Corner-transport-upwind lattice Boltzmann model for bubble cavitation

Aiming to study the bubble cavitation problem in quiescent and sheared liquids, a third-order isothermal lattice Boltzmann model that describes a two-dimensional (2D) fluid obeying the van der Waals equation of state, is introduced. The evolution equations for the distribution functions in this off-lattice model with 16 velocities are solved using the corner-transport-upwind (CTU) numerical scheme on large square lattices (up to $6144\times 6144$ nodes). The numerical viscosity and the regularization of the model are discussed for first- and second-order CTU schemes finding that the latter choice allows to obtain a very accurate phase diagram of a nonideal fluid. In a quiescent liquid, the present model allows us to recover the solution of the 2D Rayleigh-Plesset equation for a growing vapor bubble. In a sheared liquid, we investigated the evolution of the total bubble area, the bubble deformation, and the bubble tilt angle, for various values of the shear rate. A linear relation between the dimensionless deformation coefficient $D$ and the capillary number Ca is found at small Ca but with a different factor than in equilibrium liquids. A nonlinear regime is observed for $\text{Ca}\gtrsim 0.2$.

Figure 1

Evolution of the normalized peak velocity $u_{\perp }(0,t)/U$ of decaying shear waves without regularization, for various values of the relaxation time (a) $\tau =0.001$, (b) 0.010, (c) 0.100, (d) 0.200, (e) 0.300, (f) 0.500. The results obtained using the $\mathrm{CTU}n$ scheme, $n\in \{1,2\}$, and the lattice spacing $\delta s=1/S$, $S\in \{128,256\}$, for axial (A) and diagonal waves (D), are marked with $n\mathrm{A}$ $S$ and $n\mathrm{D}$ $S$, respectively.

Figure 2

Evolution of the normalized peak velocity $u_{\perp }(0,t)/U$ of decaying shear waves obtained using the CTU1 numerical scheme with and without regularization, for various values of the relaxation time (a) $\tau =0.001$, (b) 0.010, (c) 0.100, (d) 0.200, (e) 0.300, (f) 0.500. The results which carry the symbol S in the plot keys were obtained on lattices with the spacing $\delta s=1/\left(128\sqrt{2}\right)$, while the remaining results were obtained on lattices with $\delta s=1/128$. The results obtained using the regularization procedure are marked with the symbol R.

Figure 3

Evolution of the normalized peak velocity $u_{\perp }(0,t)/U$ of decaying shear waves obtained using the CTU2 numerical scheme with and without regularization, for various values of the relaxation time (a) $\tau =0.001$, (b) 0.010, (c) 0.100, (d) 0.200, (e) 0.300, (f) 0.500. The results which carry the symbol S in the plot keys were obtained on lattices with the spacing $\delta s=1/\left(256\sqrt{2}\right)$, while the remaining results were obtained on lattices with $\delta s=1/256$. The results obtained using the regularization procedure are marked with the symbol R.

Figure 4

Liquid-vapor phase diagram: Symbols refer to LB results on the liquid branch ($\blacksquare$) and on the vapor one ($•$). The full lines correspond to the results of the Maxwell construction.

Figure 5

Dependence of the numerical liquid and vapor densities ${\rho }_L$ and ${\rho }_V$, respectively, on the relaxation time $\tau$ and on the lattice spacing $\delta s$ at temperature $T=0.80$, obtained with the CTU1 and the CTU2 numerical schemes ($\delta t={10}^{-4}$). The horizontal line in each plot shows the corresponding theoretical density value computed using the Maxwell construction.

Figure 6

Values of the critical bubble radius $R_c$ from LB simulations versus the theoretical predictions of Eq. (40) represented by the full line. The marks correspond to various values of the external density ${\rho }_{\mathrm{ext}}$. The LB results were obtained with the CTU1 scheme on a lattice with spacing $\delta s=1/128$ and size $L=2048$ ($\circ$) and with the CTU2 scheme on lattices with $\delta s=1/256$ and sizes $L=4096\left(▾\right),2048\left(▵\right),1024\left(\blacksquare \right)$. Inset: Values of $R_c$ from LB simulations with the CTU2 scheme on the lattice with $\delta s=1/256$ and size $L=4096\left(▾\right)$, and from the theoretical predictions $(---)$ as a function of the external density ${\rho }_{\mathrm{ext}}$.

Figure 7

The radius $R$ of the growing bubble in a quiescent liquid as a function of time for lattice size $L=4096(•),6144\left(▴\right)$ from the lattice Boltzmann simulations. The full and dashed lines correspond to the numerical solutions of the RP equation (48) for $L=4096,6144$, respectively.

Figure 8

The deformation $D$ of the bubble as a function of the capillary number Ca in a lattice of size $L=6144$ for shear rates $\dot{\gamma }\delta t=1.67\times {10}^{-6}(•),3.33\times {10}^{-6}\left(★\right),5.00\times {10}^{-6}(*)$. The full line has slope 0.89.

Figure 9

Density plots of the bubble (top panels) at time $t/\delta t=5\times {10}^5$ in a lattice of size $L=6144$ and the density profiles (bottom panels) in the interface regions, plotted along the Cartesian axes $x(•)$ and $y(\circ )$ centered in the middle of the flow domains, for shear rates $\dot{\gamma }\delta t=1.67\times {10}^{-6}$ (left) and $5.00\times {10}^{-6}$ (right). The values of the capillary number are $\text{Ca}=0.18$ (left)and 0.61 (right).

Figure 10

The tilt angle $\theta$ of the bubble as a function of time in a lattice of size $L=6144$ for shear rates $\dot{\gamma }\delta t=1.67\times {10}^{-6}(•),3.33\times {10}^{-6}\left(★\right),5.00\times {10}^{-6}(*)$.

Figure 11

The fraction $A_{\mathrm{rel}}$ of the bubble area as a function of time in a lattice of size $L=6144$ for shear rates $\dot{\gamma }\delta t=0\left(▵\right),1.67\times {10}^{-6}(•),3.33\times {10}^{-6}\left(★\right),5.00\times {10}^{-6}(*)$.