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Wake Structure and Aerodynamic Performance of Low

Aspect-Ratio Revolving Plates at Low Reynolds number

Chengyu Li

1

and Haibo Dong

2

Department of Mechanical & Aerospace Engineering,

University of Virginia, Charlottesville, VA 22904

Understanding of vortex formation and aerodynamic loading is important for studying rotor

wake and rotary-winged micro air vehicles. In this work, direct numerical simulation (DNS) is used

to study three-dimensional flow structure and aerodynamic performance of low aspect ratio

revolving plates in low Reynolds number flows. These plates are modeled as rectangular plates with

zero thickness at a fixed 30o angle of attack. The span varies from 1 to 4 times of the chord length.

A total of five revolving cycles (

=10

radians) and a length matched tip Reynolds number of 500

are used in all cases. In general, the flow initially consists of a connected and coherent leading-edge

vortex (LEV), tip vortex (TV), and trailing-edge vortex (TEV) loop; the span-wise flow is widely

present over the plate and the wake region. At the end of first cycle, lift coefficients for different

aspect ratio cases reduce as much as 30 percent. When plate aspect ratio increases, hairpin-like

vortical structures are formed at the wing tip and further affect the stability of the wake structure

due to the wing-wake and wake-wake interaction.

Nomenclature

AR

= Aspect ratio

= Angle of attack

c = Chord length

L

C

= Lift coefficient

D

C

= Drag coefficient

P

C

= Aerodynamic power coefficient

= Rotational angle

Retip

= Tip Reynolds number

/tT

= Stroke ratio

= Angular velocity

I. Introduction

here has been substantial research aimed at understanding the unsteady fluid dynamics and forces of

rotor wake. Due to significant three-dimensional effects, the study of the unsteady flow structure

generated by low aspect ratio plates at low Reynolds number still remains as a challenging problem. For a

better understanding of the fluid dynamical mechanisms leading-edge vortex (LEV) and tip vortex (TV)

formation, it is necessary to study the formation of the whole vortex structure and aerodynamic

performance. The aerodynamic or hydrodynamic performance and vortex formation of three-dimensional

wings or plates has been extensively investigated both experimentally [1,2,3,4] and numerically[5] for the

1

Graduate Student, AIAA student member, cl2xt@virginia.edu

2

Associate Professor, AIAA Associate Fellow, haibo.dong@ virginia.edu

T

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52nd Aerospace Sciences Meeting

13-17 January 2014, National Harbor, Maryland

AIAA 2014-1453

Copyright © 2014 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

AIAA SciTech

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American Institute of Aeronautics and Astronautics

influence of Reynolds number effect [6,7,8] and velocity profile influences [9,10,11]. However, some of

the fundamental issues of low aspect ratio plates under revolving motion for low Reynolds number are

still far from being understood yet. For fixed-wing aircraft design, rotor crafts always operate under the

influence of their own wake. However, predictions methodology of the wake for rotor remains many

major challenges in fluid mechanics. Understanding of vortex structure and formation definitely will be

helpful for the prediction of rotor performance, vibratory loads, and blade-vortex interaction noise.

Computational modeling provides an opportunity to explore the relative force production and the relative

efficiency of revolving plates. The revolving plates can be synthesized relatively easily and the resulting

performance metrics correlated to flow physics over a large rand of scales.

The purpose of the present computational study is to examine the vortex formation and aerodynamic

performance about finite aspect ratio revolving plate in quiescent flow via a high-fidelity direct numerical

simulation (DNS) in-house solver. An outline of the chapter is given below. In Section II, a brief

introduction to the DNS methodology, simulation setup, and solver validation are present. In Section III,

aerodynamic force production and power consumption time history and period averaged values are given

first, followed by the flow structure for the first revolving cycle. Next, the circulations of LEV and plate

surface pressure are present. Finally, a short summary of current work is shown in Section IV.

II. Methodology

A. Governing Equation and Numerical Method

A second-order finite-difference based solver [12] for simulating flows with immersed boundaries on

fixed Cartesian grids has been developed which allows us to explore the wake structures with complex

immersed 3-D moving bodies. The biggest advantage of this method is that a Cartesian grid method

wherein flow past immersed complex geometrics can be simulated on non-body conformal Cartesian

grids and this eliminates the need for complicated re-meshing algorithms. The Eulerian form of the

Navier-Stokes equations is discretized on a Cartesian mesh and boundary conditions on the immersed

boundary are imposed through a “ghost-cell” procedure. The method also employs a second-order center-

difference scheme in space and a second-order accurate fractional-step method for time advancement. The

pressure Poisson equation is solved using the semi-coarsening multi-grid method with immersed-

boundary methodology. The details of this method and validation of the code can be found in [13,14].

The non-dimensional equations governing the flow in the numerical solver are the time-dependent,

viscous incompressible Navier-Stokes equations, written in indicial form as Eq. (1):

0

i

i

u

x

2

1

Re

ij

ii

j i j j

uu

uu

p

t x x x x

(1)

B. Simulation setup and plate kinematics

Simulations are performed in a large rectangular domain typically of size

30 30 30

in the stream-

wise (x), vertical (y) and span-wise (z) directions. Typical grid size of the dense region ranges from

186 160 186

to

240 120 240

with the smallest resolution of

0.035x

for the case of AR=1, 2

cases, and a larger size were used for simulation of flows around AR=4 plate. Grid stretching is applied in

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all directions with finer resolution near the plate to capture the wake structure shown in Fig.1. The

domain dimension and the number of grid points were determined with performing detailed domain

independence and grid refinement studies to ensure that the present choices does not influence the flow

field in a significant manner.

Plates of AR=1, 2, and 4 are considered in this studies. The plate root is extended out a distance

0.5

root

dc

away from the rotational axis at a fixed angle attack

30

. Dimensionless parameters, tip

Reynolds number (

Retip

), lift and drag coefficient (

L

C

,

D

C

), and aerodynamic power coefficient (

P

C

) are

defined shown in Eq. (2). The instantaneous aerodynamic power was calculated as

1

n

ii

i

P F v

, where n

is total number of triangular element on the plate,

i

F

is the aerodynamic force on each element and

i

v

is

the corresponding velocity of the element.

tip

U

is the velocity at wing tip, c is the chord length,

is the

kinematic viscosity, L is the lift production,

is the flow density,

is the rotation angular velocity,

and

tip

d

,

root

d

are the distances from the axis of rotation to the wing tip and the wing root. For matching

the tip Reynolds number for each cases, the constant angular velocity

=1.67, 1.0, and 0.56 respect to

plates of AR=1.0, 2.0, 4.0.

2

3

Re

,

,0.5

0.5

tip

tip

LD tip

Ptip

Uc

LD

CC US

P

CUS

(2)

(a)

(b)

Figure 1: Geometry definition for AR=2.0 revolving plate. (a) Top view of rotating plate orientation for

AR=2.0; (b) Top view of rotating plate orientation for AR=2.0

C. Validation

We compare results from the three-dimensional simulations and experimental measurements [15] [16]

for both translational and rotational studies. For all cases, the Reynolds number is fixed at 800 (for

rotational cases the tip velocity is treated as reference velocity). The plate is 160mm long in span-wise

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American Institute of Aeronautics and Astronautics

with AR=2 and 3% thickness. Angle of attack is varied with 10 degree increment. The translational

motion is performed at 1.25 chord length per second. For rotational cases, the plate is rotating about the

axis which is 60mm away from the wing root to avoid the interaction between the plate and mechanism.

The total rotational amplitude is

180

with

1.0 /srad

. In Fig. 2(b), snapshots of the span-wise

vorticity contour are shown for the simulation and experiment for the plate at

=40 o. Fig. 2(a) compares

measured lift and drag coefficients for translational and rotational cases with the simulations

for

[30 ,60 ]

. It can be observed that the two cases agreed each other very well.

(a)

(b)

Figure 2: (a) Lift and drag coefficient of rotational cases for a rectangular plate of AR=2 at Re=800. (b)

Snapshots of span-wise vorticity field at mid-span around a plate at angle of attack 40o. (The experimental

work was done by Yun Liu and Xinyan deng, Purdue University)

III. Results

The computation for different aspect ratio cases is done for five rotational cycles. Wake structures

behind the revolving plate after the impulsive start are shown in Fig. 4 with the corresponding forces and

aerodynamic power coefficient history in Fig. 3. After the initial start-up, both forces and power

coefficients are reduced by as much as 30 percent of the maximum value at the end of first cycle as shown

in Fig. 3. Depending on the aspect ratio, the force and power coefficients take different amount of period

to reach constant values. To better understand how the impulsive start affect the aerodynamic

performance, we further take a close look for AR=2 case by investigating the leading edge vortex

circulation of different cross sections and pressure mapping on the plate surface.

A. Aerodynamic performance and flow structure at Re=500

Fig. 3 gives the time courses comparison of lift coefficient (

L

C

), drag coefficient (

D

C

) and

aerodynamic power coefficient (

P

C

) of plates with

AR

=1, 2 and 4 in a completed five rotational cycles

(

=10

) under a fixed 30o angle of attach and

Retip

=500. The force and power coefficients are all

respect to its local coordinates. During the rotational motion, the instantaneous aerodynamic performance

generated on the plate for each aspect ratio case shares the similar decreasing and increasing tendency and

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American Institute of Aeronautics and Astronautics

gradually reach to a constant value once the flow reaches a nearly stationary state. At the first rotational

cycle, both aerodynamic force and power coefficients are shown a larger value because of the impulsive

start. The similar observation also presented in [6] by comparing the revolving motion and flapping

motion for a pair of hawkmoth wing model. In current work, the mean value of force coefficient, power

coefficient, lift-to-drag ratio, and lift-to-power-ratio are shown in Table 1. Although the AR=1 case owns

a relative larger lift coefficient, the drag coefficient and the aerodynamic power consumption also larger

than the other two cases. On the contrary, larger aspect ratio cases present better lift-to-drag ratio and lift-

to-power ratio.

(a)

(b)

(c)

Figure 3: Comparison of instantaneous (a) lift coefficient; (b) drag coefficient and (c) power coefficient

during five cycle for different aspect ratio.

Fig. 4 shows the vortex structure by revolving plate of AR=1, 2 and 4 in quiescent flow. Plates rotate

with respect to the positive y-axis. Iso-surface of Q=1.0 are used. Generally, a dipole structure starts to

form after revolving plate rotates 90 . The two vortex structures shedding from LEV and TV are mixed

together with different rotation velocity. This phenomenon is more obvious for AR=4 case. When plate

aspect-ratio increases, hairpin-like vortical structures are formed at the wing tip and further affect the

stability of the wake structure due to the wake-wake interaction. After a whole 360 rotation, a horse shoe

like vortex structure is formed. At far flow field, because of the rotary motion, vortex shedding from

current cycle can interact with the previous wake structure. For a matched tip Reynolds number, higher

AR plate shows much stronger interaction between the wake created in the current rotation cycle and the

previous one. The unsteady tip vortex formation is more obvious for AR=4 case.

Table 1: Comparison of mean lift coefficient (

L

C

), drag coefficient (

D

C

), aerodynamic power

coefficient (

P

C

), lift-to-drag ratio

/

LD

CC

, and lift-to-power ratio

/

LP

CC

for the last two cycles.

AR

L

C

D

C

P

C

/

LD

CC

/

LP

CC

1.0

0.375

0.502

0.282

1.34

1.78

2.0

0.313

0.463

0.224

1.48

2.06

4.0

0.315

0.473

0.220

1.50

2.15

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(a) AR=4, t/T=1/6

(d) AR=2, t/T=1/6

(g) AR=1, t/T=1/6

(b) AR=4, t/T=1/2

(e) AR=2, t/T=1/2

(h) AR=1, t/T=1/2

(c) AR=4, t/T=1.0

(f) AR=2, t/T=1.0

(i) AR=1, t/T=1.0

Figure 4: Comparison of vortex structure of first revolving cycle for AR=4 (a-c), AR=2 (d-f), and AR=1

(g-i) for different

. The three-dimensional flow structure is shown for different rotational angles through

iso-surfaces. Two surfaces are shown to highlight the inner core (Q=1.0) and outer shell (Q=3.0) of the

vortex structure.

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American Institute of Aeronautics and Astronautics

B. Velocity fields and leading edge vorticity for AR=4

To derive a quantitative description of the flow field around plate during rotary motion, we calculated

the spatial distribution of vorticity by projecting three-dimensional velocity components onto two-

dimensional planes (as shown in Fig. 5) respect to each frame of plate motion. These two-dimensional

plans are always normal to the straight line connecting leading edge of the plate. For the vorticity

calculation, the three-dimensional velocity field is constrained by two-dimensionality of the image planes,

thus ignoring any three-dimensional transport of vorticity. To quantify the strength of these vortices on

two-dimensional plans, we first visualize the vorticity field using contour line. After the leading edge

vortex is manually identified, a closed contour line is generated around this vortex with the specified

level, and then the circulation

is computed along this line. The circulation, shown in Fig. 6, is

nondimensionalized using wing tip velocity (

tip

U

) and chord length (

c

). Although the magnitude of the

circulation depends on the chosen contour level, the characteristic behavior of the vortex is not affected

by this choice.

The circulation magnitude of the leading edge vortex with specified contour level -7.5

/

tip

Uc

during

rotary motion is plotted for the two-dimensional plan at location close to the wing root, at 25%

c

, at

50%

c

, at 75%

c

, and close to the wing tip (shown in Fig. 6). The leading edge circulation results are

intuitive with respect to the lift coefficient. After the initial start-up, both forces and power coefficients

are reduced by as much as 15 to 30 percent of the maximum value at the end of first cycle for each section

cut.

Figure 5: A series of instantaneous vortex field during revolving. The contour level ranges from

-5

/

tip

Uc

to 5

/

tip

Uc

.

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American Institute of Aeronautics and Astronautics

Fig. 7 shows the surface pressure distribution on the

plate surface. During the rotary motion, the significant

low pressure area is located close to the leading edge

near the tip due to the attached LEV. Along with the

time increment, this low pressure area increases when

rotational angle

60

, however, it presents a

decreasing tendency for the rest of the first cycle. This

low pressure area changes matched with the tendency of

circulation changing within the first rotary cycle.

Figure 7: Comparison of plate surface pressure contour for the first cycle of AR=4.

Figure 6: Circulation of leading edge vortex

(LEV) for AR=4.

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IV. Summary

In this paper, revolving plates with aspect ratio 1, 2, and 4 are numerically studied at a fixed angle of

attack 30 degree with the same tip Reynolds number 500. The vortex formation and aerodynamic

performance are investigated and discussed. A stable and coherent vortex system was observed emanating

from the edges of the wing shortly after the impulsive start. At the end of first cycle, lift coefficients of

cases with different aspect ratio reduce as much as 30 percent. In addition, the increase of aspect ratio will

delay the flow reaching to a steady state. Depending on the aspect ratio, the force and power coefficients

take different amount of cycle to reach approximate constant values. The detailed analysis for AR=4 case

shows that span-wise flow development will make the leading edge vortex detach from the plate surface.

This may be one of the reasons of decrease of aerodynamic performance after the first cycle. Furthermore,

wing-wake and wake-wake interactions are much stronger for large aspect ratio case. This is mainly

because the extra span-wise flow generated cannot be feed into the main vortex ring via the shear layer.

This extra shed vorticity further form hairpin-like vortical structures and interact with each other, and

become the main source of instability. For a fixed tip Reynolds number, the larger aspect ratio cases

present more unstable vortex formation.

Acknowledgments

This is work is supported under AFRL FA9550-11-1-0058 monitored by Dr. Douglass Smith and NSF

CEBT-1313217.

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American Institute of Aeronautics and Astronautics

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