Does a revolving wing stall at low Reynolds numbers?

Advance Publication by J-STAGE Journal of Biomechanical Science and Engineering

DOI:10.1299/jbse.15-00588

Received date : 25 October, 2015 Accepted date : 1 December, 2015 J-STAGE Advance Publication date : 10 December, 2015

© The Japan Society of Mechanical Engineers

Does a revolving wing stall at low Reynolds numbers? Xiaoqian GUO*, Di CHEN* and Hao LIU*,** *Shanghai Jiao Tong University and Chiba University International Cooperative Research Center (SJTU-CU ICRC) 800 Dongchuan Road, Minhang District, Shanghai, 200240, China **Graduate School of Engineering, Chiba University 1-33, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan E-mail: [email protected]

Abstract Motivated by the dynamic, i.e., “delayed” stall mechanism in bio-inspired flapping wing aerodynamics that can achieve high lift production at high angles of attack (AoA) we utilize a simplified revolving wing model to explore the novel mechanisms in concert with stall phenomenon at low Reynolds numbers. The dynamic stall-based leading-edge vortex structure recognized as the primary reason for lift augment in insect flapping flight is also observed in revolving wings in low Reynolds (Re) number regime. However, it is still unclear why this “delayed” stall at low Res performs remarkably distinguished from that at high Res. In this study, we built a hawkmoth wing-like geometric model and a revolving kinematic model with an impulsive accelerated start as the quasi-steady model of flapping wings. We carried out a systematic parametric study on stall characteristics in terms of Res effects over a wide range from 100 up to 5.4×10 5. Our results show that in the low Res regime where most insects fly, the revolving wing likely does not stall in a conventional sense: there is no trend that lift turns to reduce at some critical AoA but keeps increasing with increasing AoAs. Moreover, the Reynolds number effect on aerodynamic forces augment is mostly pronounced at Res less than 5400 during both unsteady acceleration and steady rotation phases. Our results imply that micro air vehicles (MAV) with rotary wings will suffer a rapid drop in aerodynamic performance probably at operating Res less than 5,400, which may point to the importance of the employment of flapping wings for insect-sized MAV design.

Key words : Delayed stall, Insect flight, Low Reynolds number, Revolving wing

1. Introduction Flapping-wing aerodynamics has the characteristic of complexity for its unsteady aerodynamic mechanisms (Ellington, 1996; Willmott and Ellington, 1997a, 1997b; Liu et al., 1998). Insects such as hawkmoth can maintain high force coefficients and extraordinary stability even with high angles of attack, at which airplanes will suffer from flow separation and sudden lift reduction, eventually lead to stall. Previous research into insect flapping flight aerodynamics has revealed that the delayed or dynamic stall mechanism and associated leading edge vortex (LEV) contribute significantly to lift maintenance at high angle of attack (Usherwood and Ellington, 2002a; Liu and Aono, 2009). The revolving wing has been proved to be a quasi-steady model to reproduce similar instantaneous vertical lift and aerodynamic torque at corresponding angle of attack of the same wing in flapping motion (Dickinson, et al., 1999; Sane, 2003). The similarity in aerodynamic characteristics provides a method of less compound interaction for flapping wing performance research. Previous studies have been referred to the dynamic stall and the leading edge vortex as the primary reason for elevated lift coefficients required for insect flight. In general, a stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded, which is typically about 15 degrees in fixed-wings, but it may vary significantly depending on the fluid, foil, and Reynolds number. There is a sharp drop in lift as well as lift-to-drag ratio mostly observed at high Res greater than 100,000, which usually corresponds with some critical stall angle of


attack. At low Res less than 10,000 which most insects and small birds fly, however, merely some flexion is observed around some angle of attack and there is usually not an apparent decreasing but even a gradual increasing in lift and lift-to-drag ratio. On the other hand, revolving wings further demonstrate distinguished feature associated with stalls from the fixed wings. In the past decade, although the revolving wing has been an important subject in exploring flapping-wing aerodynamics (Usherwood and Lentink, 2002a, 2002b; Lentink and Dickinson, 2009a, 2009b; Harbig, et al., 2013; Kruyt et al., 2015), how a revolving wing stalls and performs in terms of aerodynamic forces and vortex flow structures still remain unclear yet. In this study, motivated by the dynamic stall mechanism in flapping wing aerodynamics that can achieve high lift production at high angles of attack we employ an insect wing model to study the revolving-wing aerodynamics at delayed stalls in the low Reynolds number regime. We first give a brief description of the computational fluid dynamic modeling methods used in this study, which consist of the solutions to the Navier-Stokes equations, a hawkmoth wing geometric model as well as a kinematic model undergoing revolving motions. Then we will present results of aerodynamic force coefficients and leading-edge vortex structures to demonstrate the dynamic stall characteristics over a wide range of Reynolds numbers. Finally, we will discuss extensively on the Reynolds number effect on force generation and its correlation with vortex structures.

2. Method 2.1 Numerical method The governing equations in terms of the conservation of mass and momentum are as follows:

Ñ× u = 0 ,

 where

(1)

Du  p  2u , Dt

  ( x , y , z )T represents the gradient operator, u the velocity,

(2)  the density, p the pressure, and  the

dynamic viscosity. The governing equations may be further reformed in a rotating frame of reference attaching onto a revolving wing as shown in Fig. 1, such that,

uabs  u  Ω  r ,



(3)

.

Du   Ω r   (Ω  r)   2Ω  u  p  2u , Dt

(4) .

where u abs and u are velocities respect to the global and rotating frames, r the position vector, and Ω , Ω the angular velocity and angular acceleration of the wing, respectively. To further non-dimensionalize the Eq. (4) the following representative parameters are introduced: .

. * Ut p u Ω Ω r , *  cm  , Ω  . , Ω  , r   , p  , u  , t*   R cm p0 U  where the superscript * denotes non-dimensionalization, cm the mean chord length, U the time-averaged wingtip velocity, R the wingtip radius, U  R the wingtip velocity with a constant angular velocity, and p0 the ambient pressure. Accordingly, the dimensionless equations written in a rotating frame of reference is as: *

.

2 Rcm 2 Rcm p Du  Rcm .    Ω  r   Ω  ( Ω  r )   2Ω  u   0 2 p   2 u . 2 2 2 Dt Ucm U U U U

(5)

With introduction of the Rossby number (Lentink and Dickinson, 2009a, 2009b) of Ro  R / cm and Reynolds number

Re  Ucm /  , Eq. (5) can be further simplified as: p Du 1 1 1   Ω  (Ω  r)   2Ω  u   0 2 p   2u . Dt Ro Ro Re U

(6)

These equations were solved directly using a commercial finite-volume-based code ANSYS CFX. The morphology and


kinematic characteristic of biological flyer vary from each other, results in a wide Re range from O(10 1) for tiny insects, O(102) for fruitflies, O(103) for honeybees, butterflies, beetles, and dragonflies, and up to O(10 4) for hawkmoth and large insects (Aono and Liu, 2013). In this study, simulations of the unsteady flow around a revolving wing were performed over a wide range of Reynolds numbers from 100 up to 5.4×105. Vertical and horizontal force coefficients Cv and Ch of the revolving wing model are then calculated instead of the lift and drag coefficients. Following Weis-Fogh (1973), Usherwood and Ellington (2002a), Kruyt, et al. (2014) and Noda, et al. (2014), we also use the vertical and horizontal force coefficients to assess the aerodynamic performance of the revolving wing Cv 

2Fv

 S2 2

,

(7)

2Q , (8)  S3  2 where Fv denotes the aerodynamic vertical force on a single wing, Q the torque about the rotation axis, S2 and S3 the Ch 

second and third moment of area of a single wing, respectively, S2   r 2 c(r )dr and S3   r 3c(r )dr , in which r is R

R

0

0

the local radius along wingspan.

2.2 Morphological and kinematic models We use a hawkmoth wing model based on the two-dimensional digitalized image of hawkmoth, Agrius vonvolvuli (Aono and Liu, 2006). The wing model has a mean chord length cm of 10 cm and an aspect ratio AR = 5.28 by AR  (2L )2 / S (L: single wing length; S: wing area). The wing is a rigid flat plate (Fig.1) with a uniform thickness 2% of the mean chord. The wing root is displaced from the rotation axis with an offset of 0.5cm. Lentink and Dickinson (2009b) observed that an impulsively accelerated start is beneficial for stabilization of LEV on revolving insect wings and provided good simulation for a typical insect wing at the start from rest. Follow Lentink and Dickinson (2009b) and Noda et al. (2014), we employ a simplified kinematic model of the revolving wing without consideration of unsteady wing rotation and reversal. The wing is initially at rest, achieves an impulsive start with a rapid acceleration during a very short period and then sustains a steady rotation at a constant angular velocity. The angular velocity  z of the revolving wing is defined as follow:

1   t    1  cos    , t  Taccel  z (t )   2  ,  0.084T   , t  Taccel 

(9)

where  is the rotational angular velocity, t the dimensionless time, and Taccel the time when the acceleration ends and steady rotation starts. Taccel is defined to approximate the wing kinematics of a hovering hawkmoth (Aono and Liu, 2006) where the acceleration period occupied about 10% of the flapping period. Here, Taccel is set to be 0.084T considering the calculation timestep settings, where T denotes the time of rotation period when the angular velocity is steady. The mean chord-based Reynolds number is defined as Re  U ref cm /  , where the reference velocity U ref is defines as the steady wing tip velocity, U ref  R . Simulations are all conducted up to 4 cycles (revolutions) when the flow fields are confirmed to reach a stable stage. The computation domain consists of an inner rotational zone fitting to the wing model and an outer stationary zone (Fig. 2). In the outer cuboid zone the side boundary parallel to the rotating axis is taken 300 times of cm, and the other two side boundaries both 200 times of cm. The outer boundary is initially quiescent with free-slip boundary conditions on its four faces parallel to the rotation axis, while for the top and bottom surfaces both the zero gauge pressure and zero velocity gradient conditions are taken. The inner rotating zone has a shape of sphere with a radius of 25 times of cm. The wing surface is defined as a non-slip surface and a general grid interface (GGI) connection. Both inner and outer zones are meshed with structured hexahedral meshes in which the minimum grid spacing adjacent to the wing surface is controlled by a formula  min  0.1cm / Re (Liu and Aono, 2009). The mesh for rotating domain is generated with O type grid viewed in rotation axis direction and H type grid in spanwise direction.


Fig. 1 A hawkmoth wing model with a global coordinate system (x, y, z) and a wing-fixed coordinate system (x’, y’, z’).  denotes angle of attack,  the angular velocity and R the wingtip radius.

Fig. 2 Schematic of the computational domain including a sphere-shaped inner rotational zone and a cuboid outer stationary zone.

2.3 Validation A benchmark test for validation was conducted with two sets of meshes as shown in Table 1, i.e., a coarse and a fine mesh system. The fine mesh had more grids distributed along wingspan and chord on wing surface and clustered to the wall surface.

Mesh

Table 1 Two mesh systems at Re = 5400 Spanwise and chordwise nodes Elements (million) number on wing surface

1

80×30

3.178

2

120×45

5.316

Table 2 Aspect ratios and Reynolds numbers of wing models

Present model Manduca sexta model (Usherwood and Ellington, 2002a) Bombus terrestris model (Usherwood and Ellington, 2002a)

Aspect Ratio (AR = (2L)2/S)

Reynolds number (Re = Uref cm/  )

5.28

5400

5.66

7300

6.32

5496

Usherwood and Ellington (2002a, 2002b) conducted systematic measurement of aerodynamic forces on animal wings ranging from insects to small-sized birds by means of dynamically scaled models of a hovering hawkmoth Manduca sexta and a bumblebee Bombus terrestris wing. Table 2 summaries the aspect ratios and Reynolds numbers of these models. A comparison of vertical and horizontal force coefficients between experiments and simulations is illustrated in Fig. 3, in which the aerodynamic force coefficients are plotted via angles of attack. Overall, excellent agreement in vertical coefficients is obtained with mere a difference of 3.57% lower at most while the horizontal force coefficients show no more than 6.1%. Cv shows a moderate increase with increasing angle of attack (AoA) reaching a peak approximately at 40°~50°in the ‘early’ phase and then turns to decrease, well matching the results of hawkmoth and bombus experiments. In the ‘steady’ phase, Cv vs AoA develops in a similar trend but has the peak moved forward to around 50°and a pronounced drop in magnitude. Obviously, the ‘unsteady effect’ in the ‘early’ acceleration phase leads to generating more aerodynamic forces (Cv and Ch).


Fig. 3 Comparison of a vertical and b horizontal force coefficients between experiments and simulations at Re = 5400, AR =5.28 (Usherwood and Ellington, 2002a, 2002b). Here ‘early’ means the phase of the first half-revolution from 60°to 120°and the ‘steady’ the phase from 180°to 450°from the start of the revolution, respectively.

3. Results and discussion 3.1 Stall characteristics over a wide range of low Reynolds numbers Fig. 4 shows time courses of vertical and horizontal force coefficients vs AoAs from 0°to 70°at Re = 5400, which is plotted against the angle of wing revolution. The aerodynamic force coefficients of the revolving wing obviously achieve a rapid, linear increase during the short acceleration phase immediately after the revolution starts owing to the added mass effect and the leading-edge vortex. For the vertical force coefficients, the impulsive start effect becomes more pronounced with increasing AoAs but turns to decrease when AoA exceeds 50° while the horizontal force coefficients show a monotonic increasing trend. The aerodynamic forces become more stable after a drop and milder increase in the ‘early’ phase, and start to decrease at the end of the first revolution when the downwash generated in the first revolution meets the wing. As the flow direction is modified with the downwash, the effective AoA no longer equals to, in fact, become lower than the geometric AoA. The decrease in both vertical and horizontal force coefficients is observed after revolving through 360°, corresponding to a significant shift in coefficients between early and steady phases (Fig. 5). As is postulated by Usherwood and Ellington (2002a), the development of the full vortex wake with its associated radial inflow over the wings might well shift the position of vortex breakdown inwards under ‘steady’ conditions at higher angle of attack, producing a quantitative reduction in the lift coefficient. In this study, the same trend of vertical force coefficient is observed for AoA from 0 to 50°, though the reduction is less obvious for higher angle of attack (60°- 70°). With full development of the vortex structure and the effective angle of attack more stable, and the aerodynamic forces keep steady again for the remaining revolution for lower angles of attack. But the vertical and horizontal forces show unstable fluctuation at larger angles of attack although wake is fully developed, which is observed in previous experimental studies (Dickinson et al., 1999; Usherwood and Ellington, 2002a). For further investigating the stall characteristics of the revolving wing, we define the revolution from 0°to 360°(the first revolution) as the ‘early’ phase,


and revolution from 1080° to 1440° (the fourth revolution) as the ‘steady’ phase. Average aerodynamic force coefficients for early and steady phase will be used for further discussion.

Fig. 4 Time courses of a vertical and b horizontal force coefficients vs angles of attack (10°~ 70°) at Re = 5400. Shaded areas correspond to ‘early’ (0°- 360°) and ‘steady’ (1080°- 1440°) phases.

Fig. 5 Aerodynamic force coefficients in ‘early’ and ‘steady’ phases with angles of attack varying from 0°to 70°over a range of Res from 100 up to 5.4×105: a Cv,early at early stage, b Cv,steady at steady stage, c Ch,early at early stage, and d Ch,steady at steady stage. Fig. 5 shows Cv and Ch plotted against AoA for Reynolds number varying from 100 up to 5.4×105 averaged in the ‘early’ and ‘steady’ phases. During the ‘early’ phase with unsteady acceleration effects, the Cv,early overall is


independent of Res, demonstrating apparently symmetric characteristics with a symmetric axis somehow at AoA=45°, and reaches a peak at some AoA between 40°~50°. This implies that no stall occurs here in the lift force (vertical force) because the turnover in Cv,early is due to the increasing AoA, which will reduce the vertical force component but increase the horizontal force component when the AoA is over 45°. In the ‘steady’ phase, the averaged vertical force coefficients (Fig. 5b) show more pronounced nonlinear characteristics with the peak shifting slightly (somehow 5°~10°) towards larger AoAs with increasing Res. Again, unlike the conventional stalls at high Reynolds numbers where a sharp drop in lift as well as lift-to-drag ratio is observed corresponding with some critical angle of attack, i.e., the stall angle, the vertical (lift) force keeps increasing even when AoA exceeds 45°. On the other hand, a common feature in the aerodynamic force coefficients in both phases is observed that the Reynolds number effect is obvious in the low Res regime where most insects fly, i.e., of Re is less than 5400 (Fig. 5). However, interestingly such Reynolds number effect turns to become a margin when Re is greater than 5,400. Polar diagrams and Cv/Ch are further plotted against angle of attack in Fig. 6. The polar diagrams of vertical force coefficient versus corresponding horizontal force coefficients and vertical to horizontal force coefficients ratio show coinciding tendency with the force coefficients in Fig. 5. The Cv/Ch curve shows larger values at low AoAs at all Res, e.g. a peak at AoA around 10°in Fig. 6 but also reasonable high values (>1.0) at AoA of 30°~40°, which corresponds with a stroke- averaged feathering angle of flapping wings in hawkmoth hovering. We found no dominant leading edge vortex at lower AoA of 10°, corresponding to earlier findings that at higher Reynolds numbers a strong leading edge vortex does not correspond to minimum hover power (Lentink and Dickinson, 2009b). With the existence of stalls at high Res (>105) conventional rotating wings mostly found in helicopters or more recently in multi-rotor drones operate at low AoA less than 15° (Hoffmann et al., 2007), which can achieve both large lift force generation and better aerodynamic performance with minimum power consumption. In contrast, in the low Res regime where most insects fly, the revolving wing likely does not stall at all in a conventional sense but is capable to achieve continuous increasing in the lift (vertical) force with increasing AoA. This very likely gives the low Re flyers more space (in terms of AoA) to operate their wings to ensure both large force generation and steering maneuvering ability. Therefore, hawkmoths usually operate their wings at higher AoA of 40°, at which almost twice of the vertical force can be generated compared to that at lower AoA of 10°, although the aerodynamic efficiency is not the best. Considering that hawkmoths have evolved under selective pressures for flight performance, the dominant factor for hawkmoths to flap their wing at aerodynamically suboptimal angle of attack may not be saving energy cost, but to mitigate inertial losses and optimize muscle mechanics during forward and hovering flight as postulated by Chai and Dudley (1995, 1996) and Kruyt et al. (2014).

Fig. 6 Polar diagrams in ‘early’ a and ‘steady’ b phases, and Cv/Ch in ‘early’ c and ‘steady’ d phase with angles of attack varying from 0°to 70°over a range of Res from 100 up to 5.4×105.


3.2 Relationship between flow structures and stalls To explore the relationship between flow structures and the stalls in low Reynolds number regime, we further made visualization of the leading-edge vortex structures at 25%, 50% and 75% wingspan and vortex structures around the wing surface (Fig. 7 - 8). Note that the AoA=40°is chosen here, which corresponds to the cycle-averaged feathering angle of a flapping wing in hawkmoth hovering. An intense LEV is detected coherently attached on the leading edge throughout the wingspan at all Res though they show pronounced discrepancy dependent on the spanwise location and the Reynolds numbers.

Fig. 7 Snapshots of non-dimensional spanwise vorticity (  y R / U ) contours for ‘steady’ phase at 25% (left), 50% (middle) and 75% (right) wingspan over a range of Res from 100 up to 5.4×105 at AoA=40°. (a: Re=100, b: Re=540, c: Re=5400, d: Re=5.4×104, e: Re=5.4×105)

Fig. 8 Vortex structures for ‘steady’ phase over a range of Res from 100 up to 5.4×105 at AoA=40°. Vortex structures are visualized using surface of constant Q criterion colored by spanwise vorticity to indicate direction; blue indicates negative and red is positive. (a: Re=100, b: Re=540, c: Re=5400, d: Re=5.4×104, e: Re=5.4×105)

Overall, the vortex structures show pronounced variation with increasing Res from 100 up to 5400 but then merely slight discrepancy is observed among three higher Res of 5,400, 54,000 and 5.4×105 at all three wing spans. This supports the similar trend in aerodynamic force generation at the three higher Res as observed in the vertical and horizontal force coefficients in Fig. 5 - 6. Moreover, at Re =100, the LEV is not attached closely to the leading edge and there is no flow separation along the wingspan. This is also observed at Re = 540 though the LEV is more developed and the trailing vortex seems to separate slightly at 75% wingspan. As the Res further increases, the vortex structure turns to change remarkably from wing base towards wing tip. With the Res increased, the LEV structures within the inner portion of the wing span keeps growing in size and strength, attaching coherently onto the leading edge (Fig.6 8). It may be fair to suggest that Reynolds number is responsible for the enhancement of the leading edge vortex. Therefore, the size of LEV and the averaged vertical force coefficient grow evidently with Reynolds number increases


in lower regime. At such large angle of attack at 40° for which lift force would suffer reduction for airfoils, the revolving hawkmoth still maintain LEV as well as high vertical force over a wide range of Reynolds numbers. The revolving kinematics and the hawkmoth-like wing morphology might help to maintain the leading edge vortex. The conventional stall phenomenon is not observed in the Reynolds numbers regime with the hawkmoth-like wing model.

4. Conclusion We have studied computationally the Reynolds number effects on aerodynamic performance of a hawkmoth-wing revolving model undergoing an impulsively started acceleration and steady rotation over a wide range of Res from 100 up to 5.4×105. Our results have herein confirmed that the revolving wing model does not stall in a conventional sense but is capable to achieve continuous increasing in the lift (vertical) force with increasing AoAs. The leading-edge vortices are observed attaching the wing surface at all Reynolds numbers, which are stable and intense. In addition, it is found that the vertical force coefficients in the ‘steady’ phase show a pronounced nonlinear dependency of Res, with the peak shifts towards larger AoA as Res increases; the vertical-to-horizontal force coefficient ratio achieves much better performance at AoA of 40°with the vertical force doubled at AoA of 10°where most rotary winged vehicles operate. This likely gives the low Re flyers more space (in terms of AoA) to operate their wings to ensure both large force generation and steering maneuverability.

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