A Method to Fabricate Kelp Models with Complex
Morphology to Study the Effect on Drag and Blade Motion
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Fryer, M., D. Terwagne, P. M. Reis, and H. Nepf. “Fabrication of
Flexible Blade Models from a Silicone-Based Polymer to Test the
Effect of Surface Corrugations on Drag and Blade Motion.”
Limnol. Oceanogr. Methods 13, no. 11 (July 16, 2015): 630–639.
As Published
http://dx.doi.org/10.1002/lom3.10053
Publisher
American Society of Limnology and Oceanography, Inc.
Version
Original manuscript
Accessed
Thu May 26 07:37:12 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/101872
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Detailed Terms
http://creativecommons.org/licenses/by-nc-sa/4.0/
Limnology and Oceanography: Methods
A Method to Fabricate Kelp Models with Complex
Morphology to study the Effect on Drag and Blade Motion
Journal:
Limnology and Oceanography: Methods
r
Fo
Manuscript ID:
LOM-15-02-0015
Wiley - Manuscript type:
New Methods
Date Submitted by the Author:
06-Feb-2015
Fryer, Marisa; Massachusetts Institute of Technology, Department of Civil
and Environmental Engineering
Terwagne, Denis; Massachusetts Institute of Technology, Department of
Civil and Environmental Engineering; Université Libre de Bruxelles, Faculté
des Sciences
Reis, Pedro; Massachusetts Institute of Technology, Department of Civil
and Environmental Engineering
Nepf, Heidi; Massachusetts Institute of Technology, Department of Civil
and Environmental Engineering
Keywords:
ew
vi
Re
Complete List of Authors:
kelp, morphology, drag, plant-flow-interaction
ly
On
Page 1 of 26
1
2
3
4
5
6
7
8
9
A Method to Fabricate Kelp Models with Complex
Morphology to study the Effect on Drag and Blade
Motion
M. Fryer, D.Terwagne, P.M.Reis and H.Nepf
r
Fo
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology
10
(MIT), Cambridge, Massachusetts
11
Corresponding author: mfryer@mit.edu
12
Running Head: Fabrication of and Drag on Corrugated Blades
ew
vi
13
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
1
Limnology and Oceanography: Methods
1
Acknowledgements
2
This material is based upon work supported by the National Science Foundation under grant
3
EAR-1140970.
4
r
Fo
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
2
Page 2 of 26
Page 3 of 26
1
Abstract
2
Macrocystis blades develop longitudinal corrugations in regions with strong current and
3
wave action. This study examined the effect of corrugations on blade motion and blade drag by
4
constructing flexible blades with different corrugation amplitude and a control blade with no
5
corrugation. The models were designed to be dynamically and geometrically similar to natural
6
blades. Acrylic molds were etched using a laser cutter and filled with a silicone-based polymer to
7
create flexible model blades with sinusoidal corrugations. The corrugated and flat model blades
8
were tested in a water channel using drag force measurements and video analysis. The
9
corrugated blades experienced a drag reduction of up to 60% compared to the flat blade.
r
Fo
Re
10
Additionally, the corrugated models exhibited smaller motion, as quantified by the maximum
11
vertical displacement. The reduction in drag may explain why corrugations are observed in
12
exposed regions of high current and wave action, where a reduction in drag provides important
13
protection against breakage.
ew
ly
On
14
vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
3
Limnology and Oceanography: Methods
1
Introduction
2
Kelp blades alter their morphology to adapt to changes in their environment, such as
3
variations in nutrient availability and hydrodynamic conditions, with the morphological changes
4
either enhancing nutrient availability or reducing hydrodynamic drag (Koehl et al. 2008). This
5
process is known as phenotype plasticity. For example, in exposed regions with strong current
6
and wave action Macrocystis blades are thicker and have longitudinal corrugations (Figure 1),
7
compared to sheltered regions, where the blades are thinner and exhibit little or no corrugations
8
(Hurd and Pilditch 2011, summarized here in Table 1).
r
Fo
9
The longitudinal corrugations appear in regions of high velocity which are less likely to
Re
10
be nutrient limited (e.g. Gerard 1982 a, b), so the purpose of the corrugations is unlikely to be
11
associated with nutrient flux. Rominger and Nepf (2014) hypothesized that the corrugations
12
develop to reduce hydrodynamic drag. Previous studies of corrugation in metal sheets have
13
shown that corrugation enhances rigidity (Briassoulis 1986). Rominger and Nepf (2014) showed
14
that in some flow conditions, increasing blade rigidity significantly reduces drag by limiting the
15
blade motion that occurs in response to flow oscillations (e.g. turbulence or waves). The degree
16
of blade motion, in response to flow, can be characterized by the following non-dimensional
17
force ratio of blade rigidity (resistance to bending) to fluid drag (Michelin et al. 2008):
18
(1)
19
where E is the Young’s modulus, I is the area moment of inertia, 𝜌𝑓 is the density of the fluid,
20
𝑈∞ is the streamwise velocity, and b, h and l are the width, thickness and length of the blade,
21
respectively.
ew
ly
On
22
vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
𝜂=𝜌
𝐸𝐼
2 3
𝑓 𝑏𝑈∞ 𝑙
,
Rominger and Nepf (2014) created six flat blade models with a range of η between 10−6
4
Page 4 of 26
Page 5 of 26
1
and 10 by varying the material (high and low density polyethylene, and aluminum) and the
2
model blade thickness (h). They found that below 𝜂 = 10−4, both the mean and instantaneous
3
drag forces on the model blade increased rapidly with decreasing non-dimensional rigidity (η).
4
The instantaneous peaks in drag, which can tear or dislodge blades, were associated with inertial
5
forces created by the large-amplitude flapping motions of the blade. Rominger and Nepf (2014)
6
also showed that the range of η values over which blade drag is most sensitive to blade rigidity
7
(𝜂 < 10−4) corresponds to the range of η values exhibited in real kelp blades in coastal
8
environments. Therefore, for real blades in the field, changes in morphology that increase blade
9
rigidity are expected to reduce drag.
r
Fo
Re
10
In this study, we test the drag-reducing effect of corrugations, as hypothesized in
11
Rominger and Nepf (2014). A method was developed to fabricate a set of dynamically- and
12
geometrically-scaled model blades with different degrees of realistic corrugation using etched
13
acrylic molds. We measured the drag on each model blade when exposed to the same flow
14
conditions and used video analysis to observe the effects of corrugation on the motion of the
15
blade.
16
Methods
17
Fabrication
ew
ly
On
18
vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
We designed and fabricated three model kelp blades: one flat blade and two blades with
19
sinusoidal corrugations running parallel to the streamwise length of the blade, with different
20
amplitudes. The model blades were made from a silicone-based polymer and cast using laser-cut
21
acrylic molds (more details provided below). The three models had the same length l, width b,
22
and sheet thickness h, and were fabricated from the same material, so that they had the same
23
value of E. The design parameters were chosen using three constructs: First, working under the
5
Limnology and Oceanography: Methods
1
hypothesis that kelp blades develop corrugated morphologies in regions where changes in blade
2
rigidity have the greatest effect on drag force, each blade was designed to be within the critical
3
range 𝜂 < 10−4. This, according Rominger and Nepf (2014), is the range over which variations
4
in rigidity have the most significant effect on drag. Second, the ratio of blade thickness to
5
corrugation wavelength and amplitude should mimic what has been observed in actual kelp
6
blades (Figure 1 and Table 1). Third, the model parameters were limited by the resolution and
7
total size of the fabrication tools. Specifically, the molds were fabricated using a laser cutter with
8
a bed size of 46 cm x 81 cm and an x-y resolution of 0.1 mm.
r
Fo
9
The flat blade can be approximated as a rectangular beam, for which the area moment of
Re
10
inertia about the horizontal axis perpendicular to blade length (y-axis in Figure 2) is
11
(2)
12
For the corrugated models, the area moment of inertia can be approximated as
13
(3)
14
where a is the amplitude of the corrugation, and 𝜆 is the wavelength of corrugation, as shown in
15
Figure 2b (Lau, 1981). The blades were made from a silicone-based elastomer, vinylpolysiloxane
16
(VPS), which is available in a range of elastic moduli. Specifically, we used a VPS (Zhermack
17
Elite Double 8) with a Young’s modulus of E = 0.23±0.01 MPa. This material is prepared by
18
mixing equal parts base and catalyst in liquid form, after which polymerization occurs in
19
approximately 20 minutes. To reach a non-dimensional force ratio (η) in the scaled-down model
20
blade that was comparable to real kelp blades, the elastic modulus (E) was intentionally chosen
21
to be less than that of real blades (E = 5 MPA, Table 1). The VPS has a density of 1024 ± 20
22
kg/m^3, so that the models were somewhat buoyant, which is consistent with real blades (ρ =
23
1040 kgm-3, Hale 2001).
𝐼=
𝑏ℎ3
12
𝐼=
ew
vi
𝑏𝑎2 ℎ
2
(1 +
𝜋 2 𝑎2
2𝜆2
)
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
6
Page 6 of 26
Page 7 of 26
1
The flat model was designed to have a value of η ≈ 10−5 in a current of 𝑈∞ = 20 cm/s
2
(see eqn. 1). In order to meet that target, we set length l = 30.0±0.1 cm, width b = 3.0±0.1 cm,
3
and thickness h = 0.75±0.01 mm, with the uncertainty representing the measurement uncertainty
4
of the completed blade. These dimensions were held constant for all three blades. For geometric
5
similarity, the corrugated model blades were constructed to have similar ratios of thickness to
6
amplitude and wavelength of corrugation (h:a:λ) as real blades from exposed sites. We combined
7
measurements from Hurd and Pilditch (2011, shown in Table 1) and measurements from the
8
Macrocystis blades shown in Figure 1(h = 0.45 mm, a = 1.13 mm, λ = 5.0 mm), which yielded
9
2.5 < a/h < 3 and 6 < λ/h < 11. The first corrugated model was designed with amplitude a = 2.7h
r
Fo
Re
10
(2.0±0.05 mm), which is in the middle of the range observed in the field. A second corrugated
11
model was chosen with corrugation amplitude a = 2h (1.5±0.05 mm) to produce a dimensionless
12
rigidity half way between the flat and fully corrugated models. The wavelength was set at λ = 8h
13
(= 6.0±0.05 mm) for both corrugated blades, which is also in the middle of the observed range.
14
The reported uncertainty corresponds to measurements of the final blade, which contributes to
15
the uncertainty in η. For the flat blade, η = 7.8±0.3x10-6, calculated using eqs. 1 and 2. For the
16
corrugated blades, η = 2.4±0.2x10-4 and η = 5.0±0.4x10-4, for a = 1.5 mm and 2 mm,
17
respectively, calculated from eqs. 1 and 3.
ew
ly
On
18
vi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
The templates for laser cutting the blade molds were designed in Matlab. Each mold
19
consisted of two acrylic plates out of which a sinusoidal topography was etched. Given the x-y
20
resolution of the laser cutter (0.1 mm) and the wavelength λ = 6.0 mm, the corrugations for the
21
models were created as discrete curves with 60 steps per wavelength. Visual inspection of the
22
Macrocystis blade (Figure 1) indicates that the blade thickness is uniform when measured
23
perpendicular to the local blade surface (red line in Figure 3), and this characteristic was
7
Limnology and Oceanography: Methods
1
preserved in the model blade. The surface arcs of the mold were constructed such that the perfect
2
sinusoid was at the center of the blade (dashed black curve in Figure 3). The upper and lower
3
surfaces of the blade are one-half the thickness (h) above or below the sinusoidal curve,
4
measured in the direction perpendicular to the center curve (e.g. red line in Figure 3) at each
5
discrete point.
6
The surface curves (blue lines in Figure 3) were converted to values ranging from 0 to
7
255, and the discrete vectors were extended to create 2D matrices, which were finally converted
8
to grayscale images. The grayscale images were used as templates for the laser cutter. The power
9
and speed of the laser were set to etch a depth of twice the amplitude for black pixels. The laser
r
Fo
10
then adjusted to a fraction of that power corresponding to the grayscale value (between 0 and
11
255) for non-black pixels. Photographs of an example of the resulting template and acrylic plates
12
are shown in figure 4.
ew
vi
13
Re
The acrylic molds were cut leaving 1.5 cm of flat space along three sides of the blade
14
area. This space was used to lay out metal spacers of the desired blade thickness (h = 0.75 mm).
15
The bottom mold was filled with the VPS mixture (Figure 5a), after which the top surface was
16
screwed into place (Figure 5b) and held down with weights while the silicone-based polymer
17
cured (requiring 20 minutes minimum). Once cured, the excess polymer was cut away from the
18
edges of the model blade. Separate acrylic molds were created and used for the two different
19
corrugated models, and uncut plates of acrylic were used for the flat model. Figure 6 shows all of
20
the completed blades.
21
Flume Testing
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
22
The three blades were tested in a water channel with a width of 38 cm and a water depth
23
of 18 cm above a false bottom. The blades were held in a horizontal position by a clamp at mid-
8
Page 8 of 26
Page 9 of 26
1
depth. The current speed was 20 cm/s. Following Rominger and Nepf (2014), a vortex street was
2
created by a 1-cm thick bar of height D = 2.5 cm that spanned the flume at mid-depth, which was
3
aligned with the clamp and blade. While mimicking the interaction of the blades with individual
4
turbulent eddies, the vortex street provided a single scale (bar height D) and a frequency (1.2 Hz)
5
that enabled simpler visualization and interpretation of the blade motion. The periodic eddies
6
created by the bar had a wavelength of 4D. The upstream edge of the blade was placed 4D
7
downstream from the bar.
8
9
r
Fo
Beneath the false bottom, the clamp was attached to a load cell (Futek LSB210) that
measured the streamwise forces on the clamp and blade. Mean drag was calculated as an average
10
over a 5 minute long record collected at 2,000 Hz. Additionally, we measured the force on the
11
clamp without a blade and subtracted this value to isolate the force on the blade alone. The load
12
cell was calibrated by measuring known weights incremented from 0 to 0.006 N, and the
13
measurements were recorded using Labview software. Videos were taken of each blade in the
14
channel (using a Sony DFW-X710 camera). The videos were used to measure the average
15
maximum vertical displacement of the blade tip per cycle of vortex shedding. The displacement
16
measurements were calculated from ten arbitrary one-cycle sequences for each blade. From each
17
of ten cycles, we recorded the maximum vertical displacement of the blade’s tip (in either the
18
upward or downward direction from the centerline) and then calculated the average over the ten
19
cycles. The vertical displacement was measured in Matlab by counting the number of pixels in
20
the image between the centerline and the tip position and then converting to centimeters using
21
the length of the clamp as calibration.
22
Results
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
9
Limnology and Oceanography: Methods
1
The measured drag force on the blades decreased significantly with the addition of
2
corrugations (Figure 8). Specifically, the drag on the corrugated blades was reduced relative to
3
the flat blade by 12% and 35%, respectively for a = 1.5 mm and 2 mm (Figure 8a). Note that this
4
drop in drag occurred despite an increase in surface area. The reduction in drag per surface area
5
relative to the flat blade was larger; 40% and 62%, respectively for a = 1.5 mm and 2 mm
6
(Figure 8b). The reduction in drag confirms that corrugations of the scale observed in the field
7
can raise stiffness sufficiently to reduce by about 1/3 the total drag on individual blades.
8
9
r
Fo
Over a similar change in dimensionless blade rigidity (η = 5x10-6 to η = 5x10-4, which
corresponds to the change in rigidity between the flat and a = 2 mm blades in this paper),
10
Rominger and Nepf observed a 45% reduction in drag (2014). Because surface area was constant
11
in their experiments, this reduction (45%) also extends to drag per surface area. Their observed
12
reduction is between our observed drag reduction (35%) and drag per surface area reduction
13
(62%). The agreement in the magnitude of drag reduction over a similar increase in
14
dimensionless rigidity (η) suggests that the parameter of dimensionless rigidity works universally
15
for rigidity associated with blade thickness or with blade morphology (here, corrugations).
ew
vi
On
16
Re
The dependence of the blade motion on the level of corrugation was assessed
ly
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
17
qualitatively using video images. Figure 9 shows a video sequence for each of the three blades.
18
The sequences are composed of 5 frames taken at intervals of 0.2 seconds, which represents
19
approximately one cycle of vortex shedding. For a vortex shedding frequency of 1.2 Hz, one full
20
period is ≈0.83 seconds. The flat blade (lowest η) experienced the greatest oscillation amplitude,
21
or vertical displacement from the centerline. The downstream tip of the blade exhibited a
22
maximum vertical displacement of 5.4±0.3 cm from the neutral blade position, with the
23
uncertainty representing the standard error measured over ten sequences. The blade movement
10
Page 10 of 26
Page 11 of 26
1
decreased with increasing corrugation amplitude. Specifically, the maximum tip excursion
2
decreased to 3.0±0.2 cm and 1.9±0.2 cm for a = 1.5 mm (η = 2.4x10-4) and 2 mm (η = 5.0x10-4),
3
respectively. Rominger and Nepf (2014) also observed a reduction in vertical displacement for
4
blades of higher nondimensional rigidity. Specifically, Rominger and Nepf observed a change in
5
rms-vertical displacement from 2.5 cm (η≈10-6) to 0.5 cm (η≈4x10-4) (2014).
6
Our least rigid blade (flat blade, η = 7.8x10-6) exhibited the most asymmetric oscillations,
7
meaning that the blade preferentially remained above or below the clamp centerline. For
8
example, the sequence in figure 9a shows asymmetric oscillations favoring the downward
9
direction. We quantified this asymmetry as the difference between the maximum upward and
r
Fo
10
maximum downward displacement of the tip within each vortex period, normalized by the
11
overall maximum displacement, and averaged this fraction over ten cycles. An asymmetry of
12
zero indicates equal displacement up and down (perfect symmetry) while 1 is the maximum
13
asymmetry. Using this metric, the flat blade exhibited 67±6% asymmetry in its vertical motion,
14
the a = 1.5 mm corrugated blade exhibited 56±7% asymmetry, and the a = 2 mm corrugated
15
blade exhibited 42±8% asymmetry. These observations suggest that oscillation symmetry
16
increases with blade rigidity. Rominger and Nepf (2014) also noted asymmetric blade motion for
17
η < 1x10-4, and they showed that it was associated with enhanced vertical acceleration at the tip
18
and an elevation in peak drag force.
19
Discussion
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
20
Our measurements of drag demonstrate that corrugations of the scale observed in the
21
field can raise stiffness sufficiently to reduce by one-third the drag on individual blades. The
22
reduction in drag is similar to that achieved by changes in blade thickness over a comparable
11
Limnology and Oceanography: Methods
1
range of non-dimensional rigidity. Decreasing drag protects the blade from being dislodged or
2
torn.
3
Rominger and Nepf (2014) identified the following trade-off with blade stiffness: stiffer
4
blades have the positive influence of decreasing drag, but the negative influence of also
5
decreasing flux to the blade surface, because stiffer blades exhibit less relative motion between
6
the water and blade surface. This trade off may explain why kelp selects corrugation, rather than
7
an increase in blade thickness, in order to enhance rigidity. Nutrient intake is a function of the
8
surface area of the blade. Developing corrugations maintains a constant surface area to volume
9
ratio, whereas increasing thickness decreases this ratio, which can diminish nutrient flux per
10
blade volume. Previous studies have also shown that in high-flow environments, the nutrient
11
uptake by kelp is limited by physiological controls – how fast the kelp can store or use nutrients
12
– not by hydrodynamic controls – how fast nutrients are delivered to the blade surface (Gerard
13
1982 a, b). This means that in high-flow environments, the decrease in flux to the blade surface
14
caused by increased stiffness may not be detrimental to blade survival, since the uptake by the
15
blade is not limited by hydrodynamic controls in these environments. At the same time, the
16
decrease in drag associated with corrugations will have the greatest benefit in high-flow
17
environments, where the scale of drag (which increases with current speed squared) is higher
18
than in low flow environments. This may explain why corrugations are observed only in high-
19
flow environments.
ew
vi
Re
ly
On
20
r
Fo
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Next, we consider additional complexity that can be added to the model blade
21
morphology with this fabrication method. The sinusoidal models used in these experiments were
22
developed to mimic the longitudinal corrugations observed on Macrocystis blades. However, the
23
actual corrugations observed in nature are not perfect sinusoids, but have curved peaks and
12
Page 12 of 26
Page 13 of 26
1
valleys and begin and end at random locations along the length of the blade (Figure 1). In order
2
to create templates of more realistic corrugations, we began with an image of lines representing
3
the peaks of the corrugations (Figure 10a). The distribution of peak lines was based on tracings
4
of Eisenia arborea blades made by Denny and Robertson (2002) and contour edits of the image
5
of a Macrocystis blade (Figure 1). A Matlab code filled in a linear vertical gradient where each
6
peak line ended and a horizontal sinusoidal gradient between adjacent peaks. An example of
7
peak lines converted to a grayscale template is exhibited in Figure 10b.
8
9
r
Fo
Finally, our blade fabrication method can be used to build a canopy of kelp blades to
consider how blade stiffness (gained either through blade thickness or corrugation) influences the
10
interactions between multiple blades. Kelp blades do not exist in isolation in nature, and when
11
the blades become entangled, they are at higher risk of tearing. The new models could be used to
12
explore whether increased rigidity reduces entanglement between blades. Changes in blade
13
rigidity may also impact light availability, and thus photosynthesis within a canopy. Less rigid
14
blades, which move more freely, might produce more even light conditions within the canopy, as
15
the movement of blades keeps any individual blade from being shaded for an extended period of
16
time. Alternatively, the light absorption of a submerged blade is related to its projected
17
horizontal area (Zimmerman, 2005). More rigid blades maintain a more horizontal position in
18
flow (Figure 9), which may provide an advantage in light capture and photosynthesis.
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
19
20
13
Limnology and Oceanography: Methods
1
References
2
D. Briassoulis. 1986. Equivalent orthotropic properties of corrugated sheets. Computers &
3
4
5
6
Structures 23: 129 – 138.
M. Denny and L. Roberson. 2002. Blade motion and nutrient flux to the kelp Eisenia arborea.
Biological Bulletin 203: 1 – 13.
V. Gerard. 1982a. In situ rates of nitrate uptake by giant kelp, macrocystis pyrifera (l.)
7
c. agardh: Tissue differences, environmental effects, and predictions of nitrogen-limited
8
growth. J. Exp. Mar. Biol. Ecol. 62: 211 – 224.
9
11
B. Hale. 2001. Macroalgal materials: foiling fracture and fatigue from fluid forces. Ph.D. thesis,
Stanford University.
ew
13
pyrifera. Marine Biology 69: 51 – 54.
vi
12
V. Gerard. 1982b. In situ water motion and nutrient uptake by the giant kelp Macrocystis
Re
10
r
Fo
C. Hurd and C. Pilditch. 2011. Flow-induced morphological variations affect diffusion
14
boundary-layer thickness of macrocystis pyrifera (heterokontophyta, laminariales). J.
15
Phycol. 47: 341 – 351.
16
On
M. Koehl, W. Silk, H. Liang, and L. Mahadevan. 2008. How kelp produce blade shapes suited to
ly
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
17
different flow regimes: A new wrinkle. Integrative and Comparative Biology 48: 834 –
18
851.
19
20
21
22
23
J. Lau. 1981. Stiffness of corrugated plate. Journal of the Engineering Mechanics Division
107: 271 – 275.
S. Michelin, S. L. Smith, and B. Glover. 2008. Vortex shedding model of a flapping flag.
Journal of Fluid Mechanics 617: 1 – 10.
J. Rominger and H. Nepf. 2014. Effects of blade flexural rigidity on drag force and mass
14
Page 14 of 26
Page 15 of 26
1
2
transfer rates in model blades. Limnol. Oceanogr. 59: 2028 – 2041.
Zimmerman, R., 2006. Light and photosynthesis in seagrass meadows. In: A. Larkum, C. Duarte
3
and R. Orth, editors, Seagrasses: Biology, Ecology and Conservation, pages 303 – 321.
4
Springer, Dordrecht.
r
Fo
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
15
Limnology and Oceanography: Methods
1
Table 1: Measurements of Macrocystis blade morphology from exposed and sheltered sites from
2
Hurd and Pilditch (2011). The standard deviation of each measurement is given in parentheses.
3
The elastic modulus of algal material was reported by Hale (2001).
Blade thickness, h (mm)
Elastic modulus, E (Pa)
Blade length, l (m)
Corrugation amplitude, a (mm)
Corrugation wavelength, λ (mm)
4
5
Exposed Morphology
0.46 (0.07)
5x106
0.62 (0.05)
1.4 (0.2)
3.07 (0.15)
Sheltered Morphology
0.42 (0.13)
5x106
0.52 (0.03)
-
r
Fo
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Page 16 of 26
16
Page 17 of 26
1
2
Figure 1: (a) A photo of a Macrocystis blade, showing the longitudinal corrugations running
3
along the length of the blade. (b) A cut section of a Macrocystis blade, showing the regularity of
4
the corrugation wavelength and amplitude. (Photo credit: Rominger and Nepf, 2014)
ew
vi
Re
5
r
Fo
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
17
Limnology and Oceanography: Methods
1
2
3
4
r
Fo
Figure 2: a) Sketch of blade attached to clamp from the side view. Flow in the channel moves in
the x-direction. b) Cross section of a corrugated model blade perpendicular to flow. The area
moment of inertia I is defined about the y-axis.
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
18
Page 18 of 26
Page 19 of 26
r
Fo
vi
Re
1
ew
2
Figure 3: Discrete curves representing one wavelength of the blade mold cross-section, which is
3
5*λ = 3 cm wide in total, for the model with corrugation amplitude a = 1.5 mm. A perfect
4
sinusoid (dashed line) is aligned with the center of the blade. The top and bottom curves (blue
5
lines) mark the templates for the acrylic molds. The red line shows the thickness of the blade
6
measured perpendicular to the surfaces, while the green line shows the vertical displacement
7
between the surfaces.
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
8
9
19
Limnology and Oceanography: Methods
1
2
Figure 4: (a) The grayscale templates created in Matlab with (b) the finished acrylic molds. The
3
holes in the corners of the molds were drilled after laser cutting to simplify the alignment process
4
when filling the molds. The model blades were 3 cm wide (b) x 30 cm long (l), so each mold is 6
5
cm x 31.5 cm.
6
8
ew
vi
9
Re
7
r
Fo
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
20
Page 20 of 26
Page 21 of 26
1
r
Fo
3
Figure 5: Casting process using the laser-cut molds. (a) The acrylic molds are set on a table.
4
Metal spacers are placed along the outer perimeter of the left mold to set the thickness, and (b)
5
the prepared silicone mixture is poured onto the mold. (c) The mixture is covered with the right
6
acrylic mold and held down with weights to cure (about 20 minutes). (d) A cross-sectional view
7
of a 4x4 cm prototype mold filled with the VPS mixture (pink).
ew
2
vi
Re
8
21
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
Limnology and Oceanography: Methods
1
Figure 6: Completed blades displayed here with duplicates (left to right: flat, a = 1.5 mm
3
corrugated, a = 2 mm corrugated, flat, a = 1.5 mm corrugated, a = 2 mm corrugated). Each blade
4
has an additional 1 to 2 mm of flat space at the end where the clamp attaches.
5
ew
vi
2
Re
r
Fo
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
22
Page 22 of 26
Page 23 of 26
1
a)
2
b)
3
Figure 7: (a) Schematic of experimental setup, which is the same as Rominger and Nepf (2014).
4
(b) The a = 1.5 mm corrugated blade held horizontally in the channel by a narrow metal clamp,
5
which is attached to the load cell beneath the false bottom.
r
Fo
ew
vi
Re
6
7
23
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
Limnology and Oceanography: Methods
r
Fo
ew
vi
Re
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
1
2
Figure 8: (a) The measured drag force and (b) the drag force per surface area as a function of
3
dimensionless blade rigidity. The standard error in the time-averaged drag measurements is
4
within the size of the symbols shown. The uncertainty in the η measurements is represented by
5
the black lines. The uncertainty in the flat blade’s rigidity is within the size of the symbol.
6
24
Page 24 of 26
Page 25 of 26
1
2
r
Fo
Re
3
4
Figure 9: (a) flat blade, (b) 1.5 mm amplitude corrugation, (c) 2 mm amplitude corrugation.
5
Each sequence represents one cycle of vortex shedding. The frequency of vortex shedding from
6
the D = 2.5 cm bar is 1.2 Hz, which yields a period of 0.83 seconds. The 5 frames in each of the
7
above sequences are therefore spaced 0.2 s apart (¼ the period).
ew
vi
On
8
9
ly
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Limnology and Oceanography: Methods
25
Limnology and Oceanography: Methods
r
Fo
1
Re
2
Figure 10: The image on the left shows one example of random corrugation peaks, and the image
3
on the right is the resulting grayscale template.
ew
4
vi
ly
On
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
26
Page 26 of 26