Chaotic mixing by longitudinal vorticity (original) (raw)

Abstract

Computational fluid dynamics study of the chaotic advection flow in static mixer. Mixing enhancement by using longitudinal vortex generator. Poincaré sections and the Lagrangian trajectories show that chaotic advection takes place in the alternated configuration.

Figures (15)

Fig. 1. Isometric views of the (a) in-line and (b) alternate configurations, and (c) of one tab with the main rotation angles. The flow is in the z-axis directio:

Fig. 1. Isometric views of the (a) in-line and (b) alternate configurations, and (c) of one tab with the main rotation angles. The flow is in the z-axis directio:

Fig. 2. Dimensions of a row of vortex generator (dimensions in mm). Transverse (a) and radial (#) angles of the vortex generators. Table 1

Fig. 2. Dimensions of a row of vortex generator (dimensions in mm). Transverse (a) and radial (#) angles of the vortex generators. Table 1

Fig. 3. Zoom on a single row of vortex generators showing the adapted mesh on (a) a cross-section (x, y) at Z=50 mm, (b) horizontal plane (x, z) and (c) vertical plane (y, z The Computational Fluid Dynamics CFD code Ansys Fluent is used for the present simulations. It is based on an Eulerian approach to solve Navier-Stokes equations through cell-centered finite volume discretization. The flow equations are solved sequen- tially with double precision and a second-order upwind scheme In fact, on ANSYS FLUENT, the iterations stop when all the criterions are achieved. Meanwhile, it is better to achieve a criterion of 10~® for the passive scalar equation to have an asymptotic behavior for T while this behavior is achieved at 10~° for the other equations.

Fig. 3. Zoom on a single row of vortex generators showing the adapted mesh on (a) a cross-section (x, y) at Z=50 mm, (b) horizontal plane (x, z) and (c) vertical plane (y, z The Computational Fluid Dynamics CFD code Ansys Fluent is used for the present simulations. It is based on an Eulerian approach to solve Navier-Stokes equations through cell-centered finite volume discretization. The flow equations are solved sequen- tially with double precision and a second-order upwind scheme In fact, on ANSYS FLUENT, the iterations stop when all the criterions are achieved. Meanwhile, it is better to achieve a criterion of 10~® for the passive scalar equation to have an asymptotic behavior for T while this behavior is achieved at 10~° for the other equations.

Fig. 4. Streamwise vortex intensity and streamlines on cross-section (x, y) at different z (z in mm) for one row of vortex generato1

Fig. 4. Streamwise vortex intensity and streamlines on cross-section (x, y) at different z (z in mm) for one row of vortex generato1

Fig. 5. Dimensionless helicity contours on different cross-sections for (a) in-line and (b) alternate configurations. A non-uniform unstructured three-dimensional mesh with hexahedral volumes is generated using Gambit software. Special Two flow configurations are studied; both consist of a circular pipe of radius r= 10 mm fitted by five rows of vortex generators. In the first configuration, as shown in Fig. l(a), the vortex generators are in-line in the flow direction, while in the second case, as shown in Fig. 1(b), an alternate radial rotation of 90° is applied on the angle / (Fig. 1(c)) which is presented by the values

Fig. 5. Dimensionless helicity contours on different cross-sections for (a) in-line and (b) alternate configurations. A non-uniform unstructured three-dimensional mesh with hexahedral volumes is generated using Gambit software. Special Two flow configurations are studied; both consist of a circular pipe of radius r= 10 mm fitted by five rows of vortex generators. In the first configuration, as shown in Fig. l(a), the vortex generators are in-line in the flow direction, while in the second case, as shown in Fig. 1(b), an alternate radial rotation of 90° is applied on the angle / (Fig. 1(c)) which is presented by the values

Fig. 7. (a) Probability density function at the MHER's outlet and (b) the evolution of the mixing fraction for both configurations. At the inlet (z =0 mm), as shown in Fig. 6, the upper region is set to Tmax =320 and in the lower region to Tin = 300. Fig. 6 shows the contours of the scalar T at the outlet of each of the five arrays for both cases. The strong effect of the primary CVP is

Fig. 7. (a) Probability density function at the MHER's outlet and (b) the evolution of the mixing fraction for both configurations. At the inlet (z =0 mm), as shown in Fig. 6, the upper region is set to Tmax =320 and in the lower region to Tin = 300. Fig. 6 shows the contours of the scalar T at the outlet of each of the five arrays for both cases. The strong effect of the primary CVP is

Fig. 8. Poincaré section for (a) in-line and (b) alternate configurations for initial 5000 particles injected at (xy9=5 mm, yo=0 mm). clearly highlighted by the distribution of the scalar especially for the first array. The engulfment process of Tyin into Tmax caused by the primary CVP shown in this Fig. 6 is very effective for the mixing process (Ottino, 1990; Hu et al., 2013). For the following arrays downstream, the scalar distribution in the in-line config- uration remains symmetrical and the engulfment process con- tinues similarly to the first array. However, it is intensified due to the fact that the primary CVP moves upward as it is shown in

Fig. 8. Poincaré section for (a) in-line and (b) alternate configurations for initial 5000 particles injected at (xy9=5 mm, yo=0 mm). clearly highlighted by the distribution of the scalar especially for the first array. The engulfment process of Tyin into Tmax caused by the primary CVP shown in this Fig. 6 is very effective for the mixing process (Ottino, 1990; Hu et al., 2013). For the following arrays downstream, the scalar distribution in the in-line config- uration remains symmetrical and the engulfment process con- tinues similarly to the first array. However, it is intensified due to the fact that the primary CVP moves upward as it is shown in

Fig. 9. Projections of the Lagrangian trajectories for different initial positions for the (a) in-line and (b) alternate configurations

Fig. 9. Projections of the Lagrangian trajectories for different initial positions for the (a) in-line and (b) alternate configurations

Fig. 10. Estrangement of two initially very close particles in both configurations. For the alternate configuration, the particles are found to move away from each other and the distance between them increases exponentially, while in the in-line configuration, the particles remain close. This exponential increase in the distance reported for the alternating configuration is caused by the chaotic advection flow.

Fig. 10. Estrangement of two initially very close particles in both configurations. For the alternate configuration, the particles are found to move away from each other and the distance between them increases exponentially, while in the in-line configuration, the particles remain close. This exponential increase in the distance reported for the alternating configuration is caused by the chaotic advection flow.

Fig. 11. Striation of initial circle of radius r=1 mm injected at the inlet and which is centered at (a) (¥g=0 mm, yo=O mm) and (b) (xo=5 mm, yo=0 mm) where py is the pressure at the inlet and p, is the average pressure in a given cross-section with the streamwise coordinate z. As it can be observed, when the particles pass throughout the different arrays of vortex generators, the stretching and folding of the circle become more significant, especially for the alternated case when irregular distributions are successively noticed in the flow direction. These random stretching and folding in the alternated configuration are very efficient for mixing different fluids or reagents (Ottino, 1989).

Fig. 11. Striation of initial circle of radius r=1 mm injected at the inlet and which is centered at (a) (¥g=0 mm, yo=O mm) and (b) (xo=5 mm, yo=0 mm) where py is the pressure at the inlet and p, is the average pressure in a given cross-section with the streamwise coordinate z. As it can be observed, when the particles pass throughout the different arrays of vortex generators, the stretching and folding of the circle become more significant, especially for the alternated case when irregular distributions are successively noticed in the flow direction. These random stretching and folding in the alternated configuration are very efficient for mixing different fluids or reagents (Ottino, 1989).

From this figure, it can be observed that Ap increases mono- lithically in both configurations. Downstream from the second vortex generator, the pressure drop in the alternate configuration becomes greater than that in the in-line one and reaches its maximum at the outlet. The maximum pressure drop increase in the alternate mixer does not exceed 7% relative to that in the in- line, while it was shown that the alternate configuration enhances the mixing efficiency by about 12% relative to that in the in-line configuration. This increase is caused by higher flow blockage area. mixing process within a multifunctional heat exchanger/reactor. Two configurations are studied; the first one consists of several arrays of rectangular vortex generators which are mounted in-line, while in the second one the arrays are periodically rotated by 90° with respect to the pipe axis. The flow is considered to be laminar and water is taken as the working fluid. As a first step, the flow structure is investigated showing that each vortex generator generates a counter-rotating streamwise vortex pair (CVP) which in its turn induces a secondary CVP in the very near-wall region. These CVPs are continuously rotated in the alternated configura- tion and cover a larger area in the flow volume. It is shown that the primary CVP is very efficient for the mixing process especially in the alternated configuration. However, the effect of the second- ary CVP on the mixing process is neglected. Moreover, by explor- ing the Poincaré sections and the Lagrangian trajectories of

From this figure, it can be observed that Ap increases mono- lithically in both configurations. Downstream from the second vortex generator, the pressure drop in the alternate configuration becomes greater than that in the in-line one and reaches its maximum at the outlet. The maximum pressure drop increase in the alternate mixer does not exceed 7% relative to that in the in- line, while it was shown that the alternate configuration enhances the mixing efficiency by about 12% relative to that in the in-line configuration. This increase is caused by higher flow blockage area. mixing process within a multifunctional heat exchanger/reactor. Two configurations are studied; the first one consists of several arrays of rectangular vortex generators which are mounted in-line, while in the second one the arrays are periodically rotated by 90° with respect to the pipe axis. The flow is considered to be laminar and water is taken as the working fluid. As a first step, the flow structure is investigated showing that each vortex generator generates a counter-rotating streamwise vortex pair (CVP) which in its turn induces a secondary CVP in the very near-wall region. These CVPs are continuously rotated in the alternated configura- tion and cover a larger area in the flow volume. It is shown that the primary CVP is very efficient for the mixing process especially in the alternated configuration. However, the effect of the second- ary CVP on the mixing process is neglected. Moreover, by explor- ing the Poincaré sections and the Lagrangian trajectories of

Fig. 12. Striations length variation of initial circle of radius r= 1 mm injected at the inlet and which is centered at (a) (xy=0 mm, yo=0 mm) and (b) (xo=5 mm, yo=0 mm).

Fig. 12. Striations length variation of initial circle of radius r= 1 mm injected at the inlet and which is centered at (a) (xy=0 mm, yo=0 mm) and (b) (xo=5 mm, yo=0 mm).

Fig. 13. Streamwise variation of the pressure drop for both configurations. different passive tracers, it is found that chaotic advection takes place in the alternated configuration while the flow in the in-line configuration stays regular. The future work will focus on different arrangements of the vortex generators arrays and on the effect of such geometries on the heat transfer in turbulent flow regimes.

Fig. 13. Streamwise variation of the pressure drop for both configurations. different passive tracers, it is found that chaotic advection takes place in the alternated configuration while the flow in the in-line configuration stays regular. The future work will focus on different arrangements of the vortex generators arrays and on the effect of such geometries on the heat transfer in turbulent flow regimes.

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