Marcello Lappa

Three-dimensional numerical simulation of Marangoni instabilities in non-cylindrical liquid bridges in microgravity

Instability of Marangoni convection in non-cylindrical (convex or concave) liquid bridges of low Prandtl number fluids is investigated by direct three-dimensional and time-dependent simulation of the problem. Body-fitted curvilinear coordinates are adopted; the non-cylindrical original physical domain in the (r,z,φ) space is transformed into a cylindrical computational domain in a (ξ,η,φ) space. The geometry of the domain is transformed using a coordinate transformation method by surface fitting technique. The field equations are numerically solved explicitly in time and with a finite difference technique in a staggered grid. The numerical results are analyzed and interpreted in the general context of the bifurcations theory. The computations show that for semiconductor melts the first bifurcation is characterized by the loss of spatial symmetry rather than by the onset of oscillatory flow and that it is hydrodynamic in nature. The flow field azimuthal organization related to the critical wave number, depends on the geometrical aspect ratio A=L/D of the liquid bridge and on the shape factor S (convex S>1, concave S<1) of the free surface. The critical azimuthal wave number increases when the geometrical aspect ratio of the bridge is decreased and, for a fixed aspect ratio, can be shifted to higher values by increasing the volume (convex bridges) or to lower values by decreasing the volume (concave bridges). This behavior is explained on the basis of the relation between the typology of the azimuthal disturbances and the structure of the fluid-dynamic field. A generalized law is found to correlate the critical azimuthal wave number of the instability to the geometrical aspect ratio and to the shape factor. A second oscillatory (Hopf) bifurcation occurs when further increasing the Marangoni number. Experimental results available in literature on this second bifurcation are considered for comparison. The experimental and numerical results show a good agreement.

Marangoni flotation of liquid droplets

Flotation of liquid droplets on pool surfaces, in the presence of temperature differences, is studied experimentally and numerically. Coalescence or sinking of the droplet is prevented by the thermal Marangoni motion, owing to the surface tension imbalance at the pool surface. The mechanism is the same as that investigated in previous works on coalescence and wetting prevention in the presence of temperature differences. If the droplet is colder than the liquid surface, the flow is directed radially towards the drop; this radial flow field drags the ambient air under the drop, thus creating an air film and avoiding a direct contact between the droplet and the pool molecules. The surface velocities are measured visually with a CCD camera to image the motion of tracers floating on the pool surface; the surface temperature distributions along the pool and the droplet surfaces are measured by an infrared thermocamera. The experimental results are correlated by numerical results obtained under the assumption of spherical drop and axisymmetric flow regime

Influence of buoyancy forces on Marangoni flow instabilities in liquid bridges

The influence of buoyancy forces on oscillatory Marangoni flow in liquid bridges of different aspect ratio is investigated by three‐dimensional, time‐dependent numerical solutions and by laboratory experiments using a microscale apparatus and a thermographic visualisation system. Liquid bridges heated from above and from below are investigated. The numerical and experimental results show that for each aspect ratio and for both the heating conditions the onset of the Marangoni oscillatory flow is characterized by the appearance of a standing wave regime; after a certain time, a second transition to a travelling wave regime occurs. The three‐dimensional flow organization at the onset of instability is different according to whether the bridge is heated from above or from below. When the liquid bridge is heated from below, the critical Marangoni number is larger, the critical wave number (m) is smaller and the standing wave regime is more stable, compared with the case of the bridge heated from above. For the critical azimuthal wave number, two correlation laws are found as a function of the geometrical aspect ratio A.

Combined effect of volume and gravity on the three-dimensional flow instability in noncylindrical floating zones heated by an equatorial ring

The present paper strongly extends a previous analysis dealing with the investigation of the three-dimensional Marangoni flow instability in cylindrical floating zones (straight liquid column of full-zone extent) of a low Prandtl liquid laterally heated by a ring positioned around the equatorial plane and under microgravity conditions. The new study gives insights into the combined influence of volume and gravitational effects. The deformation of the free melt–gas interface due to the gravity field is taken into account. Parallel supercalculus is used to reduce the otherwise prohibitive computational time. The prominent features of the three-dimensional field are largely dependent on geometrical parameters. The results (full zone) are heretofore unseen and show that the interplay between the upper half and lower half of the liquid domain is an essential factor for the correct description of the phenomena under investigation. They are contrasted with the case of the half zone for which a rich variety of in...

PARALLEL SOLUTION OF THREE-DIMENSIONAL MARANGONI FLOW IN LIQUID BRIDGES

SUMMARY This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three-dimensional and time-dependent Navier‐Stokes equations on distributedmemory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non-linear Navier‐Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three-dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time-consuming part of the algorithm. A grid-partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed-Up is ever smaller than the ideal linear Speed-Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier‐Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency. Copyright

Behavior of drops in contact with pool surfaces of different liquids

The behavior of small liquid drops, hanging from a circular disk and approaching a pool surface of the same liquid at different temperatures, is studied experimentally and numerically. The experiments show that if isothermal conditions prevail the drop is immediately engulfed by the liquid. On the contrary, if the temperature of the drop is sufficiently larger or sufficiently smaller than the temperature of the liquid surface, this engulfment is prevented even if the drop is pressed on the liquid surface and “enters” the liquid pool. A number of experiments have been carried out on silicone oils (with different viscosities). At the same time the problem was studied numerically with the assumption that a thin air film is formed between the drop and the liquid bath surface, due to the entrainment of the surrounding air caused by the Marangoni flow; the pressure in the air film balances the pressure necessary to keep the drop submerged in the liquid bath. The critical temperature differences for the drop eng...

Assessment of the role of axial vorticity in the formation of particle accumulation structures in supercritical Marangoni and hybrid thermocapillary-rotation-driven flows

Evidence is provided that when the so-called phenomenon of particle accumulation structure (PAS) occurs, extended regions exist where ½ of the axial component of vorticity matches the angular frequency of the traveling wave produced by the instability of the Marangoni flow. Several cases are considered in which such axial component is varied by “injecting” vorticity into the system via rotation of one of its endwalls. The results show that both the resulting PAS lines and the trajectories of related solid particles undergo significant changes under the influence of imposed rotation. By analysis of such findings, a validation and a generalization/extension of the so-called “phase-locking” model are provided.

3D numerical simulation of on ground Marangoni flow instabilities in liquid bridges of low Prandtl number fluid

The influence of gravity on the Marangoni flow instability in half zone liquid bridges in the case of liquid metals is investigated by direct 3D and time‐dependent simulation of the problem. The computations are carried out for different heating conditions and environments (zero g conditions and on ground liquid zone heated from above or from below). The case of cylindrical shape (simplified model) and of melt/air interface deformed by the effect of gravity (real conditions) are considered. The comparison among these situations gives insight into the separate (gravity) effects of buoyancy forces and of the free surface deviation with respect to straight configuration. Body‐fitted curvilinear co‐ordinates are adopted to handle the non‐cylindrical problem. The liquid bridge exhibits different behaviours according to the allowed bridge shape. If the shape is forced to be cylindrical, the flow field is stabilized in the case of heating from above and destabilized if gravity is reversed. If the deformation is taken into account, gravity always stabilizes the Marangoni flow regardless of its direction (parallel or antiparallel to the axis) and the 3D flow structure is different according to the heating condition (from above or from below). In the latter case, the critical Marangoni number is larger and the critical wave number is smaller, compared with the opposite condition. In addition, for Pr=0.02 (Gallium), a surprising heretofore unseen behaviour arises. No steady bifurcation occurs and the flow becomes unstable directly to oscillatory disturbances. This phenomenon has never been reported before in the case of low Prandtl number liquids.

On the convective disturbances induced by g-jitter on the space station

A numerical tool has been prepared to quickly predict the overall disturbances for the typical Microgravity Environment (MGE) on the ISS. A reference study case (that shows a large sensitivity to acceleration disturbances) is identified and numerical simulations are carried out to compare the results of the solution of the full Navier-Stokes equations with a time-dependent acceleration (that give the instantaneous time-dependent flow) with the solutions of the time-averaged field equations (Gershuni formulation), containing all the g-jitter terms (like those that identify the microgravity environment of the ISS) grouped in a single parameter. The paper shows that the overall disturbances of the thermo-fluid-dynamic field are easily evaluated assigning as input to the fluid-dynamic code a single (equivalent) frequency g-jitter (in the direct formulation) or an overall vibrational Rayleigh number (in the time-averaged formulation). The code is validated and applied to the problem of thermodiffusion in a typical metal alloy.

Growth and mutual interference of protein seeds under reduced gravity conditions

This analysis deals with new models and computational methods as well as with novel results on the relative importance of “controlling forces” in macromolecular crystal growth. The attention is focused in particular on microgravity fluid-dynamic aspects and on the case of the simultaneous growth of different seeds. A “kinetic-coefficient-based” volume of fraction method is specifically and carefully developed according to the complex properties and mechanisms of macromolecular protein crystal growth. It is shown that the size and the shape of the growing crystals play a “critical role” in the relative importance of surface effects and in determining the intensity of convection. Convective effects, in turn, are found to impact growth rates, macroscopic structures of precipitates, particle size and morphology as well as the mechanisms driving growth. The face growth rates in particular depend on the complex multicellular structure of the convective field and on associated “pluming phenomena.” The relative i...