Sébastien Tanguy

Benchmarks and numerical methods for the simulation of boiling flows

Comparisons of different numerical methods suited to the simulations of phase changes are presented in the framework of interface capturing computations on structured fixed computational grids. Due to analytical solutions, we define some reference test-cases that every numerical technique devoted to phase change should succeed. Realistic physical properties imply some drastic interface jump conditions on the normal velocity or on the thermal flux. The efficiencies of Ghost Fluid and Delta Function Methods are compared to compute the normal velocity jump condition. Next, we demonstrate that high order extrapolation methods on the thermal field allow performing accurate and robust simulations for a thermally controlled bubble growth. Finally, some simulations of the growth of a rising bubble are presented, both for a spherical bubble and a deformed bubble.

On the computation of viscous terms for incompressible two-phase flows with Level Set/Ghost Fluid Method

In this paper, we present a detailed analysis of the computation of the viscous terms for the simulation of incompressible two-phase flows in the framework of Level Set/Ghost Fluid Method when viscosity is discontinuous across the interface. Two pioneering papers on the topic, Kang et al. 10 and Sussman et al. 26, proposed two different approaches to deal with viscous terms. However, a definitive assessment of their respective efficiency is currently not available. In this paper, we demonstrate from theoretical arguments and confirm from numerical simulations that these two approaches are equivalent from a continuous point of view and we compare their accuracies in relevant test-cases. We also propose a new intermediate method which uses the properties of the two previous methods. This new method enables a simple implementation for an implicit temporal discretization of the viscous terms. In addition, the efficiency of the Delta Function method 24 is also assessed and compared to the three previous ones, which allow us to propose a general overview of the accuracy of all available methods. The selected test-cases involve configurations wherein viscosity plays a major role and for which either theoretical results or experimental data are available as reference solutions: simulations of spherical rising bubbles, shape-oscillating bubbles and deformed rising bubbles at low Reynolds numbers.

Solving elliptic problems with discontinuities on irregular domains - the Voronoi Interface Method

We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solutions gradients are second-order accurate and first-order accurate, respectively, in the L ∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions.

A Ghost Fluid/Level Set Method for boiling flows and liquid evaporation

The development of numerical methods for the direct numerical simulation of two-phase flows with phase change, in the framework of interface capturing or interface tracking methods, is the main topic of this study. We propose a novel numerical method, which allows dealing with both evaporation and boiling at the interface between a liquid and a gas. Indeed, in some specific situations involving very heterogeneous thermodynamic conditions at the interface, the distinction between boiling and evaporation is not always possible. For instance, it can occur for a Leidenfrost droplet; a water drop levitating above a hot plate whose temperature is much higher than the boiling temperature. In this case, boiling occurs in the film of saturated vapor which is entrapped between the bottom of the drop and the plate, whereas the top of the water droplet evaporates in contact of ambient air. The situation can also be ambiguous for a superheated droplet or at the contact line between a liquid and a hot wall whose temperature is higher than the saturation temperature of the liquid. In these situations, the interface temperature can locally reach the saturation temperature (boiling point), for instance near a contact line, and be cooler in other places. Thus, boiling and evaporation can occur simultaneously on different regions of the same liquid interface or occur successively at different times of the history of an evaporating droplet. Standard numerical methods are not able to perform computations in these transient regimes, therefore, we propose in this paper a novel numerical method to achieve this challenging task. Finally, we present several accuracy validations against theoretical solutions and experimental results to strengthen the relevance of this new method.

Effect of rising motion on the damped shape oscillations of drops and bubbles

The objective of this work is to determine the effect of the rising motion on the dynamics of inertial shape oscillations of drops and bubbles. We have carried out axisymmetric direct numerical simulations of an ascending drop (or bubble) using a level-set method. The drop is initially elongated in the vertical direction and therefore performs shape oscillations. The analysis is based on the decomposition of the interface into spherical harmonics, the time evolutions of which are processed to obtain the frequency and the damping rate of the oscillations. As the drop accelerates, its shape flattens and oscillations no longer take place around a spherical equilibrium shape. This causes the eigenmode of oscillations to change, which results in the appearance of spherical harmonics of high order that all oscillate at the same frequency. For both drops and bubbles, the frequency, which remains controlled by the potential flow, slightly decreases with the rising velocity. The damping rate of drops, which is controlled by the dissipation within boundary layers at the interface, strongly increases with the rising velocity. At terminal velocity, the damping rate of bubbles, which results from the dissipation by the potential flow associated with the oscillating motion, remains close to that of a non-rising bubble. During the transient, the rate of deformation of the equilibrium shape of bubbles can be comparable to the oscillation frequency, which causes complex evolutions of the shape. These results extend the description of shape oscillations to common situations where gravity plays a role. In particular, the present conclusions are useful to interpret experimental results where the effect of the rising motion is often combined with that of surfactant.

A time splitting projection scheme for compressible two-phase flows. Application to the interaction of bubbles with ultrasound waves

This paper is focused on the numerical simulation of the interaction of an ultrasound wave with a bubble. Our interest is to develop a fully compressible solver in the two phases and to account for surface tension effects.As the volume oscillation of the bubble occurs in a low Mach number regime, a specific care must be paid to the effectiveness of the numerical method which is chosen to solve the compressible Euler equations. Three different numerical solvers, an explicit HLLC (Harten-Lax-van Leer-Contact) solver 48, a preconditioning explicit HLLC solver 14 and the compressible projection method 21,53,55, are described and assessed with a one dimensional spherical benchmark. From this preliminary test, we can conclude that the compressible projection method outclasses the other two, whether the spatial accuracy or the time step stability are considered.Multidimensional numerical simulations are next performed. As a basic implementation of the surface tension leads to strong spurious currents and numerical instabilities, a specific velocity/pressure time splitting is proposed to overcome this issue. Numerical evidences of the efficiency of this new numerical scheme are provided, since both the accuracy and the stability of the overall algorithm are enhanced if this new time splitting is used. Finally, the numerical simulation of the interaction of a moving and deformable bubble with a plane wave is presented in order to bring out the ability of the new method in a more complex situation.

On two-phase flow solvers in irregular domains with contact line

We present numerical methods that enable the direct numerical simulation of two-phase flows in irregular domains. A method is presented to account for surface tension effects in a mesh cell containing a triple line between the liquid, gas and solid phases. Our numerical method is based on the level-set method to capture the liquid-gas interface and on the single-phase Navier-Stokes solver in irregular domain proposed in 35 to impose the solid boundary in an Eulerian framework. We also present a strategy for the implicit treatment of the viscous term and how to impose both a Neumann boundary condition and a jump condition when solving for the pressure field. Special care is given on how to take into account the contact angle, the no-slip boundary condition for the velocity field and the volume forces. Finally, we present numerical results in two and three spatial dimensions evaluating our simulations with several benchmarks.

Non-linear shape oscillations of rising drops and bubbles: Experiments and simulations

This paper focuses on shape-oscillations of a gas bubble or a liquid drop rising in another liquid. The bubble/drop is initially attached to a capillary and is released by a sudden motion of that capillary, resulting in the rise of the bubble/drop along with the oscillations of its shape. Such experimental conditions make difficult the interpretation of the oscillation dynamics with regard to the standard linear theory of oscillation because (i) amplitude of deformation is large enough to induce nonlinearities, (ii) the rising motion may be coupled with the oscillation dynamics, and (iii) clean conditions without residual surfactants may not be achieved. These differences with the theory are addressed by comparing experimental observation with numerical simulation. Simulations are carried out using Level-Set and Ghost-Fluid methods with clean interfaces. The effect of the rising motion is investigated by performing simulations under different gravity conditions. Using a decomposition of the bubble/drop shape into a series of spherical harmonics, experimental and numerical time evolutions of their amplitudes are compared. Due to large oscillation amplitude, non-linear couplings between the modes are evidenced from both experimental and numerical signals; modes of lower frequency influence modes of higher frequency, whereas the reverse is not observed. Nevertheless, the dominant frequency and overall damping rate of the first five modes are in good agreement with the linear theory. Effect of the rising motion on the oscillations is globally negligible, provided the mean shape of the oscillation remains close to a sphere. In the drop case, despite the residual interface contamination evidenced by a reduction in the terminal velocity, the oscillation dynamics is shown to be unaltered compared to that of a clean drop.

Unsteady rising of clean bubble in low viscosity liquid

When a submerged bubble is initially at rest in a stagnant low viscosity liquid such as water, buoyancy forces accelerate the bubble upwards. The increasing relative velocity of the bubble with the surrounding liquid provokes deformations on the bubble shape that affect its vertical acceleration and also induce surface tension driven oscillations. Our theoretical model, which is compared with full Navier–Stokes simulations predicts, with a reasonable accuracy, both the position of the bubble centre of mass, as well as the time varying bubble shape under those conditions for which the Reynolds number is large, the amplitude of the deformation is small, the bubble interface is free of surfactants and the bubble rises following a straight vertical path. The model can be used as a first approximation to describe the initial instants of the unsteady buoyancy driven rising of millimetre sized bubbles typically generated in water aerators.

Integration of low g sloshing models with spacecraft attitude control simulators

Very High Resolution Science and Observation missions are more and more demanding both in terms of high stability and in terms of maneuverability. But those two requirements are not compatible; indeed the rotation of the spacecraft induces sloshing of the propellant inside the tanks. And such fluid movement induces perturbation back on the platform. Given the high level of pointing stability that must be achieved such perturbation can be a major driver in the system trade-off.