*2.6. Thermal Field*

Temperature field can be added in LBM as a third set of the distribution function, in addition to the two from the two fluid components. The LBE for the temperature distribution function *hi*(*x*ˆ, *t*) can be written following the laws of diffusion and convection as follows:

$$h\_i(\mathbf{\hat{x}} + \mathbf{\hat{e}}\_i \Delta t, t + \Delta t) = h\_i(\mathbf{\hat{x}}, t) - \frac{1}{\mathbf{\hat{r}}\_T} (h\_i(\mathbf{\hat{x}}, t) - h\_i^{cpu}(\mathbf{\hat{x}}, t)) \tag{20}$$

where *τ<sup>T</sup>* and *hi equ*(*x*ˆ, *t*) are the temperature relaxation time and the equilibrium temperature distribution function. Both can be written as follows:

$$
\pi\_T = \mathfrak{A}\mathfrak{a}\_{L\!\!\!\!\/ I} + 0.5 \tag{21}
$$

$$w\_i^{c\eta u}(\mathbf{\hat{x}}, \mathbf{t}) = w\_i T \left[ 1 + 3 \frac{\mathbf{\hat{e}}\_i \cdot \mathbf{\hat{n}}'}{c^2} \right] \tag{22}$$

where *αLU*, *T* and *u*ˆ**'** are thermal diffusivity in lattice units, the macroscopic temperature evaluated from the zeroth moment of the temperature distribution function (*<sup>T</sup>* <sup>=</sup> <sup>8</sup> ∑ *i*=0 *hi*) and the macroscopic mixture velocity imported from the flow field, respectively. It is reported in the work by Guo et al. [39] that truncating the equilibrium temperature distribution function after the first order velocity term still recovers the convective diffusive temperature equation. Although it is a common practice to use a lower resolution lattice for the temperature field (e.g., D2Q4) than for the flow field [39], in the current work the typical D2Q9 as for the flow field is used for the three sets of distribution functions.

Originally, this can be seen as a one-way coupled temperature field with the flow one, although the relation to the surface tension can be locally back-coupled using a locally variable interaction parameter, and will be shown in Section 4.2.

It is worth mentioning that the implementation of the TRT is only essential for the flow field, while for the temperature field SRT is sufficient since the stability issue is not critical. If TRT was to be used, the asymmetric part of the relaxation time would be the one responsible for the thermal diffusion, not the symmetric one as in the kinematic viscosity case according to the scope of Equation (15).
