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26 pages, 5898 KiB  
Article
Research on the Impact of the Slider on the Aerodynamic Characteristics of a Terrestrial–Aerial Spherical Robot
by Dongshuai Huo, Hanxu Sun, Xiaojuan Lan and Minggang Li
Actuators 2025, 14(3), 118; https://doi.org/10.3390/act14030118 - 27 Feb 2025
Viewed by 188
Abstract
This research introduces the first design concept for a ducted coaxial-rotor amphibious spherical robot (BYQ-A1), utilizing the principle of variable mass control. It investigates whether the BYQ-A1’s variable-mass slider has a certain regularity in its impact on the aerodynamic properties of the BYQ-A1. [...] Read more.
This research introduces the first design concept for a ducted coaxial-rotor amphibious spherical robot (BYQ-A1), utilizing the principle of variable mass control. It investigates whether the BYQ-A1’s variable-mass slider has a certain regularity in its impact on the aerodynamic properties of the BYQ-A1. Utilizing the Blade Element Momentum Theory (BEM) and Wall Jet Theory, an aerodynamic calculation model for the BYQ-A1 is established. An orthogonal experimental method is used to conduct tests on the impact of the variable-mass slider on the aerodynamic properties of the ducted coaxial-rotor system and validate the effectiveness of the aerodynamic calculation model. The results show that the slider generates an internal ground effect and ceiling effect within the BYQ-A1 that enhance the lift of the upper and lower rotors when the robot is equipped with it. The increased total lift compensates for the additional aerodynamic drag caused by the presence of the slider. This novel finding provides guidance for the subsequent optimization design and control method research of the BYQ-A1 and also offers valuable references for configuration schemes that incorporate necessary devices between coaxial dual rotors. Full article
(This article belongs to the Section Actuators for Robotics)
Show Figures

Figure 1

Figure 1
<p>Subfigure (<b>a</b>) displays the BYQ-A1 and its orientation in flight mode. Subfigure (<b>b</b>) displays BYQ-A1’s orientation in land mode. Subfigure (<b>c</b>) displays the ducted coaxial-rotor drive mechanism. Subfigure (<b>d</b>) displays the gyro frame drive mechanism. Subfigure (<b>e</b>) displays the variable-mass-center drive mechanism.</p>
Full article ">Figure 2
<p>An inflow model of the ducted coaxial-rotor system with a variable-center slider.</p>
Full article ">Figure 3
<p>Subfigure (<b>a</b>) shows the overall appearance of the aerodynamic test platform. Subfigure (<b>b</b>) shows the placement of the ATI sensor when measuring the force/torque of the upper rotor. Subfigure (<b>c</b>) shows the placement of the ATI sensor when measuring the force/torque on the variable-mass-center slider. Subfigure (<b>d</b>) shows the placement of the ATI sensor when measuring the force/torque of the lower rotor. Subfigure (<b>e</b>) shows the placement of the conduction wires. Subfigure (<b>f</b>) shows a pair of flanges connected by nylon screws.</p>
Full article ">Figure 4
<p>Subfigure (<b>a</b>) shows the placement of the FLAME 60A 12S Electronic Speed Controller and T-motor F7 flight control board. Subfigure (<b>b</b>) shows the variable-mass-center slider installed on the gyro frame of the BYQ-A1 via two metal rods with scales.</p>
Full article ">Figure 5
<p>Graph showing the relationship between lower-rotor lift and eccentricity variation. This study conducted fits for the <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>l</mi> <mi>z</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of sliders, yielding the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>l</mi> <mi>z</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>18.89</mn> <msubsup> <mrow> <mover> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mn>1</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>37.55</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>19.92</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>0.67</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>−</mo> <mn>13.98</mn> </mrow> </semantics></math>,<math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.991</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>l</mi> <mi>z</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>27.44</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>56.72</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>32.21</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>1.14</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>−</mo> <mn>13.88</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.997</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>l</mi> <mi>z</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>6.29</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>6.49</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>8.98</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>2.57</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>−</mo> <mn>13.23</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.994</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>l</mi> <mi>z</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>31.48</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>45.25</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>22.53</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>6.86</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>−</mo> <mn>12.65</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.973</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Graph showing the relationship between lower-rotor torque and eccentricity variation. This study conducted fits for the <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>l</mi> <mi>y</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of sliders, yielding the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>l</mi> <mi>y</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.08</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>7.00</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>4.26</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.11</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>−</mo> <mn>0.07</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.991</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>l</mi> <mi>y</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.17</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>6.20</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>3.15</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>0.24</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>−</mo> <mn>0.06</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.990</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>l</mi> <mi>y</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>1.13</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>0.60</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>0.07</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>0.58</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>−</mo> <mn>0.04</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.995</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>l</mi> <mi>y</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>3.64</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>4.88</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>2.22</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>0.87</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>−</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>0.997</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>Graph depicting the relationship between upper-rotor lift and eccentricity variation. This study conducted fits for the <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>u</mi> <mi>x</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of sliders, yielding the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>u</mi> <mi>z</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>35.6</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>77.23</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>58.11</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>18.19</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>2.21</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mn>14.28</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.989</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>u</mi> <mi>z</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>8.88</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>5</mn> </msubsup> <mo>+</mo> <mn>17.14</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>11.27</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>2.58</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.18</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <mn>14.10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.972</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>u</mi> <mi>z</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>8.76</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>12.03</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>3.43</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>0.08</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>13.57</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.970</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>u</mi> <mi>z</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>15.80</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>30.48</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>19.72</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>5.24</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>12.49</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.974</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Relationship curve of the resistance in relation to the eccentricity. This study conducted fits for the <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>f</mi> <mi>z</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of slider, yielding the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.25</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>−</mo> <mn>4.96</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.974</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.04</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>−</mo> <mn>4.92</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.979</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.26</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>−</mo> <mn>3.53</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.951</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.26</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>−</mo> <mn>1.49</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.956</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 9
<p>Relationship curve of the torque in relation to the eccentricity. This study performed fits for the <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>f</mi> <mi>y</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of slider, resulting in the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.77</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>1.65</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>1.30</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.40</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>−</mo> <mn>0.005</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.996</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.89</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>1.76</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>1.33</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.43</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>−</mo> <mn>0.009</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.991</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.80</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>1.79</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>1.41</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.39</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>−</mo> <mn>0.005</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.985</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>f</mi> <mi>z</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>1.49</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>1.06</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mn>0.25</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>0.004</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.993</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>Relationship curve of the total lift in relation to the eccentricity variation. This study performed fits for the <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>z</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of sliders, generating the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>z</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>5.51</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>7.51</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>0.80</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>3.33</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mn>18.05</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.986</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>z</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>9.36</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>15.29</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>4.06</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>2.34</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <mn>17.97</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.957</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>z</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>13.16</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>26.97</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>21.49</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>7.33</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mn>18.28</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.952</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>z</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>11.64</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>23.86</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>19.36</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>6.59</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>+</mo> <mn>18.30</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.951</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>Relationship curve of the total torque in relation to the eccentricity variation. This study performed fits for the <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>t</mi> <mi>y</mi> <mi>i</mi> </mrow> </msub> </semantics></math> variables associated with four types of sliders, generating the following equations and <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mn>2</mn> </msup> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>t</mi> <mi>y</mi> <msup> <mn>1</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>4.91</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>9.11</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>1.42</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>4.24</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>1</mn> </msub> <mo>−</mo> <mn>0.15</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.997</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>t</mi> <mi>y</mi> <msup> <mn>2</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>13.89</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>25.05</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>10.26</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>2.67</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> <mo>−</mo> <mn>0.22</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.993</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>t</mi> <mi>y</mi> <msup> <mn>3</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>14.73</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>4</mn> </msubsup> <mo>+</mo> <mn>28.60</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>3</mn> </msubsup> <mo>−</mo> <mn>14.17</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>1.63</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>3</mn> </msub> <mo>−</mo> <mn>0.11</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.999</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>t</mi> <mi>y</mi> <msup> <mn>4</mn> <mo>′</mo> </msup> </mrow> </msub> <mo>=</mo> <mn>13.04</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>4</mn> </msubsup> <mo>−</mo> <mn>15.40</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>3</mn> </msubsup> <mo>+</mo> <mn>5.53</mn> <msubsup> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> <mn>2</mn> </msubsup> <mo>−</mo> <mn>3.07</mn> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">¯</mo> </mover> <mn>4</mn> </msub> <mo>−</mo> <mn>0.12</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.996</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 12
<p>The curve of the relationship among the experimental data, the fitted mathematical model, and the theoretical mathematical model.</p>
Full article ">
21 pages, 5167 KiB  
Article
A New Modified Blade Element Momentum Method for Calculating the Aerodynamic Performance of a Wind Turbine in Yaw
by Jiaying Wu, Zhenye Sun, Weijun Zhu, Shifeng Fu, Chang Xu and Wenzhong Shen
Energies 2025, 18(5), 1063; https://doi.org/10.3390/en18051063 - 21 Feb 2025
Viewed by 317
Abstract
The yaw state constitutes a typical operating condition for wind turbines. However, the widely used Blade Element Moment (BEM) theory, due to its adoption of planar disc assumptions, introduces certain computational inaccuracies in yaw conditions. This research aims to develop a new modified [...] Read more.
The yaw state constitutes a typical operating condition for wind turbines. However, the widely used Blade Element Moment (BEM) theory, due to its adoption of planar disc assumptions, introduces certain computational inaccuracies in yaw conditions. This research aims to develop a new modified BEM method by replacing the momentum theory in traditional BEM with the Madsen analytical linear two-dimensional actuator disc model in order to enhance the accuracy in calculating the aerodynamic performance of yawed wind turbines. Two approaches are introduced to determine the variable parameters in the new modified model: one based on traditional BEM predictions in non-yaw conditions and the other using empirical values determined using experimental data. The new modified model is evaluated against experimental data, CENER FAST, and HAWC2 for the MEXICO rotor. From the comparisons, the new modified method demonstrates closer agreements with experimental values, particularly in the mid and outer parts of the blades. At a wind speed of 15 m/s and a yaw angle of 30°, the discrepancies between computation and measurement are reduced by at least 2.33, 1.22, and 3.25 times at spanwise locations of 60%Radius (R), 82%R, and 92%R, respectively, compared to CENER FAST or HAWC2, demonstrating the feasibility of the proposed methodology. Full article
(This article belongs to the Section A3: Wind, Wave and Tidal Energy)
Show Figures

Figure 1

Figure 1
<p>Flowchart for the new modified model.</p>
Full article ">Figure 2
<p>Axial forces on the MEXICO rotor at wind speeds of (<b>a</b>) <math display="inline"><semantics> <mrow> <mn>10</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mn>15</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mn>24</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 3
<p>Tangential forces on the MEXICO rotor at wind speeds of (<b>a</b>) <math display="inline"><semantics> <mrow> <mn>10</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mn>15</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mn>24</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Axial force variation with azimuth angle on the MEXICO rotor at a wind speed of <math display="inline"><semantics> <mrow> <mn>15</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, a yaw angle of 30<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math> and the span station (<span class="html-italic">r/R</span>) of (<b>a</b>) 25%, (<b>b</b>) 35%, (<b>c</b>) 60%, (<b>d</b>) 82%, and (<b>e</b>) 92%.</p>
Full article ">Figure 5
<p>Tangential forces variation with azimuth angle on the MEXICO rotor at a wind speed of <math display="inline"><semantics> <mrow> <mn>15</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, a yaw angle of 30<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math> and the span station (<span class="html-italic">r/R</span>) of (<b>a</b>) 25%, (<b>b</b>) 35%, (<b>c</b>) 60%, (<b>d</b>) 82%, and (<b>e</b>) 92%.</p>
Full article ">Figure 6
<p>Axial force variation with azimuth angle on the MEXICO rotor at a wind speed of <math display="inline"><semantics> <mrow> <mn>24</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, a yaw angle of 15<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math> and the span station (<span class="html-italic">r/R</span>) of (<b>a</b>) 25%, (<b>b</b>) 35%, (<b>c</b>) 60%, (<b>d</b>) 82%, and (<b>e</b>) 92%.</p>
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<p>Tangential forces variation with azimuth angle on the MEXICO rotor at a wind speed of <math display="inline"><semantics> <mrow> <mn>24</mn> <mo> </mo> <mi mathvariant="normal">m</mi> <mo>/</mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>, a yaw angle of 30<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math> and the span station (<span class="html-italic">r/R</span>) of (<b>a</b>) 25%, (<b>b</b>) 35%, (<b>c</b>) 60%, (<b>d</b>) 82%, and (<b>e</b>) 92%.</p>
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24 pages, 9637 KiB  
Article
Determining Quasi-Static Load Carrying Capacity of Composite Sandwich Rotor Blades for Copter-Type Drones
by Chien Wei Jan and Tai Yan Kam
Drones 2024, 8(8), 355; https://doi.org/10.3390/drones8080355 - 30 Jul 2024
Viewed by 1057
Abstract
The development of light composite rotor blades with acceptable load carrying capacity is an essential issue to be dealt with in the design of relatively large copter-type drones. In this paper, a method is established to determine the quasi-static blade load carrying capacity [...] Read more.
The development of light composite rotor blades with acceptable load carrying capacity is an essential issue to be dealt with in the design of relatively large copter-type drones. In this paper, a method is established to determine the quasi-static blade load carrying capacity which is vital to drone reliability. The proposed method, which provides a systematic procedure to determine blade load carrying capacity, consists of three parts, namely, a procedure to determine the distributed quasi-static blade aerodynamic load via the Blade Element Momentum (BEM) approach, a finite element-based failure analysis method to identify the actual blade failure mode, and an optimization method to determine the actual blade load carrying capacity. The experimental failure characteristics (failure mode, failure thrust, failure location) of two types of composite sandwich rotor blades with different skin lamination arrangements have been used to verify the accuracy of the theoretical results obtained using the proposed load carrying capacity determination method. The skin lamination arrangement for attaining the optimal blade-specific load carrying capacity and the blade incipient rotational speed for safe drone operation has been determined using the proposed method. Full article
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Figure 1
<p>Quadcopter drone.</p>
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<p>Rotor blade: (<b>a</b>) Dimensions; (<b>b</b>) NACA 4418 Airfoil showing skin and core.</p>
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<p>Lamination arrangement of composite sandwich blade (<b>a</b>) Type 1 blade, (<b>b</b>) Type 2 blade.</p>
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<p>Lamination arrangement of composite sandwich blade (<b>a</b>) Type 1 blade, (<b>b</b>) Type 2 blade.</p>
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<p>Blade element.</p>
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<p>Elemental airfoil aerodynamics.</p>
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<p>Experimental setup for rotor blade thrust measurement.</p>
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<p>Iterative procedure for updating vertical uplifting force.</p>
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<p>Locations of blade elements and resultant thrust.</p>
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<p>Rotational speed vs. rotor blade thrust.</p>
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<p>Finite element mesh for composite sandwich blade. (<b>a</b>) Type 1 blade, (<b>b</b>) Type 2 blade.</p>
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<p>Iterative procedure for updating incipient failure rotational speed.</p>
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<p>Finished rotor blade product.</p>
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<p>Experimental setup.</p>
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<p>Experimental failure pattern of composite blade (<b>a</b>) Type 1 blade, (<b>b</b>) Type 2 blade.</p>
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<p>Failure analysis results for Type 1 blade under resultant thrust. (<b>a</b>) Failure index for Maximum stress criterion (Failure location: x = −0.48, y = 10.99). (<b>b</b>) Failure index for Tsai–Wu criterion (Failure location: x = −0.48, y = 10.59). (<b>c</b>) Buckling mode shape (Failure location: x = −0.323, y = 10.7).</p>
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<p>Failure analysis results for Type 1 blade under resultant thrust. (<b>a</b>) Failure index for Maximum stress criterion (Failure location: x = −0.48, y = 10.99). (<b>b</b>) Failure index for Tsai–Wu criterion (Failure location: x = −0.48, y = 10.59). (<b>c</b>) Buckling mode shape (Failure location: x = −0.323, y = 10.7).</p>
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<p>Failure analysis results for Type 2 blade under resultant thrust (<b>a</b>) Failure index for Maximum stress criterion (Failure location: x = 0.12, y = 3.5). (<b>b</b>) Failure index for Tsai–Wu criterion (Failure location: x = 0.12, y = 3.5). (<b>c</b>) Buckling mode shape (Failure location: x = −0.398, y = 4.88).</p>
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<p>Failure analysis results for Type 2 blade under resultant thrust (<b>a</b>) Failure index for Maximum stress criterion (Failure location: x = 0.12, y = 3.5). (<b>b</b>) Failure index for Tsai–Wu criterion (Failure location: x = 0.12, y = 3.5). (<b>c</b>) Buckling mode shape (Failure location: x = −0.398, y = 4.88).</p>
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<p>Failure index distribution of composite sandwich blade under elemental thrusts, drag forces, and centrifugal force. (<b>a</b>) Type 1 blade, (<b>b</b>) Type 2 blade.</p>
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<p>Failure index distribution of composite sandwich blade under elemental thrusts, drag forces, and centrifugal force. (<b>a</b>) Type 1 blade, (<b>b</b>) Type 2 blade.</p>
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<p>Effects of Region 2 length on blade load carrying capacity and weight.</p>
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<p>Effects of Region 2 on failure rotational speed and specific load carrying capacity.</p>
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<p>Relation between blade rotational speed and displacement.</p>
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18 pages, 3489 KiB  
Article
Development and Measurement of a Very Thick Aerodynamic Profile for Wind Turbine Blades
by Alois Peter Schaffarczyk, Brandon Arthur Lobo, Nicholas Balaresque, Volker Kremer, Janick Suhr and Zhongxia Wang
Wind 2024, 4(3), 190-207; https://doi.org/10.3390/wind4030010 - 12 Jul 2024
Cited by 1 | Viewed by 2203
Abstract
We designed 60% thick airfoil to improve the aerodynamic performance in the root region of wind turbine rotor blades, taking into account current constraints. After an extensive literature review and patent research, a design methodology (including the considerations of simple manufacturing) was set [...] Read more.
We designed 60% thick airfoil to improve the aerodynamic performance in the root region of wind turbine rotor blades, taking into account current constraints. After an extensive literature review and patent research, a design methodology (including the considerations of simple manufacturing) was set up, and extensive 2D- and 3D-CFD investigations with four codes (Xfoil, MSES, ANSYS fluent, and DLR-tau) were performed, including implementation inside a generic 10 MW test-blade (CIG10MW). Comparison with results from Blade Element Momentum (BEM) methods and the estimation of 3D effects due to the rotating blade were undertaken. One specific shape (with a pronounced flat-back) was selected and tested in the Deutsche WindGuard aeroacoustic Wind Tunnel (DWAA), in Bremerhaven, Germany. A total of 34 polars were measured, included two trailing edge shapes and aerodynamic devices such as vortex generators, gurney flaps, zig-zag tape, and a splitter plate. Considerable changes in lift and drag characteristics were observed due to the use of aerodynamic add-ons. With the studies presented here, we believe we have closed an important technological gap. Full article
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<p>Earlier known profiles of 60% thickness.</p>
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<p>Geometric interpolations between a circle (at hub) and last profile (DU00-W2-401).</p>
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<p>Negative lift for profile (aerovide 0.7) from <a href="#wind-04-00010-f002" class="html-fig">Figure 2</a> at moderate AOAs.</p>
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<p>cL vs. cD (polar) for profile from <a href="#wind-04-00010-f002" class="html-fig">Figure 2</a>.</p>
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<p>Outline of BAL-shapes. BAL003 emphasized.</p>
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<p>CFD for BAL003: <math display="inline"><semantics> <msub> <mi>c</mi> <mi>L</mi> </msub> </semantics></math> vs. AOA. Errors bars for tau calculation indicate amplitude of oscillation due to periodic vortex shedding.</p>
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<p>CFD for BAL003: <math display="inline"><semantics> <msub> <mi>c</mi> <mi>D</mi> </msub> </semantics></math> vs. AOA. Errors bars for tau calculation indicate amplitude of oscillation due to periodic vortex shedding.</p>
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<p>Proposed D60B-V5 airfoil for wind-tunnel testing. With and without tilted TE. Shape not to scale.</p>
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<p>Smooth implementation of new profile with tilted TE (green) along a generic blade. Shape not to scale.</p>
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<p>Chord distribution of CIG10MW and IEA-22MW reference wind turbine.</p>
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<p>Twist distribution of CIG10MW and IEA-22MW reference wind turbine.</p>
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<p>Location of newly developed t/c = 0.6 profile for generic CIG10MW blade, radius (r) in m, and thickness as t/c. KSS is the name of our BEM Code. Rotor diameter = 100 m. Compared to IEA-22 MW reference wind turbine [<a href="#B17-wind-04-00010" class="html-bibr">17</a>].</p>
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<p>Complete set of polars (<math display="inline"><semantics> <msub> <mi>c</mi> <mi>L</mi> </msub> </semantics></math> vs. AOA) for CIG10MW blade. D60B-V5 is presented with simulated and measured data, see <a href="#wind-04-00010-t002" class="html-table">Table 2</a>.</p>
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<p>Experimental set-up.</p>
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<p>Sample velocity field at <math display="inline"><semantics> <mrow> <msup> <mn>0</mn> <mo>°</mo> </msup> <mi>A</mi> <mi>O</mi> <mi>A</mi> </mrow> </semantics></math>, simulated with ANSYS fluent.</p>
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<p>Sample velocity field at <math display="inline"><semantics> <mrow> <msup> <mn>6</mn> <mo>°</mo> </msup> <mi>A</mi> <mi>O</mi> <mi>A</mi> </mrow> </semantics></math>, simulated with ANSYS fluent.</p>
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<p>Pressure (coefficient) of clean configuration at <math display="inline"><semantics> <msup> <mn>0</mn> <mo>°</mo> </msup> </semantics></math> AOA. Results from panel code Xfoil and RANS codes ANSYS fluent and DLR tau.</p>
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<p>Pressure (coefficient) of clean configuration at <math display="inline"><semantics> <msup> <mn>6</mn> <mo>°</mo> </msup> </semantics></math> AOA. Results from panel code Xfoil and RANS codes ANSYS-fluent and DLR-tau.</p>
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<p>Drag reduction by using a splitter plate; 2D-CFD results for clean case included.</p>
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<p>Selected measured polars, description of configuration is shown in <a href="#wind-04-00010-t003" class="html-table">Table 3</a>.</p>
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<p>Comparison of radial distribution of tangential (driving) forces. BEM (KSS and WZX-A) vs. 3D-RANS. Especially in the part from where the 60% profile (<math display="inline"><semantics> <mrow> <mi>r</mi> <mo>&lt;</mo> <mn>17</mn> </mrow> </semantics></math> m) starts, pronounced (positive) changes are visible. om = <math display="inline"><semantics> <mi>ω</mi> </semantics></math> is the angular velocity.</p>
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<p>Comparison of simulated 2D polar from three codes (Xfoil, fluent, and tau) for the final design: <math display="inline"><semantics> <msub> <mi>c</mi> <mi>L</mi> </msub> </semantics></math> vs. AOA.</p>
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<p>Comparison of simulated 2D polars from three codes applied for the final design: <math display="inline"><semantics> <msub> <mi>c</mi> <mi>D</mi> </msub> </semantics></math> vs. AOA.</p>
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<p>RN variation for clean airfoil D60B-V5 measured in DWAA.</p>
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24 pages, 15676 KiB  
Article
Numerical Simulation Method for the Aeroelasticity of Flexible Wind Turbine Blades under Standstill Conditions
by Xianyou Wu, Rongxiang Liu, Yan Li, Pin Lv, Chuanqiang Gao and Kai Feng
Energies 2024, 17(14), 3395; https://doi.org/10.3390/en17143395 - 10 Jul 2024
Viewed by 1099
Abstract
With the trend towards larger and lighter designs of wind turbines, blades are progressively being developed to have longer and more flexible configurations. Under standstill conditions, the separated flow induced by a wide range of incident flow angles can cause complex aerodynamic elastic [...] Read more.
With the trend towards larger and lighter designs of wind turbines, blades are progressively being developed to have longer and more flexible configurations. Under standstill conditions, the separated flow induced by a wide range of incident flow angles can cause complex aerodynamic elastic phenomena on blades. However, classical momentum blade element theory methods show limited applicability at high angles of attack, leading to significant inaccuracies in wind turbine performance prediction. In this paper, the geometrically accurate beam theory and high-fidelity CFD method are combined to establish a bidirectional fluid–structure coupling model, which can be used for the prediction of the aeroelastic response of wind turbine blades and the analysis of fluid–structure coupling. Aeroelastic calculations are carried out for a single blade under different working conditions to analyze the influence of turbulence, gravity and other parameters on the aeroelastic response of the blade. The results show that the dominant frequency of the vibration deformation response in the edgewise direction is always the same as the first-order edgewise frequency of the blade when the incoming flow condition is changed. The loading of gravity will make the aeroelastic destabilization of the blade more significant, which indicates that the influence of gravity should be taken into account in the design of the aeroelasticity of the wind turbine. Increasing the turbulence intensity will change the dominant frequency of the vibration response in the edgewise direction, and at the same time, it will be beneficial to the stabilization of the aeroelasticity response. Full article
(This article belongs to the Section I: Energy Fundamentals and Conversion)
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Figure 1
<p>Schematic diagram of the coupling principle of fluid–structure coupling calculation.</p>
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<p>Comparison of dynamic stall aerodynamic response.</p>
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<p>Comparison of blade root torque in the edgewise direction.</p>
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<p>Comparison of blade root torque in the flapwise direction.</p>
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<p>Photo of measurement turbine.</p>
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<p>Implementation flow of mesh interpolation and deformation technology.</p>
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<p>Integral mapping of aerodynamic force from grid to beam node.</p>
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<p>(<b>a</b>) The selection method of the corresponding points of blade elements on the beam nodes of the surface grid. (<b>b</b>,<b>c</b>) Different selection policies for O3.</p>
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<p>Trajectory of a certain point on the blade.</p>
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<p>Blade model.</p>
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<p>Surface grid at blade root.</p>
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<p>Surface grid at blade tip.</p>
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<p>Schematic diagram of simulation case conditions.</p>
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<p>Comparison between Bladed and the CFD-CSD method of this paper for the calculation of blade root flapwise moment for B operating condition.</p>
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<p>Comparison between Bladed and the CFD-CSD method of this paper for the calculation of blade root flapwise moment for C condition.</p>
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<p>Time–frequency diagram of blade tip edgewise deformation under condition A.</p>
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<p>Time–frequency diagram of tip edgewise deformation under condition B.</p>
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<p>Time–frequency diagram of tip edgewise deformation under condition C.</p>
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<p>Comparison of tip edgewise deformation under conditions B and C.</p>
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<p>Time–frequency diagram of blade tip flapwise deformation under condition A.</p>
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<p>Time–frequency diagram of blade tip flapwise deformation under condition B.</p>
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<p>Time–frequency diagram of blade tip flapwise deformation under condition C.</p>
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<p>Comparison of tip flapwise deformation under conditions B and C.</p>
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<p>Time–frequency diagram of edgewise-direction force of blade root under condition A.</p>
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<p>Comparison of edgewise-direction force of blade root under conditions B and C.</p>
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<p>Time–frequency diagram of edgewise-direction moment of blade root under condition A.</p>
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<p>Comparison of edgewise-direction moment of blade root under conditions B and C.</p>
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<p>Calculated working conditions of a three-bladed rotor.</p>
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<p>Time-domain plot of the edgewise deformation of the three-blade tip.</p>
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<p>Time–frequency plot of blade tip edgewise deformation. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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<p>Time-domain plot of the flapwise deformation of the three-blade tip.</p>
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<p>Time–frequency plot of blade tip flapwise deformation. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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<p>Time–frequency plot of blade tip flapwise deformation. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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<p>Time-domain plot of edgewise-direction moments at the root of three blades.</p>
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<p>Time–frequency plot of the edgewise moment at the blade root. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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<p>Time–frequency plot of the edgewise moment at the blade root. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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<p>Time-domain plot of flapwise-direction moments at the root of three blades.</p>
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<p>Time–frequency plot of the flapwise moment at the blade root. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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<p>Time–frequency plot of the flapwise moment at the blade root. (<b>a</b>) blade 1; (<b>b</b>) blade 2; (<b>c</b>) blade 3.</p>
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33 pages, 15478 KiB  
Article
Use of Dampers to Improve the Overspeed Control System with Movable Arms for Butterfly Wind Turbines
by Yutaka Hara, Hiroyuki Higami, Hiromitsu Ishikawa, Takeshi Ono, Shigenori Saito, Kenichiro Ichinari and Katsushi Yamamoto
Energies 2024, 17(11), 2727; https://doi.org/10.3390/en17112727 - 3 Jun 2024
Viewed by 796
Abstract
To reduce the cost of small wind turbines, a prototype of a butterfly wind turbine (6.92 m in diameter), a small vertical-axis type, was developed with many parts made of extruded aluminum suitable for mass production. An overspeed control system with movable arms [...] Read more.
To reduce the cost of small wind turbines, a prototype of a butterfly wind turbine (6.92 m in diameter), a small vertical-axis type, was developed with many parts made of extruded aluminum suitable for mass production. An overspeed control system with movable arms that operated using centrifugal and aerodynamic forces was installed for further cost reduction. Introducing this mechanism eliminates the need for large active brakes and expands the operating wind speed range of the wind turbine. However, although the mechanism involving the use of only bearings is simple, the violent movement of the movable arms can be a challenge. To address this in the present study, dampers were introduced on the movable arm rotation axes to improve the movement of the movable arms. To predict the behavior of a movable arm and the performance of the wind turbine with the mechanism, a simulation method was developed based on the blade element momentum theory and the equation of motion of the movable arm system. A comparison of experiments and predictions with and without dampers demonstrated qualitative agreement. In the case with dampers, measurements confirmed the predicted increase in the rotor rotational speed when the shorter ailerons installed perpendicularly to the movable arms were used to achieve the inclination. Field experiments of the generated power at a wind speed of 6 m/s (10 min average) showed relative performance improvements of 11.4% by installing dampers, 91.3% by shortening the aileron length, and 57.6% by changing the control target data. The movable arm system with dampers is expected to be a useful device for vertical-axis wind turbines that are difficult to control. Full article
(This article belongs to the Special Issue Vertical Axis Wind Turbines: Current Technologies and Future Trends)
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<p>(<b>a</b>) Photo of prototype of butterfly wind turbine with diameter of 6.92 m and three looped blades; (<b>b</b>) Schematic of a movable arm with an aileron as overspeed control system, which works as aerodynamic brake when rotor rotational speed is high or wind speed is strong.</p>
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<p>Schematic illustrating main size of prototype of butterfly wind turbine, coordinate, vertical distribution of wind speed used in analytical prediction of turbine performance here.</p>
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<p>Schematic of horizontal arm including movable arm and aileron. Cross-section of vertical blade is depicted in red in this figure. Blue arrow depicts centrifugal force acting on section (<span class="html-italic">i</span>) of aileron when it is inclined.</p>
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<p>Schematic of movable arm and aileron inclined at slant angle <span class="html-italic">η</span>. Cross-section of fixed arm is shown in light brown. Rotational axis of movable arm is horizontally shifted by <span class="html-italic">aad</span> (axis-axis-distance) from rotational axis of wind turbine rotor.</p>
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<p>(<b>a</b>) Schematic of installation position of dampers; (<b>b</b>) Graph illustrating rotational angular velocity dependence of total resistance moment of six dampers installed in one movable arm.</p>
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<p>Schematic of quadruple-multiple streamtube (QMS) model [<a href="#B28-energies-17-02727" class="html-bibr">28</a>]. In analysis, azimuth angle <span class="html-italic">Ψ</span> is defined as rotor revolving counterclockwise; origin is set at 90° from the most upstream direction. However, origin of integer parameter <span class="html-italic">I</span> is defined at opposite side of that of <span class="html-italic">Ψ</span>.</p>
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<p>Flow speed distributions predicted in two vertical levels (<b>a</b>) <span class="html-italic">W</span> = 5 and (<b>b</b>) <span class="html-italic">W</span> = 10 under condition of <span class="html-italic">V</span><sub>∞</sub> = 6 m/s, <span class="html-italic">N</span> = 70 rpm, <span class="html-italic">λ</span> = 4.224. Black symbol depicts flow speed at outer rotor in upwind side, red symbol depicts that at outer rotor in downwind side, blue symbol depicts that at inner rotor in upwind side, and green symbol depicts that at inner rotor in downwind side. Analytical prediction of movement of movable arm assumed flow speed at outer rotor in equator level (<span class="html-italic">W</span> = 10) as flow speed which movable arm and aileron encounter.</p>
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<p>Flowchart of prediction of rotor performance (force, torque, and power) of butterfly wind turbine including behavior of movable arms. Prediction calculation assumes stationarity. However, in this flowchart, integer parameters <span class="html-italic">I</span> and <span class="html-italic">n</span> indicate azimuth and virtual rotor revolution number in calculation to obtain converged state.</p>
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<p>Prediction of variations in slant angle of movable arm in cases with and without dampers under condition of <span class="html-italic">V</span><sub>∞</sub> = 6 m/s, <span class="html-italic">N</span> = 70 rpm, <span class="html-italic">λ</span> = 4.224, and <span class="html-italic">span</span><sup>(ail)</sup> = 600 mm. Horizontal axis of this figure depicts accumulated azimuth angle until 16 virtual rotor revolutions at which convergence of slant angle was obtained for case with dampers. However, behavior of movable arm in case without dampers did not reach the convergence even at 30 rotor revolutions in this condition.</p>
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<p>Prediction of power coefficient of prototype of butterfly wind turbine equipped with movable arms with dampers. Span length of aileron <span class="html-italic">span</span><sup>(ail)</sup> is assumed to be 600 mm in this prediction.</p>
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<p>Schematic of measurement system in field experiments of prototype. Data of tilt sensor installed in movable arm are transmitted by radio wave to receiver. Signal generated by photoelectric pulse generator was used to measure azimuth angle and rotor rotational speed.</p>
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<p>Representative experimental data of slant angle of movable arm, rotor rotational speed, and wind speed in case without dampers. Although decrease in rotor rotational speed was observed when movable arms inclined, simultaneously, violent movements of movable arms and noise generated by their impact on stoppers that limit slant angle range were observed.</p>
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<p>Representative experimental data of slant angle of movable arm, rotor rotational speed, and wind speed in case with dampers. Movements of movable arms became gentry and noise is gone by effects of dampers.</p>
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<p>Comparison of variation in slant angle of movable arm during one rotor revolution between cases with and without dampers. Horizontal axis is azimuth angle defined in <a href="#energies-17-02727-f006" class="html-fig">Figure 6</a>.</p>
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<p>Prediction of rotor torque of prototype equipped with movable arms without dampers. Span length of aileron <span class="html-italic">span</span><sup>(ail)</sup> is assumed to be 600 mm in this prediction. Black solid line is control target (load torque), and intersections with rotor torque curves give expected operating states.</p>
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<p>Comparison of wind speed dependence of rotor rotational speed between prediction and experiment data, which was averaged for 20 s, in case without dampers and <span class="html-italic">span</span><sup>(ail)</sup> of 600 mm.</p>
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<p>Prediction of rotor torque of prototype equipped with movable arms with dampers. Span length of aileron <span class="html-italic">span</span><sup>(ail)</sup> is assumed to be 600 mm in this prediction. Black solid line is control target (load torque), and intersections with rotor torque curves give expected operating states.</p>
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<p>Comparison of wind speed dependence of rotor rotational speed between prediction and experiment data, averaged for 20 s, in case with dampers and <span class="html-italic">span</span><sup>(ail)</sup> of 600 mm.</p>
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<p>Prediction of rotor torque of prototype equipped with movable arms with dampers. Span length of aileron <span class="html-italic">span</span><sup>(ail)</sup> is assumed to be 400 mm in this prediction. Black solid line is control target (load torque), and intersections with rotor torque curves give expected operating states.</p>
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<p>Comparison of wind speed dependence of rotor rotational speed between prediction and experiment data, which was averaged for 20 s, in case with dampers and <span class="html-italic">span</span><sup>(ail)</sup> of 400 mm.</p>
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<p>Prediction of rotor torque shown above is the same as that in <a href="#energies-17-02727-f019" class="html-fig">Figure 19</a> presenting the case with dampers and short ailerons of <span class="html-italic">span</span><sup>(ail)</sup> = 400 mm. Control target (load torque) depicted in solid black line is different from that used in cases shown in <a href="#energies-17-02727-f015" class="html-fig">Figure 15</a>, <a href="#energies-17-02727-f017" class="html-fig">Figure 17</a> and <a href="#energies-17-02727-f019" class="html-fig">Figure 19</a>.</p>
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<p>Comparison of wind speed dependence of rotor rotational speed between prediction and experiment data (averaged for 20 s), for case with dampers and <span class="html-italic">span</span><sup>(ail)</sup> of 400 mm after changing control target data.</p>
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<p>Comparison among each turbine condition for electric generation obtained experimentally (averaged for 10 min). Dashed-dotted line shows prediction after further improving control target data.</p>
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<p>Schematic of relative wind velocities that <span class="html-italic">i</span>-th section of aileron encounters when it is under condition of azimuth <span class="html-italic">Ψ</span> with slant angle of <span class="html-italic">η</span>.</p>
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<p>Schematic of correction of velocity component <span class="html-italic">V</span><sub>t</sub><sup>(ail)</sup> by considering self-rotation movement of aileron and derivation of aerodynamic force components.</p>
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<p>Schematic of relative wind velocities that <span class="html-italic">j</span>-th section of movable arm encounters when it is under condition of azimuth <span class="html-italic">Ψ</span> with slant angle of <span class="html-italic">η</span>. Difference between aerodynamic center position and reference position of movable arm is ignored because it is small.</p>
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<p>Schematic of correction of velocity component <span class="html-italic">V</span><sub>t</sub><sup>(ma)</sup> by considering self-rotation movement of movable arm and the derivation of aerodynamic force components.</p>
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<p>Variations in 10-minute-averaged rotor rotational and wind speeds under the same time span shown in <a href="#energies-17-02727-f018" class="html-fig">Figure 18</a> [from 12:00 on 24 January 2023 to 12:00 on 25 January 2023]. Wind turbine with dampers and ailerons of <span class="html-italic">span</span><sup>(ail)</sup> = 600 mm was used.</p>
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<p>Variations in 20-second-averaged rotor rotational and wind speeds under the same time span shown in <a href="#energies-17-02727-f018" class="html-fig">Figure 18</a> [from 12:00 on 24 January 2023 to 12:00 on 25 January 2023]. Wind turbine with dampers and ailerons of <span class="html-italic">span</span><sup>(ail)</sup> = 600 mm was used.</p>
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<p>Comparison of wind speed dependence of rotor rotational speed between prediction and experiment data: (<b>a</b>) 10-minute-averaged data corresponding to <a href="#energies-17-02727-f0A5" class="html-fig">Figure A5</a>; (<b>b</b>) 20-second-averaged data corresponding to <a href="#energies-17-02727-f0A6" class="html-fig">Figure A6</a> (same as <a href="#energies-17-02727-f018" class="html-fig">Figure 18</a>).</p>
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<p>Enlarged view of <a href="#energies-17-02727-f0A6" class="html-fig">Figure A6</a> for time range from 5300 to 7300 s. Time variations of 20-second-averaged rotor rotational and wind speeds.</p>
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<p>Enlarged view of <a href="#energies-17-02727-f0A6" class="html-fig">Figure A6</a> for time range from 5300 to 7300 s. Time variations in rotor rotational and wind speeds obtained based on sampling interval of 1.53 s, for which no moving average was processed.</p>
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18 pages, 6073 KiB  
Article
Development of Individual Rotor Mutual Induction (IRMI) Method for Coaxial Counter-Rotating Rotor
by Shigeo Yoshida, Haruto Fuchiwaki and Koji Matsuoka
Appl. Sci. 2024, 14(11), 4782; https://doi.org/10.3390/app14114782 - 31 May 2024
Viewed by 577
Abstract
A coaxial counter-rotating rotor (CCRR), which has two rotors rotating in the opposite directions on the same axis, is seen as a promising technology for low-cost floating tidal current/ocean current power generation using single-point mooring, as the torques of the front and rear [...] Read more.
A coaxial counter-rotating rotor (CCRR), which has two rotors rotating in the opposite directions on the same axis, is seen as a promising technology for low-cost floating tidal current/ocean current power generation using single-point mooring, as the torques of the front and rear rotors are cancelled. In the evaluation and design of such turbines, there is a need for an accurate analysis method with a low computational load that considers the strong mutual induction between the two rotors placed close together. An individual rotor mutual induction (IRMI) method was developed in this study, aiming to significantly reduce the calculation time of conventional computational fluid dynamics (CFD), considering the mutual induction that are not considered in conventional modified blade element and momentum methods. In this method, the basic characteristics of the front and rear rotors are calculated in advance using full-model CFD. In calculations for the CCRR, in addition to these individual characteristics of each rotor, the interaction between the rotors is considered using the actuator disk model CFD calculated in advance. The condition where the torques of the front and rear rotors are cancelled is determined at the same time. This method was used to analyze models in which the front and rear rotors were approximately the same diameter and placed close together (10% of the rotor diameter). A comparison with the mixing plane model CFD revealed that they agree quite well when mutual induction is considered, although both the power and thrust are overcalculated when it is ignored. The simulation time of the IRMI would be almost counter-proportional to the numbers of TSR conditions to solve as compared with the CFD with the MP model. Full article
(This article belongs to the Section Fluid Science and Technology)
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<p>Flowchart of the IRMI method.</p>
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<p>Outlines of the CCRR blades [<a href="#B15-applsci-14-04782" class="html-bibr">15</a>].</p>
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<p>Outlines of the CCRR: three-bladed FR and five-bladed RR.</p>
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<p>CFD domain for the CCRR with the MP method.</p>
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<p>CFD domains for the individual rotor CFD.</p>
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<p>Mesh around the blade section.</p>
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<p>Individual rotor thrust coefficients.</p>
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<p>Individual rotor power coefficients.</p>
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<p>Individual rotor torque coefficients.</p>
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<p>Normalized induced velocity distribution around an actuator disk.</p>
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<p>Mutual induction factors at each rotor position.</p>
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<p>Average mutual induction factor at each rotor position.</p>
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<p>Examples of convergence: TSR of FR = 2.2.</p>
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<p>IRMI results with mutual induction ignored.</p>
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<p>IRMI results when considering mutual induction.</p>
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19 pages, 23333 KiB  
Article
Research on the Calculation Method of Propeller 1P Loads Based on the Blade Element Momentum Theory
by Wenhui Yan, Xiao Tian, Junwei Zhou and Kun Zhang
Aerospace 2024, 11(5), 332; https://doi.org/10.3390/aerospace11050332 - 23 Apr 2024
Viewed by 2761
Abstract
Aircraft propellers produce relatively large in-plane loads, called propeller 1P loads, during maneuvers such as turning, diving, and lifting, and these loads can negatively affect the flight and control of the aircraft. In order to study the change rule of 1P aerodynamic loads, [...] Read more.
Aircraft propellers produce relatively large in-plane loads, called propeller 1P loads, during maneuvers such as turning, diving, and lifting, and these loads can negatively affect the flight and control of the aircraft. In order to study the change rule of 1P aerodynamic loads, in this paper, a mathematical model of the propeller 1P aerodynamic loads has been developed based on the blade element momentum theory. This mathematical model was then corrected using the Pitt–Peters incoming flow correction method, the Prandtl tip correction method, and the propeller root flow correction method. Based on this mathematical model, a calculation procedure for the propeller 1P aerodynamic loads was developed using MATLAB software, and the accuracy of the procedure was verified by comparing the results with CFD simulation results. Numerical simulations show that the results calculated based on the proposed mathematical model for the coefficients of thrust, power, bending moment, and the tangential force of the propeller have an error of less than ±6.00% compared to the CFD simulation results. For a small inflow angle, the coefficients of bending moment and tangential force of the whole propeller fluctuate in a small range. But, as the inflow angle increases, the fluctuation range of the aerodynamic characteristic parameters of the propeller increases and the fluctuation becomes more complicated. Through numerical calculations, it has been shown that the mathematical model presented herein is reliable and accurate. In addition, it greatly shortens the calculation time and improves the calculation efficiency. It is expected that the proposed model can provide a certain help for the strength design of the propeller structure and the study of the aerodynamic performance of the whole aircraft. Full article
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Figure 1
<p>BEM theoretical velocity diagram.</p>
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<p>Definition of the propeller coordinate system and decomposition of the incoming velocity.</p>
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<p>Blade element force diagram.</p>
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<p>Propeller wake deflection.</p>
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<p>Variation in <span class="html-italic">F<sub>cl</sub></span> with <span class="html-italic">r/R</span>.</p>
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<p>Flowchart of calculation procedure for propeller 1P load.</p>
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<p>Geometry of the propeller model.</p>
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<p>Propeller grid.</p>
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<p>Variation in <span class="html-italic">C<sub>P</sub></span> with propeller rotation speed for different grid numbers.</p>
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<p>Variation in propeller <span class="html-italic">C<sub>T</sub></span> and <span class="html-italic">C<sub>P</sub></span> with rotational speed at zero inflow angle.</p>
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<p>Propeller pressure contour plot and streamline distribution: (<b>a</b>) propeller pressure contour; (<b>b</b>) propeller streamline distribution.</p>
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<p>Schematic diagram of the azimuthal angle.</p>
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<p>Variation in propeller aerodynamic parameters with azimuth at an inflow angle of 9°: (<b>a</b>) <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span>; (<b>b</b>) <span class="html-italic">C<sub>T</sub></span> and <span class="html-italic">C<sub>P</sub></span>.</p>
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<p>Variation in the deviation with azimuth at an inflow angle of 9°: (<b>a</b>) <span class="html-italic">e C<sub>B</sub></span> and <span class="html-italic">e C<sub>F</sub></span>; (<b>b</b>) <span class="html-italic">e C<sub>T</sub></span> and <span class="html-italic">e C<sub>P</sub></span>.</p>
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<p>Variation in propeller aerodynamic parameters with azimuth at an inflow angle of 12°: (<b>a</b>) <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span>; (<b>b</b>) <span class="html-italic">C<sub>T</sub></span> and <span class="html-italic">C<sub>P</sub></span>.</p>
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<p>Variation in the deviation with azimuth at an inflow angle of 12°: (<b>a</b>) <span class="html-italic">e C<sub>B</sub></span> and <span class="html-italic">e C<sub>F</sub></span>; (<b>b</b>) <span class="html-italic">e C<sub>T</sub></span> and <span class="html-italic">e C<sub>P</sub></span>.</p>
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<p>Variation in propeller aerodynamic parameters with azimuth at an inflow angle of 15°: (<b>a</b>) <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span>; (<b>b</b>) <span class="html-italic">C<sub>T</sub></span> and <span class="html-italic">C<sub>P</sub></span>.</p>
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<p>Variation in the deviation with azimuth at an inflow angle of 15°: (<b>a</b>) <span class="html-italic">e C<sub>B</sub></span> and <span class="html-italic">e C<sub>F</sub></span>; (<b>b</b>) <span class="html-italic">e C<sub>T</sub></span> and <span class="html-italic">e C<sub>P</sub></span>.</p>
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<p>Variation in <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> of the whole propeller with azimuth for different inflow angles: (<b>a</b>) <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> at an inflow angle of 9°; (<b>b</b>) <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> at an inflow angle of 12°; and (<b>c</b>) <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> at an inflow angle of 15°.</p>
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<p><span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> of the whole propeller at different inflow angles: (<b>a</b>) <span class="html-italic">C<sub>B</sub></span> of the whole propeller; (<b>b</b>) <span class="html-italic">C<sub>F</sub></span> of the whole propeller.</p>
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<p>Variation in <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> with advance ratio: (<b>a</b>) variation in <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> of single-blade propeller with advance ratio; (<b>b</b>)variation in <span class="html-italic">C<sub>B</sub></span> and <span class="html-italic">C<sub>F</sub></span> of three-blade propeller with advance ratio.</p>
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18 pages, 8358 KiB  
Article
Wind Tunnel Investigation of Transient Propeller Loads for Non-Axial Inflow Conditions
by Catharina Moreira, Nikolai Herzog and Christian Breitsamter
Aerospace 2024, 11(4), 274; https://doi.org/10.3390/aerospace11040274 - 30 Mar 2024
Cited by 3 | Viewed by 1836
Abstract
Recent developments in electrical Vertical Take-off and Landing (eVTOL) vehicles show the need for a better understanding of transient aero-mechanical propeller loads for non-axial inflow conditions. The variety of vehicle configurations conceptualized with different propellers in terms of blade geometry, number of blades, [...] Read more.
Recent developments in electrical Vertical Take-off and Landing (eVTOL) vehicles show the need for a better understanding of transient aero-mechanical propeller loads for non-axial inflow conditions. The variety of vehicle configurations conceptualized with different propellers in terms of blade geometry, number of blades, and their general integration concept results in aerodynamic loads on the propellers which are different from those on conventional fixed-wing aircraft propellers or helicopter rotors. Such varying aerodynamic loads have to be considered in the vehicle design as a whole and also in the detailed design of their respective electric propulsion systems. Therefore, an experimental approach is conducted on two different propeller blade geometries and a varying number of blades with the objective to explore the characteristics at non-axial inflow conditions. Experimental data are compared with calculated results of a low-fidelity Blade Element Momentum Theory (BEMT) approach. Average thrust and side force coefficients are shown to increase with inflow angle, and this trend is captured by the implemented numerical method. Measured thrust and in-plane forces are shown to oscillate at the blade passing frequency and its harmonics, with higher amplitudes at higher angles of inflow or lower number of blades. Full article
(This article belongs to the Special Issue Gust Influences on Aerospace)
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Figure 1
<p>Components of the experimental propeller test bench for the wind tunnel campaign. Propeller blades (not shown on this picture) attached to the rotor fixture on the left.</p>
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<p>(<b>a</b>) Test bench with Type A (APC) propeller blades positioned on the rotating platform in the wind tunnel, including axes adopted for load measurements. (<b>b</b>) Outside view of the TUM-AER Wind Tunnel A which was used for these experiments.</p>
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<p>Different propeller types investigated. Type B (Ramoser) propeller enabled the measurement of the same blade geometry in 2-, 3-, 4-, and 5-bladed configurations.</p>
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<p>Comparison of radial blade geometry of APC 18×8E, APC 18×12E, and RAM 18×12 for extracted and published (pub) data by APC; In the figure, (<b>a</b>) radial pitch-to-diameter ratio, (<b>b</b>) radial blade twist, and (<b>c</b>) chord-to-diameter ratio; all data over non-dimensional blade radius <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>/</mo> <mi>R</mi> </mrow> </semantics></math>; published data for chord-to-diameter ratio of both APC blades overlap.</p>
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<p>Three representative airfoil sections of APC 18×8E blade shape.</p>
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<p>Section-wise polar input obtained from 2D-RANS simulations for different Reynolds Numbers for APC 18×8E blade.</p>
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<p>Thrust and power coefficients (<math display="inline"><semantics> <msub> <mi>C</mi> <mi>T</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>C</mi> <mi>P</mi> </msub> </semantics></math>) over advance ratio <span class="html-italic">J</span>; REF data from [<a href="#B23-aerospace-11-00274" class="html-bibr">23</a>].</p>
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<p>Thrust and power coefficients, <math display="inline"><semantics> <msub> <mi>C</mi> <mi>T</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>C</mi> <mi>P</mi> </msub> </semantics></math>, over <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>o</mi> <mi>A</mi> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mo>∞</mo> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and 25 m/s with 5000 RPM; Type B propellers with 2, 3, and 4 blades.</p>
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<p>Experimental force coefficients <math display="inline"><semantics> <msub> <mi>C</mi> <mi>x</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>C</mi> <mi>y</mi> </msub> </semantics></math> over <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>o</mi> <mi>I</mi> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mo>∞</mo> </msub> <mo>=</mo> <mrow> <mo>[</mo> <mn>10</mn> <mo>,</mo> <mn>25</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> m/s at 5000 RPM. Type B propellers with 2, 3, and 4 blades.</p>
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<p>Lateral force coefficients <math display="inline"><semantics> <msub> <mi>C</mi> <mi>X</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>C</mi> <mi>Y</mi> </msub> </semantics></math> for Type B (Ramoser) propeller blades, including variations in pitch, RPM, wind speed, and blade count. Experimental data.</p>
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<p>Experimental forces over one rotation for two Type B propellers at different incidence angles.</p>
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<p>Experimental and BEMT results recorded over a single rotation for in-plane side force coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <msub> <mi>F</mi> <mi>x</mi> </msub> </msub> </semantics></math> and roll moment coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <msub> <mi>M</mi> <mi>x</mi> </msub> </msub> </semantics></math> for Type B propeller as a two-bladed variant (<b>left</b>) and a three-bladed variant (<b>right</b>); 5000 RPM at an inflow of <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mo>∞</mo> </msub> <mo>=</mo> <mn>25</mn> </mrow> </semantics></math> m/s; (<b>a</b>,<b>b</b>) at <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>o</mi> <mi>I</mi> <mo>=</mo> <msup> <mn>15</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>; (<b>c</b>,<b>d</b>) at <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>o</mi> <mi>I</mi> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>.</p>
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<p>Experimental RMS values for the forces acting on the RAM propeller; measured for different propeller blade count and varying <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>o</mi> <mi>I</mi> </mrow> </semantics></math>, colored by experimental mean <math display="inline"><semantics> <msub> <mi>C</mi> <mi>T</mi> </msub> </semantics></math>. Results include different RPM, <math display="inline"><semantics> <msub> <mi>V</mi> <mo>∞</mo> </msub> </semantics></math> and blade pitch configurations.</p>
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<p>Power Spectral Density of lateral force <math display="inline"><semantics> <msub> <mi>F</mi> <mi>y</mi> </msub> </semantics></math> for Type B propeller with 3 blades, pitch 12 inches, 5000 RPM. Experimental data.</p>
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<p>Type B (Ramoser) with two blades, 5000 RPM, pitch 12 inches, wind speed 25 m/s.</p>
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<p>Type B (Ramoser) with three blades, 5000 RPM, pitch 12 inches, wind speed 25 m/s.</p>
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25 pages, 11489 KiB  
Article
Effect of Turbulent Wind Conditions on the Dynamic Characteristics of a Herringbone Planetary Gear System of a Wind Turbine
by Wei-qiang Zhao, Wenhui Zhao, Jie Liu and Na Yang
Machines 2024, 12(4), 227; https://doi.org/10.3390/machines12040227 - 28 Mar 2024
Cited by 1 | Viewed by 1104
Abstract
Due to complex environmental factors, the gear transmission systems of wind turbines are continuously affected by large torque load excitation with periodic and random properties. This paper shares the load-sharing and dynamic characteristics of a herringbone planetary gear system applied in a wind [...] Read more.
Due to complex environmental factors, the gear transmission systems of wind turbines are continuously affected by large torque load excitation with periodic and random properties. This paper shares the load-sharing and dynamic characteristics of a herringbone planetary gear system applied in a wind turbine. The gear dynamic model is established using a typical lumped parameter method, in which the nonlinear transmission errors of the gear pairs and left and right-side coupling stiffness of the herringbone gears are included. With the help of the blade element momentum theory, the precise calculation of the hub load of the wind turbine, which is the external excitation of the gear system, is implemented, in which the wind shear, tower shadow, turbulent effect, and tip loss correction are taken into consideration. The nonlinear dynamic characteristics of the system are obtained using the Runge-Kutta method and then discussed. The results show that the turbulent effect plays a major role in the impact on the load-sharing characteristics, and a reasonable set of the support stiffness of rotational components can improve the load-sharing characteristics of the system. The purpose of this research is to provide some useful references in numerical modelling and methods for designers and researchers of wind turbine transmission systems. Full article
(This article belongs to the Section Turbomachinery)
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<p>Overall research framework diagram.</p>
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<p>The dynamic model of the gear transmission system.</p>
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<p>Meshing dynamic model, (<b>a</b>) sun gear and planetary gears, (<b>b</b>) ring gear and planetary gears, (<b>c</b>) carrier and planetary gears.</p>
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<p>Wind shear coefficient.</p>
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<p>Tower shadow coefficient.</p>
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<p>Wind velocity description diagram, (<b>a</b>) divided wind farm nodes, and (<b>b</b>) the wind velocity of node 1.</p>
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<p>Turbulent wind conditions at <span class="html-italic">t</span> = 3, 6, and 9 s.</p>
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<p>Definition of the position coordinate of the blade element for the load calculation of the wind turbine.</p>
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<p>Aerodynamic load with the wind shear effect of the different blades and hub at <span class="html-italic">t</span> = 10 s, (<b>a</b>) blades, and (<b>b</b>) hub.</p>
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<p>Aerodynamic load with the wind shear and tower effect of the different blades and hub at <span class="html-italic">t</span> = 10 s, (<b>a</b>) blades, and (<b>b</b>) hub.</p>
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<p>Aerodynamic load with the wind shear, tower effect, and turbulence of the different blades and hub at <span class="html-italic">t</span> = 10 s, (<b>a</b>) blades, and (<b>b</b>) hub.</p>
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<p>Global load-sharing coefficient between the ring gear and planetary gear with the change in <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>p</mi> </msub> </mrow> </semantics></math>, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Global load-sharing coefficient between the sun gear and planetary gear with the change in <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>p</mi> </msub> </mrow> </semantics></math>, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Global load-sharing coefficient between the ring gear and planetary gear with the change in <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>r</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>p</mi> </msub> </mrow> </semantics></math>, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Global load-sharing coefficient between the sun gear and planetary gear with the change in <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>r</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>p</mi> </msub> </mrow> </semantics></math>, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Analysis of the proportion of each influencing factor on the time-vary load-sharing coefficient between the ring gear and planetary gear 1 (<math display="inline"><semantics> <mrow> <mi>s</mi> <msubsup> <mi>l</mi> <mrow> <mi>r</mi> <mi>p</mi> <mn>1</mn> </mrow> <mi>q</mi> </msubsup> </mrow> </semantics></math>), (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Analysis of the proportion of each influencing factor on the time-vary load-sharing coefficient between the ring gear and planetary gear 1 (<math display="inline"><semantics> <mrow> <mi>s</mi> <msubsup> <mi>l</mi> <mrow> <mi>s</mi> <mi>p</mi> <mn>1</mn> </mrow> <mi>q</mi> </msubsup> </mrow> </semantics></math>), (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Analysis of the meshing forces between the ring gear and planetary gear 1, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Analysis of the meshing forces between the sun gear and planetary gear 1, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Analysis of the axial displacement from the ring gear, (<b>a</b>) left side, (<b>b</b>) and right side.</p>
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<p>Analysis of the torsional displacement from the ring gear, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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<p>Analysis of the axial displacement from the sun gear, (<b>a</b>) left side, (<b>b</b>) and right side.</p>
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<p>Analysis of the torsional displacement from the sun gear, (<b>a</b>) left side, and (<b>b</b>) right side.</p>
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30 pages, 10193 KiB  
Article
A Comprehensive Investigation of Linear and Nonlinear Beam Models on Flexible Wind Turbine Blade Load Calculations
by Xinwen Ma, Xianghua Peng, Jingwei Sun, Yan Chen and Zhihong Huang
J. Mar. Sci. Eng. 2024, 12(4), 548; https://doi.org/10.3390/jmse12040548 - 25 Mar 2024
Cited by 1 | Viewed by 1701
Abstract
This study was performed to investigate the effects of structural nonlinearity and large deformations on the aeroelastic loads of flexible wind turbine blades. First, a blade structural analysis model was established using the geometrically exact beam (GEB) theory. Subsequently, the blade element momentum [...] Read more.
This study was performed to investigate the effects of structural nonlinearity and large deformations on the aeroelastic loads of flexible wind turbine blades. First, a blade structural analysis model was established using the geometrically exact beam (GEB) theory. Subsequently, the blade element momentum (BEM) theory was corrected using the geometrically exact method leading to the development of a geometrically exact blade element momentum (GE-BEM) model. The results from the GE-BEM model indicated that flapwise deformations always reduce blade fatigue loads, while torsional deformations decrease fatigue loads under low wind speeds but increase them under high wind speeds. Finally, the linear Euler–Bernoulli beam and the GEB were compared to explore the influence of geometric nonlinearity on the blade aeroelastic loads, which revealed that the Euler beam model underestimates the blade loads. The simulations that used the GEB model produced torsional root twist fatigue loads that were 57.49% greater than those generated when the Euler beam model was used. Furthermore, the flapwise bending moment fatigue loads at the root were 8.24% greater than those obtained by the Euler beam model. The smallest discrepancy between the results of the two models was 7.26%, and it corresponded to the edgewise fatigue load. Full article
(This article belongs to the Topic Wind, Wave and Tidal Energy Technologies in China)
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<p>Normalized flapwise deformation at the blade tip for different wind speeds [<a href="#B27-jmse-12-00548" class="html-bibr">27</a>].</p>
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<p>(<b>a</b>) Wind turbine coordinate systems [<a href="#B24-jmse-12-00548" class="html-bibr">24</a>]. (<b>b</b>) Initial and deformed configurations of the GEB.</p>
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<p>Schematic of the blade deformation and the deformed rotor plane frame.</p>
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<p>Airflow velocity vectors and aerodynamic forces in the deformed rotor frame.</p>
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<p>Flowchart of the aeroelastic module algorithm.</p>
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<p>Deformation diagram and calculation conditions for the 45° pre-curved beam.</p>
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<p>Tip displacement and rotation of the 10 MW RWT blade under flapwise static tip loading: (<b>a</b>) flapwise tip displacement, (<b>b</b>) edgewise tip displacement, (<b>c</b>) torsional tip rotation, and (<b>d</b>) longitudinal tip displacement compared with hGAST software [<a href="#B25-jmse-12-00548" class="html-bibr">25</a>].</p>
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<p>Tip displacement and rotation of the 10 MW RWT blade under flapwise static tip loading: (<b>a</b>) flapwise tip displacement, (<b>b</b>) edgewise tip displacement, (<b>c</b>) torsional tip rotation, and (<b>d</b>) longitudinal tip displacement compared with hGAST software [<a href="#B25-jmse-12-00548" class="html-bibr">25</a>].</p>
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<p>(<b>a</b>) Loading and being of the 45° pre-curved cantilever beam [<a href="#B41-jmse-12-00548" class="html-bibr">41</a>] and (<b>b</b>) beam free end time–history responses compared with reference [<a href="#B41-jmse-12-00548" class="html-bibr">41</a>].</p>
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<p>(<b>a</b>) Aerodynamic loads at 0.8 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> for the 10 MW RWT and (<b>b</b>) blade root moments of the 10 MW RWT.</p>
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<p>(<b>a</b>) Blade tip response under turbulent wind conditions and (<b>b</b>) blade tip flapwise velocity.</p>
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<p>(<b>a</b>) Axial inflow wind speed at 0.99 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> in the time domain and (<b>b</b>) attack angle at 0.99 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> in the time domain.</p>
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<p>PSDs of the aerodynamic forces per unit length at 0.8 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> under turbulent wind conditions with an average wind speed of 11.4 m/s: (<b>a</b>) normal aerodynamic force PSD, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </semantics></math>, and (<b>b</b>) tangential aerodynamic force PSD, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>t</mi> </msub> </mrow> </semantics></math>.</p>
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<p>PSDs of the root moments under turbulent wind conditions with an average wind speed of 11.4 m/s: (<b>a</b>) flapwise direction and (<b>b</b>) edgewise direction.</p>
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<p>PSDs of the aerodynamic forces per unit length, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </semantics></math>, at 0.8 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> under turbulent wind conditions with average wind speeds of (<b>a</b>) 7 m/s and (<b>b</b>) 24 m/s.</p>
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<p>(<b>a</b>) Data for the FFA-W3-241 airfoil. (<b>b</b>) Attack angle distribution of the blade.</p>
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<p>Relative differences in the DELs of the aerodynamic forces per unit length at 0.8 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> calculated by the six correction models and the baseline model under turbulent wind fields with mean wind speeds of (<b>a</b>) 7 m/s, (<b>b</b>) 11.4 m/s, and (<b>c</b>) 24 m/s.</p>
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<p>Results from six different simulation models for a turbulent wind field with a mean wind speed of 11.4 m/s: (<b>a</b>) DELs of the normal aerodynamic forces per unit length at 0.8 <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> </mrow> </semantics></math> and (<b>b</b>) PSDs of the flapwise root bending moments.</p>
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<p>Results from six simulation models for a turbulent wind field with a mean wind speed of 11.4 m/s: (<b>a</b>) DELs of the tangential aerodynamic forces per unit length at 0.8 and (<b>b</b>) PSDs of the edgewise root bending moments.</p>
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<p>DELs of the blade root loads calculated by the six different models: (<b>a</b>) flapwise and edgewise root moments and (<b>b</b>) torsional root moments.</p>
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29 pages, 8718 KiB  
Article
Rotor Performance Predictions for Urban Air Mobility: Single vs. Coaxial Rigid Rotors
by Jason Cornelius, Sven Schmitz, Jose Palacios, Bernadine Juliano and Richard Heisler
Aerospace 2024, 11(3), 244; https://doi.org/10.3390/aerospace11030244 - 20 Mar 2024
Cited by 4 | Viewed by 2512
Abstract
This work details the development and validation of a methodology for high-resolution rotor models used in hybrid Blade Element Momentum Theory Unsteady Reynolds Averaged Navier–Stokes (BEMT-URANS) CFD. The methodology is shown to accurately predict single and coaxial rotor performance in a fraction of [...] Read more.
This work details the development and validation of a methodology for high-resolution rotor models used in hybrid Blade Element Momentum Theory Unsteady Reynolds Averaged Navier–Stokes (BEMT-URANS) CFD. The methodology is shown to accurately predict single and coaxial rotor performance in a fraction of the time required by conventional CFD methods. The methodology has three key features: (1) a high-resolution BEMT rotor model enabling large reductions in grid size, (2) a discretized set of momentum sources to interface between the BEMT rotor model and the structured URANS flow solver, and (3) leveraging of the first two features to enable highly parallelized GPU-accelerated multirotor CFD simulations. The hybrid approach retains high-fidelity rotor inflow, wake propagation, and rotor–rotor interactional effects at a several orders of magnitude lower computational cost compared to conventional blade-resolved CFD while retaining high accuracy on steady rotor performance metrics. Rotor performance predictions of thrust and torque for both single and coaxial rotor configurations are compared to test the data that the authors obtained at the NASA Langley 14- by 22-ft. Subsonic Tunnel Facility. Simulations were run with both fully turbulent and free-transition airfoil performance tables to quantify the associated uncertainty. Single rotor thrust and torque were predicted on average within 4%. Coaxial thrust and power were predicted within an average of 5%. A vortex ring state (VRS) shielding phenomenon for coaxial rotor systems is also presented and discussed. The results support that this hybrid BEMT-URANS CFD methodology can be highly parallelized on GPU machines to obtain accurate rotor performance predictions across the full spectrum of possible UAM flight conditions in a fraction of the time required by conventional higher-fidelity methods. This strategy can be used to rapidly create look-up tables with hundreds to thousands of flight conditions using a three-dimensional multirotor CFD for UAM. Full article
(This article belongs to the Special Issue Recent Advances in Applied Aerodynamics)
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<p>Computational cost vs. model fidelity for various rotor analysis approaches [<a href="#B29-aerospace-11-00244" class="html-bibr">29</a>].</p>
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<p>Variables parametrized to create the ~800-flight-condition CFD coaxial rotor performance table [<a href="#B80-aerospace-11-00244" class="html-bibr">80</a>].</p>
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<p>Rotor stand in NASA Langley 14- by 22-ft. Subsonic Tunnel Facility. (<b>a</b>) Fully assembled test stand; (<b>b</b>) motor and load cell arrangement.</p>
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<p>Schematic of KDE 30.5” x 9.7 DUAL-EDN rotor blades and hub [<a href="#B82-aerospace-11-00244" class="html-bibr">82</a>].</p>
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<p>Verification of KDE 30.5” rotor blade geometry. (<b>a</b>) CAD model; (<b>b</b>) scan-generated point cloud (lower surface).</p>
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<p>KDE 30.5” rotor blade—blade discretization by airfoil, Mach, and Reynolds number.</p>
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<p>Fully turbulent vs. free-transition airfoil performance comparison: M = 0.38; Re = 446 k. (<b>a</b>) Lift coefficient vs. angle of attack. (<b>b</b>) Drag coefficient vs. angle of attack.</p>
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<p>C81 grid generation of a KDE airfoil section. (<b>a</b>) Structured airfoil-fitted O-grid; (<b>b</b>) close-up of blunt trailing-edge grid.</p>
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<p>Lift and drag vs. angle of attack for station 8: Re = 426 k; M = 0.43. (<b>a</b>) Lift coefficient vs. angle of attack; (<b>b</b>) drag coefficient vs. angle of attack.</p>
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<p>Lift and drag vs. angle of attack for station 9 at two (Re and Mach) combinations. (<b>a</b>) Lift coefficient vs. angle of attack; (<b>b</b>) drag coefficient vs. angle of attack.</p>
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<p>Pictures of the RotCFD model and grid for the KDE 30.5” rotor. (<b>a</b>) Model front view; (<b>b</b>) model side view; (<b>c</b>) rotor and grid.</p>
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<p>Definition of shaft angle. (<b>a</b>) Axial climb (SA = −90 deg); (<b>b</b>) edgewise (SA = 0 deg); (<b>c</b>) axial descent (SA = 90 deg). Red dot signifies upper rotor.</p>
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<p>Single rotor thrust and torque of free-transition vs. fully turbulent airfoil tables, V = 3.81 m/s. (<b>a</b>) Thrust. (<b>b</b>) Torque.</p>
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<p>Single rotor thrust and torque of free-transition vs. fully turbulent airfoil tables, V = 7.62 m/s. (<b>a</b>) Thrust. (<b>b</b>) Torque.</p>
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<p>Coaxial rotor thrust and torque of free-transition vs. fully turbulent airfoil tables, V = 6.1 m/s. (<b>a</b>) Thrust, (<b>b</b>) Torque.</p>
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<p>Single and coaxial rotor thrust coefficient vs. shaft angle. (<b>a</b>) 75% throttle; (<b>b</b>) 25% throttle.</p>
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<p>Disk plots of rotor lift coefficient in descent conditions at a shaft angle = 90 deg at 25% throttle. (<b>a</b>) 2.29 m/s; (<b>b</b>) 4.57 m/s; (<b>c</b>) 6.10 m/s.</p>
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<p>Velocity magnitudes (<b>left</b>) and vectors (<b>right</b>) at a shaft angle = 90 deg at 25% Throttle. (<b>a</b>) 2.29 m/s; (<b>b</b>) 4.57 m/s; (<b>c</b>) 6.10 m/s.</p>
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26 pages, 11579 KiB  
Article
Algorithm for Propeller Optimization Based on Differential Evolution
by Andry Sedelnikov, Evgenii Kurkin, Jose Gabriel Quijada-Pioquinto, Oleg Lukyanov, Dmitrii Nazarov, Vladislava Chertykovtseva, Ekaterina Kurkina and Van Hung Hoang
Computation 2024, 12(3), 52; https://doi.org/10.3390/computation12030052 - 6 Mar 2024
Cited by 7 | Viewed by 2671
Abstract
This paper describes the development of a methodology for air propeller optimization using Bezier curves to describe blade geometry. The proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span. The goal [...] Read more.
This paper describes the development of a methodology for air propeller optimization using Bezier curves to describe blade geometry. The proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span. The goal of optimization is to identify the appropriate geometry of a propeller that reduces the power required to achieve a given thrust. Because the proposed optimization problem is a constrained optimization process, the technique of generating a penalty function was used to convert the process into a nonconstrained optimization. For the optimization process, a variant of the differential evolution algorithm was used, which includes adaptive techniques of the evolutionary operators and a population size reduction method. The aerodynamic characteristics of the propellers were obtained using the similar to blade element momentum theory (BEMT) isolated section method (ISM) and the XFOIL program. Replacing the angle of geometric twist with the angle of attack of the airfoil section as a design variable made it possible to increase the robustness of the optimization algorithm and reduce the calculation time. The optimization technique was implemented in the OpenVINT code and has been used to design helicopter and tractor propellers for unmanned aerial vehicles. The development algorithm was validated experimentally and using CFD numerical method. The experimental tests confirm that the optimized propeller geometry is superior to commercial analogues available on the market. Full article
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<p>The Bezier curves determine the variation of the design variables.</p>
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<p>Airfoil construction using cubic Bezier curves.</p>
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<p>Drawing of a blade from Bezier curves and the geometric twist curve.</p>
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<p>Aerodynamic coefficients of the CLARK Y profile by different methods.</p>
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<p>Calculation and interpolation of aerodynamic characteristics of the profile by propeller span.</p>
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<p>NACA 5868-9 propeller calculation in ANSYS CFX, (<b>a</b>) the problem statement, (<b>b</b>) the domain dimensions.</p>
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<p>Example of NACA 5868-9 propeller calculation for 1500 rpm and φ<sub>0.75</sub> = 25° case, (<b>a</b>) velocity in stationary domain, (<b>b</b>) pressure at propeller wall.</p>
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<p>Comparison by thrust coefficient (<b>a</b>) and power coefficient (<b>b</b>).</p>
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<p>Convergence of the metrics in the tests corresponding to case 1.</p>
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<p>Top view of the propeller blades obtained for case 1.</p>
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<p>Examples of the cross-sections of the blades obtained for case 1. The airfoils are normalized with chord of length 1.</p>
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<p>Convergence of the metrics in the tests corresponding to case 2.</p>
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<p>Top view of the propeller blades obtained for case 2.</p>
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<p>Examples of the cross-sections (airfoils) of the blades obtained for case 2. The airfoils are normalized with chord of length 1.</p>
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<p>(<b>a</b>) Injection mold for the manufacture of the propellers; (<b>b</b>) manufactured propellers.</p>
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<p>Experimental setup.</p>
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<p>The designed (<b>a</b>) and commercial (<b>b</b>) propellers installed in the experimental setup.</p>
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<p>Experimental data of thrust from required power dependence for propellers: (<b>a</b>) designed, (<b>b</b>) commercial DJI propeller.</p>
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<p>Mean measurement values and scatter of propeller thrust from power requirement: (<b>a</b>) comparison for designed propeller between experimental values and numerical test, calculated by ISM, (<b>b</b>) comparison between experimental values for designed and commercial DJI propeller.</p>
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25 pages, 8998 KiB  
Article
An Investigation of Tidal Stream Turbine Wake Development Using Modified BEM–AD Model
by Chee M. Pang, David M. Kennedy and Fergal O’Rourke
Energies 2024, 17(5), 1198; https://doi.org/10.3390/en17051198 - 2 Mar 2024
Viewed by 948
Abstract
Tidal stream turbines (TST) are a promising option for electricity generation to meet the ever-increasing demand for energy. The actuator disk (AD) method is often employed to represent a TST, to evaluate the TST operating in a tidal flow. While this method can [...] Read more.
Tidal stream turbines (TST) are a promising option for electricity generation to meet the ever-increasing demand for energy. The actuator disk (AD) method is often employed to represent a TST, to evaluate the TST operating in a tidal flow. While this method can effectively reduce the computational cost and provide accurate prediction of far-wake flow conditions, it falls short of fully characterising critical hydrodynamics elements. To address this limitation, a hybrid method is implemented by coupling AD with the blade element momentum (BEM) theory, using detailed performance data, such as thrust, to enhance the prediction of the wake effects. This work focuses on the development of a hybrid BEM–AD method using Reynolds-Averaged Navier–Stokes (RANS) turbulence models within computational fluid dynamics (CFD). Two variations and a hybrid modification of an AD model are presented in this paper. The first modified variation is a velocity variation that takes into account velocity profile inflow into the disk’s configuration. The second modified variation is a radial variation that integrates the blade element theory into the disk’s configuration. The hybrid modified model combines both the velocity profiles influenced and blade element theory in the design and analysis of the actuator disk. Several key investigations on some of the pre-solver parameters are also investigated in this research such as the effect of changing velocity and radial distance on the porosity and loss coefficient of the actuator disk performance. Importantly, this work provides an improved method to evaluate the key wake effects from a TST array which is crucial to determine the power performance of the TST array. Full article
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<p>Simple illustration of flow passing through an actuator disk.</p>
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<p>Geometry of the fluid domain: (<b>a</b>) Front view; (<b>b</b>) Side cross-section view.</p>
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<p>Fluid domain inlet profile: (<b>a</b>) Velocity profile; (<b>b</b>) Turbulence intensity profile.</p>
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<p>Disk domain for case study view from front: (<b>a</b>) Disk dimension; (<b>b</b>) Disk mesh.</p>
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<p>Comparison of numerical prediction with experimental measurements [<a href="#B8-energies-17-01198" class="html-bibr">8</a>] for a range of tip speed ratios: (<b>a</b>) Power coefficient; (<b>b</b>) Thrust coefficient.</p>
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<p>The effect of changing normalised velocity on the (<b>a</b>) Porosity and (<b>b</b>) Resistance coefficient of the velocity variation.</p>
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<p>The radially changing (<b>a</b>) Porosity and (<b>b</b>) Resistance coefficient on the actuator disk with respect to normalised radius of the body region of radial variation.</p>
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<p>The effect of changing normalised velocity on the (<b>a</b>) Porosity and (<b>b</b>) Resistance coefficient on the disk’s base.</p>
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<p>The effect of changing normalised velocity on the (<b>a</b>) Porosity and (<b>b</b>) Resistance coefficient on the disk’s tip.</p>
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<p>The effect of changing normalised velocity and blade section on the (<b>a</b>) Porosity and (<b>b</b>) Resistance coefficient on the disk’s body.</p>
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<p>Isometric sectional view of the concentrated mesh fluid domain.</p>
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<p>Vertical water column comparison between the normal mesh and the concentrated mesh fluid domain in terms of downstream (<b>a</b>) Normalised velocity and (<b>b</b>) Turbulence intensity at 5-disk diameters downstream from the actuator disk.</p>
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<p>Velocity at six different point locations plotted against different mesh setups, with an increasing number of elements.</p>
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<p>Comparison of downstream centreline (<b>a</b>) Velocity and (<b>b</b>) Turbulence intensity (bottom) of the velocity variation BEM–AD model against experimental measurements.</p>
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<p>Comparison of (<b>A</b>) Vertical velocity and (<b>B</b>) Vertical turbulence intensity (right) of the velocity variation BEM–AD model with experimental measurements at downstream distances of (<b>a</b>) 5D, (<b>b</b>) 8D, and (<b>c</b>) 10D.</p>
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<p>Comparison of downstream centreline (<b>a</b>) Velocity and (<b>b</b>) Turbulence intensity (bottom) of the radial variation BEM–AD model against experimental measurements.</p>
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<p>Comparison of (<b>A</b>) Vertical velocity and (<b>B</b>) Vertical turbulence intensity (right) of the radial variation BEM–AD model with experimental measurements at downstream distances of (<b>a</b>) 5D, (<b>b</b>) 8D, and (<b>c</b>) 10D.</p>
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<p>Comparison of downstream centreline (<b>a</b>) Velocity and (<b>b</b>) Turbulence intensity (bottom) of the modified hybrid BEM–AD model against experimental measurements.</p>
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<p>Comparison of (<b>A</b>) Vertical velocity and (<b>B</b>) Vertical turbulence intensity (right) of the modified hybrid BEM–AD model with experimental measurements at downstream distances of (<b>a</b>) 5D, (<b>b</b>) 8D, and (<b>c</b>) 10D.</p>
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<p>Comparison of downstream centreline (<b>a</b>) Velocity and (<b>b</b>) Turbulence intensity (bottom) for the velocity variation, radial variation, and hybrid modification against experimental measurements.</p>
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<p>Velocity contour of (<b>a</b>) Radial variation model and (<b>b</b>) Hybrid modified model from the side view.</p>
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<p>Turbulence intensity contour of (<b>a</b>) Radial variation model and (<b>b</b>) Hybrid modified model from the side view.</p>
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<p>Disk Power density contour of (<b>a</b>) Velocity variation model, (<b>b</b>) Radial variation model, and (<b>c</b>) Hybrid modified model.</p>
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<p>Power density contour of the hybrid modified model from the side view.</p>
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28 pages, 1228 KiB  
Article
Comparison of Unsteady Low- and Mid-Fidelity Propeller Aerodynamic Methods for Whirl Flutter Applications
by Christopher Koch, Nils Böhnisch, Hendrik Verdonck, Oliver Hach and Carsten Braun
Appl. Sci. 2024, 14(2), 850; https://doi.org/10.3390/app14020850 - 19 Jan 2024
Cited by 2 | Viewed by 1979
Abstract
Aircraft configurations with propellers have been drawing more attention in recent times, partly due to new propulsion concepts based on hydrogen fuel cells and electric motors. These configurations are prone to whirl flutter, which is an aeroelastic instability affecting airframes with elastically supported [...] Read more.
Aircraft configurations with propellers have been drawing more attention in recent times, partly due to new propulsion concepts based on hydrogen fuel cells and electric motors. These configurations are prone to whirl flutter, which is an aeroelastic instability affecting airframes with elastically supported propellers. It commonly needs to be mitigated already during the design phase of such configurations, requiring, among other things, unsteady aerodynamic transfer functions for the propeller. However, no comprehensive assessment of unsteady propeller aerodynamics for aeroelastic analysis is available in the literature. This paper provides a detailed comparison of nine different low- to mid-fidelity aerodynamic methods, demonstrating their impact on linear, unsteady aerodynamics, as well as whirl flutter stability prediction. Quasi-steady and unsteady methods for blade lift with or without coupling to blade element momentum theory are evaluated and compared to mid-fidelity potential flow solvers (UPM and DUST) and classical, derivative-based methods. Time-domain identification of frequency-domain transfer functions for the unsteady propeller hub loads is used to compare the different methods. Predictions of the minimum required pylon stiffness for stability show good agreement among the mid-fidelity methods. The differences in the stability predictions for the low-fidelity methods are higher. Most methods studied yield a more unstable system than classical, derivative-based whirl flutter analysis, indicating that the use of more sophisticated aerodynamic modeling techniques might be required for accurate whirl flutter prediction. Full article
(This article belongs to the Collection Structural Dynamics and Aeroelasticity)
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Figure 1

Figure 1
<p>Simplified pylon model used in this work (<b>a</b>) together with a basic stability map (<b>b</b>) showing parameter ranges for whirl flutter, static divergence and stable areas, obtained by varying <math display="inline"><semantics> <msub> <mi>K</mi> <mi>θ</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>K</mi> <mi>ψ</mi> </msub> </semantics></math>.</p>
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<p>Sensitivity of whirl flutter with respect to the individual propeller derivatives for varying scale factors.</p>
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<p>Flow chart for the blade lift and wake equilibrium.</p>
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<p>Chord and twist distributions of the blade (R = <math display="inline"><semantics> <mrow> <mn>1.25</mn> </mrow> </semantics></math><math display="inline"><semantics> <mi mathvariant="normal">m</mi> </semantics></math>).</p>
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<p>Propeller model and coordinate system used in this study.</p>
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<p>Comparison of steady blade loading for operating point <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>V</mi> <mo>=</mo> <msub> <mi>V</mi> <mi>D</mi> </msub> </mrow> </semantics></math>. The top plot shows the out-of-plane force distribution (<math display="inline"><semantics> <msub> <mi>F</mi> <mi>x</mi> </msub> </semantics></math>, in N/m) along the blade span. The bottom plot shows the corresponding tangential or in-plane blade loading (<math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> </mrow> </msub> </semantics></math>).</p>
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<p>Distribution of variation in out-of-plane blade forces and angle of attack during one revolution with steady disc pitch angle. (<b>a</b>) Local blade angle of attack perturbation due to disc pitch <math display="inline"><semantics> <mi>θ</mi> </semantics></math>; (<b>b</b>) Variation in out-of-plane force in UPM simulation.</p>
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<p>Comparison of the four in-plane hub load derivatives under steady disc pitch angle. The top row shows the y- and z-forces, and the second row, the y- and z-moments.</p>
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<p>Frequency-dependent transfer function from disc pitch to in-plane hub loads. The first row shows the real (solid lines) and imaginary (dashed lines) part of the force components, while the second row shows the moments. For better visibility, only UPM results are shown, representative of mid-fidelity codes.</p>
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<p>Comparison of the linearized derivatives. The dashed horizontal line marks the reference derivatives from the Houbolt/Reed method.</p>
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<p>Comparison of whirl flutter stability map predictions with different aerodynamic methods and for different pylon lengths. For the long pylon, the quasi-steady Houbolt/Reed results are outside the range at higher frequencies (5.5 Hz for equal stiffness).</p>
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<p>Comparison of stability measure relative to Houbolt/Reed among different methods at three operating points (rows) and for three pylon lengths (columns).</p>
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<p>Comparison of different linearization strategies for frequency-dependent transfer functions.</p>
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<p>Transfer function from disc pitch to hub in-plane loads for operating points with either no or high thrust setting, calculated with UPM.</p>
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<p>Comparison of whirl flutter stability map predictions with UPM for low and high thrust.</p>
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<p>Comparison of 1P hub loads due to disc pitch of different BEM formulations.</p>
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<p>Comparison of whirl flutter stability predictions of different BEM formulations.</p>
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