† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 10972010, 11028206, 11371069, 11372052, 11402029, and 11472060), the Science and Technology Development Foundation of China Academy of Engineering Physics (CAEP), China (Grant No. 2014B0201030), and the Defense Industrial Technology Development Program of China (Grant No. B1520132012).
Motivated by inconveniences of present hybrid methods, a gradient-augmented hybrid interface capturing method (GAHM) is presented for incompressible two-phase flow. A front tracking method (FTM) is used as the skeleton of the GAHM for low mass loss and resources. Smooth eulerian level set values are calculated from the FTM interface, and are used for a local interface reconstruction. The reconstruction avoids marker particle redistribution and enables an automatic treatment of interfacial topology change. The cubic Hermit interpolation is employed in all steps of the GAHM to capture subgrid structures within a single spacial cell. The performance of the GAHM is carefully evaluated in a benchmark test. Results show significant improvements of mass loss, clear subgrid structures, highly accurate derivatives (normals and curvatures) and low cost. The GAHM is further coupled with an incompressible multiphase flow solver, Super CE/SE, for more complex and practical applications. The updated solver is evaluated through comparison with an early droplet research.
Accurate and efficient simulation of interface evolution plays an important role for a wide range of computational fluid dynamics (CFD) problems, especially for multiphase flow. It is even more crucial for problems involving derivatives such as normals or curvatures. For example, the accurate calculation of the interfacial tension, which is highly sensitive to curvatures, is essential for the simulation of droplets or bubbles. Various numerical methods have been proposed to describe the evolution of the interface. These methods are further divided into eulerian capturing methods (ECMs) (e.g. the level set method (LSM)), which implicitly capture the interface, and lagrangian tracking methods (LTMs) (e.g. the front tracking method (FTM)), which explicitly track the interface.
ECMs capture smooth interface with high accuracy and automatically handle interface topology change such as merging or breaking. Nevertheless, ECMs suffer inconveniences in the following aspects: losses of subgrid structures, inaccurate approximations of normals and curvatures, and large computational stencils required by high order numerical methods. The LTM provides sharp interface with high mass conservation. However, the application to three-dimensional (3D) problems is difficult. Main inconveniences include the inefficient interface reconstruction and complicated calculation of interface topology change.
In order to combine advantages of different methods, popular and effective hybrid methods, such as the hybrid particles level set method (HPLSM), have been proposed. However, they are based on ECMs and inherit common inconveniences of ECMs. Moreover, key procedures of these schemes include dissipative interpolation on fixed grids. This kind of interpolation may smooth out important sharp details.
Another possible strategy for improving interface resolution is to employ gradient information. Gradient-augmented methods (GAM), including the compact scheme and the space time conservation element and solution element (CE/SE) method, show that gradients can significantly improve efficiency, accuracy and compactness. The compactness also provides a simple treatment of adaptive mesh refinement and boundary conditions. GAMs achieve success in a wide range of computational fluid dynamics problems.[6–12] As far as we know, few papers[13,14] report applications of this strategy for capturing interface.
Motivated by inconveniences of hybrid methods, a gradient-augmented hybrid method (GAHM) is presented for capturing interfaces in an incompressible two-phase flow. The method uses a modified FTM as its skeleton and combines the LSM and cubic Hermit interpolation (CHI). Gradients are coupled in all steps of the GAHM to maintain high resolution within a single spatial cell. Smooth eulerian level set values are calculated from the interface. They are used for physical simulation as done in the LSM and also for a local eulerian interface reconstruction. The reconstruction avoids marker particles redistribution and enables an automatic treatment of interfacial topology change.
In the next section, we will briefly present the numerical approach used for simulating an incompressible two-phase flow. Then we describe the GAHM in Section 3. Finally, we demonstrate properties of the GAHM including accuracy, convergence and cost.
Consider two incompressible fluids, fluid 1 with density ρ1 and viscosity μ1 and fluid 2 with density ρ2 and viscosity μ2. Let σ denote the surface tension coefficient. The dimensionless governing equations for two-phase incompressible flow with interfacial forces are
In Eq. (
By employing the continuum surface force (CSF) model, The interface force is reformulated as a volume force κ(φ)φδ(φ)/We with δ(φ) = dH(φ)/dφ and κ(φ) = (φ/|φ|). H(φ) is written as
Pressure can be evaluated form Eq. (
As shown in Fig.
In the GAHM, the lagrangian interface is composed of points and cubic curves. A curve is governed by a CHI equation and stored as a line segment. The two end points of the line segments are called markers. Unlike the FTM, the GAHM also evolves level set value φ and its gradient
In the GAHM, the interface is evolved by solving the lagrangian level set equation
Eulerian level set values are calculated from the tracked interface. Figure
if (PM is in line segment P0P1)
According to the above code, an eulerian grid node has several level set values corresponding to different line segments. Refer to Fig.
As the interface deforms, the distances of neighboring markers are non-uniform. This may cause extra resources, where markers are crowded, or loss of the subgrid structures, where markers are insufficient. Numerical errors may also lead to oscillations on the tracked interface. This problem is hard to handle by the FTM, particularly difficult in 3D applications. Du et al. presented three methods: grid free tracking (GF), grid-based tracking (GB), and locally grid-based tracking (LGB). The GF reconstructs an interface with information about markers. It is accurate, but expensive and inefficient. The GB reconstructs an interface with crossings of the interface with the fixed grid edges. It is robust but dissipative. The LGB combines GF with GB. It uses the GF for interface propagation, and the GB for local interface reconstruction in bifurcating regions. The concept of the LGB is borrowed and combined with the CHI. We call the modified LGB the gradient-augmented LGB (GALGB).
where subscripts nm and ip denote variables on new marker and intermediate point, respectively. Finally all new markers are connected and then added to the interface. Since the tracked interface is reconstructed with eulerian datum, the interfacial topology changes are naturally and automatically handled.
A difficult case is used to examine the capability of capturing subgrid structures. The initial interface is a circle of radius 0.15 centered at (0.5, 0.75) in a unit square with grid size 100 × 100. The prescribed velocity field is defined as
This velocity field stretches the circle into a very thin, spiral filament, which is very difficult to capture on the fixed grids. For example, the area obtained by the LSM on a grid of size 64 × 64 completely disappears. In order to analyze the accuracy, the velocity field is reversed at t = 8 when the interface reaches a maximal deformation. Then the circle is recovered at t = 16. In this test, the reconstruction of the interface is an essential part of the scheme.
To further investigate grid dependence and accuracy, the values of considered grid cell spacing l are 1/50, 1/100, 1/200, and 1/400, respectively. Table
A symmetrical and laminar 2D droplet falling case is employed to evaluate the performance of the GAHM for incompressible multiphase flows. The computational domain is a rectangle with 100 × 200 uniform grid cells. The boundary condition is non-slip wall. The three numbers are Re = 100, Fr = 1, and We = 50. Density and viscosity ratios are 1.125 and 50. The interface is captured with the HPLSM or GAHM. An initial stationary circular droplet with a radius of 1 is centered at (4.5, 16.5) inside a fluid. Driven by the gravity, the droplet falls toward the bottom and induces several vortices. During the falling, it deforms and stretches into a dumbbell shape. The handle of the final shape at the time of 105 is much thinner than the size of a grid cell and it is very difficult to capture.
Based on a combination of the accurate FTM, smooth LSM and compact CHI, a gradient-augmented interface capturing method is presented for an incompressible two-phase flow. Gradient is of great help to reduce numerical error and to optimize computational stencil. Smooth and sharp level set contours are projected from the tracked interface. As a result, normals are approximated more easily and accurately. The local eulerian interface reconstruction eliminates marker redistribution and automatically handles interfacial topology changes. Test results demonstrate significant improvements on mass loss, highly accurate derivatives, clear subgrid structures, and low cost.
The GAHM is further coupled with an incompressible two-phase flow, Super CE/SE, for more practical evaluations. The updated solver maintains a sharp dumbbell-shaped interface, which ensures correct interfacial tension and physical characteristics. Results reveal that the GAHM is capable of accurately simulating the incompressible two phase flow with complex interface deformation.