In this study, a numerical model for simulating two-phase flow is developed. The Cartesian grid with Adaptive Mesh Refinement (AMR) is adopted to reduce the computational cost. An explicit projection method is used for the time integration and the Finite Difference Method (FDM) is applied on a staggered grid for the discretization of spatial derivatives. The Volume of Fluid (VOF) method with Piecewise-Linear Interface Calculation (PLIC) is extended to the AMR grid to capture the gas-water interface accurately. A coarse-fine interface treatment method is developed to preserve the flux conservation at the interfaces. Several two-dimensional (2D) and three-dimensional (3D) benchmark cases are carried out for the validation of the model. 2D and 3D shear flow tests are conducted to validate the extension of the VOF method to the AMR grid. A 2D linear sloshing case is considered in which the model is proved to have 2nd-order accuracy in space. The efficiency of applying the AMR grid is discussed with a nonlinear sloshing problem. Finally, 2D solitary wave past stage and 2D/3D dam break are simulated to demonstrate that the model is able to simulate violent interface problems.

Two-phase flow is very important as it is common in many engineering fields, ranging from naval hydrodynamics, geophysics to ocean engineering. The experimental study is a widely used approach for analyzing physical problems. However, due to the existence of small-scale structures and violent velocity change, experimental measurements on violent two-phase flow problems could be extremely difficult or even impossible. With the improvement of hardware and numerical algorithms, numerical simulation could be a powerful alternative for predicting the behaviors of these problems. However, there still exist many difficulties in numerical simulating two-phase flow.

Among the difficulties, the computational cost could be one of the major challenges in Computational Fluid Dynamics (CFD). Traditionally, a fixed structured grid is usually adopted for simulating two-phase flow problems. One of the remarkable features of this grid is that mesh refinement needs to be applied in priority on the range of all the caring regions, which is inefficient or even impossible when the caring regions are unpredictable. Adaptive Mesh Refinement (AMR) grid was proposed by Berger et al. [

For most of the research with AMR, single-phase flow is considered. As the method was proposed for hyperbolic partial differential equations, a natural implementation of AMR is the solution of the Euler equation [

Research that applying AMR in two-phase flow is also numerous, relevant research can be found in surface-tension-driven interfacial flows such as bubbles [

The objective of this paper is to develop a 3D numerical model for simulating two-phase flow with AMR. The model is the foundation of future work for wave-body interactions in the ship and ocean engineering field. To develop such a model, the difficulties discussed above should be considered. The first one is the complex interface topology, or in other words, to capture the gas-water interface accurately. For this problem, widely used methods include the front tracking method [

Large density ratios usually exist in two-phase flow, such as in bubble flow in which the density ratio could reach 10^{6}. This large ratio can result in numerical instability which may come from gas velocities being mixed with liquid densities [

A 3D numerical model for simulating two-phase flow with AMR is developed in this paper. The VOF method is adopted to capture the gas-water interface accurately. A coarse-fine interface treatment method for two-phase flow is developed to preserve the flux conservation at the coarse-fine interfaces. An inconsistent transport scheme is adopted for the advection of mass and momentum. The work is based on Zhang et al. [

The single fluid formulation for multiphase flow is adopted in this model [

in which

The VOF method is utilized to capture the free surface. In the method, a scalar function

Appropriate initial and boundary conditions for

In this section, the numerical methods on a single grid are introduced. For simplicity, the discretization schemes of the spatial derivatives are given in the 2D form. There is no extra difficulty for their extension to 3D.

The incompressible version of the projection method proposed by Xiao [

Advection step

Non-advection Step I

Non-advection Step II

Here the corner mark ^{n}, ^{n+1}, *, and ** indicate the values of corresponding quantities at the current time step, the next time step, and the two predicted values between the two steps. This procedure was demonstrated to yield the first-order accuracy in time integration [

By applying the continuity equation in

The pressure is calculated by solving the equation after non-advection Step I and followed by non-advection Step II.

To ensure the temporal stability of the explicit projection method in the model, the time step

The Finite Difference Method (FDM) on the Cartesian grid with a staggered arrangement of variables is adopted. A 2D case is illustrated in

For the discretization of the advection step (

The values of

The central difference method is adopted for all the spatial derivatives in

The discretization of

For the solution of

With the

In the discretizations of the two non-advection steps and the PPE equation (

where

As mentioned before, the AMR grid is employed in this model. The computation domain is defined as a quadtree (2D) or octree (3D) structure with all the nodes of the tree to indicate a grid block with the same grid number (see

A fixed time step strategy is adopted due to the incompressibility of the fluid. The single grid solver is applied on all the grid blocks without too many changes, except for the interpolation between blocks to provide boundary conditions.

After each operation (advection, diffusion, and pressure correction) within each time step, interpolations are necessary as the grid blocks need ghost cells at their boundaries to complete the discretization stencils. Two layers of ghost cells are used for each block. Take the grid in

Due to the staggered arrangements of the variables, the interpolations for variables

Other interpolations can be categorized into the expressions above. The interpolations in 3D are similar.

A refining/de-refining process is conducted based on the distance of a grid block from the free surface at the end of each time step. During the process, the divergence-free condition must be preserved. Here, the face averaging method is adopted for the de-refining process as it is a simple way to preserve the divergence of the velocity field on a staggered grid as well as global 2nd-order accuracy. For the refining process, the method proposed by Tóth et al. [

A careful treatment of the coarse-fine interface is necessary during the solution process. Due to the staggered grid, velocities at the interface will share the same face for the coarse and fine grid as shown in

A simple method to avoid the situation is to force the velocities to satisfy the flux conservation condition after the pressure correction step automatically. A pressure boundary condition method is proposed by Vanella et al. [

Then the flux conservation condition will be automatically satisfied.

The complete algorithm of the model is as follows:

Initialize the flow field with the initial topology of the water phase (

Calculate the position of the interface at the new time step to obtain

Obtaining the density and dynamic viscosity coefficient of the cell (

Compute the provisional velocity field

Compute the provisional velocity field

Calculate the pressure field

Compute the velocity field

Compute the distance from free surface for each grid block, refine/de-refine the blocks under the defined strategy.

Proceed to Step (2) for the next time step.

In this section, three selected benchmark cases are carried out for the validation of the model. Shear flow tests are conducted to validate the extension of the VOF method to the AMR grid. A linear sloshing case is considered to check the accuracy of the model. A nonlinear sloshing problem is simulated to validate the accuracy of the model as well as discuss the efficiency with the AMR grid.

The shear flow test is a widely used benchmark to validate the VOF method [

The zone scale is 1.0 for both 2D and 3D on all the dimensions. The initial shape of the scalar field is given as a circle in 2D and a sphere in 3D. So the corresponding volume fraction is set as:

A 4-level AMR grid is used with the CFL number set to be 0.1 for all the simulations. The minimum scale of the grid is set as 1/128. The results at selected moments are shown in

2D linear free sloshing is considered to validate the accuracy of the model. In this problem, a tank is filled with water and gas and the initial wave profile is given. The liquid is initially at rest and evolves submitted to the gravity force. An illustration of the problem is presented in

When the wave amplitude is small, a linear analytic solution for 2D viscous waves in a rectangular tank was derived by Wu et al. [

where

in which

Here the width of the tank, the water depth, and the initial wave amplitude are set as 1.0 m, 0.5 m, and 0.01 m, respectively. A 4-level AMR grid is adopted for this case. The grids are uniform around the interface in both directions with the spacing of 1/128 m. A constant time step of 0.00001 s is used for all the simulations. Reynolds numbers of 20 and 200 are considered here to validate the model and both cases run up to 10 s. The results are shown in

The spatial accuracy of the model is measured with the problem. The _{2} and _{2} norm error is nearly 2nd-order while the

A 2D nonlinear sloshing resonance problem is considered with this model. In this problem, a partially filled tank submitted to the gravity force and an external horizontal force is simulated. The computational model with corresponding parameters consistent with Liu et al.’s [_{n} =

A 4-level AMR grid is adopted with the finest grid to be 0.0045 m in both directions. A constant time step of 0.001 s is used. Time histories of the free surface height at the three gauges for the time before 6.7 s are shown in

With time going on, the wave amplitude will increase continuously until wave breaking occurs. Two selected moments are given in

The efficiency of the AMR grid is discussed with this problem. An AMR grid with more levels usually leads to a smaller computational cost, if the minimum scale of the grid is set to be the same. To discuss it in quantitative, the stable phase of the nonlinear sloshing problem is selected (

The mean block number and corresponding CPU time for the four grids are shown in

Mean block | Mean grid | Grid save (%) | CPU time (s) | CPU save (%) | Grid save—CPU save (%) | |
---|---|---|---|---|---|---|

1 | 1024 | 16384 | – | 1964 | – | – |

2 | 388.86 | 6222 | 62.02 | 879 | 55.24 | 6.78 |

3 | 277.74 | 4444 | 72.88 | 731 | 62.78 | 10.10 |

4 | 264.04 | 4225 | 74.21 | 727 | 62.98 | 11.23 |

In this section, two selected cases are utilized to show that the model is able to simulate violent interface problems.

To demonstrate the model’s ability in dealing with wave breaking problems, a solitary wave past a submerged stationary stage is considered with the model. The computational model of the problem is shown in

A 4-level AMR grid is employed and a fixed time step of 0.001 s is adopted for the simulations. The minimum grid scales of the medium grid is

Wave profiles at selected moments are shown in

Dam break is a typical violent free-surface flow problem, as complicated phenomena including merging, overturning, and air entrainment occur during the process. The problem was studied by Martin et al. [

Both 2D and 3D simulations are conducted, with the minimum grid scale to be 1/128 and 1/64 for 2D and 3D, respectively. A constant time step of 0.001 s is used. The water is initially at rest (

When the wavefront reaches the vertical wall on the right side, the water will run up and overturn. The process is shown in

A numerical model for simulating two-phase flow is developed with AMR. An explicit projection method is used to decouple the velocity and pressure. The free surface is captured with the VOF method combined with PLIC. A coarse-fine interface treatment method is developed to preserve the flux conservation at the interfaces. The adaptive grid is managed with the open-source PARAMESH library.

Several benchmark cases are considered to validate the model. The VOF method is validated with the shear flow test. The accuracy of the model is validated with a linear sloshing problem, for which analytic solutions can be derived. The model is demonstrated to have 2nd-order accuracy in space. The efficiency of AMR is demonstrated with a nonlinear sloshing problem.

The ability of the model to simulate more complex problems is demonstrated with two selected cases including solitary wave breaking and dam break. Good agreements are obtained between the current numerical results and experimental or numerical results from other sources. It is demonstrated that the model is able to simulate interface problems with wave breaking.

The authors are grateful for the important and valuable comments of anonymous reviewers.