Numerical simulation of finite-amplitude spatial perturbations of advective thermocapillary flow in a weakly rotating liquid layer under microgravity conditions

Abstract

The behavior of spatial finite-amplitude perturbations of the advective flow in a weakly rotating layer of incompressible fluid with free non-deformable boundaries under microgravity conditions that arise when the Marangoni number is above the critical value is investigated. Convective heat exchange under Newton’s law is present on the boundaries. The temperature of the fluid near the boundaries is a linear function of the coordinates. The study is based on the equations of convection in the Boussinesq approximation in a rotating reference frame in a Cartesian coordinate system. The axis of rotation coincides with the vertical z -axis. The two limiting cases are considered: spatial perturbations in the form of rolls with axes perpendicular to the x -axis and spatial perturbations of the second type in the form of rolls with axes parallel to the x -axis. In the presence of rotation, there are all three velocity components that depend on the time and two spatial coordinates x and z or y and z . The behavior of the finite-amplitude perturbations arising in a rotating liquid layer is studied numerically by the grid method (two-field method) on the basis of a nonlinear problem for various values of Grashoff number and Taylor number and fixed values of Prandtl number (Pr=6,7) and Biot number (Bi=0,1). An explicit finite-difference scheme with central differences is used. The Poisson equation for perturbation stream functions is solved by the successive over-relaxation (SOR) method. Behind the stability threshold in the case of monotonic linear instability, steady-state periodic x and y periodic-amplitude perturbations of the velocity in the form of a system of spatial vortices that rotate in opposite directions and temperatures in the form of alternating cold and warm temperature spots are generated in the case of vibrational instability. For oscillatory instability, the period of oscillation of perturbations is determined, with the increase in the variable parameters of the problem, the time period for the repetition of the disturbance pattern decreases. As the Marangoni number increases, the maximum temperature increases, the form of the finite-amplitude perturbations changes, and the character of the motion becomes more complicated.