Axisymmetric lattice Boltzmann simulation of droplet impact on solid surfaces

Droplet-solid interaction is a ubiquitous fluid phenomenon that underpins a wide range of applications. To further the understanding of this important problem, we use an axisymmetric lattice Boltzmann method (LBM) to model the droplet impact on a solid surface with different wettability. The method applies a popular free-energy LBM developed by Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)] to simulate incompressible binary fluids with physical density and viscosity contrasts. The formulation is recast in cylindrical coordinates for modeling the normal impact of a three-dimensional (3D) droplet in the no-splashing regime, in which an axisymmetric flow is considered. The droplet deposits on or rebounds from the surface, governed by three key parameters: Weber number, Ohnesorge number, and equilibrium contact angle, which quantifies the surface wettability. We elucidate the distinct impact dynamics by probing droplet morphology and contact line behavior in great detail, which are quantitatively characterized by spreading factor, droplet aspect ratio, and dynamic contact angle. The simulations also resolve fluid velocity field inside and outside the droplet, which provides additional insight into the morphological evolution and mass-momentum transfer during impact. Explicit comparison between axisymmetric and conventional 2D LBM highlights the importance of axisymmetric terms in governing equations for reproducing physical impact behavior. The axisymmetric LBM significantly reduces computational cost as compared with 3D LBMs and offers an effective means to study droplet impact in applicable conditions.

Figure 1

Schematic plot of the simulation domain used in droplet impact study.

Figure 2

Comparison of droplet spreading factor $\left(D^{*}\right)$ and aspect ratio $\left(H^{*}\right)$ variations between the present study and previous experiments and simulations. The droplet normally impacts on a hydrophobic surface (${\theta }^{\mathrm{eq}}={107}^{\circ }$) at $\mathrm{We}=103$ and $\mathrm{Oh}=0.015$.

Figure 3

Instantaneous morphologies of droplet impacting at $\mathrm{We}=20$ (black solid lines) and $\mathrm{We}=96$ (red dashed lines) on the solid surfaces with ${\theta }^{\mathrm{eq}}={40}^{\circ }$ (left portion of each frame) and ${\theta }^{\mathrm{eq}}={110}^{\circ }$ (right portion). The Oh number is fixed at 0.0218 for all simulations.

Figure 4

We-dependent variations of droplet spreading factor (upper panel), aspect ratio (middle panel), and dynamic contact angle (lower panel) in nondimensional time at $\mathrm{Oh}=0.0218$ on hydrophilic (${\theta }^{\mathrm{eq}}={40}^{\circ }$, left panel) and hydrophobic (${\theta }^{\mathrm{eq}}={110}^{\circ }$, right panel) surfaces. The insets in (e) are the magnified views of the spreading and recoiling stages.

Figure 5

Instantaneous morphologies of droplet impacting at $\mathrm{Oh}=0.0218$ (red dashed lines) and $\mathrm{Oh}=0.074$ (black solid lines) on the solid surfaces with ${\theta }^{\mathrm{eq}}={40}^{\circ }$ (left portion of each frame) and ${\theta }^{\mathrm{eq}}={110}^{\circ }$ (right portion). The We number is fixed at 20 for all simulations.

Figure 6

Oh-dependent variations of droplet spreading factor (upper panel), aspect ratio (middle panel), and dynamic contact angle (lower panel) in nondimensional time at $\mathrm{We}=20$ on hydrophilic (${\theta }^{\mathrm{eq}}={40}^{\circ }$, left panel) and hydrophobic (${\theta }^{\mathrm{eq}}={110}^{\circ }$, right panel) surfaces. The inset in (e) is the magnified view of the recoiling stage.

Figure 7

Instantaneous velocity vector fields of droplet impacting on a hydrophilic solid surface (${\theta }^{\mathrm{eq}}={40}^{\circ }$) at $\mathrm{We}=20$ and $\mathrm{Oh}=0.074$.

Figure 8

Normalized kinetic energy and viscous dissipation energy of the droplet versus nondimensional time at various Oh numbers on (a) hydrophilic, ${\theta }^{\mathrm{eq}}={40}^{\circ }$ and (b) hydrophobic, ${\theta }^{\mathrm{eq}}={110}^{\circ }$ surfaces. The insets are the detailed views of the normalized kinetic energy variation. The We number is fixed at 20 for all simulations. The symbols are $❋$, $\diamond$, and $◯$, respectively, for KE* at $\mathrm{Oh}=0.0245$, 0.0520, and 0.0740, while the symbols $▹$, $\square$, and $▵$, respectively, for DE* at $\mathrm{Oh}=0.0245$, 0.0520, and 0.0740.

Figure 9

(a) Spreading factor, (b) aspect ratio, and (c) dynamic contact angle versus nondimensional time at $\mathrm{We}=20$, $\mathrm{Oh}=0.0218$ for different equilibrium contact angles. The symbols are $❋$, $▹$, $◯$, $\diamond$, $◃$, $●$, $▵$, and $\square$, respectively, for ${\theta }^{\mathrm{eq}}=30$, 40, 50, 60, 70, 80, 100, and ${110}^{\circ }$.

Figure 10

Instantaneous velocity vector fields of droplet impacting on a hydrophobic solid surface (${\theta }^{\mathrm{eq}}={110}^{\circ }$) at $\mathrm{We}=96$ and $\mathrm{Oh}=0.0218$.

Figure 11

Droplet morphology comparison between axisymmetric and 2D LBMs at $\mathrm{We}=20$, $\mathrm{Oh}=0.0218$, and ${\theta }^{\mathrm{eq}}={110}^{\circ }$.