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Treating Curvilinear and Moving Surfaces in
a Cartesian Framework
One of the main disadvantages of the use of
a marker and cell approach as opposed to the use of a boundary-fitted
coordinate approach is the fact that curvilinear walls cannot
be represented precisely. To date, the ELMMC method has made use
of rectangular cells to represent curved solid boundaries, and
in the process of doing so, wall effects have largely been approximated.
Methods that rely on a curvilinear grid mapping approach, however,
are not flexible enough to handle wave breaking and large-scale
solid motion. In order to avoid grid skewness and other sources
of error, every time the domain shape changes, the grid mapping
has to be redone, and the mapping metrics and Jacobian have to
be recalculated. These calculations are very costly in a transient
solution, and the interpolations necessitated by the continually
changing grid can introduce undesirable errors where the solution
is most highly sought.
The idea then arose that if free surfaces undergoing
severe deformations could be so well handled with surface markers,
then surely solid walls could be handled in a similar fashion!
This has lead the SMU team to introduce "boundary markers"
to represent solid boundaries, both stationary and moving. The
concept of boundary markers builds on the successful handling
of free surfaces with surface markers, and preserves the important
capabilities of the original method in handling severely contorted
free surfaces. Velocity and pressure boundary conditions are prescribed
accurately on solid boundaries by ensuring that these conditions
are satisfied on the surface of a given boundary as opposed to
on the center of the cells that make up the boundary. Under the
current NSF grant, a proof of concept has been successfully completed
by investigating well known flows, such as flow around a cylinder
and flow around a NACA 0018 hydrofoil. It has been shown that
the new method can successfully handle the difficult task of predicting
the separation point around a bluff body, providing accurate results
and giving confidence in the method's success.
An example simulation for flow around a cylinder
As fundamental a fluid dynamics problem as flow
around a cylinder is, it remains an important test case for demonstrating
the accuracy of a simulation technique, especially when it comes
to the difficult task of predicting the separation point. In the
case considered here, the Reynolds number is 1,800 based on the
diameter of the cylinder, and hence the flow is laminar. The results
are presented in Figure 7. Two animations are
provided, one involving the total velocity vectors and the other
the pressure contours. Than, two plots showing the corresponding
pressure coefficient around the cylinder and the time history
of the lift force, respectualy. The vortex shedding behind the
cylinder is well simulated by the numerical technique, as evidenced
in the two animations. More importantly, the method predicts that
the separation point is at 82º (minimum Cp value in last
plot in Figure 7), which matches perfectly with
the established value for laminar flow! The transient results
for the lift force provide an opportunity to check the temporal
accuracy of the method by calculating the Strouhal number of shedding.
The value calculated from the simulation is St=0.212, which compares
very favorably with published results. These basic calculations
and comparisons provide confidence in the potential of the new
unmapped approach to represent curvilinear boundaries in the context
of the Cartesian marker and micro cell technique.
Figure 7: Flow
around a cylinder
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An example simulation of solid body motion induced
by hydrodynamic flow
The transport of rigid bodies (e.g., rocks,
cars, tree trunks, debris) is essential to the simulation of realistic
tsunami scenarios. Surveys of past tsunamis have suggested that
the debris generated from the early stages of the wave impact
become water-born projectiles, thus increasing dramatically the
destructive potential of the wave. Indeed, debris from the first
row of houses carried within the wave can proceed to destroy the
remainder of the town in its path. Similarly to the methods for
treating the free surface and curvilinear solid boundaries, a
method has been developed to handle moving boundaries. This capability
will be important in the investigations of moving solids carried
by tsunamis as well as of scouring effects behind structures submerged
by incident and reflected tsunamis. To illustrate the potential
of the new method, a simulation is presented for the induced motion
of a hinged ellipsoidal foil in response to flow in a horizontal
channel (Figure 8). The domain is 10 cm long
and 6 cm wide. The foil is 1.8 cm long and 0.6 cm wide, and is
hinged 0.2 cm from its upper tip. The flow enters from the left
side with a uniform velocity of 8 cm/s and exits on the right.
The flow is accelerated from rest at t=0 s to t=1.5 s with the
foil held normal to the channel walls. Once released, the foil
rotates due to the pressure differences between its front and
wake. As the foil moves, a trace of the shedding in its wake is
generated as a result of the phase lag between the position of
the body and the vortex generation in the flow. The foil does
not comply precisely with the fluid motion, but rather interacts
with it, both affecting the flow and being affected by it. As
expected, the final alignment of the foil in the absence of gravity
is parallel to the walls of the channel. This representative simulation
provides a proof of concept of the novel unmapped approach. It
is anticipated that his enabling capability will prove to be invaluable
in the modeling of solidfluid interactions and solid transport
over large distances.
Figure 8 Flow around a cylinder
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