Rankine vortex

Figure 17.4. Rankine vortex model For the Rankine model, we deduce ...

The rotating flow is of vortex type in the hydrocyclone. This is modeled by a Rankine vortex of radius a and circrrlation T, using equation [17.18]. The radius a depends directly on the geometrical corrtraction ratios Ri/R and R /R between the radius of the upper cavity and those of the extraction apertures. 

A real swirling flow normally has a core of near solid-body rotation surrounded by a region of near loss-free rotation as sketched in Fig. 2.1.3. This is called a Rankine vortex . 

To the right in Fig. 3.1.1 the radial profiles of the axial and tangential gas velocity components are sketched. The former shows the outer region of downwardly directed axial flow and the inner one of upwardly directed flow. As mentioned, the downward velocity at the wall is the primary mechanism for particle transport out the dust outlet. The axial velocity often shows a dip aroimd the center hne. Sometimes this is so severe that the flow there is downwardly directed. The tangential velocity profile resembles a Rankine vortex a near loss-free swirl surrounding a core of near solid-body rotation. 

He considered that the rapid flame propagation could be achieved with the same mechanism as vortex breakdown. Figure 4.2.2 schematically shows his vortex bursting mechanism [4,5]. When a combustible mixture rotates, Ihe pressure on the axis of rotation becomes lower than the ambient pressure. The amount of pressure decrease is equal to max in Rankine s combined vor-fex, in which p denotes fhe unburned gas density and Vg denotes the maximum tangential velocity of the vortex. However, when combustion occurs, the pressure on the axis of rofafion increases in the burned gas owing to the decrease in the density, and becomes close to the ambient pressure. Thus, there appears a pressure jump AP across the flame on fhe axis of rotation. This pressure jump may cause a rapid movement of the hot burned gas. By considering the momentum flux conservation across the flame, fhe following expression for the burned gas speed was derived ... 

Figure 7.5. Tangential velocity and pressure distributions based on Rankine s combined vortex model.

Naturally, the results of dimensional analysis discussed above and their consequences were not known to the ship builders of the 19th century. Since the time of Rankine, the total drag resistance of a ship has been divided into three parts the surface friction, the stern vortex and the bow wave. However, the concept of Newtonian mechanical similarity, known at that time, only stated that for mechanically similar processes the forces vary as F p l2 v2. Scale-up was not considered for assessing the effect of gravity. 

To avoid difficulties linked to the singularity at r = 0, an eddy- (or vortex-) type flow is commonly modeled using Rankine s model ... 

The Rankine model is of great practical interest, although it is not an exact solution of the Navier-Stokes equations. In the presence of viscosity, the vorticity discontinuity at r = o is removed. The radius a of the vortex widens in time if a secondary flow does not counteract the effect of viscosity. 

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