Evolution of the diffusion-induced flow over a disk, submerged in a stratified viscous fluid

In the present paper the results of mathematical modeling of the evolution of the $\mathrm{3D}$ diffusion-induced flow over a disk (with diameter $d$ and thickness $H = 0.76\cdot d$), immersed in a linearly density stratified incompressible viscous fluid (described by the Navier–Stokes equations in the Boussinesq approximation), are shown. The disk rests at the level of the neutral buoyancy (which coincides with its axis of symmetry $z$) and disturbs the homogeneity of the background diffusion flux in the fluid, forming a complex system of the slow currents (gravitational internal waves). Over time, two thin horizontal convection cells are formed at the upper and lower parts of the disk, stretching parallel to the $z$ axis and adjacent to the base cell with thickness $d/2$. For the first time the fundamental mechanism for the formation of each new half-wave near the vertical axis $x$ (passing through the center of the disk) during half the buoyancy period of the fluid $T_b$ is analyzed in detail. This mechanism is based on gravitational instability. The beginning of this instability was fixed at $0.473\cdot T_b$ at a height of $3.9\cdot d$ above the center of the disk. The same mechanism is also realized over the place where the body moves in the horizontal direction. The $\mathrm{3D}$ vortex structure of the flow is visualized by the isosurfaces of the imaginary part of the conjugate eigenvalues of the velocity gradient tensor. The method SMIF with an explicit hybrid finite difference scheme for the approximation of the convective terms of the equations (second-order approximation, monotonicity), which has proved itself over the past $30$ years, is used for the mathematical modeling.