Vortex formation, instabilities

Fig. 2.2 Schematic of the/i-vortex formation and definition scotch Initially the vorticity is organized spanwise at it is uniformly distributed in flow direction. By an instability process, the vorticity starts to wrap up into a vortex which by stretching takes the idealized form of a A composed mainly of two side-vortex-rods with an internal flow behaviour as described by eq.(2.2). This process ends when the vorticity cannot be concentrated anymore and viscous diffusion processes start to dominate the flow field in the event, which starts from thereon to decay. Since this model also has to account for the non-slip condition, the wall near flow can be described by a viscous tornado. Therefore the question arises whether by incorporating the model of the viscous tornado into the 1-vortex model it would be possible to describe the flow field completely.

As Re increases, skin friction becomes proportionately less and, at values greater than about 20, flow separation occurs with the formation of vortices in the wake of the sphere. At high Reynolds numbers, the size of the vortices progressively increases until, at values of between 100 and 200, instabilities in the flow give rise to vortex shedding. The effect of these changes in the nature of the flow on the force exerted on the particle is now considered. 

Figure 4.10 shows a close look at the boundary layer for two cases with Re q = 500, but different aspect ratios and RCa values. Panel (a) shows the start of the formation of smaller vortices due to the growth of the instabilities in the shear layer. It is observed that as the Reynolds number increases (for a constant aspect ratio), the point at which the separated shear layer becomes unstable moves upstream. Panel (b) shows that the distance between the two points of separation on the ellipse are located farther in the vertical direction due to the decrease in the aspect ratio, which creates a wider vortex-shedding area. For AR = 0.25, the shear layers roll up much closer to the tips of the ellipse and occasional interactions between the vortices separated from the upper and lower tips are observed, which is similar to flow past normal plates. At AR = 0.25, shortly downstream of the flow separation point, the shear layer mixing leads to the reattachment of the boundary layer, which is similar to that of higher Reynolds number flows past... 

The DPF method [5-7] assumes spray formation is a combination of random and nonrandom processes. An instability analysis is used to describe primary breakup, which is uniquely determined for a given set of initial conditions (fluid physical properties and atomizer parameters) and a model of the breakup mechanism. The drop size distribution arises from fluctuations in the initial conditions due to such factors as gas and liquid turbulence, atomizer passage surface roughness, vortex shedding, liquid mixture composition, etc. 

The instability of liquid films subjected to surface tension and/or density gradients causes convection flow forming vortex cells known as Benard cells. The Marangoni (Ma) and the Rayleigh (Ra) numbers are often used for determining the conditions for the onset of the formation of convection cells caused by the surface tension gradient and density gradient, respectively. 

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