Drops and bubbles: deformation, break-up, atomization

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The breaking of a powerful ocean wave in gigantic sprays, the myriad droplets ejected by a flying liquid drop when it impacts and splashes on an obstacle, the delicate flutter of microscopic droplets emitted by a bubble bursting on a free liquid surface are fascinating events. The numerical simulation of such droplets, bubbles and waves is the focal point of the research reported in these pages. These numerical simulations are a fairly complex problem of applied mathematics and computational mechanics. They require the research, development and coding of specific methods for free surfaces and fluid interfaces. Many researchers of d'Alembert working as a team have contributed to the development of this methodology. The numerical treatment of evolving interfaces, or surface tracking that we chose to perform using principally volume-of-fluid (VOF) methods is now a maturing subject, with however many theoretical and coding improvements still to come. Computational calculations allow, under certain conditions, a fair reproduction of the reality, so called numerical experiments can be performed, useful both for curiosity-driven and application-driven science.


In what follows we dwell on the following topics

Atomization processes involve the development of the instabilities at the interface of a high-speed jet. The combustion of liquid fuels requires their initial breakup into small droplets. Combustion technology thus provides a powerful incentive to study atomization. Injection devices present in both petrol and diesel car engines control the atomisation of fuels and thus the typical droplet size. This eventually determines the quality of the combustion, such as engine efficiency or pollution rate. We have performed numerous calculations of atomization.

Droplet impacts on solid or liquid surface have become a cultural icon. It is also an important issue in engineering. In agriculture, the penetration of rainwater depends strongly upon the humidity of the soil. Impact of aerosol drops, containing agricultural treatment products is also of considerable interest as well. In internal combustion engines, the rebound on combustion chamber or pipe walls modifies the size of the droplets and affects the combustion processes.

Bubble dynamics and other topics: Bubble dynamics has extremely varied applications, in power plant technology, chemical engineering, biology and oceanography. A striking observation is sonoluminescence: when excited by strong acoustic waves, bubbles emit a brief light flash, visible to the naked eye. Bubble oscillations relate to several other important problems: the stability of oscillating bubbles, or jet formation in a cavity for instance. Bubble dynamics is important in other engineering issues such as the damage brought to hydraulic circuits or turbine vanes by cavitation.

Numerical methods: we have developped numerical methods based on the idea of volume-of-fluid interface tracking and height functions. Volume-of-fluid is a method of interface tracking somewhat similar to level-sets, it provides explicit tracking of the interface which is robust with respect to topology changes and conserves mass to high accuracy and in some cases to machine accuracy. It has been combined with height-function methods and octree mesh refinement in the Gerris code.


 Gerris simulation of Diesel Jet. Click on image to enlarge

We have studied both the two and three-dimensional instabilities of a liquid-gas interface in a shear flow. These instabilities account in particular for liquid jet atomisation near the injection pipe. They are therefore fundamental in the determination of processes for liquid combustion.

Linear stability theory of small perturbations for a parallel two-phase flow (Orr-Sommerfeld theory in variable density case) is the first step in a systematic study of the instability. This theory is unfortunately impossible to formulate analytically and numerical simulations of the Orr-Sommerfeld equations are needed. P. Yecko and T. Boeck have shown the existence of new unstable modes in the dynamics, unknown in the literature (Yecko, Zaleski & Fullana 2002, Boeck & Zaleski 2005, Boeck et al 2007, Bague et al. 2010, see references on atomization. Transient three-dimensional growth was investigated by Yecko & Zaleski (2005).

Beyond linear instability growth, experiments and numerical simulations are particularly revealing of the structures and mechanisms of atomization. For instance, sheet and fiber-like objects are universally observed at sufficiently high jet speed (Lasheras or Hopfinger, Raynal and Villermaux).

This work also applied to the study of atomisation in Diesel engine jets. In this context, we have used adaptave mesh refinement on an oct tree, using the gerris code, resulting in the image above.

Droplet impact and splashing

 Gerris simulation of a splash performed by Pascal Ray. Click on image to enlarge

We have studied both the fully three-dimensional and axisymmetric impacts of a droplet on a solid surface, see references on droplet impact and splashing. Droplet impacts and collisions are found in all areas of science and engineering. For instance in the droplet model of atomic nuclei, collisions between nucleons may be modelled as droplets collisions. Droplet impacts on the wall of combustion chambers play an important role in determining the eventual size of the droplets. Impact of agricultural sprays and rain droplet on soil and plants is also noteworthy.

Axisymmetric simulation of droplet impact on a solid surface (Stéphane Popinet, Zhen Jian, Jie Li et Guy-Jean Michon)

The theory of axisymmetric droplet impact is rather difficult, despite some remarkable advances of 2D potential flow theory. The actual impact involve viscous flow and potential theory cannot apply except as an approximation in some regions of the flow. Surface tension is also involved in an important way. Finally, compressible and molecular scale effect are probably important in the thin air layer below the impacting droplet.

Bubble dynamics and other topics

 Front tracking simulation of a bubble bursting at a free surface performed by Laurent Duchemin. Click on image to enlarge

Bubbles bursting at a free surface offer fascinating dynamics, and play a rôle in cloud forming as well as in the taste of champagne. Oscillating bubbles in a liquid bulk present complex dynamics such as sonoluminescence. (For our work on the subject see references on bubble dynamics.) An important point for understanding the mechanism controlling sonoluminescence is that bubbles cannot remain spherical. The calculation of the deformation of oscillating bubbles is a well-posed example of a free surface problem. A systematic approach allowed to find the role of the viscosity in the formation of the liquid jet observed in a cavitation bubble near a wall.

Central jet formed after a bubble bursts on a free surface (Laurent Duchemin)

A droplet forms at the tip of the jet, usually an order of magnitude, or approximately ten times smaller than the initial bubble. It has high velocity and accounts for the tickling sensation felt when approaching a glass of bubbly liquid with large bubbles (such as some bubbly natural water brands) to one's face.

Numerical methods and schemes

  A typical discretization for velocities and viscous stresses around a cell cut by an interface, extracted from Tryggvason G., Scardovelli R. and Zaleski S., Direct Numerical Simulations of Gas-Liquid Multiphase Flows, Cambrigge University Press, to appear (February 2011) . Click on image to enlarge

The main difficulty of numerical methods for the simulation of interfaces is the tracking of a curve or surface on the computational grid. Another difficulty is the discretization of the singular forces involved : surface tension is an obvious example but gravity also creates problems. It can be obtained through the single equation or immersed boundary method: it amounts to write the Navier-Stokes equations for the two fluids (liquid and gas) as if there was only one fluid, but with variable viscosity and density. The surface tension term is alro added on the grid, using a for instance a weighted distribution on a few neighbouring grid sites.  

Volume-of-fluid methods

Our team has developed (see VOF references) an original version of this method (Volume of Fluid/Piecewise Linear Interface Construction or VOF/PLIC). The original part of this approach lies in the reconstruction of the interface through non-continuous segment and to propagate them with a Lagrangian method. It is this Lagrangian dynamics that differs from other methods through the elementary flux calculation. In particular, we have shown that the method amounts to an area-preserving mapping of the computational cells (a geometrical transformation) and that the volume is conserved to machine accuracy. The scheme has been tested on different theoretical and experimental standard test cases. Notably, one obtains a very good agreement with the capillary wave theory of small amplitudes, which confirms the correct calculation of the surface tension terms. In addition, the topology changes are well described in the case of the detachment of a drop by gravity.

Codes for the VOF method have been developped in planar and axisymmetric geometries in 2D and in 3D, The SURFER code that has been developped by several DALEMBERT coworkers developed may be found here It is however not maintained anymore apart for some infrequent additions. The Gerris Flow Solver code developped by Stephane Popinet also uses the VOF method.

Front tracking: marker chains

We have also developed a surface or marker method for interface tracking. Our method is only two-dimensional, but has the advantage of higher accuracy by using third-order splines and a "pressure-correction" methods to improve the computation of surface tension and remove spurious currents. (See front-tracking references and book). The interface is then reconstructed by splines which connect virtual marker particles. This method allows a much better accuracy, for instance for a capillary-wave test case.The corresponding code may be found here.




Institut Jean Le Rond d'Alembert
UPMC - Université Pierre et Marie Curie, CNRS, UMR 7190
4 place Jussieu,
75005 Paris

tel +33 1 44 27 87 23

email: zaleski (aa tt) ida.upmc.fr

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