An Alternative High-Density Ratio Pseudo-potential Lattice Boltzmann Model with Surface Tension Adjustment Capability

Abstract

In this paper, an alternative model is presented to simultaneously achieve thermodynamic consistency, leading to high-density ratios, and surface tension adjustment capability based on the pseudo-potential lattice Boltzmann model (LBM). These features are achieved by adding two relatively simple terms, each with an adjustable coefficient, selected from the Taylor expansion of the original interaction force, one for achieving and maintaining thermodynamic consistency, and the other for adjusting the surface tension independent of the density ratio. The model also takes advantage of being independent of using SRT, MRT, 2D or 3D lattice models. The capability of the model is evaluated and verified by performing several static and dynamic benchmark test cases. First, the coexistence densities extracted for a flat interface from the model are compared with those of the Maxwell equal area rule in a wide temperature range. Results show that the thermodynamic consistency is well achieved and the coexistence densities are independent of the surface tension strength as well. Next, the well-known Laplace law for a droplet is evaluated and satisfied with the model. In addition, a wide range of surface tension values is achievable at a fixed temperature by adjusting the surface tension adjustment coefficient. After that, the ellipsoidal droplet oscillation is simulated with a good agreement between the analytical and numerical results for the oscillation period. The spurious velocities around the droplet are then evaluated and shown to be reasonable, and comparatively low even at a relatively high-density ratio around of 2400. It was found that the magnitude of these velocities can be further reduced, especially at high-density ratios, by reducing the liquid to gas dynamic viscosity ratio or equivalently increasing the gas to liquid kinematic viscosity ratio. Next, the stability, and the fascinating phenomenon of symmetric bifurcation of a rotating planar droplet are simulated and formation of two- and four-lobed shapes along with rigid rotation is presented depending on the value of the surface tension. Finally, the droplet impact on a liquid film is simulated at a relatively high-density ratio around of 720, and Reynolds number of 1440. However, the upper limit of physical parameters for simulating this phenomenon is found to be a density ratio of 720 and Re and We numbers of 6600 and 275, respectively. Effects of the surface tension strength are shown as well. The crown spread radius from the model agrees well with the analytical solution reported in the literature.

Appendices

Appendix 1

In this section, the model is analyzed mathematically based on the discrete pressure tensor proposed by Shan [39]. The discrete pressure tensor of the pseudopotential model using the original interaction force given by Eq. (12), is obtained as follows [12]:

$$ \int {(\nabla .{\text{ P}})d} \varOmega = \int {\nabla .(\rho c_{s}^{2} {\text{I}})d} \varOmega - \int {\text{F}} d\varOmega $$

(a1)

where $d\varOmega$ is a finite volume. Using the Gauss integration theorem:

$$ \int {{\text{ P}} .d} {\text{A}} = \int {\rho c_{s}^{2} {\text{I}} .d} {\text{A}} - \int {\text{F}} d\varOmega $$

(a2)

Then in discrete form:

$$ \sum {{\text{P}} . {\text{A}}} { = }\sum {\rho c_{s}^{2} {\text{I}} . {\text{A}}} - \sum {\text{F}} $$

(a3)

$$ \to \varvec{P}_{d} = \rho c_{s}^{2} \varvec{I} + \frac{G}{2}\psi \left( {\varvec{x},t} \right)\left[ {\mathop \sum \limits_{\alpha } w\left( {\left| {\varvec{e}_{\alpha } } \right|^{2} } \right)\psi \left( {\varvec{x} + \varvec{e}_{\alpha } ,t} \right)\varvec{e}_{\alpha } \varvec{e}_{\alpha } } \right] $$

(a4)

The Taylor series expansion of the above equation leads to the following expression:

$$ \varvec{P}_{\varvec{d}} = \left( {\rho c_{s}^{2} + \frac{{Gc^{2} }}{2}\psi^{2} } \right.\left. { + \frac{{Gc^{4} }}{12}\psi \nabla^{2} \psi } \right)\varvec{I} + \frac{{Gc^{4} }}{6}\psi \nabla \nabla \psi $$

(a5)

The modified interaction force is again presented below:

$$ \begin{aligned} \varvec{F}_{{\varvec{new}}} = \varvec{F} + \lambda \frac{{Gc^{4} }}{12}\nabla \left( {\left| {\nabla \psi } \right|^{2} } \right) - \kappa \frac{{Gc^{4} }}{6}\nabla^{2} \psi \nabla \psi \hfill \\ = \varvec{F} + \lambda \frac{{Gc^{4} }}{12}\nabla \left( {\left| {\nabla \psi } \right|^{2} } \right) - \kappa \frac{{Gc^{4} }}{6}\left[ {\nabla \cdot \left( {\nabla \psi \nabla \psi } \right) - \frac{1}{2}\nabla \left| {\nabla \psi } \right|^{2} } \right] \hfill \\ \end{aligned} $$

(a6)

Therefore, the new discrete pressure tensor is simply given by:

$$ \to \varvec{P}_{d}^{new} = \left( {\rho c_{s}^{2} + \frac{{Gc^{2} }}{2}\psi^{2} } \right.\left. { + \left( { - \lambda - \kappa } \right)\frac{{Gc^{4} }}{12}\left| {\nabla \psi } \right|^{2} + \frac{{Gc^{4} }}{12}\psi \nabla^{2} \psi } \right)\varvec{I} + \frac{{Gc^{4} }}{6}\left( {\kappa \nabla \psi \nabla \psi + \psi \nabla \nabla \psi } \right) $$

(a7)

Notice that the added terms to the original interaction force in Eq. (a6) are in the gradient form just like the well-known $\nabla \left( {\rho c_{s}^{2} } \right)$ term, thus they are directly absorbed in the discrete pressure tensor as well.

The normal component of the tensor is obtained as follows:

$$ P_{n}^{d} = \rho c_{s}^{2} + \frac{{Gc^{2} }}{2}\psi^{2} + \frac{{Gc^{4} }}{12}\left[ {\left( { - \lambda + \kappa } \right)\left( {\frac{d\psi }{dn}} \right)^{2} + 3\psi \frac{{d^{2} \psi }}{{dn^{2} }}} \right] $$

(a8)

Thus, $\varepsilon_{d} = \frac{ - 2\alpha }{\beta } = - \frac{2}{3}\left( { - \lambda + \kappa } \right)$.

As observed, the same as the continuum analysis, defining $\lambda = \lambda_{0} + \kappa$ leads to:

$$ \varepsilon_{d} = \frac{2}{3}\lambda_{0} $$

(a9)

Furthermore, considering a flat interface along y direction, the surface tension relation can be obtained as follows:

$$ P_{n}^{d} = P_{xx}^{d} = \rho c_{s}^{2} + \frac{{Gc^{2} }}{2}\psi^{2} + \frac{{Gc^{4} }}{12}\left[ {\left( { - \lambda + \kappa } \right)\left( {\frac{d\psi }{dx}} \right)^{2} + 3\psi \frac{{d^{2} \psi }}{{dx^{2} }}} \right] $$

(a10)

$$ P_{\tau }^{d} = P_{yy}^{d} = \rho c_{s}^{2} + \frac{{Gc^{2} }}{2}\psi^{2} + \frac{{Gc^{4} }}{12}\left[ {\left( { - \lambda - \kappa } \right)\left( {\frac{d\psi }{dx}} \right)^{2} + \psi \frac{{d^{2} \psi }}{{dx^{2} }}} \right] $$

(a11)

$$ \sigma_{d} = \int\limits_{ - \infty }^{ + \infty } {\left( {P_{xx} - P_{yy} } \right)} dx = \frac{{Gc^{4} }}{6}\int\limits_{ - \infty }^{ + \infty } {\left[ {\kappa \left( {\frac{d\psi }{dx}} \right)^{2} + \psi \frac{{d^{2} \psi }}{{dx^{2} }}} \right]} dx $$

(a12)

$$ \int_{ - \infty }^{ + \infty } {\psi \frac{{d^{2} \psi }}{{dx^{2} }}} dx = \left. {\psi \frac{d\psi }{dx}} \right|_{ - \infty }^{ + \infty } - \int_{ - \infty }^{ + \infty } {\frac{d\psi }{dx}} d\psi $$

(a13)

Knowing $\frac{d\psi }{dx}$ is zero at $\pm \infty$ and $\frac{d\psi }{dx} = \frac{d\psi }{d\rho }\frac{d\rho }{dx} = \psi^{\prime}\frac{d\rho }{dx}$:

$$ \to \sigma_{d} = - \frac{{Gc^{4} }}{6}\left( {1 - \kappa } \right)\int\limits_{{\rho_{g} }}^{{\rho_{l} }} {\psi^{\prime 2} \frac{d\rho }{dx}} d\rho $$

(a14)

which is exactly the same relation as Eq. (36) derived from the continuum pressure tensor.

Appendix 2

In this appendix, the current model, the multirange potential [8] and the Li and Luo model [24] are further discussed and compared.

The multirange interaction force proposed by Sbragaglia et al. is given by [8]:

$$ \varvec{F} = - \psi \left( {\varvec{x},t} \right)\mathop \sum \limits_{\alpha } w\left( {\left| {\varvec{e}_{\alpha } } \right|^{2} } \right)\left[ {G_{1} \psi \left( {\varvec{x} + \varvec{e}_{\alpha } ,t} \right)\varvec{ + }G_{2} \psi \left( {\varvec{x} + 2\varvec{e}_{\alpha } ,t} \right)} \right]\varvec{e}_{\alpha } $$

(b1)

This interaction force was claimed to be able to adjust the EOS through G₁ and the surface tension through G₂, independently. In part III of their paper [24], Li and Luo made an analysis of the Sbragaglia et al. model to obtain the corresponding relation for the ε parameter and found the following relation (see Eq. (16) in [24]):

$$ \varepsilon = \frac{{4G_{2} }}{{G_{1} + 6G_{2} }} $$

(b2)

where it is clear that the parameter ε is affected by both G₁ and G₂ coefficients implementing the multirange potential model of Sbragaglia et al. [8]. It should be mentioned that the interaction potential in the Li and Luo analysis of the multirange model is taken to be $\psi = \sqrt {\frac{{2\left( {P_{EOS} - \rho c_{s}^{2} } \right)}}{{(G_{1} + 2G_{2} )c^{2} }}}$. Therefore, Li and Luo showed that the multirange interaction force is not capable of adjusting the EOS and surface tension independently, implementing real EOSs like the Carnahan-Starling.

In terms of achieving thermodynamic consistency, our approach is considerably different from the Li and Luo one. The parameter ε in Li and Luo model is given by [12]:

$$ \varepsilon = - 2\left( {\alpha + 24G\sigma } \right)/\beta $$

(b3)

where σ is a parameter adjusted to numerically achieve the thermodynamic consistency, achieving by applying the following model proposed earlier by Q. Li et al. [11]:

$$ \varvec{F}_{\alpha } = \omega_{\alpha } \delta t\left( {1 - \frac{1}{2\tau }} \right)\left[ {\frac{{\varvec{e}_{\alpha } - \varvec{V}^{\varvec{'}} }}{{c_{s}^{2} }} + \frac{{\left( {\varvec{e}_{\alpha } \cdot \varvec{V}^{\varvec{'}} } \right)}}{{c_{s}^{4} }}\varvec{e}_{\alpha } } \right] \cdot \varvec{F} $$

(b4)

In other words, they modified the velocity in the Guo force scheme as follows, using the SRT LB equation [11]:

$$ {\varvec{V}^{\prime}} = {\varvec{V}} + \sigma {\mathbf{F}}/\left( {\upsilon \psi ^{2} } \right) $$

(b5)

where $\varvec{V} = \varvec{u} + {\mathbf{F}}\delta t/2\rho$ with $\rho \varvec{u} = \mathop \sum \nolimits f_{\alpha } e_{\alpha }$ and in the denominator, $\nu = (\tau - 0.5)/3$ being the kinematic viscosity. Therefore, our approach for adjusting the mechanical stability condition ε, i.e., adding the term $+ \lambda \frac{{Gc^{4} }}{12}\nabla \left( {\left| {\nabla \psi } \right|^{2} } \right)$ to the original interaction force, is different and novel, although both models add a source term to the LB equation.

In addition, when developed to the MRT version, the Li and Luo model summarized in Eqs (b4) and (b5), is affected by other relaxation times, especially in Eq. (b5), thus some considerations and modifications are carried out on the force term in the momentum space and, as reported in Li and Luo [12], thus mathematical calculations are needed. Whereas, in the current model the source term in the momentum space is computed by a simple code multiplying the force term by the transformation matrix due to no modifications on the Guo force scheme.

In terms of the approach for adjusting the surface tension independently, again the source term should be calculated mathematically in the Li and Luo model, considering the discrete lattice effects on the source term whereas, computing a Laplacian term is the only needed task to make the surface tension adjustable in the current model. As mentioned before, the multirange interaction force by Sbragaglia et al. is based on modifying the interaction force as it is in the current model. Therefore, modification of the interaction force can be viewed at least a different mathematical modeling and approach. The superiority of the current model over the multirange model includes the independent adjustment of the EOS and surface tension, considering only the nearest neighbors interactions and, therefore, easier treatment of boundary conditions.

The value of ε for achieving thermodynamic consistency in the Li and Luo model is indeed a numerical finding as it is in our model, both models use the same interaction potential given by Eq. (13). However, it is not a hard task to find the consistent ε value and is found to be almost independent of the EOSs implemented. Similarly, the parameter λ for achieving thermodynamic consistency in the current model is found by performing several simulations, but when recognized, its value is almost the same implementing other EOSs. For example, we also implemented the Peng–Robinson (P-R) EOS and the consistent λ is found to be in the range of 2.6 to 2.65, which is almost the same value of 2.65 found for the C-S EOS. The consistent σ in the Li and Luo model is reported to be in the range of 0.11–0.12. Xu et al. [18] extended their model to the D3Q15 lattice and σ = 0.12 was reported for both C-S and P-R EOSs. Therefore, in this regard, the obtained thermodynamically consistent value of ε in the current model, like the Li and Luo one, is almost independent of the EOS and thus it is applicable in the general framework of the pseudopotential LBM. The thermodynamically consistent value of ε for the current model is, therefore, $\varepsilon = 1 + \lambda_{0} = 1 + 2.65 = 3.65$, almost independent of implementing different equation of states.