Mathematical Modeling of the Internal Waves Emergence in the Stratified Viscous Fluid

The paper presents the results of the mathematical modeling and 3D visualization of the linearly density stratified 3D incompressible viscous fluid flows uniformly moving in a horizontal direction from the left to the right around a disk (along the axis of symmetry of the disk). If we will consider the fluid flows in the reference frame connected with fluid then we can investigate the nonlinear fundamental 3D mechanism of the formation of the 3D gravitational internal waves over the place M of the impulse start of the back side of the disk in the horizontal direction from the right to the left. At the present paper this 3D mechanism is analyzed in detail (for the first time). For the visualization of the 3D vortex structures of the fluid flows the isosurfaces of beta were drawing, where beta is the imaginary part of the complex-conjugate eigen-values of the velocity gradient tensor. During each half of the buoyancy period a small deformed vortex ring is generated over the point M (due to shear and gravitational instabilities), gradually grows in size and shifts down to the point M. The left half of ring is transformed in the internal half-wave. The right half of ring is compressed by next right halves of rings generated later. Thus the number of the internal waves between the back side of the disk and point M is always equal to the number of the buoyancy periods past since the disk start. This flows are described by the Navier-Stokes equations in the Boussinesq approximation. For the mathematical modeling the numerical method SMIF with an explicit hybrid finite difference scheme (second-order approximation, monotonicity) has been used.