Stationary turbulent flow regime

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

For the turbulent flow regime, the flux of bubbles of volume co toward the test bubble of volume V can be considered as a diffusion flux with the effective coefficient of diffusion Dr. Consider a bubble of volume V, placed in a turbulent flow of liquid containing bubbles of volume co with the number concentration n. Assuming that the process is stationary and spherically symmetric, we have the following equation describing the distribution n(r) ... 

However, several flow transition regimes have been identified between laminar and fully turbulent flow. The cessation of laminar Couette flow is marked by the appearance of Taylor vortices in the gap between the two cylinders. For the case of stationary outer cylinder, the critical angular velocity, C0crit> of inner cylinder at which these flow instabilities first appear can be estimated by using the following equations [102] ... 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

For flow around submerged objects, a = 1 in the creeping flow regime, and a = 0 for turbulent flow. Since the hydrodynamic drag force exerted by fluid B on solid particle A acts in the opposite direction of va when the fluid is stationary, and the gravitational force acts downward, these two forces are balanced ... 

Differential equations (3.33) and (3.34) are easily solved for stationary regime of axial symmetric non-swirling turbulent flow of incompressible Newton two-phase medium without consideration of inter-phase heat- and mass-transfer. That is why the force of inter-phase interaction... 

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