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

A Multi-Domain Method for Computation of Wake Flows

Yasuo Osawa, Vinay Kalro, and Tayfun Tezduyar (AHPCRC-UM/Rice)

With the recent advances in flow simulation and modeling methods and the advanced parallel supercomputers, we now have powerful computational tools capable of simulating complex flow problems in applied fluid mechanics. Methods designed for a general class of challenging flow problems sometimes need to be re-designed and/or optimized for more specific classes of problems which pose their own specific challenges. One of these specific classes of problems is 3D simulation of unsteady wake flow in the far downstream of an object. The analysis of the far wake behavior is important to understand its effect on secondary objects. Examples of this class of applications are a small aircraft in the wake of a larger aircraft and a parachute crossing the wake flow of an aircraft. These are not easy to compute, because the two objects are separated by a large distance compared to the length scales of the objects, and the wake regions are rather long.

We have developed a multi-domain parallel computational method to carry out this class of simulations. With this method, we simulate unsteady flow past the primary object, unsteady wake flow in the long region between the two objects, and the influence of this wake flow on the secondary object. The base method is a finite element formulation with the streamline-upwind/Petrov-Galerkin [1] and pressure-stabilizing/Petrov-Galerkin [2] stabilizations. The overall formulation and the solution techniques have been implemented on parallel computing platforms by using the MPI programming environment. The parallel platforms used include the CRAY T3E 1200, recently acquired by the AHPCRC.

Figure 1. Arrangement of the objects and the subdomains.

The multi-domain computation approach is based on dividing the entire simulation domain into an ordered sequence of overlapping subdomains (see Figure 1).

In case of simulations involving objects in tandem, Subdomain-1 (SD-1) and Subdomain-3 (SD-3) would be used for computation of the unsteady flow around the first and second objects, respectively. Because these objects would have complex geometries, such as an aircraft or a parachute, these domains are discretized with unstructured meshes, and the flow solver is based on a general-purpose finite element implementation. Subdomain-2 (SD-2), which connects SD-1 and SD-3, would be used for computation of the unsteady wake flow generated by the first object. SD-2 is discretized with highly-refined structured meshes. A special-purpose finite element implementation for structured meshes can be optimized to yield much higher computational speeds compared to a general-purpose implementation. In fact, the computation over SD-2 can be accomplished by methods other than the finite element method, such as the spectral method, that might be more desirable for computations over structured meshes. Thus SD-2 serves as an accurate, low cost information channel from SD-1 to SD-3.

Example: Cylinder Wake Computation
3D numerical simulation of cylinder wake problem at Re=140 as far as 300 diameters (d) downstream of the cylinder was performed. This example verifies the multi-domain method and shows its capability. The second phase vortex shedding in the far wake of a cylinder at this Reynolds number was observed in laboratory experiments. The entire simulation domain is divided into three subdomains. SD-1 is for around the cylinder. SD-2 starts at 2d downstream from the center of the cylinder and goes until 155d. SD-3 starts at 150d and goes until 300d to capture the second phase vortex shedding. SD-2 and SD-3 have more than 4 million nodes each and hexahedral elements, and result in more than 17 million coupled nonlinear equations each. The computation over each wake subdomain requires, for every time step, about 36 seconds on 128 processors of the CRAY T3E 1200. The free-stream velocity is set to 1.0, and the time step size to 0.1. The simulation requires at least 6,000 time steps, since the domain is 600 units long (the diameter of the cylinder is 2 units).

Figure 2. Magnitude of the vorticity.

Figure 2 shows the magnitude of the vorticity at the horizontal center plane in these subdomains. SD-3 successfully captures the second phase vortex shedding, in addition to the Karman vortex street in the near wake. Figure 3 shows close-up view of SD-2 and SD-3. The spacing between the vortices in the second phase vortex shedding is about twice the spacing we see in the first phase, and this is in agreement with laboratory experiments.

Subdomain-2

Subdomain-3

Figure 3. Close-up view of the magnitude of the vorticity.

Example: Flow Past Wings in Tandem
In this 3D simulation, we compute unsteady flow past a leading large wing and two trailing small wings placed in the far wake of the larger one and affected by its wing tip vortices (see Figure 4). The Reynolds number is 1,000 for the leading wing. We assume symmetry with respect to the plane passing through the middle of the primary wing and the two trailing ones. SD-1 contains half of the primary wing and is handled with a general-purpose finite element implementation. SD-2 is the wake region and is handled with a special-purpose implementation and a structured mesh. SD-3 contains one of the trailing wings and is discretized with an unstructured mesh and handled by a general-purpose implementation. The leading and trailing wings both have rectangular shapes, an aspect ratio of 8, NACA0012 cross-section, and an angle of attack of 8.0 degrees. The trailing wing has half the cord-length of the leading wing.

Figure 4. Arrangement of the wings and subdomains.

Figure 5 shows the results for these three subdomains. For the leading wing, we see the tip vortices as well as vortex shedding from the center of the wing that quickly dissipates downstream because of the coarse mesh. For the wake region, we see the wing tip vortices and the vortex shedding. For the trailing wing, the left wing tip has more exposure to the wing tip vortices from the leading wing.

Figure 5. Iso-surfaces of the streamwise and spanwise components of the vorticity.

References
  1. A. Brooks and T. Hughes, Streamline-upwind/ Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations , Computer Methods in Applied Mechanics and Engineering, 32 (1982) 199-259.
  2. T. Tezduyar, Stabilized Finite Element Formulations for Incompressible Flow Computations , Advances in Applied Mechanics, 28 (1991) 1-44.