Steady flow regimes and mixing performance in arrow-shaped micro-mixers

The different flow regimes occurring for increasing Reynolds numbers (Re) in an arrow-shaped micro-mixer operated for liquid mixing are investigated by the synergic use of experimental flow visualizations and direct numerical simulations. The tilting angle of the arrow-mixer inlet channels with respect to those of the T-shaped mixer is $\alpha ={20}^{\circ }$. Consistent with previous studies in the literature, it is found that the onset of the engulfment regime, and consequently the sharp increase of the degree of mixing between the two flow streams, occurs at a lower Reynolds number than for a T-mixer operating in the same conditions. However, in contrast to what is observed for T-mixers, the degree of mixing does not increase monotonically with Re. In fact, at approximately $\text{Re}=150$, based on the inlet bulk velocity and the hydraulic diameter of the mixing channel, there is a drop of the mixing degree. This is due to a change in the topology of the vorticity field and, in particular, to the presence of a unique vortical structure in the mixing channel instead of two corotating ones typical of the engulfment regime. By further increasing the Reynolds number, the vortical structure in the mixing channel grows in size and eventually starts to oscillate, leading again to an increase of the degree of mixing, starting from $\text{Re}\simeq 170$. Additional simulations have been carried out in order to investigate the sensitivity to the tilting angle of the inlet channels, namely for $\alpha ={10}^{\circ },{15}^{\circ },{25}^{\circ }$. As expected, it is found that the value of Re at which the engulfment regime occurs decreases with increasing $\alpha$. Then, for $\alpha ={10}^{\circ }$ and ${15}^{\circ }$, the degree of mixing monotonically increases with the Reynolds numbers and the flow topology in the whole engulfment regime is practically the same as for T-mixers. Conversely, the configuration with $\alpha ={25}^{\circ }$ shows a behavior of mixing and of flow features with increasing Reynolds number very similar to those observed for $\alpha ={20}^{\circ }$.

Figure 1

Experimental model (a) and sketch of the arrow mixer with $\alpha ={20}^{\circ }$ (b).

Figure 2

Experimental (a) and numerical (b) dye concentration fields and isosurface of the ${\lambda }_2$ vortex indicator (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\mathrm{Re}=120$.

Figure 3

Flow features in the top part of the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=120$: (a) isosurface of the ${\lambda }_2$ vortex indicator and velocity streamlines (top left), contours of the in-plane velocity magnitude, velocity streamlines, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the sections at $Z=0.375$ (top right), $Z=0.4875$ (bottom left), and $Z=0.6$ (bottom right); (b) contours of the $y$ velocity, in-plane velocity vectors, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the cross sections at (from top to bottom) $Y=0.7$, $Y=0.5$, $Y=0.3$, and $Y=0.1$.

Figure 4

Isosurface of the ${\lambda }_2$ vortex indicator from a different view (a), $y$ vorticity and in-plane velocity (b), and dye concentration fields (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=120$. Considered cross sections (from top to bottom): $Y=-0.7$, $Y=-1$, $Y=-2.5$, and $Y=-3.5$. All the results are from numerical simulations.

Figure 5

Experimental (a) and numerical (b) dye concentration fields and isosurface of the ${\lambda }_2$ vortex indicator (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=140$.

Figure 6

Flow features in the top part of the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=140$: (a) isosurface of the ${\lambda }_2$ vortex indicator and velocity streamlines (top left), contours of the in-plane velocity magnitude, velocity streamlines, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the sections at $Z=0.375$ (top right), $Z=0.4875$ (bottom left), and $Z=0.6$ (bottom right); (b) contours of the $y$ velocity, in-plane velocity vectors, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the cross sections at (from top to bottom) $Y=0.7$, $Y=0.5$, $Y=0.3$, and $Y=0.1$.

Figure 7

Isosurface of the ${\lambda }_2$ vortex indicator (a), $y\text{−}$ vorticity fields (b), and dye concentration fields (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=140$. Considered cross sections (from top to bottom): $Y=-0.7$, $Y=-1$, $Y=-2.5$, and $Y=-3.5$. All the results are from numerical simulations.

Figure 8

Experimental (a) and numerical (b) dye concentration fields and isosurface of the ${\lambda }_2$ vortex indicator (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=150$.

Figure 9

Flow features in the top part of the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=150$: (a) isosurface of the ${\lambda }_2$ vortex indicator and velocity streamlines (top left), contours of the in-plane velocity magnitude, velocity streamlines, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the sections at $Z=0.375$ (top right), $Z=0.4875$ (bottom left), and $Z=0.6$ (bottom right); (b) contours of the $y$ velocity, in-plane velocity vectors, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the cross sections at (from top to bottom) $Y=0.7$, $Y=0.5$, $Y=0.3$, and $Y=0.1$.

Figure 10

Isosurface of the ${\lambda }_2$ vortex indicator (a) $y\text{−}$ vorticity fields (b) and dye concentration fields (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=150$. Considered cross sections (from top to bottom): $Y=-0.7$, $Y=-1$, $Y=-2.5$, and $Y=-3.5$. All the results are from numerical simulations.

Figure 11

Experimental (a) and numerical (b) dye concentration fields and isosurface of the ${\lambda }_2$ vortex indicator (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=170$.

Figure 12

Flow features in the top part of the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=170$: (a) isosurface of the ${\lambda }_2$ vortex indicator and velocity streamlines (top left), contours of the in-plane velocity magnitude, velocity streamlines, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the sections at $Z=0.375$ (top right), $Z=0.4875$ (bottom left), and $Z=0.6$ (bottom right); (b) contours of the $y$ velocity, in-plane velocity vectors, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the cross sections at (from top to bottom) $Y=0.7$, $Y=0.5$, $Y=0.3$, and $Y=0.1$.

Figure 13

Isosurface of the ${\lambda }_2$ vortex indicator (a), $y\text{−}$vorticity fields, (b) and dye concentration fields (c) at $\text{Re}=170$. Considered cross sections (from top to bottom): $Y=-0.7$, $Y=-1$, $Y=-2.5$, and $Y=-3.5$. All the results are from numerical simulations.

Figure 14

Experimental (a) and numerical (b) dye concentration fields and isosurface of the ${\lambda }_2$ vortex indicator (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=190$.

Figure 15

Isosurface of the ${\lambda }_2$ vortex indicator (a), $y\text{−}$vorticity fields, (b) and dye concentration fields (c) for the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=190$. Considered cross sections (from top to bottom): $Y=-0.7$, $Y=-1$, $Y=-2.5$, and $Y=-3.5$. All the results are from numerical simulations.

Figure 16

Flow features in the top part of the arrow mixer with $\alpha ={20}^{\circ }$ at $\text{Re}=190$: (a) Isosurface of the ${\lambda }_2$ vortex indicator and velocity streamlines (top left), contours of the in-plane velocity magnitude, velocity streamlines, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the sections at $Z=0.375$ (top right), $Z=0.4875$ (bottom left), and $Z=0.6$ (bottom right); (b) contours of the $y$ velocity, in-plane velocity vectors, and trace of the isosurface of the ${\lambda }_2$ (thick line) for the cross sections at (from top to bottom) $Y=0.7$, $Y=0.5$, $Y=0.3$, and $Y=0.1$.

Figure 17

Mixing degree as a function of the Reynolds number for the T-shaped micro-mixer and for the arrow-mixer with $\alpha ={20}^{\circ }$.

Figure 18

Mixing degree as a function of the Reynolds number for the T-shaped micro-mixer and for the arrow mixers with different angles.

Figure 19

Isosurface of the ${\lambda }_2$ vortex indicator for the arrow mixer with $\alpha ={10}^{\circ }$. Simulations carried out at (a) $\text{Re}=180$ and (b) $\text{Re}=200$.

Figure 20

Isosurface of the ${\lambda }_2$ vortex indicator for the arrow mixer with $\alpha ={25}^{\circ }$. Simulations carried out at (a) $\text{Re}=130$, (b) $\text{Re}=140$, (c) $\text{Re}=150,$ and (d) $\text{Re}=160$.