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AHPCRC Bulletin: Winter/Spring 1997 - Volume 7 Number 1-2

EDICT for Computation of Unsteady Flows with Interfaces

Tayfun Tezduyar, Shahrouz Aliabadi and Marek Behr (AHPCRC-UM)

When we solve an unsteady flow problem with interfaces, such as fluid-fluid interfaces or free-surface flows, the location of the interface is also an unknown and needs to be determined as part of the overall solution. Depending on the complexities of the interface, one can use a fixed mesh or moving mesh method. An interface-tracking method typically requires a moving mesh method, whereas an interface-capturing method can very efficiently be implemented with a fixed mesh approach.

The Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, a moving mesh method, was first introduced in 1990 by the first author and his team. In the DSD/SST method, the finite element formulation of the problem is written over its space-time domain, and therefore motion of the interfaces are taken into account automatically. The changes of the spatial domain are managed by updating the mesh with the combined approaches of moving the mesh and remeshing it whenever the mesh distortion becomes too high. Moving the mesh is accomplished by special methods for specific problems and by automatic methods for more general cases. In 3D simulations, reducing the frequency of automatic remeshing becomes essential because the cost of frequent 3D automatic mesh generation could become prohibitive. With parallel implementation of the DSD/SST formulation, we carried out 3D simulations for a number of problems, such as sloshing in a vibrating container, a gas stream impinging on a liquid, and flow past hydraulic structures.

When the interface is complex and very unsteady, especially in 3D simulations, reducing the frequency of remeshing becomes a difficult task. In such cases, interface-capturing methods with fixed meshes remain as practical alternatives. Typically these interface-capturing methods are more flexible but yield less accurate representation of the interface.

The Enhanced-Discretization Interface-Capturing Technique (EDICT) was first introduced by these authors in [1] and described in more detail in [2]. In EDICT, we start with the basic approach of a volume of fluid method. The Navier-Stokes equations are solved over a non-moving mesh. An interface function with two distinct values serves as a marker identifying the two fluids. The interface function is governed by an advection equation.

To increase the interface accuracy, we use functions corresponding to enhanced discretization at and near the interface. We start with a base mesh, Mesh-1, with the set of elements denoted by E1. A subset of E1, (E1)2 is identified as at or near the interface (see Figure 1). A more refined Mesh-2 is constructed by patching together second-level meshes generated over each element in (E1)2 (see Figure 2). The set (E1)2 will change from one time level to other, depending on which elements the interface is passing through. An element which is in (E1)2 now, may be out of it some time later, and come back in again some time after that. For each element in (E1)2 there will be a unique second-level mesh. If an automatic mesh generator is used to generate that, the mesh will be generated only once and stored, to be used later if that element needs a second-level mesh again. The functions for velocity and pressure will all have two components each: one coming from Mesh-1 and the second one coming from Mesh-2. The set of elements in Mesh-2 is denoted by E2.

To further increase the interface accuracy, we identify a subset of the elements in E2, (E2)3, as those at or very near the interface (see Figure 2). Mesh-3 is constructed by patching together third- level meshes generated over each element in (E2)3 (see Figure 3). The set (E2)3 will change from one time level to another, depending on where the interface is. The construction of Mesh-3 from Mesh-2 will be very similar to the construction of Mesh-2 from Mesh-1. The functions for the interface function will have three components, each coming from one of these three meshes.

Mesh-2 and Mesh-3 will not be re-defined every time step but frequently enough to keep the interface in zones covered with these higher level meshes. In parallel implementation, re-defining these meshes will involve some load balancing cost, but we do not expect this to be a major cost. We will of course have the option of not having a third-level mesh for the interface function. We also have the option of having zones covered by Mesh-2 and Mesh-3 to match. An advantage in keeping them non-matching, however, is that we can keep Mesh-2 wider than Mesh-3, target keeping the interface within Mesh-3, and solve for the interface function only over the part of the computational domain covered by Mesh-2.

The EDICT has already been implemented on the shared-memory parallel computing platform of the SGI multi-processor systems, and this was reported in [1, 2].

Example: 2D sloshing in a container

A container with dimensions 10 cm 7.5 cm is filled 2/3 with water and 1/3 with air. It is suddenly subjected to the gravitational acceleration (g = 9.8 m/s) and a horizontal acceleration of magnitude 0.2g.

We first compute this problem with the DSD/SST formulation using a highly-refined moving mesh. The mesh has 30,000 quadrilateral elements and 30,351 nodes. We will refer to this solution, obtained with the Interface-Tracking Technique, as Solution-IT.

Next, we compute the problem with the Interface-Capturing Technique using triangular meshes. The base mesh, Mesh-1, consists of 13,739 elements and 7,021 nodes. Mesh-2 is obtained by subdividing each element in (E1)2 into four elements. Mesh-3 is obtained by subdividing each element in (E2)3 into four elements. We set (E2)3 = E2; this means that we have the Mesh-2 and Mesh-3 zones matching. We carry out the computations with a time step size of 0.005 s. We redefine (E1)2 at every five time steps, with all elements in E1 within a distance of 0.5 cm from the interface declared to be in (E1)2.

Solution-1 is obtained with the base discretization, where all trial and weighting functions come only from Mesh-1. Solution-2 is obtained with the EDICT, where all functions come from Mesh-1 Mesh-2. Solution-3 is obtained by using a more enhanced discretization, where the functions for velocity and pressure come from Mesh-1 Mesh-2, and for the interface function from Mesh-1 Mesh-2 Mesh-3.

Figure 4 shows Mesh-1 together with Mesh-2 and Mesh-3 (both shown on top of Mesh-1) at t = 0.5 s. Refinement of 2,221 elements of Mesh-1 results in Mesh-2 having 8,884 elements and Mesh-3 having 35,536 elements. The graphs in Figure 5 show the time histories of the horizontal forces exerted on the container for all four solutions. We assume that the target solution is Solution-IT, obtained with the Interface-Tracking Technique, using a highly-refined mesh. It is clear from these graphs that Solution-1 has significant frequency and amplitude errors. Solution-2 is superior to Solution-1, but it is Solution-3 that is in best agreement with Solution-IT.

Example: Axisymmetric filling/impact

We have a circular cylinder with both the radius and height of 10 cm. The lower half of the cylinder is filled with a liquid and the upper half with air. At t = 0.0 s we start injecting the same liquid through a circular section with radius 2.15 cm and positioned concentrically at the top of the cylinder. The injection stream has a uniform flow speed of 1.0 m/s.

A finite element mesh with triangular elements is used. Mesh-1 consists of 30,000 elements and 15,251 nodes. Mesh-2 and Mesh-3 are generated in the same way as they were generated in the previous example. However, this time, at every redefinition of (E1)2, all elements in E1 within a distance of 0.7 cm from the interface are declared to be in (E1)2. The solution is obtained with the EDICT, where the trial and weighting functions for velocity and pressure come from Mesh-1 Mesh-2, and for the interface function from Mesh-1 Mesh-2 Mesh-3.

Figure 6 shows Mesh-2 (on top of Mesh-1) at t = 0.15 s. Refinement of 8,473 elements of Mesh- 1 results in Mesh-2 having 33,892 elements and Mesh-3 having 135,568 elements. Figure 7 shows a sequence of air-liquid interactions seen at different instants during the simulation of this problem. The pictures show the injection stream impacting the still liquid, formation of surface waves, and entrapment of air in the liquid.

References

[1] T. Tezduyar, S. Aliabadi and M. Behr, "Enhanced-Discretization Interface-Capturing Technique," to appear in Proceedings of the ISAC '97 High Performance Computing on Multiphase Flows, Tokyo.
[2] T. Tezduyar, S. Aliabadi and M. Behr, "Parallel Finite Element Computing Methods for Unsteady Flows with Interfaces," to appear in Computational Fluid Dynamics Review 1997 (eds. M. Hafez and K. Oshima).