Parallel solution of three-dimensional Marangoni flow in liquid bridges

Abstract

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 distributed-memory, 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 behaviour, can effectively be parallelized on a distributedmemory parallel machine by remapping the distributed data structure. The numerical code is based on a three-dimensional Simplified Marker and Cell primitive variable method applied to a staggered finitedifference grid. Using this method, the problem is split in 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.

3D Analysis of Crystal/Melt Interface Shape and Marangoni Flow Instability in Solidifying Liquid Bridges

“…For cylindrical liquid bridges previous numerical computations ( [7][8][9][10][11]) have shown that the flow structure of the supercritical state depends on the value of the aspect ratio of the liquid bridge. The lower the aspect ratio, the higher the azimuthal wave number m resulting in the more complex flow organization.…”

“…Integrating over the generic control volume and using the Gauss theorem to transform volume integrals in surface integrals, the equations read The problem is solved with the well known Marker and Cell method (see e.g. Lappa and Savino [10] and Fletcher [25]). …”

“…Lappa and Savino [10] and Lappa et al [11] used parallel supercalculus to study the azimuthal structure (characterized by the appropriate value of the azimuthal wave number of the instability) of the flow pattern that is established after the first Marangoni flow bifurcation. They analyzed the influence of the aspect ratio and of the non-cylindrical (convex or concave) interface shape.…”

On the variety of particle accumulation structures under the effect of g-jitters

“…Forward differences in time and central-differencing schemes in space (second order accurate) have been used to discretize the energy and momentum governing equations. The related solution strategy is not discussed here, the interested reader being referred to various books and articles in the literature for an exhaustive treatment (for the implementation of this method on parallel machines, the reader may consider, e.g., Lappa and Savino, 1999;Lappa, 2003a and2004). The above statement, however, does not apply to the part concerning the computation of PAS with equations (14), which was not validated in earlier studies of the present author (both Lappa, 2013a and2013b, in fact, were based on a simplified version of the Maxey-Riley equation, the so-called inertial equation).…”

On the onset of multi-wave patterns in laterally heated floating zones for slightly supercritical conditions

“…More precisely, at each time step, an intermediate velocity field is determined without the knowledge of the correct pressure field, and therefore no incompressibility condition is enforced. The intermediate velocity field is then modified by a second step in which a pressure equation is solved and then the related results are used to produce a divergence-free vector field 42 .…”