Secondary flow pattern

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

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 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. 

Unsteady-state mass transfer caused by excessively fast current or potential ramps. This is especially likely to occur in measurements involving laminar flow past elongated surfaces and in free-convection studies, in which the establishment of secondary flow patterns may require long times. A compromise between the time sufficient to reach steady-state transport and the time necessary to avoid bulk depletion and surface roughening (in metal deposition) is required, and is found most reliably by preliminary experimentation. 

One of the benefits of static mixers like those mentioned above are the secondary flow patterns set up by the swirling helical flow. 

Figure 35. Secondary flow patterns in the cross-section of a helically coiled tube...

Air approaching the tower is displaced, resulting in flow separation and wake formation. A strong negative pressure and secondary flow patterns in the wake are created, the naturally buoyant exhaust plume is drawn down and ground fog forms. This is illustrated in Figure 6.13(A). 

Figure 9.41 presents the predicted secondary flow patterns that result from the vicoelastic flow effects. The Giesekus model with one relaxation time was used for the solution presented in the figure. For the simulation, a relaxation time, A, of 0.06 seconds was used along with a viscosity, r], of 8,000 Pa-s and a constant a of 0.80. Similar results were achieved using the Phan-Thien Tanner-1 model. As expected, when the White-Metzner model was used, a flow without secondary patterns was predicted. This is due to the fact that the White-Metzner model has a second normal stress difference, N2 of zero. 

The flow pattern remains the same for total flow rates from 10 to 2000 ml h 4, corresponding to residence times of 4.3-1140 ms (see Figure 1.89). [20]. Thus pure laminar flow without any secondary flow patterns applies. 

For increased flow rates, however, the CFD simulations show more and more deviations from an ideal SAR multi-lamination pattern [7]. Since inertial forces come into play, a secondary flow pattern is superposed on the SAR velocity profile of the creeping flow regime. Streamlines seeded at the initial lamellae interface in top view for various Reynolds numbers. The lamellae pattern right at the outlet for the same set of Reynolds numbers were also given. Further simulations showed that for Re above -15 the center lamellae are thinned out until they detach from the top and bottom walls for Re 30 (see center image of Figure 1.128). 

The continuous change of flow direction in zig-zag channels can induce secondary flow patterns at sufficiently high Re, which besides diffusion can act as a mixing mechanism. By means of recirculation patterns, material is transported transverse to the flow direction and improves the mixing. 

Figure 17. Secondary flow pattern caused by flow through 90 elbows. (Reproduced with permission from reference 55. Copyright 1987.)...

The three dimensional flow field in a tank characterized by secondary flow patterns was long inaccessible to theoretical treatment. It is therefore not surprising that it was first tackled by the statistical theory of turbulence [20, 57, 209, 289]. 

For Taylor numbers exceeding Tc, the flow develops a secondary flow pattern in which ur and uz are both nonozero. A sketch of the stability criteria given by (3-86) is shown in Fig. 3 8. The reader who is interested in a detailed description of the stability analysis that leads to the criterion (3-86) is encouraged to consult Chap. 12 or one of the standard textbooks on hydrodynamic stability theory (see Chandrashekhar [1992] for a particularly lucid discussion of the instability of Couette flows).12... 

Fig. 7.7 Geometry of a staggered herringbone micromixer (left) and a curved channel to induce Dean Vortexes with corresponding secondary flow patterns (right). Arrow indicates flow [92].

Figure 3.6. Secondary flow patterns and velocity profiles in coiled tube at (a) low and b) high flow velocities. (According to Ref. 73, by courtesy Elsevier Scientific Publishing Co.)...

Figure 3.12. Secondary flow pattern in an imprinted Z-formed channel. Note the change in flow pattern as the streamlines turn alternatively from clockwise to anticlockwise at right and left bend. (t> is the angle of the bend and / is the distance between the bends.

Kilner, P.J., Yang, G.Z., Mohiaddin, R.H., Firmin, D.N., and Longmore, D.B. 1993. Helical and retrograde secondary flow patterns in the aortic arch studied by three-directional magnetic resonance velocity mapping. Circulation 88[paitl] 2235-2247. 

Figure 2 Secondary flow patterns in a rectangular duct based on Prandtl (45) cross sectional view of flow.

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. 

Figure 8.14 Schematic double-celled secondary flow pattern...

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