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Fall 1997 - Volume 7 Number 4
Winter/Spring 1998 - Volume 8 Number 1-2

The Shear Slip Mesh Update Method

Marek Behr and Tayfun Tezduyar (AHPCRC UM/Rice)

Flows with moving boundaries and interfaces can be encountered in many real-world fluid mechanics applications, including fluid-object and fluid-structure interactions and flows with moving mechanical components. The Deformable-Spatial-Domain/Stabilized Space-Time (DSD/ SST) formulation was introduced earlier [1,2] for this class of flow problems. In the DSD/SST formulation, the finite element formulation of a problem is written over its associated space-time domain, and this automatically takes into account the changes in the spatial domain. The finite element functions are continuous in space and discontinuous in time. The discontinuity in time allows the computations to be carried out one space-time slab at a time. This way we are not adding to the computations the full burden of a fourth dimension.

As the spatial domain changes, the finite element mesh needs to be updated to accommodate the change. In general one can count on using automatic or special mesh moving. This is the approach most frequently used so far in our DSD/SST computations. When needed but not too frequently the mesh movement can be followed by remeshing, defined as creating a new finite element mesh with a new node connectivity. Frequent remeshing with automatic mesh generators can be prohibitively expensive in 3D simulations. Remeshing also requires projection of the solution from the old mesh to the new one, and this introduces projection errors. In some cases the frequency of remeshing can be reduced substantially. Sometimes remeshing can be totally eliminated. Several cases in this category have been successfully simulated using the DSD/SST approach. In all these cases, the changes in the spatial domain can be characterized as small or medium. At such levels, we do not face a major problem in terms of the remeshing cost.

For flow problems with spinning geometries, on the other hand, the mesh-update requirements become substantially different. The Shear-Slip Mesh Update Method (SSMUM), introduced previously [3] was developed for this purpose. The DSD/SST formulation, enhanced with the SSMUM, can be very effectively applied to modeling of geometries that undergo rotations or straight-line translations. This is accomplished by letting, at each time step, the elements in a thin zone undergo shear deformation. This zone is remeshed frequently via regeneration of element connectivity. In this approach the nodal positions for the old (deformed) and new (good-quality) meshes are matched, so that projection is not necessary. Since only a small part of the overall connectivity is being regenerated, the computational cost associated with the regeneration is kept low. Application of this method to a 3D computation involving translating geometries and structured meshes was presented by the second author and his coworkers. This approach was used in 3D simulation of two high-speed trains passing each other in a tunnel. Here we focus on computations involving spinning geometries and mixed structured/unstructured meshes.

Mesh Update Method
Figure 1. The SSMUM concept. Regions of deforming elements are shown in gray.
The SSMUM strategy is based on special-purpose mesh designs that combine regions of rigid non-deforming elements with layers of shear-absorbing deforming elements. A translating object is embedded in a strip (in 2D) or a tube (in 3D) of rigid elements that move glued to that object. Similarly, a rotating object is embedded in a disk of rigid elements that rotate glued to that object. These non-deforming regions are immersed in another set of non-deforming elements spanning the exterior boundaries, as shown in Figure 1.

Figure 2. Shear-absorbing element layer: a 2D view.
A single-element shear-slip layer is illustrated in Figure 2 in the context of a 2D spatial domain with a translating object. The quadrilateral elements on the right, coded blue, belong to the stationary exterior mesh, while the elements on the left, coded purple, belong to the translating mesh which follows the moving object. The direction of the movement is upward. Evolution of a single shear-absorbing element during the two time steps illustrated here is highlighted in red. The spatial outline of that element deforms during the first step, and then re-connects to new purple nodes, which moved into proper positions. This process is carried out repeatedly, with changes in element connectivity, but with no projection. Note that the spatial mesh at the lower level of the second time step is of the same quality as the one at the lower level of the first time step.

It may not be necessary to re-connect the elements in the shear-slip layer at every time step. Choosing a smaller time step, for a given mesh resolution and rotational or translational speed, is still possible, leading to lower frequency of re-connects. Conversely, selecting a wider shear-slip layer, more than one element thick, will allow the use of larger time steps. These possibilities, although slightly more complicated in implementation, lend considerable flexibility to the SSMUM.

Example: 2D Flow Past Two Counter-rotating Square Cylinders
Figure 3. 2D flow past two counter-rotating square cylinders: computational domain.
We begin with this test computation to illustrate the capability of the method to handle flows past multiple rotating objects.

The arrangement of the squares is shown in Figure 3. Both squares have a side dimension of 2.0 units. The top and bottom squares are rotating counter-clockwise and clockwise, respectively, each with a rotational velocity magnitude of 0.25 P. The axes of rotation of the two objects are 4.0 units apart and are placed symmetrically about the centerline of the domain. The shear-slip layers surrounding each cylinder have an inner radius of 1.5 units and are 0.05 units thick.

Figure 4. 2D flow past two counter-rotating square cylinders: vorticity field at t = 125.0, 126.0 and 127.0.
The mesh consists of 31,928 space-time nodes and 31,492 triangular elements, and each shear-slip layer is one element thick and has 160 segments in the circumferential direction. Joining the moving and fixed regions of the mesh, it shears during each time step and, at the end of a time step, re-connects to the new nodes belonging to the rotating interior disk. A free-stream velocity of 1.0 is imposed. The Reynolds number based on the free-stream velocity and the size of the squares is 400.

Figure 4 shows, at three equally-spaced instants spanning one full revolution, the vorticity field in the vicinity of the squares. We observe a regular vortex shedding from the corners of the cylinders as they move opposite to the flow direction and enter the gap, and also a lower frequency shedding cor-responding to flow past a compound object. The effect of the layer of deforming elements is hardly visible in the solution. This computation was carried out on an IBM SP2.

Example: 3D Flow Past a Rotating Propeller
Figure 5. 3D flow past a rotating propeller: selected boundaries of the initial mesh (left),and close-up view of the inner boundary of the shear-slip layer and the propeller (right).
Extending the method to 3D simulations, we compute here a high-Reynolds number flow past a rotating propeller. The propeller is rotating in clockwise direction as seen from the inflow side, with a rotational velocity magnitude of 0.125 P. Although a similar simulation can be performed by using a rotating frame of reference and a non-deforming mesh, we view our computation as a first step towards modeling of more complex geometries in which the rotating part interacts with other stationary or moving objects.

The propeller consists of a conical afterbody, a spherical tip, and seven blades with radius of 2.9. The overall length of the propeller fairing is 5.0. The shear-slip layer is an axisymmetric, closed shell with interior radii 3.5 and 1.5 and a uniform thickness of 0.1, co-axial with the propeller fairing.

Figure 6. 3D flow past a rotating propeller: pressure and iso-surface p = -0.025 at t = 51.2.
The mesh consists of 308,698 space-time nodes and 948,420 tetrahedral elements, and is shown in Figure 5. The shear-slip layer, which is one element thick, has 64 segments in the circumferential direction and 46 segments in the remaining direction. This layer goes through shear deformation during each time step, and at the end of a time step re-connects to the new nodes belonging to the rotating interior disk. The unstructured meshes in both the inner (rotating) and the outer (stationary) rigid regions of the domain were generated using an automatic mesh generator, while the structured mesh, which fills the shear-slip region, was generated manually.

A free-stream velocity with magnitude 1.0 is imposed, leading to a Reynolds number of approximately 1106 based on the propeller diameter.

Figure 6 shows, at t = 51.2, the pressure distribution on the propeller and the pressure iso-surface corresponding to p = -0.025. Blade wakes and the core of a ring vortex are clearly visible. This computation was carried out on the CRAY T3E 1200, acquired recently by the AHPCRC.

References
  1. T. Tezduyar, M. Behr and J. Liou, A new strategy for finite element computations involving moving boundaries and interfaces the DSD/ST Procedure: I. The concept and the preliminary tests , Computer Methods in Applied Mechanics and Engineering, 94 (1992) 339 351.
  2. T. Tezduyar, M. Behr, S. Mittal and J. Liou, A new strategy for finite element computations involving moving boundaries and interfaces the DSD/ST Procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders , Computer Methods in Applied Mechanics and Engineering, 94 (1992) 353 371.
  3. M. Behr and T. Tezduyar, A note on Shear-Slip Mesh Update Method , in Lecture Notes of the Workshop on Parallel Computing in Applied Fluid Mechanics, Associazione Amici Scuola Normale Superiore, Pisa, Italy (1997).
  4. M. Behr and T. Tezduyar, The Shear-Slip Mesh Update Method , to appear in Computer Methods in Applied Mechanics and Engineering, (1998).