| |
Flow Through Porous Media
In order to be relevant to actual
tsunami mitigation, the modeling of tsunami interactions on a
shore-line must account for the local topography, including the
sand, rocks, and vegetation. The local topography may be natural
or man-made, but in either case, the need exists to determine
its effects on the behavior of the tsunami and the overall pattern
of destruction. There is also the real potential of altering the
local topography to minimize damage to life and property. On the
one hand, the use of dikes, breakwaters, and even fortified walls
is not uncommon in tsunami-prone regions. On the other hand, surveys
of recent tsunami-stricken regions have suggested that the local
vegetation and natural topography can play a significant role
in altering the flow behavior and consequences of a tsunami. There
have even been discussions in the tsunami scientific community
of the potential use of "tsunami forests" in high-risk
regions. Since it is not computationally feasible to account for
specific units of trees and rocks, it is more practical (and usual)
to consider simulating breakwaters, forests, and ground vegetation
as regions of porous media of given porosity and permeability.
However, accounting for these types of obstructions in the simulations
requires that the numerical technique be capable of handling porous
media in a general manner.
For this purpose, the SMU team has set out to develop this capability
within the framework of their ELMMC-3D technique. Many difficulties
had to be overcome because research in porous media flow has been
essentially limited to saturated media, has not addressed the
issue of wave impact with a dry porous medium, and has not yet
established the range of applicability of the governing equations.
To address these issues, small-scale experiments have been conducted
at SMU to guide the theoretical developments of the governing
equations and the impact boundary condition.
Simulations of flow through a porous dike compared
to flow through a solid dike
In the two simulations that follow, a single
large wave is generated by releasing a water mass from behind
an infinitesimally thin gate. The purpose of this comparison is
to investigate the effects of a porous structure on the progress
of a large wave, with the specific intent of answering the question:
Could porous media, such as breakwaters and forests, be used effectively
to mitigate the impact force of a tsunami? These examples also
demonstrate the capability of the ELMMC method as a potentially
powerful analysis tool that can be used in the assessment and
design of mitigation schemes. In both simulations, the domain
is 187.5 cm long, 100 cm wide, and 125 cm tall, and is discretized
with cubical macro cells of dimension 3.125 cm. The resulting
computational domain is composed of 60 x 32 x 40 cells. Surface
cells are subdivided into 27 micro cells, three in each spatial
direction. The water initially contained behind the gate occupies
20 x 32 x 15 cells. The dike, which is 6 x 12 x 10 cells, is placed
20 cells downstream of the gate and centered between the sidewalls
of the tank. A no-slip boundary condition is imposed on all solid
walls. The dike in Figure 3 is impermeable, while
the dike in Figure 4 is made of aluminum foam
with a porosity j=0.87, permeability
K=5x10-6 m2, and Forchheimer coefficient
CF=0.316.
Figure 3: Simulation of the impact of
a
single large wave with a solid dike
Figure 4: Wave impact with a porous dike
(selected walls removed to facilitate visualization)
If you get an error message or need help
with '.rm' files, please refer to the help
section.
At time t=0 s, the water mass is instantaneously released from
behind the gate, and begins to flow toward the dike under the
influence of gravity. In the case of the impermeable dike, the
water builds up upstream of the dike, topples over it, and splashes
violently over and behind the dike. In the porous case, however,
water flows both around the dike and through it, and the large
wave that over toppled the impermeable dike never develops.
Because of the finite permeability of the dike, the velocity
of the fluid going through the dike is smaller than that of
the flow around it. Furthermore, because of the pressure field,
a lateral flow develops from the clear fluid region toward the
porous dike. The right and left streams have coalesced with
the central stream, and are rising up the downstream wall. The
waves that build up on the sidewalls are seen to interact in
the last frames, creating a complex wave structure that has
the potential to scour the base of either of the two dikes.
A delay is observed between the height of the water around and
within the dike. The delay is attributable to the damping effect
generated by the viscous and pressure drag within the porous
matrix. The drag effect also manifests itself in the smoothness
of the free surface inside the dike, indicating that the large
eddies surrounding the structure have a small influence on the
flow inside it. By comparing the two solutions, it is clear
that the porous dike diverts the flow away from a sheltered
wake region, prevents the formation of the overtopping central
wave, and significantly lessens the impact velocities.
Modeling impact of a fluid wave
front with a porous medium
The relevance of the simulation
of flow though dry porous media is heavily dependent on the
accuracy of the mathematical boundary condition used to enforce
the impact. As previously noted, this research area has not
been addressed in the literature. The SMU team has just developed
an impact boundary condition, which has made it possible to
correctly simulate the interaction of waves with dry porous
media. In this simulation, the computational domain consists
of a rectangular reservoir with dimensions 0.75 m x 0.75 m x
0.9 m, discretized with cubical macro cells of dimension 0.03
m. The square inlet is 0.15 m on each side, is centered along
the back wall at an elevation of 0.675 m from the floor, and
has a velocity of 1.6 m/s. A horizontal porous plate with thickness
0.12 m is located at 0.15 m from the floor of the tank. The
physical properties of the porous material are identical to
those in the previous problem. The corespondent animation for
the numerical simulation is presented in Figure
5. After impact, part of the jet goes through the porous
matrix, while the rest spreads along the top. A splash-up appears
when the flow reaches the front solid wall of the tank. A very
complex flow develops through the porous plate until the lower
part of the tank fills up with water. When the water level rises
well above the top of the porous layer, the damping effect of
the porous matrix diminishes to the point that sloshing develops
and free surface oscillations driven by the inlet flow become
evident once again.
Figure5: Modeling free surface impact
with a porous layer
If you get an error message or need help
with '.rm' files, please refer to the help
section.
|
|
