Numerical Simulation of the Advantages of the Figure-Eight Flapping Motion of an Insect on Aerodynamics under Low Reynolds Number Conditions
<p>Computational model of an (<b>a</b>) insect and (<b>b</b>) wing. Only the two wings shown in gray were analyzed without body reproduction. The red line represents the wing length.</p> "> Figure 2
<p>Definition of the flapping motion of the insect in (<b>a</b>) bird’s eye view, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>–</mo> <mi>z</mi> </mrow> </semantics></math> plane, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>–</mo> <mi>z</mi> </mrow> </semantics></math> plane. The wings, which are the analysis objects, are shown in gray. On the other hand, the body, which is not the analysis object, is shown by the dotted line.</p> "> Figure 3
<p>Representation of insect flapping motion. (<b>a</b>) Time history of the positional angle, feathering angle, and elevation angle. (<b>b</b>) Trajectory of wing tips. In <a href="#biomimetics-09-00249-f003" class="html-fig">Figure 3</a><b>b</b>,The gray solid and dotted lines represent the trajectory of the wings tips and the body of the insect, respectively.</p> "> Figure 4
<p>3D27V model for the three-dimensional lattice Boltzmann method. The numbers (0~26) represent the 27 directions of the 3D27Vmodel.</p> "> Figure 5
<p>Virtual boundary points. The gray and white areas represent the interior of the object and fluid, respectively.</p> "> Figure 6
<p>Schematic view of the physical quantity calculation method at the virtual boundary point. The gray and white areas represent the interior of the object and fluid, respectively.</p> "> Figure 7
<p>Schematic view of the computational model: (<b>a</b>) bird’s eye view, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>–</mo> <mi>z</mi> </mrow> </semantics></math> plane, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>–</mo> <mi>y</mi> </mrow> </semantics></math> plane. The representative length <math display="inline"><semantics> <mrow> <mi>L</mi> </mrow> </semantics></math> is the mean chord length <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mi>m</mi> </mrow> </msub> </mrow> </semantics></math> shown in <a href="#sec2dot1-biomimetics-09-00249" class="html-sec">Section 2.1</a>. In <a href="#biomimetics-09-00249-f007" class="html-fig">Figure 7</a><b>a</b>, the wings and grid model are shown in green and black or gray lines, respectively. In <a href="#biomimetics-09-00249-f007" class="html-fig">Figure 7</a><b>b</b>,<b>c</b>, the wings are shown in gray areas, and block areas of grid model consisting of different grid sizes are represented by black, green, red, and blue lines.</p> "> Figure 8
<p>Time history of the lift coefficient in seven cycles with <math display="inline"><semantics> <mrow> <mi>U</mi> <mo>=</mo> <mn>0.04</mn> </mrow> </semantics></math>.</p> "> Figure 9
<p>Time history of the (<b>a</b>) lift coefficient, (<b>b</b>) pressure component of lift coefficient, and (<b>c</b>) viscous stress component of lift coefficient for three resolutions.</p> "> Figure 10
<p>Schematic view of the movement of an oscillating plate.</p> "> Figure 11
<p>Time history of the lift coefficient of an oscillating plate. The result was compared with those of Trizila [<a href="#B36-biomimetics-09-00249" class="html-bibr">36</a>] and Wang et al. [<a href="#B37-biomimetics-09-00249" class="html-bibr">37</a>].</p> "> Figure 12
<p>Trajectory of the wing tip in each motion. (<b>a</b>) With figure-eight motion and (<b>b</b>) without figure-eight motion. The gray line represents the trajectory of the wing tip.</p> "> Figure 13
<p>Vortex structures and normalized helicity density over one stroke cycle for each motion. (<b>a</b>) With and (<b>b</b>) without figure-eight motions at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.25</mn> <mi>T</mi> </mrow> </semantics></math>; (<b>c</b>) with and (<b>d</b>) without figure-eight motions at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.50</mn> <mi>T</mi> </mrow> </semantics></math>; (<b>e</b>) with and (<b>f</b>) without figure-eight motions at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.75</mn> <mi>T</mi> </mrow> </semantics></math>; and (<b>g</b>) with and (<b>h</b>) without figure-eight motions at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1.00</mn> <mi>T</mi> </mrow> </semantics></math>.</p> "> Figure 14
<p>Time histories of (<b>a</b>) lift coefficient, (<b>b</b>) thrust coefficient, and (<b>c</b>) power coefficient over one stroke cycle for each motion.</p> "> Figure 15
<p>Vortex structures (<math display="inline"><semantics> <mrow> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>) and normalized pressure coefficient for each motion in the downstroke at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mi>T</mi> </mrow> </semantics></math>. Vortex structures (<math display="inline"><semantics> <mrow> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>) and normalized pressure coefficients from (<b>a</b>) with and (<b>b</b>) without figure-eight motions. Normalized pressure coefficient at the wing tip from (<b>c</b>) with and (<b>d</b>) without figure-eight motions.</p> "> Figure 16
<p>Trajectory of the wing tip in each motion: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>180</mn> <mo>°</mo> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>225</mn> <mo>°</mo> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>270</mn> <mo>°</mo> </mrow> </semantics></math>; and (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>φ</mi> </mrow> <mrow> <mi>e</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>315</mn> <mo>°</mo> </mrow> </semantics></math>. The gray line represents the trajectory of the wing tip.</p> "> Figure 17
<p>Relation between the cycle-averaged lift coefficient and power coefficient for each motion at each Reynolds number. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>33.5</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>67</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>134</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>268</mn> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>=</mo> <mn>546</mn> </mrow> </semantics></math>; (<b>f</b>) all Reynolds numbers. Light blue areas shown in the figure (<span class="html-fig-inline" id="biomimetics-09-00249-i001"><img alt="Biomimetics 09 00249 i001" src="/biomimetics/biomimetics-09-00249/article_deploy/html/images/biomimetics-09-00249-i001.png"/></span>) indicate more efficiency than without figure-eight motions.</p> "> Figure 18
<p>Relation between the initial phase of elevation angle of the most efficient motion in generating lift and Reynolds number.</p> "> Figure 19
<p>Schematic view of the elliptic approximation.</p> "> Figure 20
<p>Relation between the aspect ratio and Reynolds number.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Model of an Insect
2.2. Motion of the Insect
2.3. Governing Equation
2.4. Virtual Flux Method
2.5. Computational Model
2.6. Evaluation Parameters
3. Results and Discussion
3.1. Grid Independence and Flapping Cycle Convergence Tests
3.2. Example Test for Analyzing the Three-Dimensional Flapping Motion
3.3. Effect of Figure-Eight Motion
3.4. Effect of Various Figure-Eight Motions and Reynolds Number
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
References
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Name | Symbol | Value |
---|---|---|
Body length | ||
Mean chord length | ||
Wing length | ||
Mean velocity of wing tip | ||
Wingbeat frequency | ||
Kinematic viscosity of air | ||
Density of air |
0 | |||
, , | |||
, , | |||
Cycle | |
---|---|
1 | 0.484 |
2 | 0.379 |
3 | 0.390 |
4 | 0.387 |
5 | 0.383 |
6 | 0.382 |
7 | 0.383 |
Resolution | |||
---|---|---|---|
0.373 | 0.408 | 0.038 | |
0.382 | 0.427 | 0.048 | |
0.392 | 0.442 | 0.052 |
Case | ||||
---|---|---|---|---|
(A) | 0.382 | 0.380 | 1.005 | |
(B) | 0.302 | 0.352 | 0.857 |
Case | Re | 33.5 | 67 | 134 | 268 | 536 | 33.5 | 67 | 134 | 268 | 536 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
(A) | 0 | 0.270 | 0.303 | 0.354 | 0.396 | 0.421 | 0.509 | 0.726 | 0.961 | 1.165 | 1.335 | |
45 | 0.350 | 0.398 | 0.463 | 0.522 | 0.558 | 0.613 | 0.849 | 1.075 | 1.246 | 1.358 | ||
90 | 0.401 | 0.458 | 0.520 | 0.581 | 0.613 | 0.624 | 0.838 | 1.019 | 1.159 | 1.223 | ||
135 | 0.383 | 0.448 | 0.516 | 0.572 | 0.589 | 0.542 | 0.733 | 0.895 | 1.014 | 1.069 | ||
180 | 0.296 | 0.363 | 0.417 | 0.460 | 0.483 | 0.413 | 0.594 | 0.741 | 0.851 | 0.922 | ||
225 | 0.203 | 0.252 | 0.312 | 0.341 | 0.336 | 0.306 | 0.463 | 0.637 | 0.757 | 0.803 | ||
270 | 0.177 | 0.202 | 0.246 | 0.274 | 0.278 | 0.306 | 0.447 | 0.623 | 0.766 | 0.864 | ||
315 | 0.202 | 0.226 | 0.265 | 0.294 | 0.310 | 0.384 | 0.558 | 0.768 | 0.956 | 1.116 | ||
(B) | - | 0.220 | 0.251 | 0.302 | 0.350 | 0.379 | 0.429 | 0.623 | 0.857 | 1.065 | 1.207 |
[−] | [μN] | [μW] | [−] |
---|---|---|---|
33.5 | 0.205 | 0.308 | 0.665 |
67 | 0.933 | 1.932 | |
134 | 4.495 | 13.53 | |
268 | 20.84 | 101.0 | |
536 | 90.32 | 772.3 | 0.117 |
33.5 | (0.623, 0.284) | 0.131 | 0.079 | 54 | 1.658 |
67 | (0.511, 0.329) | 0.154 | 54 | 2.110 | |
134 | (0.468, 0.385) | 0.176 | 53 | 2.588 | |
268 | (0.437, 0.429) | 0.197 | 53 | 2.897 | |
536 | (0.415, 0.446) | 0.208 | 0.071 | 53 | 2.930 |
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Yoshida, M.; Fukui, T. Numerical Simulation of the Advantages of the Figure-Eight Flapping Motion of an Insect on Aerodynamics under Low Reynolds Number Conditions. Biomimetics 2024, 9, 249. https://doi.org/10.3390/biomimetics9040249
Yoshida M, Fukui T. Numerical Simulation of the Advantages of the Figure-Eight Flapping Motion of an Insect on Aerodynamics under Low Reynolds Number Conditions. Biomimetics. 2024; 9(4):249. https://doi.org/10.3390/biomimetics9040249
Chicago/Turabian StyleYoshida, Masato, and Tomohiro Fukui. 2024. "Numerical Simulation of the Advantages of the Figure-Eight Flapping Motion of an Insect on Aerodynamics under Low Reynolds Number Conditions" Biomimetics 9, no. 4: 249. https://doi.org/10.3390/biomimetics9040249