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2008,20(1):17-22

NUMERICAL ANALYSIS OF THE GROUND EFFECT ON INSECT HOVERING*

GAO Tong, LIU Nan-sheng, LU Xi-yun Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, China, E-mail: xlu@ustc.edu.cn

(Received March 26, 2007, Revised September 28, 2007)

Abstract: The ground effect on insect hovering is investigated using an immersed boundary-lattice Boltzmann method to solve the two-dimensional incompressible Navier-Stokes equations. A virtual model of an elliptic foil with oscillating translation and rotation near a ground is used. The objective of this study is to deal with the ground effect on the unsteady forces and vortical structures and to get the physical insights in the relevant mechanisms. Two typical insect hovering modes, i.e., normal and dragonfly hoveringmode, are examined. Systematic computations have been carried out for some parameters, and the ground effect on the unsteady forces and vortical structures is analyzed.

Key words: insect hovering, ground effect, unsteady forces, vortex dynamics, immersed boundary-lattice Boltzmann method

1. Introduction Insect flying through air has developed the

superior and complete performance of flying in complex environments. Some work on the unsteady mechanisms of force generation in insect flight has been carried out and reviewed comprehensively by Wang [1]. In nature, flying insect usually perches on some bodies and the ground effect will play a significant influence on the flying performance [2].However, to our knowledge, the ground effect on the insect flying behaviors has never been studied and is highly desired. Thus, we will investigate the ground effect on the insect hovering in the present study.

Based on the measurement of a foil started impulsively at a high angle of attack, lift acting on the foil is enhanced by the presence of a dynamic stall vortex, or Leading-Edge Vortex (LEV) [3]. Using an

* Project supported by the National Natural Science Foundation of China (Grant No. 10332040), the Innovation Project of the Chinese Academy of Sciences, and Program for Changjiang Scholars and Innovative Research Team in University. Biography: GAO Tong (1980- ), Male, Ph. D. Student Corresponding author: LU Xi-yun, E-mail: xlu@ustc.edu.cn

analysis of the momentum imparted to the fluid by the vortex wake, the LEV can explain the high lift on the insect wings. This high lift mechanism is called the delayed stall mechanism [4,5]. In the experiment of a model of the fruit fly [5,6], large lift and drag peaks occur at the beginning and the end of the stroke in the case of advanced rotation, in addition to the large lift and drag during the translatory phase of a stroke. The force peaks at the beginning of the stroke can be explained by the wake capture mechanism. On the other hand, the unsteady mechanisms in insect flying have also been investigated by solving the Navier-Stokes equation around a moving wing. Usually, a two-dimensional (2-D) simulation of the problem can be taken to study the basic mechanisms in flapping flight. The unsteady flow and force behavior around a flapping foil were investigated [7,8].

In the present study, a virtual model, which is an elliptic foil with oscillating translation and rotation near a ground, is employed to deal with the ground effect on the insect hovering motion for the first time. The unsteady forces and flow structures are investigated by solving the 2-D incompressible Navier-Stokes equations using an immersed boundary-lattice Boltzmann method.

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2. Mathematical formulation To investigate the flow around an elliptic foil

with oscillating translation and rotation near a ground, as shown in Fig.1, the incompressible Navier-Stokes equations are used and given as

2+ = +ptu u u u (1)

= 0u (2)

where is the velocity, the pressure, u p the density of the fluid, and the dynamic viscosity.

The flapping motion of insect wing can be described as [2]

2( ) = cosm

tA t A

T (3)

02( ) = + sin( + )m

tt

T (4)

where is the amplitude of oscillating translation, mA

0 and m are the initial value and amplitude of oscillating rotation, is the flapping period, and Tis the phase difference between the oscillating rotation and translation.

Fig.1 Sketch of an oscillating translation and rotation foil near a ground

We use the chord length of the foil and the velocity related to the oscillating translation

cU

= mAU

T as the length and velocity scales,

respectively. The Reynolds number is thus defined as

= UcRe . The corresponding non-dimensional

variables shown in Eqs.(3) and (4) are still represented by the same symbols for writing convenience. In addition, as shown in Fig.1, an additional parameter

is introduced to represent the distance between the foil center and the surface. No-slip boundary condition is used on the foil and ground surface.

D

3. Numerical methodsTo solve Eqs.(1) and (2), an Immersed

Boundary-Lattice Boltzmann Method (IB-LBM) [9] is used. In the IB method, an external force is introduced into the right-hand-side of Eq. (1) to medel the effect of boundary immersed in the fluid flow, and can be obtained by distributing the Lagrangian force density to the surround fluid grid points by a 2-D Dirac delta function [9]. The Lattice Boltzmann Method (LBM) is an approach to solve the fluid dynamics problem based on microscopic kinetic models[10]. To simulate Eqs.(1) and (2), a forcing term is included in the discrete lattice Boltzmann equation [11] and given as,

1( + , + ) ( , ) =i i if t t t f tx e x

( , ) ( , ) +eqi i if t f t tFx x (5)

where

= ii

f ,1= +2i i

i

f tu e f ,

2

1= +2 sc t

,

2

2 4

: ( )= 1+ +2

eq i i ii i

s s

cf w

c cs Ie u uu e e

,

2 4

( )1= 12

i ii i

s s

F wc c i

e u e u e f

Then, the D2Q9 model is used in the present calculation[11]. On the far-field boundary, the density distribution function is set as its equilibrium state.

4. Results and discussion In the normal hovering mode, the wing

undergoes a symmetric stroke along a horizontal plane with and o

0 = 90 = 0 . In the dragonfly hovering mode, the insect wing hovers along an inclined stroke

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plane with different choices of 0 and . Based on previous work [1,2], some variables in Eqs. (3) and (4) are , and = 1.25mA o= 45m = 0o, the Reynolds number is , and the thickness ratio of the elliptic wing is 0.25. The present code has been extensively validated by Sui et al.

= 100Re

[9].4.1 Normal hovering mode

We first deal with the ground effect on the normal hovering mode (i.e., and o

0 = 90 = 0 ).The distance D is the only governing parameter based on the chosen parameters shown above.

Fig.2 Vorticity contours in the first-half stroke for normal hovering mode at = 1D

Fig.3 Vorticity contours in the first-half stroke for normal hovering mode at = 3D

Figure 2 shows the vorticity contours at two sequential instants in the first-half stroke period for

= 1D . A pair of Leading-Edge Vortex (LEV) and

Trailing-Edge Vortex (TEV) is generated. The LEV is attached to the foil during the translation, in accordance with the stall-delayed mechanism, which is of benefit to generating a higher force peak shown later. The TEV is stretched and elongated in the gap, and then dissipates quickly. The vortices shed from the foil are swept away in the horizontal direction, which indicates that the self-induced jet is dominant in the horizontal direction due to the ground effect. In the second-half stroke, the time development of vortex structures with an opposite direction is the same as the above description. When D increases, as shown in Fig.3 for , the vortex pair moves downward and dissipates slowly. Further, the newly generated vortices interact with the ones shed previously, leading to complex vortical structures over the ground and the relevant force behavior acting on the foil.

= 3D

Fig.4 Time-dependent horizontal ( HC ) and vertical coefficients ( ) in one stroke for normal hovering mode

VC

Figure 4 shows the time-dependant vertical and horizontal force coefficients during one typical stroke after reaching quasi-periodic state, where the vertical and horizontal force coefficients are defined as

2=0.5

yV

FC

U c, 2=

0.5x

H

FC

U c

respectively, with yF and xF the vertical and horizontal forces due to the friction and pressure. As shown in Fig. 4 for D =1, both the forth and back strokes generate almost equal magnitudes of the vertical force and those of horizontal force, respectively. The peak value is related to the

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stall-delayed mechanism with the attached LEV in Fig.2. When 2D , due to both the interaction of the LEV and TEV as shown in Fig.3, there exist two peaks of in each half stroke. VC

Fig.5 Mean horizontal ( HC ) and vertical ( VC ) coefficients for normal hovering mode

Figure 5 shows the time-average vertical ( VC )

and horizontal ( HC ) force coefficients after reaching

quasi-periodic state, where HC is calculated by

averaging the absolute value of HC . At D =1, both

VC and HC have higher values around 0.68 and 1.95, respectively, due to the effective ground effect. When increases, D VC and HC decrease quickly.

VC reaches its minimum value 0.18 around D =2,

and HC reaches its minimum 1.21 around D =3. Then, both the force coefficients go up slightly and almost keep to be unchanged when , where the ground effect becomes small.

5D

Based on the previous study on the normal hovering mode without the ground effect [2], it is found that the self-induced flow forms a downward jet, which penetrates a distance of several wing chords. However, as shown above, the downward jet is blocked when the ground surface occurs, and the flow-surface interactions lead to complex vortical structures, which consequently affect the forces behaviors of the flapping foil. 4.2 Dragonfly hovering mode

The ground effect on the dragonfly hovering mode is further investigated. Here, we set 0 =60o

and mainly analyze the influence of and D on the flow behaviors. 4.2.1 Effect of the distance D

Figure 6 shows the vorticity contours in one stroke at . A pair of vortices is generated in the downstroke. The LEV attaches to the foil before the vortices shed when the foil turns upward. Then, each

vortex combines with the one of the opposite sign generated in the upstroke to form two vortex-dipoles.

= 4D

Fig.6 Vorticity contours in one stroke for dragonfly hovering mode at = 4D

Fig.7 Time-dependent horizontal ( HC ) and vertical coefficients ( ) in one stroke for dragonfly hovering mode at VC

o= 60

Figure 7 shows the time-dependant and VC

HC during one typical stroke. It is seen that the positive vertical force is mainly produced during the downstroke. The differences of HC mainly occur in the upstroke with the increase of , and the variations of are relatively small, indicating that

DVC

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the ground effect mainly affects the horizontal force. The mean force coefficients VC and HC are

shown in Fig.8. The variation of HC is relatively

small, and the values of HC are around 0.6. As D

increases, HC decreases first to its minimum 0.58, then goes up somewhat from D =4, and gradually reaches to a constant value 0.603 for , where the ground effect becomes negligible. Compared with

6D

HC , VC drops smoothly from its maximum 0.31 at D =2, and then keeps nearly a constant 0.19 when

.5D

Fig. 8 Mean horizontal ( HC ) and vertical ( VC ) coefficients

for dragonfly hovering mode at o= 60

Fig.9 Time-dependent horizontal ( HC ) and vertical coefficients ( ) in one stroke for dragonfly hovering mode at

VC

= 2D

4.2.2 Effect of the stroke plane angle To deal with the effect of the stroke plane angle

, we mainly discuss the results at D =2. Figure 9

shows the time-dependant and VC HC in one stroke at several stroke plane angles. Generally, the positive lift force is mainly contributed in the downstroke. The magnitude of increases withVC ,

while the magnitude of HC decreases. In Fig.9(b), it

is noted that the distribution of at is significantly different from others in the downstroke, since the foil during the downstroke interacts with the previously shed negative vortex. At large stroke plane angles, this kind of vortex evolution is not observed. From the averaged force coefficients

VC o= 30

VC and HC

in Fig.10, the HC decreases with and VC first

reaches its minimum value around , then goes up with the increase of

o= 45. This suggests that

the insect may take a benefit from the flapping with a large stroke plane angle in the dragonfly mode.

Fig.10 Mean horizontal ( HC ) and vertical ( VC ) coefficients for dragonfly hovering mode at = 2D

5. Concluding remarks The ground effect on the insect hovering has

been investigated based on a virtual model of an elliptic foil undergoing oscillating translation and rotation near the ground. The immersed boundary-LBM is used to solve the 2-D Navier-Stokes equations. Two typical hovering modes, i.e., normal hovering mode and dragonfly hovering mode, are investigated. In the normal hovering mode, the symmetry of the flow field is destroyed when the foil moves away from the ground, and the back stroke contributes major part to the vertical force. In the dragonfly hovering mode, the ground effect mainly affects the horizontal force when the foil moves away from the ground. The large stroke plane angle is useful to obtain a high vertical force and low horizontal force. The results obtained in this study are helpful in understanding the mechanisms involved in the insect hovering near a ground.

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