Mathematical modeling of the incompressible fluid flows

Abstract

The aim of the present paper is the demonstration of the opportunities of the mathematical modeling of the separated flows of the river and the sea water around blunt bodies on the basis of the Navier-Stokes equations (NSE) in the Boussinesq approximation. In the river water only the wakes of the obstacles are observed. In the sea water along with wakes the internal waves (IWs) are generated. The 3D homogeneous (river) and density stratified (sea) incompressible viscous fluid flows around a sphere and a square cylinder have been investigated by means of the direct numerical simulation on supercomputers and the visualization of the 3D vortex structures in the wake. For solving of NSE the Splitting on physical factors Method for Incompressible Fluid flows (SMIF) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, capable for work in wide range of the Reynolds (Re) and the internal Froude (Fr) numbers and monotonous) has been developed and successfully applied. The different transitions in sphere wakes (2D-3D, laminar-turbulent and others) with increasing of Re (1 < Re < 500000) and decreasing of Fr (0.005 < Fr < 100) have been investigated in details. Thus the classifications of the viscous fluid flow regimes around a sphere have been refined. The original classification of the 2D stratified viscous fluid flow regimes around a square cylinder has been obtained at Re < 200, 0.1 < Fr < 100. At Fr =0.1, Re = 50 the formation process of the two symmetric wavy hanging (soaring) sheets of density (starting from two hanging vortices) is demonstrated. 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 Tf (for mode A, 191 < Re ≤ 300) and 0.015-0.030 Tf (for mode B, 300 ≤ Re ≤ 400), where the time Tf is the period of the flow.

Simulation and study of stratified flows around finite bodies

“…For example, in the simplest case of an axisymmetric flow past a sphere at Re = 100, two vortex structures can be identified in the wake: a vortex ring in the recirculation zone of the wake (RZ) and a vortex shell enclosing the ring [17][18][19][20][21][22][23] (Fig. 2a).…”

“…2a). At the same time, the streamlines in a frame of reference tied to the sphere visualize only the ring, the contour lines of the vorticity ω = 0.5curl v reveal only the shell, while the lines of constant pressure do not visualize anything at all [17][18][19][20][21][22][23]. Similarly, for a three-dimensional flow of complex geometry, the level surfaces of the modulus of vorticity show only some of the vortex structures occurring in the flow.…”

“…2a); at Re = 200, the axial symmetry in RZ is lost, but the flow remains steady up to Re = 270 (Figs. 2b, 3a); for 270 < Re ≤ 400, vortex loops of the same orientation are periodically formed in RZ; and, starting at Re = 400, vortex loops are formed with alternating oppositely directed orientations [21][22][23]. In [23], the basic mechanisms of vortex formation in the wake behind a sphere were identified for the first time on the basis of β-visualization.…”