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 (LB) 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 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 non-linear regime is observed for $Ca \gtrsim 0.2$.