*2.7. Units and Non-Dimensional Groups*

In order to maintain the physical similarity between the numerical simulations in lattice units, the following non-dimensional groups shall be introduced:

1. Reynolds Number:

$$\text{Re} = \frac{\sqrt{\mathcal{g}L}L}{\upsilon} \tag{23}$$

where *g* and *L* are the gravity and the characteristic length, respectively. Characteristic length will be considered as the domain width or bubble diameter depending on the study case.

2. Atwood Number:

$$\mathbf{At} = \frac{\rho\_{\mathrm{I}}|\_{2} - \rho|\_{1}}{\rho\_{\mathrm{I}}|\_{2} + \rho|\_{1}} \tag{24}$$

where *ρl*|<sup>2</sup> and *ρ*|<sup>1</sup> are the liquid phase of the high density component 2 and the low density component 1. It is worth mentioning that, for the short-range interactions, the Atwood number (density ratio) main controller is the interaction parameter *G*22.

3. Bonds Number:

$$\text{Bo} = \frac{\rho \text{g} L^2}{\gamma} \tag{25}$$

where *ρg* and *γ* represent the force field applied on the fluid component and the surface tension, whose evaluation with respect to the SC LBM will be discussed in Section 3.1, respectively. It is worth mentioning that, for the short-range interactions, the surface tension main controller is the interaction parameter *G*<sup>12</sup> = *G*21.

#### 4. Prandtl Number:

To relate the flow and the thermal relaxation times, the Prandtl number shall be introduced for the two fluid components, as follows:

$$\text{Pr} = \frac{v}{\mathfrak{a}}\tag{26}$$

#### **3. Model Verification and Validation**

#### *3.1. Young–Laplace Test*

In order to adjust the surface tension with the SC interaction parameters *G*, hence making the connection between the numerical interaction forces and the physical macroscopic property, the Young–Laplace test is always used as a key for model verification. A liquid or gas bubble is initiated in a fully periodic computational domain with a reasonable density for both components accordingly with the chosen interaction parameters. The bubble is then left to reach the steady state, the pressure inside and outside the bubble are evaluated according to Equation (9) or (19), bubble diameter (radius) is measured and the 2D surface tension can be evaluated according to the Young–Laplace equation as follows:

$$
\Delta p = p\_{inner} - p\_{outer} = \frac{\gamma}{R} \tag{27}
$$

where *pinner*, *pouter* and *R* are the inner and outer static pressure and bubble diameter, respectively. The adopted criteria to estimate the interface was half the density of the heavier component liquid phase.

The common procedure is to perform several tests for each combination of interaction parameters and evaluate the surface tension according to the linear regression of the Laplace pressure Δ*p* and 1/*R*. Figure 2 shows two values of interaction parameters for single phasetwo components with density ratio = 1. The two lines represent the two limiting cases: separation between components and numerical stability. The chosen interaction parameters equivalent to the resulting surface tensions in LU are shown on the figure legends. Figure 3 also shows the two limiting cases of surface tension for a two phase two component system with density ratio ≈ 10 between the liquid phase of the high density component 2 and the low density component 1 (At ≈ 0.8). The chosen combination of interaction parameters which led to the resulting density ratio and surface tension in LU are shown on the legend. For the higher surface tension case in Figure 3, due to a stability issues, TRT with magic parameter Λ = 1/4 was required.

For both Figures 2 and 3, the cases of lower surface tension show a non-zero density inside the bubble and a higher interfacial thickness. This is an inherent behavior of the SC model, since it is a diffuse interface approach and does not offer perfectly immiscible components [14]. Density contours in LU are shown for the four Young–Laplace test samples.

**Figure 2.** Young–Laplace test for the two limiting surface tension cases for density ratio ≈ 1 (single phase two components). Surface tension and density contours are presented in LU.

**Figure 3.** Young–Laplace test for the two limiting surface tension cases and density ratio ≈ 10 (two phase two component). Surface tensions and density contours are presented in LU.
