# Leaky Flow through Simplified Physical Models of Bristled Wings of Tiny Insects during Clap and Fling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Physical Models

#### 2.2. Dynamically Scaled Robotic Model

#### 2.3. Kinematics

#### 2.4. Test Conditions

#### 2.5. Particle Image Velocimetry (PIV)

#### 2.5.1. PIV Along Chordwise Direction

#### 2.5.2. PIV along Spanwise Direction

#### 2.5.3. Post Processing

#### 2.6. Force Measurements

#### 2.6.1. Data Acquisition

#### 2.6.2. Processing

#### 2.7. Definitions for Calculated Quantities

#### 2.7.1. Vorticity

#### 2.7.2. Lift and Drag Coefficients

#### 2.7.3. Leakiness

_{viscous}is the volumetric flow rate per unit width calculated from the experiments (Figure 8). Q

_{ideal}denotes the volumetric flow rate per unit width of leaky, inviscid flow through the same geometry, calculated as:

_{viscous}) for a bristled wing model was defined based on the amount of fluid passing through the bristles along the wing span, in the direction opposite to the wing motion. Q

_{viscous}was calculated using the equation:

_{solid}is the volumetric flow rate per unit width of fluid displaced by a solid wing in the direction of wing motion, and Q

_{bristled}is the volumetric flow rate per unit width of fluid displaced by a bristled wing model (with the same span as solid wing) in the direction of wing motion.

_{solid}and Q

_{bristled}were calculated along spanwise line L located at a distance of 10% chord length from the right side face of the solid membrane (Figure 8A), when viewing the wing along the x-z plane (Figure 2). The length of line L used is equal to the vertical extent of field of view used in PL-PIV which is about 1.5 times wing span. The volumetric flow rate per unit width displaced by solid wing was calculated from experimental data, as opposed to theoretical estimation, to: (a) account for change in volumetric flow rate per unit width along span due to reverse flow structures at the LE and TE (LEV and TEV); and (b) use the same field of view as bristled wing models.

_{wing}) was calculated using the equation below in a custom script in MATLAB (The Mathworks, Inc., Natick, MA, USA).

#### 2.7.4. Average Shear Stress

## 3. Results

#### 3.1. Chordwise Flow in Clap and Fling: LEV and TEV

#### 3.2. Force Generation during Clap and Fling

#### 3.3. Inter-Bristle Flow during Clap and Fling: Leakiness

## 4. Discussion

#### 4.1. Implications for Aerodynamic Performance

#### 4.2. Comparison with Previous Studies

#### 4.3. Inter-Bristle Flow and Leakiness

#### 4.4. Considerations for Physical Model Studies of Bristled Wings

#### 4.5. Use of Reduced Surface Area for Evaluation of Bristled Wing Force Coefficients

#### 4.6. Interaction of Inter-Bristle Flow with LEV and TEV

#### 4.7. Limitations of the Study

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Validation of Force Measurement Methodology

#### Appendix A.1. Experimental Setup

#### Appendix A.2. Physical Model

#### Appendix A.3. Prescribed Kinematics

#### Appendix A.4. Test Conditions

**Figure A1.**Experimental setup used for validation of aerodynamic force measurement methodology. A single rectangular solid wing (aspect ratio = 3) was prescribed to move in constant translation at a fixed angle of attack ($Re$ = 12), to replicate one of the experiments reported in Sunada et al. [32]: (

**A**) the top view; and (

**B**) the front view of the setup, with coordinate definitions. Drag force acting on the wing was measured for angle of attack ranging from −20 degrees to 50 degrees, using strain gauges bonded to an L-bracket attached to the wing. The L-bracket design and assembly with the wing was identical to Figure 2D. x, y, z are fixed coordinates with respect to the tank; X’, Y’, Z’ are relative coordinates that move with respect to the wing. $\alpha $ = angle of attack; LE = leading edge; TE = trailing edge.

**Figure A2.**Drag force calculation and coefficient of drag (${C}_{D}$) versus angle of attack ($\alpha $): (

**A**) Drag force (${F}_{D}$) was defined as force acting in the direction opposite to wing motion, and calculated using measured forces in tangential direction (${F}_{T}$). (

**B**) Coefficient of drag (${C}_{D}$) versus angle of attack ($\alpha $), where filled squares represent ${C}_{D}$ obtained from this study, empty squares represent data points extracted from Figure 5B of Sunada et al. [32] using ImageJ software [43]. Axes limits in B are retained the same as in Figure 5B of Sunada et al. [32] to not alter uncertainty associated with data point extraction in ImageJ. LE = leading edge; TE = trailing edge.

#### Appendix A.5. Data Acquisition and Processing

**Figure A3.**Time-varying forces in clap and fling at $Re$ = 10: (

**A**,

**B**) dimensional drag (${F}_{D}$) and dimensional lift (${F}_{L}$), respectively, for all wing models. Shading around each curve is used to indicate range of ±1 standard deviation for that particular data (across 30 cycles). W/S = 0 (—); W/S = 0.05 (....); W/S = 0.15 (- -), W/S = 0.23 (– - –), membrane only (thin green line).

**Figure A4.**Time-varying force coefficients in clap and fling at $Re$ = 10: (

**A**,

**B**) scaled coefficients of drag (${{C}_{D}}^{\prime}$) and lift (${{C}_{L}}^{\prime}$), respectively. Scaled force coefficients for the bristled wing model of W/S = 0.23 in (

**A**,

**B**) were evaluated using scaled (or reduced) bristled wing surface area given by Equation (11). Scaled lift and drag coefficients for the solid wing model (W/S = 0) are the same as ${C}_{L}$ and ${C}_{D}$ given by Equations (8) and (9), respectively. W/S = 0 (—); W/S = 0.23(– - –).

#### Appendix A.6. Results

**Figure A5.**Magnified views of inter-bristle flow during fling (${\theta}_{f}$ = 37.5%), visualized from 2D PL-PIV for $Re$ = 10: (

**A**) W/S = 0.05; (

**B**) W/S = 0.15; and (

**C**) W/S = 0.23. Out-of-plane vorticity fields are overlaid with velocity vector fields. Approximate positions of the bristles are shown as filled black circles. Inset shows the relative wing positions in chordwise view along with approximate position of laser plane VP1 used for 2D PL-PIV measurements (see Figure 7 for PL-PIV setup). Recirculating flow around the bristles can be observed for W/S = 0.23.

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**Figure 1.**Physical models of solid and bristled wings tested in this study: (

**A**) solid wing model: W/S = 0; number of bristles (n) = 0; and (

**B**–

**D**) the bristled wing models of W/S∼0.05 (n = 30), W/S∼0.15 (n = 12), and W/S∼0.23 (n = 8), respectively. Note that n refers to number of bristles on both sides of solid membrane. Gap widths between bristles: W ≈ 4.7 mm (for W/S∼0.05), W ≈ 13.7 mm (for W/S∼0.15), and W ≈ 21.2 mm (for W/S∼0.23). Chord (c) = 45 mm; span (S) = 90 mm; solid membrane width (${c}_{\mathrm{s}}$) = 8.5 mm; diameter of bristle (D) ≈ 1.3 mm; length of bristle on either side of solid membrane (${c}_{\mathrm{h}}$) = 18.3 mm. LE = leading edge; TE = trailing edge.

**Figure 2.**Robotic model used in this study: (

**A**) the front view; and (

**B**) the right side view of the experimental setup showing a bristled wing pair mounted on a drag bracket (as an example). (

**C**,

**D**) The front and side views of drag and lift brackets, respectively (strain gauges mounted on both sides of the bracket). A single bristled wing model (W/S = 0.05) is shown in (

**C**,

**D**) as a representative example. (

**E**) The bottom view of a single bristled wing model with directions of measured tangential (${F}_{T}$) and normal forces (${F}_{N}$) experienced during rotation by angle $\alpha $. Lift and drag forces were measured using lift and drag bracket, respectively, by taking components of ${F}_{T}$ and ${F}_{N}$ in the vertical (${F}_{L}$) and horizontal (${F}_{D}$) directions. ${F}_{L}$ = lift force; ${F}_{D}$ = drag force; LE = leading edge; TE = trailing edge; x,y,z are fixed coordinate definitions. The views in (

**A**,

**E**) are referred to in this paper as “spanwise” and “chordwise” views, respectively.

**Figure 3.**Non-dimensional motion profile prescribed for wing motion. Clap and fling kinematics based on Miller and Peskin [28]. Thin line represents non-dimensional linear translational velocity and the thick line indicates non-dimensional rotational velocity. Translational and rotational velocities were both non-dimensionalized by the steady translational velocity of the wing (${U}_{ST}$). Shaded gray region for $\tau $ = 0.8–1.2 indicates the motion profile of a continuous clap and fling cycle where experimental data were acquired. During fling, wing translation started at 50% of wing rotation (50% overlap). During clap, wing translation occurred throughout the wing rotation (100% overlap). Motion profile is shown for only one wing. The motion of the other wing of the pair was identical to this profile but in the opposite direction. Both wings moved along a straight line during both rotation and translation, with no changes to the stroke plane and elevation angles. U = instantaneous wingtip velocity; U during rotation motion is given by ${U}_{\mathrm{rot}}$ and was calculated as the product of chord (c) and angular velocity of the wing (${\omega}_{\mathrm{rot}}$); U during translation is given by ${U}_{\mathrm{trans}}$; $\tau $ = dimensionless time defined by Equation (2).

**Figure 4.**Prescribed wing kinematics. Chordwise view (x-y coordinates, see Figure 2E) of the wing pair at different non-dimensional time points in the cycle ($\tau $ in Figure 3) are shown. (

**A**) $\tau $ = 0–0.2: fling half-stroke—the wings rotate with respect to their trailing edges and translate away starting at 50% of rotation ($\tau $ = 0.1 in Figure 3); $\tau $ = 0.2–0.4: each wing purely translates at 45 degrees angle in opposite directions; $\tau $ = 0.4–0.7: stroke reversal consisting of 90 degrees rotation of each wing in the direction opposite to fling rotation. (

**B**) $\tau $ = 0.8–1: clap half-stroke—the wings rotate with respect to leading edge (LE) and simultaneously translate closer. Maximum angle of each wing is 45 degrees in both clap and fling half-strokes. A video showing the resultant wing motion for the prescribed kinematics is provided (see Supplementary Materials, Video S1).

**Figure 5.**Inter-wing angle variation with dimensionless time ($\tau $): (

**A**) Clap (${\theta}_{c}$); and (

**B**) fling (${\theta}_{f}$). Inter-wing angle for a particular half-stroke is specified both in terms of degrees (left ordinate) and percentage of wing rotation (right ordinate).

**Figure 6.**Setup used for 2D time-resolved particle image velocimetry (TR-PIV) measurements along horizontal plane (HP): (

**A**–

**C**) front view, side view, and bottom view (herein referred to as chordwise view), respectively, of the 2D TR-PIV setup. For a bristled wing model, the laser sheet location was adjusted so that the plane HP passed through the bristles that were closest to mid-span in the chordwise direction. TR-PIV = time-resolved PIV; LE = leading edge; TE = trailing edge; x, y, z are fixed coordinate definitions. Two videos showing representative 2D TR-PIV data in clap and fling are provided (see Supplementary Materials, Videos S2 and S3).

**Figure 7.**Setup used for 2D phase-locked particle image velocimetry (PL-PIV) measurements of inter-bristle flow: (

**A**,

**B**) the front and side views, respectively, of 2D PL-PIV setup. Inter-bristle flow during fling was visualized for 8 equally spaced angular points ranging 12.5–100% fling time at vertical plane VP1 located at 10% chord length from the LE. Inter-bristle flow during clap was visualized for eight equally spaced angular points ranging 0–87.5% clap time at vertical plane VP2 located at 10% chord length from the TE. The setup was rotated using a turn table (located at the base of the acrylic tank) so that the wing span being illuminated was always normal to the fixed laser sheet and fixed camera focal plane (across every angular position where PL-PIV data were acquired). LE = leading edge; TE = trailing edge; x, y, z are fixed coordinate definitions.

**Figure 8.**Volumetric flow rate and leakiness of flow through bristled wings. (

**A**) Diagram of bristled wing (W/S = 0.23) in spanwise view (visualized using 2D PL-PIV), showing the line L where volumetric flow rate was calculated (located at a distance of 10% chord length from the span, in the direction of wing motion). Line “L” is always parallel to the wing span during wing motion. The length of line L is equal to the field of view in PL-PIV (Figure 7), which is about 1.5 times of the span (S). (

**B**) Expected velocity profiles of leaky /reverse flow (opposite to wing motion) for viscous and ideal/inviscid conditions [34]. (

**C**) Calculation of wing tip velocity ${U}_{\mathrm{tip}}$ using Equation (14), where ${U}_{\mathrm{rot}}$ represents instantaneous rotational velocity, ${U}_{\mathrm{trans}}$ represents instantaneous translational velocity, and $\alpha $ represents the instantaneous angle made by a single wing with respect to vertical. ${U}_{\mathrm{tip}}$ denotes the wingtip velocity in the direction normal to the instantaneous wing position. The setup was rotated using a turn table located at the base of the tank to acquire PL-PIV data of inter-bristle flow at different positions (% of ${\theta}_{c}$, ${\theta}_{f}$). LE = leading edge; TE = trailing edge.

**Figure 9.**Chordwise flow during clap visualized from 2D TR-PIV. Velocity vector fields overlaid on out-of-plane z-vorticity (${\omega}_{\mathrm{z}}$) contours at $Re$ = 10 for: (

**A**) solid wing pair (W/S = 0); (

**B**): W/S = 0.05; (

**C**) W/S = 0.15; and (

**D**) W/S = 0.23. Rows I–IV show chronological progression of clap half-stroke for each column (

**A**–

**D**), in increments of 25% clap angle (${\theta}_{c}$): (I) ${\theta}_{c}$ = 12.5%; (II) ${\theta}_{c}$ = 37.5%; (III) ${\theta}_{c}$ = 62.5%; and (IV) ${\theta}_{c}$ = 87.5%. Row V shows x-distribution of ${\omega}_{\mathrm{z}}$ at ${\theta}_{c}$ = 87.5%, extracted at two locations at approximately 5% chord from the LE (y1) and TE (y2). y1 and y2 are shown in Row IV of Column A. Wing positions are superimposed for every time point, where circles indicate leading edge positions. LE = leading edge; TE = trailing edge. Note that the reference vectors used between solid and bristled wings are dissimilar.

**Figure 10.**Chordwise flow during fling visualized from 2D TR-PIV. Velocity vector fields overlaid on out-of-plane z-vorticity (${\omega}_{\mathrm{z}}$) contours at $Re$ = 10 for: (

**A**) solid wing pair (W/S = 0); (

**B**): W/S = 0.05; (

**C**) W/S = 0.15; and (

**D**) W/S = 0.23. Rows I–IV show chronological progression of fling half-stroke for each column (

**A**–

**D**), in increments of 25% fling angle (${\theta}_{f}$): (I) ${\theta}_{f}$ = 25%; (II) ${\theta}_{f}$ = 50%; (III) ${\theta}_{f}$ = 75%; and (IV) ${\theta}_{f}$ = 100%. Row V shows x-distribution of ${\omega}_{\mathrm{z}}$ at ${\theta}_{f}$ = 100%, extracted at two locations at approximately 5% chord from the LE (y1) and 10% chord from the TE (y2). y1 and y2 are shown in Row IV of Column A. Wing positions are superimposed for every time point, where circles indicate leading edge positions. Note that the reference vectors used between solid and bristled wings are dissimilar.

**Figure 11.**Time-varying force coefficients during clap and fling at $Re$ = 10. Shading around each curve is used to indicate range of ±1 standard deviation for that particular data (across 30 cycles). (

**A**,

**B**) The drag coefficient (${C}_{D}$) and lift coefficient (${C}_{L}$), respectively, evaluated using identical surface area for solid and bristled wings. Solid wing, W/S = 0 (—); bristled wings: W/S = 0.05 (....); W/S = 0.15 (- -); W/S = 0.23(– - –). Standard deviation estimates for cycle-averaged ${C}_{L}$ and ${C}_{D}$ across all W/S for $Re$ = 10 were ≤0.12 and ≤0.6, respectively.

**Figure 12.**Variation of peak force coefficients with Reynolds number. Error bars indicating range of ±1 standard deviation are included for every datapoint: (

**A**,

**B**) maximum drag coefficient (${C}_{D,\mathrm{max}}$) and maximum lift coefficient (${C}_{L,\mathrm{max}}$), respectively. Peak force coefficients were calculated from the absolute values of time-varying ${C}_{L}$ and ${C}_{D}$. Identical wing surface areas used in calculation of force coefficients of all wing models. Solid wing, W/S = 0 (- -●- -); bristled wings: W/S = 0.05 (- -⬛- -); W/S = 0.15 (- -▲- -); W/S = 0.23 (- -⧫- -). Standard deviation estimates for cycle-averaged ${C}_{L}$ and ${C}_{D}$ across all W/S and $Re$ were ≤0.16 and ≤0.9, respectively.

**Figure 13.**Variation in ratios of peak force coefficients with Reynolds number: (

**A**,

**B**) differences in peak force coefficients of a bristled wing model to those of the solid wing model for all W/S and $Re$ tested ($\Delta {C}_{F,\mathrm{max}}$; see Equation (10) for definition). $\Delta {C}_{D,\mathrm{max}}$ and $\Delta {C}_{L,\mathrm{max}}$ are shown in (

**A**,

**B**), respectively. (

**C**) The variation in ratio of peak lift to peak drag (${C}_{L,\mathrm{max}}/{C}_{D,\mathrm{max}})$ for all the wing models with $Re$. Solid wing(W/S = 0) (- -●- -); bristled wings: W/S = 0.05 (- -⬛- -); W/S = 0.15 (- -▲- -); W/S = 0.23 (- -⧫- -). ${C}_{D,\mathrm{max}}$ = peak drag coefficient; ${C}_{L,\mathrm{max}}$ = peak lift coefficient. Identical wing surface areas were used in calculations of force coefficients for all the wing models.

**Figure 14.**Horizontal velocity profiles, leakiness, and average shear stress for bristled wing models at $Re$ = 10: (

**A**,

**B**) velocity profiles along the x-direction for solid wing (W/S = 0) and bristled wing models of W/S = 0.05 and 0.23 at ${\theta}_{f}$ = 37.5%, extracted along a vertical line L located at 10% chord length from the right side face of the solid membrane (Figure 8A), when viewing the wing along the x-z plane (Figure 2A). Wing sketches are overlaid on the profiles, where dark black circles indicate approximate bristle positions. Velocity profile for W/S = 0.23 showed recirculating flow around the bristles (not observed in W/S = 0.05). However, presence of gaps showed decrease in the velocity in both bristled wing models. (

**C**) $Le$ as a function of dimensionless time ($\tau $) for $Re$ = 10, calculated at line L via Equations (12)–(16); $Le$ was observed to peak at $\tau $ ≈1.08 (${\theta}_{f}$ = 37.5% fling angle; see Figure 5D). (

**D**) Average shear stress $\overline{{\tau}_{\mathrm{xz}}}$, calculated at line L using Equation (17), as a function of dimensionless time ($\tau $) for $Re$ = 10. W/S = 0.05 (- -⬛- -); W/S = 0.15 (- -▲- -); W/S = 0.23 (- -⧫- -) in (

**C**,

**D**). An inverse relation between $\overline{{\tau}_{\mathrm{xz}}}$ and $Le$ can be observed, i.e., increasing shear stress decreases leakiness and vice-versa.

**Figure 15.**Inter-bristle flow during clap and fling visualized from 2D PL-PIV for bristled wing model of W/S = 0.05 at $Re$ = 10. Out-of-plane vorticity fields are overlaid with velocity vectors to show inter-bristle flow for different ${\theta}_{c}$ and ${\theta}_{f}$ (Figure 5). Insets on top right corners show the position of the wings in chordwise view and laser sheet orientation (VP1 for fling and VP2 for clap). Wing sketch was overlaid on top of flow fields to display the wing position and bristles. White circles with dark outlines represent approximate positions of bristles. During fling, the strength of the shear layers increased with increasing ${\theta}_{f}$ until ${\theta}_{f}$ = 62.5% and then decreased for ${\theta}_{f}$ = 87.5%. During clap, shear layers form around each bristle at all positions. Note that the reference vector lengths (and corresponding velocities) and the vorticity scale bars between clap and fling are dissimilar.

**Figure 16.**Inter-bristle flow during clap and fling visualized from 2D PL-PIV for bristled wing model of W/S = 0.15 at $Re$ = 10. Out-of-plane vorticity fields are overlaid with velocity vectors to show inter-bristle flow for different ${\theta}_{c}$ and ${\theta}_{f}$ (Figure 5). Insets on top right corners show the position of the wings in chordwise view and laser sheet orientation (VP1 for fling and VP2 for clap). Wing sketch was overlaid on top of flow fields to display the wing position and bristles. Dark black circles represent approximate positions of bristles. Velocity vectors in clap and fling do not show fluid moving in reverse direction to the wing motion. However, magnifying the x-z scales (Figure A5B) for W/S = 0.15 at ${\theta}_{f}$ = 37.5% shows onset of small amount of recirculation around a bristle. This recirculation was observed to be smaller due to shear layers around the bristles. The strength of the shear layers in fling increased with increasing ${\theta}_{f}$ until ${\theta}_{f}$ = 62.5% and then decreased for ${\theta}_{f}$ = 87.5%. During clap, shear layers with non-zero vorticity form around each bristle at all angular positions of observation. Note that the reference vector lengths (and corresponding velocities) and the vorticity scale bars between clap and fling are dissimilar.

**Figure 17.**Inter-bristle flow during clap and fling visualized from 2D PL-PIV for bristled wing model of W/S = 0.23 at $Re$ = 10. Out-of-plane vorticity fields are overlaid with velocity vectors to show inter-bristle flow for different ${\theta}_{c}$ and ${\theta}_{f}$ (Figure 5). Insets on top right corners show the position of the wings in chordwise view and laser sheet orientation (VP1 for fling and VP2 for clap). Wing sketch was overlaid on top of flow fields to display the wing position and bristles. Dark black circles represent approximate positions of bristles. At ${\theta}_{f}$ = 37.5%, recirculating flow around the bristles can be observed (Figure A5C). The strength of the shear layers in fling increased with increasing ${\theta}_{f}$ until ${\theta}_{f}$ = 62.5% and then decreased for ${\theta}_{f}$ = 87.5%. Recirculating flow around bristles was not observed during clap, but shear layers were seen to form around each bristle at all angular positions of observation. Note that the reference vector lengths (and corresponding velocities) and the vorticity scale bars between clap and fling are dissimilar.

**Figure 18.**Drag and lift coefficients at peak leakiness as a function of $Re$. Left-side ordinate in (

**A**,

**B**) shows drag coefficient at peak leakiness (${C}_{D}{\mid}_{{Le}_{max}}$) and lift coefficient at $L{e}_{max}$ (${C}_{L}{\mid}_{{Le}_{max}}$), respectively. Peak leakiness ($L{e}_{max}$) is shown in the right-side ordinate of (

**A**,

**B**). Filled symbols correspond to ${C}_{D}{\mid}_{L{e}_{max}}$ in (

**A**), and ${C}_{L}{\mid}_{L{e}_{max}}$ in (

**B**). Open symbols correspond to $L{e}_{max}$ in both (

**A**) and (

**B**). W/S = 0.05 (- -⬛- -); W/S = 0.15 (- -▲- -); W/S = 0.23 (- -⧫- -) for ${C}_{D}{\mid}_{{Le}_{max}}$ and ${C}_{L}{\mid}_{{Le}_{max}}$. W/S = 0.05 (- -□- -); W/S = 0.15 (- -▵- -); W/S = 0.23 (- -◊- -) for $L{e}_{max}$.

**Figure 19.**Diagrammatic representation of flow patterns in a bristled wing pair during fling: (

**A**) Chordwise view—equivalent diagram for a solid wing pair is also shown in the bottom for comparison. LEV and TEV for a bristled wing pair are diffuse as compared to more coherent LEV and TEV in a solid wing pair. Wing rotation about the TE is shown via dark yellow arrows. Inter-bristle flow (indicated via $Le$) is in opposite direction of wing rotation. (

**B**) 3D view of inter-bristle flow in fling (rotation about TE shown via dark yellow arrows) of W/S = 0.23 wing pair. Leaky, inter-bristle flow pattern is shown via blue arrows moving toward the cavity between the rotating wings. Shear layers formed around the bristles are indicated as colored regions (pink: counterclockwise recirculation; blue: clockwise recirculation). LEV = leading edge vortex; TEV = trailing edge vortex; $Le$ = leakiness; LE = trailing edge; TE = trailing edge.

**Table 1.**Experimental conditions used for PIV and force measurements comparing solid and bristled wing pairs during clap and fling. Camera recording rates used for time-resolved PIV (TR-PIV) measurements in chordwise plane (${\mathrm{FR}}_{\mathrm{TR}-\mathrm{PIV}}$) and phase-locked PIV (PL-PIV) measurements in spanwise plane (${\mathrm{FR}}_{\mathrm{PL}-\mathrm{PIV}}$) were varied to obtain data at identical non-dimensional time points ($\tau $) across every $Re$ tested. Cut-off frequencies used for filtering strain gauge data (${f}_{\mathrm{cut}-\mathrm{off}}$) were varied for every $Re$. ${U}_{ST}$ = steady translational velocity of the wing, $Re$ = chord-based Reynolds number given by Equation (3), T = time taken to complete one cycle, which includes one fling half-stroke, stroke reversal, and one clap half-stroke (Figure 4).

$\mathit{Re}$ | ${\mathit{U}}_{\mathit{ST}}$ (cm/s) | Clap/Fling Duration (ms) | T (ms) | ${\mathbf{FR}}_{\mathbf{TR}-\mathbf{PIV}}$ (frames/s) | ${\mathbf{FR}}_{\mathbf{PL}-\mathbf{PIV}}$ (image pairs/s) | ${\mathit{f}}_{\mathbf{cut}-\mathbf{off}}$ (Hz) |
---|---|---|---|---|---|---|

5 | 9.1 | 860 | 4300 | 116 | 0.12 | 12 |

8 | 14.6 | 540 | 2700 | 185 | 0.19 | 19 |

10 | 18.2 | 430 | 2150 | 233 | 0.23 | 23 |

12 | 21.9 | 360 | 1810 | 278 | 0.28 | 28 |

15 | 27.4 | 290 | 1450 | 345 | 0.35 | 35 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kasoju, V.T.; Terrill, C.L.; Ford, M.P.; Santhanakrishnan, A.
Leaky Flow through Simplified Physical Models of Bristled Wings of Tiny Insects during Clap and Fling. *Fluids* **2018**, *3*, 44.
https://doi.org/10.3390/fluids3020044

**AMA Style**

Kasoju VT, Terrill CL, Ford MP, Santhanakrishnan A.
Leaky Flow through Simplified Physical Models of Bristled Wings of Tiny Insects during Clap and Fling. *Fluids*. 2018; 3(2):44.
https://doi.org/10.3390/fluids3020044

**Chicago/Turabian Style**

Kasoju, Vishwa T., Christopher L. Terrill, Mitchell P. Ford, and Arvind Santhanakrishnan.
2018. "Leaky Flow through Simplified Physical Models of Bristled Wings of Tiny Insects during Clap and Fling" *Fluids* 3, no. 2: 44.
https://doi.org/10.3390/fluids3020044