About the Prospects for Passage to Instability

Abstract

The results of the direct numerical integration of the Navier-Stokes equations are evaluated against experimental data for problem on a flow around bluff bodies in an unstable regime. Experiment records several stable medium states for flow past a body. Evolution of each of these states, after losing the stability, inevitably goes by periodic vortex shedding modes. Calculations based on the Navier-Stokes equations satisfactorily reproduced all observed stable medium states. They were, however, incapable of reproducing any of a vortex shedding modes recorded experimentally. The solutions to the classic hydrodynamics equations successfully reach the boundary of instability field. However, classic solutions are unable to cross this boundary. Most likely, the reason for this is the Navier-Stokes equations themselves. The classic hydrodynamics equations directly follow from the Boltzmann equation and naturally contain the error involved in the derivation of classic kinetic equation. Just the Boltzmann hypothesis, which closed kinetic equation, allowed us to construct classic hydrodynamics on only three lower principal hydrodynamic values. The use of the Boltzmann hypothesis excludes higher principal hydrodynamic values from the participation in the formation of classic hydrodynamics equations. The multimoment hydrodynamics equations are constructed using seven lower principal hydrodynamic values. The numerical integration of the multimoment hydrodynamics equations in the problem on flow around a sphere shows that the solutions to these equations cross the boundary and enter the instability field. The boundary crossing is accompanied by appearance of very uncommon acts in scenario of system evolution.

Multimoment Hydrodynamics in Problem on Flow around a Sphere: Entropy Interpretation of the Appearance and Development of Instability

“…However, the calculation is incapable of producing anything that corresponds to seven unstable regimes observed along the three directions of instability development. The analysis of numerous divergences between the results of numerical integration of the Navier-Stokes equations and the experiment [1]- [3] led to the following conclusion. Solutions to the classic hydrodynamics equations successfully reach the border of the instability field represented by the dashed slanting line in Figure 1 from [3].…”

“…The analysis of numerous divergences between the results of numerical integration of the Navier-Stokes equations and the experiment [1]- [3] led to the following conclusion. Solutions to the classic hydrodynamics equations successfully reach the border of the instability field represented by the dashed slanting line in Figure 1 from [3]. As Reynolds number grows, these solutions move along the border of the field.…”

“…Four coefficients, namely 1 C , 2 C , 3 C , and 4 C , appear in the distribution of particle density (4.4). Seven coefficients 5 C , 6 C , 7 C , 8 C , 9 C , 18 C , and 19 C , are responsible for the distributions of pressure and stress (4.6), (4.7).…”

“…The basic property of pair distribution functions (17) from [3] subdivides Equations (2.10) and (2.11). As recast in terms of the spherical coordinates, the …”

“…Here, n is the local density of the number of particles; U is the hydrodynamic velocity; q are given by Equation (21) from [3] with the NavierStokes accuracy. In terms of spherical coordinates, these values for cylindrically symmetric flows considered in this study (i.e., for flows invariant with respect to rotation through an arbitrary angle ϕ about the Z axis) assume the form …”