Counter-rotating vortices

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

As mentioned earlier, in curved channels a secondary flow pattern of two counter-rotating vortices is formed. Similarly to the situation depicted in Figrue 2.43, these vortices redistribute fluid volumes in a plane perpendicular to the main flow direction. Such a transversal mass transfer reduces the dispersion, a fact reflected in the dependence in Eq. (108) at large Dean numbers. For small Dean numbers, the secondary flow is negligible, and the dispersion in curved ducts equals the Taylor-Aris dispersion of straight ducts. 

The modelling of aerodynamic entrainment is based on the close link between particles take-off and turbulent coherent structures above the surface. In fact, some authors [6,7] have experimentally observed that a particle take-off can be associated to the ejection of fluid from the wall region due to the presence of streamwise counter rotating vortices. If it is assumed that the presence of two streamwise counter rotating vortices produces only one ejection, each pair of streamwise vortices is considered as a possibility that a particle takes-off. Thus, for each of these possibilities, a take-olf criterion is tested. 

When guiding fluids through curved channels, the maximum in the velocity profile is displaced towards the outer channel wall and Dean vortices form (as reported, e.g., in [152]). The latter are typically characterized by two counter-rotating vortices above and below the symmetry plane of the channel coinciding with its plane of curvature. Fluid is transported outwards in this plane by means of centrifugal forces. By recirculation, back transport along the channel walls is induced. 

A Dean number of -140 is a kind of threshold value [47,152], For lower values, two counter-rotating vortices are found, whereas for higher values, two additional counter rotating vortices appear which are close to the center of the outer channel wall. Means to achieve this are changes in the flow velocity, the hydraulic diameter and the radius of curvature. 

Of significant interest are the strong counter-rotating vortices observed in lateral cross-sections. In the region of the sweep, these vortices rotate such that the central area between them is one of downward flow, as shown in Figure 4.11. Following the vortex pair in the streamwise direction, the centres of rotation are observed to be tilted upwards. 

Figure 2.4 Streamlines of the steady cellular flow composed of an array of counter-rotating vortices (top row) and the spreading in time of a weakly diffusing passive concentration field. Time increases from top to bottom.

Another set of experiments, by Paoletti et al. (2006), investigated the synchronization of stirred oscillatory BZ reaction over distances larger than the characteristic lengthscale of the flow. In this experiment the flow was composed of an annular ring of counter-rotating vortices with a superimposed additional oscillatory azimuthal flow. A simplified model of the corresponding velocity field can be written as... 

Botros and Brzustowski [77] studied the velocity field of propane TDFCF experimentally and numerically. Their study revealed a pair of counter-rotating vortices in the flame. Gollahalli and coworkers [78-80] measured the flow field and turbulent characteristics of gas jet flames in crossflow. At very low values of R, the effects of crossflow stream are more dominant and the jet fluid burns in the wake of a model stack. A recirculation vortex is created and the flame stabilizes on the wall of the recirculation bubble. Figure 29.17a presents the velocity vector field and streamlines obtained by Huang and Wang [45] for down-wash... 

Similarly to the structure of the flow fleld, heat transfer has also been studied in curved channel geometries. The complicated branch structure with competing patterns of two and four counter-rotating vortices in channels of square cross section is reflected in the Nusselt number [34]. When plotting the Nusselt number as a function of Dean number, different branches are found corresponding to symmetric and asymmetric secondary flow patterns with two and four vortices. However, the relative difference between the different branches is not very pronounced and should be hard to measure experimentally. For a Dean number of 210 and a Prandtl number of 0.7 a heat-transfer-enhancement factor of about 2.8 was determined, thus showing that curved channels as well as other channels with specific periodically varying cross sections may be used for applications where rapid heat transfer is desired. 

The simulations are performed for six different Reynolds numbers of 150, 500, 1,000, 2,000, 4,000, and 8,000. As the Reynolds number increases from low values, steady flow occurs around the body without any vortex shedding. Further increase in the Reynolds number causes the formation of a pair of symmetrical counter-rotating vortices about the centerline of the wake for RCeq<46 (for AR =1). At Re,... 

By increasing the channel size circulation patterns do not change significantly. Typical counter-rotating vortices also appear at the top and bottom of the channel in the cases of 0.5 and 1 mm ID. In Fig. 5.6 circulation patterns in two different channel sizes at the same mixture velocity of 0.06 m s (flow rate ratio equal to 1) are observed. In the 0.5 mm channel the vortices are not symmetrical around the channel centreline, with the stagnation point in the top half close to the back cap and that in the bottom part closer to the front cap. However, as the channel size further increases to 1 mm ID the vortex cores move towards the centre, and the two vortices (upper and lower haU) lie almost on a vertical line. 

In Fig. 5.8 a comparison of the circulation patterns within the aqueous phase when water and ionic liquid are the carrier phases, respectively, in the 0.2 mm ID channel at mixture velocity of 0.0037 m (flow rate ratio equal to 1) is shown for both cases. Two counter rotating vortices are observable, where the stagnation points at the vortex cores are pushed towards the rear interface for the shorter plug case (ionic liquid as carrier phase). 

As shown in the previous section, during plug flow in small channels, counter rotating vortices form in the phases with closed streamlines and a pattern symmetrical about the channel axis (Fig. 5.9). The rate of mixing inside the plug is quantified through the dimensionless circulation time, x, which relates the time for... 

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