At the present paper the homogeneous (at 200 ≤ Re ≤ 106, fig. 1 (a-b)) and stratified (at 50 ≤ Re ≤ 1000, 0.005 ≤ Fr ≤ 1, fig. 1 (c)) viscous incompressible fluid flows around a sphere are investigated by means of the direct numerical simulation (DNS) and the visualization of the 3D vortex structures in the wake (Reynolds number Re = Ud/v, where U is the free-stream velocity, d is the diameter of the sphere, and v is the kinematic viscosity; Froude number Fr = U/(N·d), where N is the buoyancy frequency). For DNS the Splitting on physical factors Method for Incompressible Fluid flows (SMIF-MERANGE) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, monotonous) is used [
]. For the visualization of the 3D vortex structures in the sphere wake the isosurfaces of Im(σ1,2) are drawing, where Im(σ1,2) is the imaginary part of the complex-conjugate eigen-values of the velocity gradient tensor (fig. 1). In spite of the set of the papers devoted to the homogeneous fluid flows around a sphere the detailed formation mechanisms of vortices (FMV) in the sphere wake are still unclear [
]. At the present paper for the homogeneous fluid flows the six basic FMV have been selected; the detailed FMV for the different unsteady periodical flow regimes are explained (270 < Re ≤ 290, 290 < Re ≤ 320, 320 < Re ≤ 400, 400 < Re < 700 and Re > 700); at 290 < Re ≤ 320 a new flow regime has been found; the laminar-turbulent transition in the boundary layer on the sphere is discussed (9·104 < Re < 4·105). The numerical studies of the stratified fluid flows past a sphere are very rare. At the present paper four different flow regimes of the stratified fluid flows have been simulated: 1) “Two steady twodimensional attached vortices” (Re < 120, Fr < 0.2, fig. 1 (c)); 2) “Unsteady two-dimensional attached vortices” (Re > 120, Fr < 0.15); 3) “Lee-wave instability” (0.2 < Fr ≤ 0.4); 4) “Nonaxisymmetric attached vortex” (Re < 500, Fr > 0.4). The high gradient sheets of density have been observed near the poles of the resting sphere [
] and of the moving sphere (Fr ≤ 0.02). The lee waves, the recirculating zone and other vortex structures of the wake have been visualized (fig. 1 (c)). This work is supported by Russian Foundation for Basic Research (grant № 05-01-00496); by the program “Mathematical Modeling” of the Presidium of the Russian Academy of Sciences (RAS); by the program № 3 for Basic Research of the Department of the Mathematical Sciences of RAS.