In this paper, we develop a three-dimensional multiple-relaxation-time lattice Boltzmann method (MRT-LBM) based on a set of nonorthogonal basis vectors. Compared with the classical MRT-LBM based on a set of orthogonal basis vectors, the present non-orthogonal MRT-LBM simplifies the transformation between the discrete velocity space and the moment space and exhibits better portability across different lattices. The proposed method is then extended to multiphase flows at large density ratio with tunable surface tension, and its numerical stability and accuracy are well demonstrated by some benchmark cases. Using the proposed method, a practical case of a fuel droplet impacting on a dry surface at high Reynolds and Weber numbers is simulated and the evolution of the spreading film diameter agrees well with the experimental data. Furthermore, another realistic case of a droplet impacting on a super-hydrophobic wall with a cylindrical obstacle is reproduced, which confirms the experimental finding of Liu et al. ["Symmetry breaking in drop bouncing on curved surfaces," Nat. Commun. 6, 10034 (2015)] that the contact time is minimized when the cylinder radius is comparable with the droplet radius.
To accurately simulate the bubble motions in the turbulent flow liquids, two way coupled Euler–Lagrange method is adopted in this work. The continuous phase is solved with the Navier–Stokes equations based on the Euler grid, while the individual bubble is tracked by using the Lagrange frame of reference. Two–way coupling is realized by transferring the interaction forces to the momentum equation of the continuous phase at each fluid flow timestep and in turn influencing the bubble motion. The interaction forces including the drag force, lift force, wall lubrication force and virtual mass force are carefully selected. Turbulent models are of significance to capture the fluid flow conditions. Comparing the different cases treated by the k-ε, PANS and LES models with the experimental data, the results of the PANS with RWM and BIT, and LES can dynamically display the oscillation of the bubble plume and the calculated time averaged axial liquid velocity reasonably agrees with the measured data. In the case of bubble injective flow conditions in large containers, the bubble induced turbulent effect can be almost neglected. However, the RWM describes the fluctuating velocity should be considered applied with the PANS model. In addition, the saturated vapor–liquid system under high pressure of 6.9 MPa is numerically simulated by implementing the PANS with R WM and BIT.
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