eggers [Phys. Rev. Lett. 93 (2004) 094502] of the critical capillary number for the Landau–Levich–Derjaguin forced dewetting transition. We generalize this prediction to large θ e and arbitrary q and express the critical capillary number as a function of θ e and r m . This implies also a prediction of the critical capillary number for the numerical case as a function of θ Δ and ϕ. The theory involves a logarithmically small parameter ϵ = 1 / ln u2061 ( l c / r m ) and is thus of moderate accuracy. The numerical results are however in approximate agreement in the general case, while good agreement is reached in the small θ Δ and q case. An analogy can be drawn between the numerical contact angle condition and a regularization of the Navier–Stokes equation by a partial Navier-slip model. The analogy leads to a value for the numerical length scale r m proportional to the slip length. Thus the microscopic length found in the simulations is a kind of numerical slip length in the vicinity of the contact line. The knowledge of this microscopic length scale and the associated gauge function can be used to realize grid-independent simulations that could be matched to microscopic physics in the region of validity of Coxs theory.

Model Skill and Sensitivity for Simulating Wave Processes on Coral Reefs Using a Shock-Capturing Green-Naghdi Solver

ABSTRACT Beetham, E.; Kench, P.S., and Popinet, S., 2018. Model skill and sensitivity for simulating wave processes on coral reefs using a shock-capturing Green-Naghdi solver. Wave-flume data from published benchmark experiments were used to extensively evaluate numerical model skill and sensitivity for applying a shock-capturing Green-Naghdi (GN) model to simulate nonlinear wave-transformation processes on complex coral reefs. Boussinesq-type models that utilise nonlinear shallow-water equations (NSWEs) to represent wave breaking and dissipation hold significant potential for understanding coastal hazards associated with global environmental change and sea-level rise. These fully nonlinear phase-resolving models typically require a threshold condition to switch from dispersive equations to shock-capturing NSWEs in areas of active wave breaking. However, limited information exists regarding how this splitting approach influences the behaviour of different surf-zone processes that contribute to wave runup and inundation on reef environments. This paper presents a comprehensive analysis of model sensitivity to explore how input parameters that control wave breaking and dissipation influence the behaviour of sea-swell waves, infragravity waves, wave setup, runup, and solitary waves on coral reefs. Results show that each wave process exhibits unique sensitivity to the free-surface slope threshold (B) that is used to represent areas of active wave breaking by locally switching from the weakly dispersive GN equations to the shock-capturing NSWEs. Accurate representation of all wave processes, however, can be achieved if the wave-face steepens to at least 35° (B ≥ 0.7) before breaking is initiated. Results from this research support and encourage the use of nonlinear phase-resolving wave models as tools for academic research, coastal management, coastal engineering, and hazard forecasting on atoll and fringing reef environments.