T-mixer operating with water at different temperatures: Simulation and stability analysis

In this paper we investigate the transition from the vortex to the engulfment regime in a T-mixer when the two entering flows have different viscosity. In particular we consider as working fluid water entering the two inlet channels of the mixer at two different temperatures. Contrary to the isothermal case, at low Reynolds numbers the vortex regime shows only a single reflectional symmetry, due to the nonhomogeneous distribution of the viscosity. Increasing the Reynolds number, a symmetry-breaking bifurcation drives the system to a new steady flow configuration, usually called the engulfment regime, similar to what it is possible to observe in an isothermal case. This flow regime is associated with an increase of the mixing between the two inlet streams. It is shown by direct numerical simulation (DNS) and by stability analysis that the engulfment regime is promoted by the temperature difference. Starting from the DNSs, the resulting flow fields are analyzed in detail considering different temperature jumps between the two inlet boundaries. Furthermore, dedicated linear stability analyses are carried out to investigate the instability mechanism associated with the occurrence of the engulfment regime. In particular, similarly to the case without temperature differences, the onset of engulfment is driven by the momentum equation, and the temperature field does not lead to any additional instability mechanism. However, the existence of a temperature field leads to quantitative changes of the stability characteristics and of the resulting flow fields via a variation of the viscosity coefficient.

Figure 1

Flow configuration. (a) Computational domain used for DNSs; (b) computational domain used for stability analysis.

Figure 2

Effect of the variations of temperature on the (a) dynamic viscosity and (b) Prandtl number. Red dots are extracted from Refs. [20, 35] and the continuous line corresponds to the analytical approximation used here.

Figure 3

Flow regimes (vortex or engulfment regime) identified using DNSs for different values of the Reynolds number ${\text{Re}}_1$ at different values of the temperature jump between the two streams entering the T-mixer $\mathrm{\Delta }T$.

Figure 4

Degree of mixing ${\delta }_m$ evaluated in the outlet channel at $y=-5$ for different temperature jumps.

Figure 5

Top and lateral views of the 3D vortical structures identified by the $\lambda 2$ criterion [42] for the vortex regime at (a) $\mathrm{\Delta }T=0{}^{\circ }\mathrm{C}$ and $\text{Re}=135$ and (b) $\mathrm{\Delta }T=40{}^{\circ }\mathrm{C}$ and $\text{Re}=110$.

Figure 6

Distribution of temperature in the vortex regime for different values of the temperature jump $\mathrm{\Delta }T=T_2-T_1$ at three selected sections in $xz$ plane at $y=0.5,-1,\text{and}-3$.

Figure 7

Distribution of the velocity in $y$ direction in the vortex regime for different values of the temperature jump $\mathrm{\Delta }T=T_2-T_1$ at three selected sections in $xz$ plane at $y=0.5,-1,\text{and}-3$.

Figure 8

Distribution of temperature in the vortex regime for different values of the temperature jump $\mathrm{\Delta }T=T_2-T_1$ at two selected sections in $xy$ plane, located at $z=h/2$ and at $z=0.95h.$

Figure 9

Top and lateral views of the 3D vortical structures identified by the $\lambda 2$ criterion [42] for the vortex regime at (a) $\mathrm{\Delta }T=0{}^{\circ }\mathrm{C}$ and $\text{Re}=160$ and (b) $\mathrm{\Delta }T=40{}^{\circ }\mathrm{C}$ and $\text{Re}=120$; the color on the isosurfaces indicates the value of temperature (or of the passive scalar in the case $\mathrm{\Delta }T=0{}^{\circ }\mathrm{C})$.

Figure 10

Distribution of temperature in the engulfment regime for different values of the temperature jump $\mathrm{\Delta }T=T_2-T_1$.

Figure 11

Distribution of the velocity in $y$ direction in the engulfment regime for different values of the temperature jump $\mathrm{\Delta }T=T_2-T_1$ at three selected sections in $xz$ plane at $y=0.5,-1,\text{and}-3$.

Figure 12

Leading direct global mode for the case $\mathrm{\Delta }T=10{}^{\circ }\mathrm{C}$ and $\text{Re}=120$ at three different sections: (a)–(d) $y=-0.5$, (b)–(e) $y=-1.5,$ and (c)–(f) $y=-3$. Left: the color contours represent the $y$ component of the velocity field and the vectors the in-plane velocity; right: temperature perturbation field.

Figure 13

DNS velocity $(y$ component) and temperature fields at $y=-0.5$ before (a)–(e) and after (b)–(f) the engulfment instability for $\mathrm{\Delta }T=10{}^{\circ }\mathrm{C}$; the velocity and temperature fields in subfigures (c)–(g) are the difference between the fields (b)–(f) and those in (a)–(e), i.e., the difference fields between the postengulfment and the pre-engulfment ones; velocity $(y$ component) and temperature eigenmodes obtained by stability analysis are depicted in subfigures (d) and (h) and normalized so as to ease comparison with (c) and (g).

Figure 14

Representative sections (top) and iso-surfaces (bottom) of the sensitivity maps (a) $S_M\left({\mathbf{x}}_{*}\right)$ and (b) $S_T\left({\mathbf{x}}_{*}\right)(\mathrm{\Delta }T=10{}^{\circ }\mathrm{C}$ and $\text{Re}=120)$.

Figure 15

Eigenmodes related to the engulfment instability at $\mathrm{\Delta }T=10{}^{\circ }\mathrm{C}$ and $\text{Re}=120$ resulting from the two tests described in the text: (a) velocity field in the $y$ direction obtained in both Tests no. 1 and 2 (fields are identical so only one is reported) and (b) temperature field obtained in Test no. 2.