Development of numerical code for the study of marangoni convection

A numerical code for solving the time-dependent incompressible 3D Navier-Stokes equations with finite volumes on overlapping staggered grids in cylindrical and rectangular geometry is developed. In the code, written in FORTRAN, the momentum equation for the velocity is solved by projection method and Poisson equation for the pressure is solved by ADI implicit method in two directions combined with discrete fast Fourier transform in the third direction. A special technique for overcoming the singularity on the cylinder's axis is developed. This code, taking into account dependence upon temperature of the viscosity, density and surface tension of the liquid, is used to study the fluid motion in a cylinder with free cylindrical surface (under normal and zero-gravity conditions); and in a rectangular closed cell with a source of thermocapillary convection (bubble inside attached to one of the cell's faces). They are significant problems in crystal growth and in general experiments in fluid dynamics respectively. Nevertheless, the main study is dedicated to the liquid bridge problem.<p><p>The development of thermocapillary convection inside a cylindrical liquid bridge is investigated by using a direct numerical simulation of the 3D, time-dependent problem for a wide range of Prandtl numbers, Pr = 0.01 - 108. For Pr > 0.08 (e.g. silicon oils), above the critical value of temperature difference between the supporting disks, two counter propagating hydrothermal waves bifurcate from the 2D steady state. The existence of standing and traveling waves is discussed. The dependence of viscosity upon temperature is taken into account. For Pr = 4, 0-g conditions, and for Pr = 18.8, 1-g case with unit aspect ratio an investigation of the onset of chaos was numerically carried out. <p><p>For a Pr = 108 liquid bridge under terrestrial conditions ,the appearance and the development of thermoconvective oscillatory flows were investigated for different ambient conditions around the free surface.<p><p>Transition from 2D thermoconvective steady flow to a 3D flow is considered for low-Prandtl fluids (Pr = 0.01) in a liquid bridge with a non-cylindrical free surface. For Pr < 0.08 (e.g. liquid metals), in supercritical region of parameters 3D but non-oscillatory convective flow is observed. The computer program developed for this simulation transforms the original non-rectangular physical domain into a rectangular computational domain.<p><p>A study of how presence of a bubble in experimental rectangular cell influences the convective flow when carrying out microgravity experiments. As a model, a real experiment called TRAMP is numerically simulated. The obtained results were very different from what was expected. First, because of residual gravity taking place on board any spacecraft; second, due to presence of a bubble having appeared on the experimental cell's wall. Real data obtained from experimental observations were taken for the calculations.<p>Doctorat en sciences appliquées