For more information:
tezduyar@gmail.com
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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.
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| 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
- 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.
- T. Tezduyar,
Stabilized Finite Element Formulations for Incompressible Flow Computations
, Advances in Applied Mechanics, 28 (1991) 1-44.
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