The appearance and development of turbulence in a flow past a sphere: Problems and the existing approaches to their solution

Abstract

The results of the direct numerical integration of the Navier-Stokes equations and integration of the pair functions theory equations are evaluated against experimental data for the problem of a flow past a hard sphere at rest in an unstable regime. Calculations based on the Navier-Stokes equations satisfactorily reproduced three stable medium states observed for a flow past a sphere. In agreement with experiment, each of these three states begins to develop after stability loss in its own direction different from those characteristic of the other states. Calculations were, however, incapable of reproducing any of the three directions of turbulence development recorded experimentally. Most likely, the reason for this is the Navier-Stokes equations themselves. The possibility is discussed that the assumption made in the derivation of the Boltzmann equation, namely, the molecular chaos hypothesis (Stosszahlansatz), may be responsible for the failure of classic hydrodynamics. This assumption is a closure to the Boltzmann equation that allows hydrodynamics to be constructed on the lower three hydrodynamic values. The inaccuracy mentioned introduced no substantial error into stable flow calculations. The error, however, increased rapidly after stability loss. We suggest the use of hydrodynamic equations based on pair functions theory as an alternative to the Navier-Stokes equations for unstable modes. These equations are derived without invoking any additional assumptions such as the Stosszahlansatz hypothesis. As distinct from classic hydrodynamics equations, pair functions theory equations predict the direction of turbulence development close to that observed experimentally.