Mathematical modling of the 3D separated viscous fluid flows

Abstract

The 3D homogeneous and density stratified viscous fluid flows around a sphere and a circular cylinder have been investigated by means of the direct numerical simulation on supercomputers and the visualization of the 3D vortex structures in the wake. These flows have been simulated on the basis of the Navier-Stokes equations (NSE) in the Boussinesq approximation. The different transitions in sphere wakes (2D-3D, laminar-turbulent and others) with increasing of Reynolds number Re (1<Re<500000) and decreasing of Froude number Fr (0.005<Fr<1000) have been investigated in details. The classifications of flow regimes have been refined. It was found also that the values of the maximum phase difference along the circular cylinder axis are approximately equal to 0.1-0.2 T (for mode A, 191<Re≤300) and 0.015-0.030 T (for mode B, 300≤Re≤400), where the time T is the period of the flow. Direct numerical simulation, viscous fluid, visualization of vortex structures, flow regime, sphere, cylinder 47.11.-j, 61.20.Ja Unsteady 3D separated and undulatory fluid flows around a moving blunt body are very wide spread phenomena in the nature. The understanding of such flows is very important both from theoretical and from practical points of view. In the experiments [1-2] the 2D internal waves structure in the vertical plane and the 3D vortex structure of the wake are observed. Using direct numerical simulation (DNS) the full 3D vortex structures of the flow (the 3D internal waves and the 3D wake) can be observed. Besides the numerical studies of the non-homogeneous (stratified) fluids are very rare. In this connection at the present paper the stratified viscous fluid flows around a sphere are investigated by means of DNS on the basis of the Navier-Stokes equations in the Boussinesq approximation on the supercomputers at the wide ranges of the main flow parameters.Let ρ(x, y, z) = ρ 0 (1 -x/(2C) + S(x, y, z)) be the density of the linearly stratified fluid where x, y, z are the Cartesian coordinates; z, x, y are the streamwise, lift and lateral directions (x, y, z) have been non-dimensionalized by d/2, d is a sphere diameter); C = Λ/d is the scale ratio, Λ is the buoyancy scale, which is related to the buoyancy frequency N and period T b (N = 2π/T b , N 2 = g/Λ); g is the scalar of the gravitational acceleration; S is a dimensionless perturbation of salinity. The density stratified viscous fluid flows have been simulated on the basis of the Navier-Stokes equations in the Boussinesq approximation (1)- (3)