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The setup is shown in Fig. 8, where a square flag of length L initially lies flat in the xyplane, and the center of its leading edge is positioned at the origin. The leading edge is
pinned, and the other three edges are free boundaries. The uniform freestream velocity U0 is
in the x-direction.
In the present simulations, the computational domain for the flow is a rectangular box
extending from (−2L, −4L, −2L) to (8L, 4L, 2L). The simulations are performed on a 164 ×
124 × 120 nonuniform grid with points clustering around the flag and in the near-field wake.
The minimum spacing is x = y = z = 0.02L. Two solid models are used for the flag to
test the different modules in the structure solver. One is a plate model where the thickness is
assumed to be infinitesimal (labeled as Flag 1), and the other is a 3D body with finite
thickness h = 0.01L (labeled as Flag 2). For the flow, the two models are nearly equivalent
since the Reynolds number considered here is moderate. As in Section 4.3, to treat an
infinitely thin body in the current immersed-boundary flow solver, an artificial thickness is
introduced so that the treatment approach of such boundaries is essentially the same as that
for general 3D bodies [19].
For Flag 1, a triangular mesh of 3200 elements are used to discretize the flag surface, and for
Flag 2, a 20-node hexahedral mesh of 25 × 25 × 2 elements are used to discretize the flag
body. In both cases, the Reynolds number is Re = U0L/ = 200, the mass ratio is sh/( fL) =
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1.0, the bending rigidity is
A time step of t = 0.002L/U0 is used.
, and Poisson’s ratio is
s
= 0.4.
Fig. 9(a) shows the time histories of the transverse displacement of point B on the trailing
edge of the flag (as indicated in Fig. 8) computed from both flag models. For comparison,
the result by Huang and Sung [24] is also plotted in this figure. The peak-to-peak excursion
amplitude, A/L, and the Strouhal number defined as St = fL/U0, f being the oscillation
frequency, all quantities measured for point B, are tabulated in Table 2. It can be seen that
the two flag models produce very similar results, and both are very close to the model in
Huang and Sung [24]. Fig. 9(b, c) further shows that the drag and lift coefficients, which are
defined with
, are not very different for the two present flag models. Note that the
drag and lift coefficients are not available in Huang and Sung [24] but are included here for
future reference. These forces contain noticeable oscillations, which are due to the likely
noise from prediction of the thin plate. A refined method as recommended by Bathe [43]
will be considered in the future.
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Fig. 10 shows a snapshot of the vortical structures for Flag 1, which is characterized by the
isosurface using the Q-criterion defined by Hunt et al. [44]. Flag 2 has almost the same
vortical structures and is thus not shown here. In this figure, the vortex shed from the trailing
edge and the vorticies from the side edges are joined together to form a hairpin-type
structure, and this hairpin may be connected with the downstream hairpin with two
antennae. The flow pattern seen here is qualitatively consistent with the wake structure
shown in Huang and Sung [24].
4.6. FSI validation: bending of a flexible plate in cross flow
In the last validation test, we consider a flexible plate that is bent by a cross flow. The
simulation results will be compared directly with those from a recent experimental study by
Luhar and Nepf [45]. As pointed out in the reference, this problem has been used as a model
to study the deformation of aquatic plants by flow of water.
As shown in Fig. 11, an elastic plate of length L, width b, and thickness h is vertically placed
in a uniform flow of velocity U0. One end of the plate is clamp-mounted and the other end is
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Figure 12.
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Bending of a flexible plate in cross flow. (a) The present baseline simulation, and (b) an
image of the plate obtained from photogrammetry in Luhar and Nepf [45].
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Table 3
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Bending of a flexible plate in cross flow: quantitative comparison between the present result and the
experimental data [45].
CD
Dx/b
Dz/b
Present simulation
1.03
2.12
0.54
Luhar and Nepf [45]
1.15 (with 10% error)
2.14
0.59
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Table 4
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Bending of a flexible plate in cross flow: deflection and the drag coefficient in the absence of gravity and
buoyancy.
Re
CD
Dx/b
Dz/b
100
1.02
2.34
0.67
400
0.94
2.34
0.68
800
0.94
2.39
0.71
1200
0.93
2.41
0.73
1600
0.94
2.45
0.75
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