Sahin Eppler

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Direct Numerical Simulation of Separated

Low-Reynolds Number Flows around

an Eppler 387 Airfoil

Mehmet SAHIN, Jeremiah HALL, Kamran MOHSENI

Department of Aerospace Engineering Sciences,

University of Colorado at Boulder,

Boulder, Colorado, 80309, USA

Koen HILLEWAERT

CENAERO, Rue des Freres Wright 29, B-6041 Gosselies, BELGIUM

Abstract

Low Reynolds number aerodynamics is important for various applications including micro-aerial

vehicles, sailplanes, leading edge control devices, high-altitude unmanned vehicles, wind turbines and

propellers. These ows are generally characterized by the presence of laminar separation bubbles.

These bubbles are generally unsteady and have a signicant effect on the overall resulting aerodynamic

forces. In this study, the time-dependent unsteady calculations of low Reynolds number ows are

carried out over an Eppler 387 airfoil in both two- and three-dimensions. Various instantaneous and

time-averaged aerodynamic parameters including pressure, lift and drag coefficients are calculated in

each case and compared with the available experimental data. An observed anomaly in the pressure

coefficient around the location of the separation bubble in two-dimensional simulations is attributed

to the lack of spanwise ow due to three-dimensional instabilities.

Keywords : Low Reynolds number aerodynamics, Laminar separation bubble, Direct numerical

simulations, Unstructured methods, Parallel computing.

1 INTRODUCTION

Low Reynolds number aerodynamics is important for various applications including micro-aerial vehicles,

sailplanes, leading edge control devices, high-altitude unmanned vehicles, wind turbines and propellers.

Different from the familiar high Reynolds number aerodynamics, the low Reynolds number airfoil aero-

dynamics is generally characterized by the existence of the laminar separation bubbles shown in Fig. 1

which involve the separation of the laminar boundary layer from the surface due to a strong adverse

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pressure gradient and the reattachment of the shear layer shortly downstream. The region between the

separation and the reattachment point is called the separation bubble. These bubbles are generally un-

steady and have a signicant effect on the overall resulting aerodynamic forces. In particular they are

responsible from the increase in pressure drag due to signicant increase in boundary layer thickness over

the separation bubble.

Extensive experimental studies have been conducted in order to determine the performance charac-

teristics of airfoils at low Reynolds numbers. McGhee et al. [14] conducted wind-tunnel experiments in

the Langley Low-Turbulence Pressure Tunnel (LTPT) in order to determine the performance charac-

teristics of the Eppler 387 airfoil at Reynolds numbers from 60,000 to 460,000. The authors computed

lift and pitching moment data from the airfoil surface pressure distributions and drag data from wake

surveys. Oil ow visulization was also used to determine laminar-separation and turbulent-reattachment

locations. Cole and Mueller [2] performed similar experiments on the Eppler 387 airfoil and gave pressure

distributions similar to that of McGhee et al. Selig and McGranahan [18] documented the aerodynamic

characteristics of six different airfoils at Reynolds numbers of 100,000, 200,000, 350,000 and 500,000

including the Eppler 387 airfoil. The data taken on the E387 was compared with results from NASA

Langley in the Low-Turbulence Pressure Tunnel for surface oil ow visualization, lift data, moment data

and drag polars. Burgmann et al. [1] used scanning PIV measurement technique to investigate the span-

wise structure and dynamics of the roll-up of vortices within the separation bubble over an SD7003 airfoilat Reynolds numbers of 20,000-60,000. The authors reported non-regular half-moon shaped vortices

which extend in the spanwise direction. Montelpare and Ricci [15] analyzed the separation of the laminar

boundary layer over the Eppler 387 airfoil at low Reynolds numbers by means of infrared thermography.

Various computational approaches have been explored for the prediction of low-Reynolds number aero-

dynamics including inviscid potential ow simulations with viscous boundary layer corrections, Reynolds-

Averaged Navier-Stokes (RANS) simulations, Large Eddy Simulations (LES) and Direct Numerical Sim-

ulations (DNS). Eppler and Somers [5] developed an airfoil analysis code based on the solution of invis-

cid potential ow combined with the integral boundary layer method. Pauley et al. [16] and Lin and

Pauley [11] investigated the unsteady laminar boundary layer separation from an Eppler 387 airfoil at

low Reynolds numbers using two-dimensional direct numerical simulations. For a relatively mild pressure

gradient they found a closed steady separation bubble. When a stronger pressure gradient was applied, a

limit cycle oscillations formed in which periodic vortex shedding occurred due to Kelvin-Helmholtz insta-

bilities in the shear layer. The Strouhal number was determined by inviscid linear stability analysis to be

the most amplied instability wave of the shear layer. Their computed time-average pressure coefficient

distribution showed a region of nearly constant pressure followed by an abrupt decrease in surface pres-

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sure just before reattachment. Tang [19] used RANS simulations to predict low Reynolds number airfoil

aerodynamics. In order to predict the transition point, the author rst performed a laminar Navier-Stokes

computation and based on this laminar solution, the separation induced transition is determined as the

point where the tangential velocity adjacent to the solid surface reverses its direction for the second time

after the laminar separation. Then a RANS computation is performed with zero production term in

the selected turbulence model before the transition point. Tatineni and Zhong [20] and Windte et al.

[21] also considered the numerical simulation of low Reynolds number compressible ows over airfoils by

solving RANS equations. Yuan et al. [22] conducted a parametric study of LES at a Reynolds number

of 60,000 for the ow past an SD7003 airfoil. The authors investigated the effects of grid resolution and

sub-grid scale models. Jovicic and Breuer [10] employed large eddy simulations applying the dynamic

model by Germano as a subgrid scales model in order to predict and analyze the turbulent ow past a

NACA4415 airfoil at high angle of attacks. Hoarau et al. [8] investigated the three-dimensional transition

to turbulence around a NACA0012 wing at Reynolds numbers from 800 to 10,000. The authors presented

three-dimensional undulated large-scale vortices row with a regular spanwise wavelength which is very

similar to that of bluff body wakes. Deng [4] conducted direct numerical simulations for ow separation

and transition around a NACA0012 airfoil with an attack angle of 4 and a Reynolds number of 100,000

and the details of ow separation, formation of detached shear layer, Kelvin-Helmholtz instability, vortex

shedding, interaction of non-linear waves, breakdown and reattachment investigated.In the current paper, time-dependent both two- and three-dimensional direct numerical simulations

are carried out in order the investigate the ow structure around an Eppler 387 airfoil at a Reynolds

number of 60,000. There are two reasons to chose the Eppler 387 airfoil. The rst one is that it reveals

relatively larger laminar separation bubble at low Reynolds numbers. The second one is the availability

of detailed experimental measurements. The ow structure is examined using instantaneous and mean

vorticity contours as well as surface pressure and skin friction plots. The initial two-dimensional time-

average pressure coefficient distribution indicates a region of nearly constant pressure followed by an

abrupt decrease in the surface pressure just before reattachment as in the work of Pauley et al. [16].

However, three-dimensional simulations indicate signicant decrease in the size of the abrupt decrease

in the surface pressure which is in accord with the experimental result of McGhee et al. [14]. In our

opinion this may be attributed to the importance of the spanwise ow and vorticity destruction induced

by three-dimensional instabilities. The three-dimensional simulations seem to conrm the formation of

half-moon type vortices from the laminar shear layer with no regular spanwise structure as observed

experimentally of Burgmann et al. [1]. These vortices interact with each other and have tendency to

burst in to the outer ow causing a signicant uid motion from airfoil surface into the mean ow.

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The paper is organized as follows: In Section 2 the computational technique employed in the present

work is briey described. In Section 3 both two- and three-dimensional simulations are presented for the

Eppler 387 airfoil at a Reynolds number of 60,000. The emphasis given to the detailed comparison of

ow structure between two- and three-dimensional simulations. Conclusions are presented in Section 4.

2 NUMERICAL ALGORITHM

The numerical simulations are performed using the ARGO code [7] developed at CENAERO. The ARGO

code is based on edge-based hybrid nite element- nite volume dened on unstructured P1 tetrahedral

meshes. The original nite element formulation is reformulated into a nite volume formulation for

computational efficiency and to allow for convective stability enhancements. In accordance to the nite

element discretisation, the convective terms use central, kinetic-energy preserving ux functions; however

this ux is blended with a small amount (typically 5%) of a velocity-based upwind ux for stability; the

diffusive uxes and source terms retain the original nite element formulation at all times.

The time integration method is the three-point backward difference scheme. Since the numerical

schemes are implicit, the ow solver must solve at each time-step a system of nonlinear equations. For

this purpose, it relies on an damped inexact Newton method; the resulting linear equations are solved

iteratively with the matrix-free (nite difference) GMRES algorithm, preconditioned by the minimum

overlap RAS (restricted additive Schwarz) domain decomposition method [3].The solver uses the AOMD (Algorithm Oriented Mesh Database) library [17] for the management of

the topological mesh entities across the processors. In addition, it relies on the message passing interface

(MPI) for exchanging data between the nodes and the Autopack library [13] for handling non-deterministic

asynchronous parallel communications.

3 NUMERICAL RESULTS

In this section two- and three-dimensional time-dependent direct numerical simulations are carried outfor the Eppler 387 airfoil at a Reynolds number of 60,000 and an angle of attack 6 . For the present

calculations the relative residual is set to 10 8 . The calculations have been performed on an IBM SMP

Cluster available at NCAR and Phantom Linux Cluster in the authors group at CU.

3.1 Two-Dimensional Simulations

Initial two-dimensional calculations are carried out for the assessment of solution accuracy as well as the

code validation. The computation domain far eld is set to 20 c where c is the cord length. In order to

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investigate the mesh dependency of the solutions, three different meshes are employed: a coarse mesh M1

with 168,942 vertices, a medium size mesh M2 with 336,603 vertices and a ne mesh M3 with 664,824

vertices. The successive meshes are generated by multiplying the mesh sizes by 1 / 2 in each direction;details of the meshes are given in Table I. These successive mesh renements are also used to adapt the

both two- and three-dimensional meshes in order to reduce the computational cost. The computational

coarse mesh M1 is shown in Fig. 2. The dark band near the airfoil surface is caused by boundary layer

elements, which is formed in 17 layers with an initial layer thickness of 1 .4c 10 4 and a total thickness

of 0.025c. The boundary layer elements continues smoothly into the wake region in order to capture

the vortex shedding process. Near the airfoil leading edge the mesh along the airfoil surface is rened

due to strong changes in the airfoil leading edge curvature. Renement is also applied the outside the

boundary layer so that the element sizes matches better with the elements at the far eld. The meshes

are partitioned into 16 sub-domains apriori for parallel computation.

The calculations are carried out at a chord-based Reynolds number of 60,000 and an angle of attack 6

similar to that of Ref. [6]. The speed of the free stream velocity is set to 24 .9m/s ; the constant viscosity

is modied in order to achieve the above Reynolds number. The pressure and the temperature are also

set consistent with air at standard pressure and temperature. In order to speed up the calculations

the energy equation is set to isothermal ow. In addition, large time steps are used during the initial

calculations in order to reach vortex shedding regime with less number of iterations. Then the calculationsare continued with a time step of 10 4 s. The computed time-averaged pressure distribution is given in

Fig. 3. The computed results indicate convergence towards the mesh independent results. The results

on meshes M2 and M3 are relatively very close to each other. Although the calculations on ne mesh M3

are desirable due to its higher accuracy, its computational cost is signicantly higher particularly for the

three-dimensional computations. Therefore, based on the numerical results on meshes M1 to M3 we made

one more mesh adaptation in order to reduce the required number of vertices further. This adaptation

level leads to a new adaptive mesh with 392,250 vertices. The results obtained on this adaptive mesh

are very close to the results on the ne mesh M3 as seen in Fig. 3. Therefore, we will continue our

calculations with the new adaptive mesh since the numerical results between the new adaptive mesh and

the ne mesh M3 are almost identical, but with approximately half the computational cost.

The time variation of lift and drag coefficients is given in Fig. 4 on the adaptive mesh for the above

ow parameters. An approximate period of T = 0 .0152s is observed in the lift and drag coefficients. The

ow parameters are non-dimensionalized by the airfoil chord length c and the free-stream velocity U .

The non-dimensional Strouhal number St = c/TU is 2.64 with a non-dimensional mean lift coefficient

of 0.9752 and a drag coefficient of 0 .0425. Several snapshot of vortex shedding at times t = 0, t = T/ 4,

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t = 2 T/ 4 and t = 3 T/ 4 are presented in Fig. 5. In addition, airfoil surface pressure coefficient and

skin friction coefficient are also given for the same time levels in Fig. 6. The vorticity contours indicate

separation of the shear layer over the airfoil upper surface, after which Kelvin-Helmholtz type instabilities

develop into vortices. Fig. 5 shows two large vortices over the airfoil upper surface which enlarge in size

by absorbing vorticity from the shear layer. This process continues until the downstream vortex reaches

a critical size. When the downstream primary vortex reaches its critical size, it begins to advect towards

the trailing edge. The strengthening of the upstream vortex is observed by deepening in the C p and C f

curves while the the strength of the convected vortex is gradually weakened. In addition to these large

vortices, other small size vortices may be observed next to the airfoil surface due to vortex solid-wall

interactions. The location of the large vortices and their transportation can be observed more clearly

from the pressure coefficient distribution in Fig. 6 by the dip in the C p curves. As the primary vortex

leaves the trailing edge of the airfoil, a secondary vortex is formed on the lower part of the airfoil and

orbit around the more stronger primary vortex once they clear the airfoil. During this motion the weaker

vortex is stretched around the primary vortex and it loses its strength rapidly due to vortex stretching.

In addition, the shedding of the primary vortex over the airfoil surface signicantly reduces the bound

vortcity of the airfoil and causes a temporary drop in the lift coefficient.

The time averaged vorticity contours, pressure contours and streamlines are given in Fig. 7 around

the Eppler 387 airfoil. The streamlines indicate a relatively larger separation bubble over the airfoilupper surface. A small secondary bubble is also observed beneath the primary separation bubble. The

separation points may be identied more clearly from the time mean skin friction coefficient in Fig. 8.

The separation point is also revealed by a start of plateau in the pressure coefficient distribution and the

reattachment is indicated by the end of bump. The plateau corresponds the stable part of the separated

laminar shear layer. The bump in the time averaged pressure curve is related to where the vortices spend

the most time as they develop at the end of laminar separation layer. The positive spike in average skin

friction coefficient around x = 0 .6c is due to secondary vortices. The separation bubble causes negative

skin friction over the upper surface of the airfoil which reduces the skin friction drag. However, the bump

observed in the pressure coefficient causes signicant pressure drag and it is signicantly larger than the

skin friction reduction.

The comparison of the present two-dimensional numerical results with the experimental results of

McGhee et al. [14] in Fig. 8 indicates signicant discrepancies particularly for the pressure coefficient

distribution. The values of the pressure distribution seem to be signicantly off by a constant factor

up to the the bump at the end of the pressure plateau. In addition, the bump almost disappears in

the experimental results. As described in the following section, we believe three-dimensional effects are

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responsible for that.

3.2 Three-Dimensional Simulations

Three-dimensional numerical simulations are carried out for the same ow parameters. The computational

mesh is created by sweeping the two-dimensional cross-section of the adaptive mesh in the third dimension

with 65 uniform node points between 0 .0 z/c 0.5. The periodic boundary conditions are applied atz/c = 0 and z/c = 0 .5 planes. The three-dimensional adapted mesh consists of 8,498,750 vertices and it

is partitioned into 360 sub-domains apriori . Therefore, each processor reads its own mesh data belongs to

its sub-domain number when the calculations start. Each iteration on 360 nodes at the NCAR Bluevista

machine takes approximately 120 seconds. For the present three-dimensional calculations we use 2000

iterations in order to obtain the time-averaged statistics after initial transient calculations.

The three-dimensional numerical results are neither as well organized nor as periodic as in two dimen-

sions. This may be clearly seen from the lift and drag coefficient history shown in Fig. 9. There is no

repeated pattern in either C l or C d . The time average lift and drag coefficients are computed as 0 .8128

and 0 .0772, respectively. Although the lift coefficient is 17% lower compared to the two-dimensional

simulations the drag coefficient is approximately 82% larger. The sequence of snapshots for the vorticity

magnitude iso-surfaces around an Eppler 387 airfoil is shown in Fig. 10 at several time levels along with

the vorticity magnitude contours at z/c = 0 .25 plane in Fig. 11. The stable laminar shear layer detachesfrom the airfoil upper surface due to adverse pressure gradient and Kelvin-Helmholtz instabilities grow in

the separated shear layer. These instabilities in the shear layer lead to formation of rather short curved

vortex tubes in the spanwise direction with no regular structure. These vortex structures are similar to

the half-moon type vortices observed in the experimental work of Burgmann et al. [1]. Further down-

stream, these vortices create more complex three-dimensional vortex structures. As it may be seen there

is no regular structure in the vortex structures due to the roll-up of vortices and very strong interactions

between these vortices. The strong interaction between the vortices causes vortex burst into the outer

ow causing a signicant uid motion from airfoil surface into the mean ow. This mechanism also causes

a signicant transport of vorticity into the mean ow which may explain the lower lift coefficient in the

three-dimensional simulations. Different from the two-dimensional simulation, the formation of vortices

over the airfoil upper surface is more continuous and vortices move with almost a constant speed towards

the airfoil trailing edge. Therefore, the strength of vorticity in the primary vortices shed from the shear

layer is weaker compared to that of two-dimensional primary vortices. In addition, the shedding vortices

follow a path rather far away from the airfoil upper surface compared to the two-dimensional simulations.

The formation of vortices as a consequence of the shear layer roll-up will take place further downstream

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as well. Another difference is that the shear layer in the two-dimensional simulations is parallel to the

airfoil chord while it is parallel to the mean ow in the three-dimensional simulations. However, the

main difference in the three-dimensional simulations is the existence of a very large inviscid ow region

between the stable part of the shear layer and the airfoil upper surface as seen in Fig. 11. This is

mainly due to the vortex roll-up process in the separated shear layer which prevents the vortices moving

further upstream close to the airfoil surface. The sequence of surface pressure coefficient distribution at

z/c = 0 .25 is given in Fig. 12 at the same time levels with Fig. 10 and Fig. 11. It may be seen that

the amplitude of the oscillation in the pressure coefficient distribution towards the airfoil trailing edge

is relatively weaker compared to that of two-dimensional simulations. This is due to more continuous

but weaker shed vortices from the shear layer which follow a path rather far away from the airfoil upper

surface.

The comparison of mean pressure and skin friction coefficients are given in Fig. 13 and the the

pressure coefficient is compared with the experimental results of McGhee et al. [14] as well as the two-

dimensional numerical results. The comparison shows that there is a signicant difference between two-

and three-dimensional pressure distributions and the three-dimensional numerical results are relatively in

good agreement with the experimental results of McGhee et al. [14]. An observed anomaly in the pressure

coefficient around the location of the separation bubble in two-dimensional simulations is attributed to

the lack of spanwise ow due to three-dimensional instabilities. From the skin friction coefficient curve inFig. 13, we observe that the separation point moves towards the leading edge for the three-dimensional

simulations. The size of the primary separation bubble and the size of the secondary weaker separation

bubble beneath the primary bubble are signicantly increased. The early separation of the laminar shear

layer and the larger separation region in the three-dimensional simulations may be seen more clearly in

Fig. 14. In addition, the mean shear layer moves further away from the airfoil surface compared to the

two-dimensional simulations indicating larger boundary boundary layer thickness. This may explain the

higher drag coefficient observed in the three-dimensional simulations.

Spectral analysis is also performed in order to nd the wavelengths present in the kinetic energy in the

spanwise direction. The spectral analysis indicates that the dominant wavelengths seen from Fig. 15 are

0.5c and 0 .25c for points (0 .75c, 0.1c) and (1 .0c, 0.1c), respectively. In both cases a signicant part of the

energy is still contained in a wavelength of 0 .5c, which is the width of the computational domain. This

suggests that the size of the computational domain in the spanwise direction ought to further increase

in order to reach full mesh-independence. Therefore, the computation domain in the spanwise direction

is increased to 1 .0c. However, we had to use the same number of grid points in the spanwise direction

due to the available computer limitations. The computed spectral analysis of the kinetic energy is given

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in Fig. 16 for the same locations. The new spectral analysis also indicates that the domain wavelength

dominates the energy specturum. However, the computed pressure coefficient distribution does not show

any difference compared to the previous results as seen in Fig. 17. Nevertheless, the results clearly

indicate a much better representation of the physics compared to the 2D computations given the much

improved correspondence to the measured Cp distributions and the conrmation of the half-moon shaped

structures.

4 CONCLUSIONS

In this study, the time-dependent unsteady calculations of low Reynolds number ows are carried out

over an Eppler 387 airfoil in both two- and three-dimensions. Various instantaneous and time-averaged

aerodynamic parameters including pressure, lift, and drag coefficients are calculated in each case and

compared with the available experimental data. In our simulations we demonstrate that there is a

signicant difference between two- and three-dimensional pressure coefficient distributions over the airfoil

surface. This is particularly due to three-dimensional instabilities leading the ow to move in the third-

dimension. Eventually, the three-dimensional structure of the ow leads signicant difference for overall

aerodynamic characteristic of the airfoil. The present three-dimensional simulations are shown to be

in relatively good agreement with the experimental results of McGhee et al. [14]. In addition, these

numerical simulations provide us very detailed information for the laminar separation bubble which isnot directly possible with the wind tunnel experiment due to relatively large level of free stream turbulence

uctuations at the inow.

5 ACKNOWLEDGMENTS

This work was supported by NSF-ITR grant number CN50427947. The authors acknowledge the use of

the Bluevista machine at NCAR.

References

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bubble. Exp. Fluids 41 , (2006), 319326.

[2] G. M. Cole and T. J. Mueller, Experimental measurements of the laminar separation bubble on an

Eppler 387 airfoil at low Reynolds number. UNDAS-1419-FR, (1990).

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[3] X.C. Cai, C. Farhat and M. Sarkis, A minimum overlap restricted additive Schwarz preconditioner

and applications in 3D ow simulations. Contemp. Math. 218 , (1998), 478484.

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2006-1269, (2007).

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dimensional transition to turbulence in the incompressible ow around a NACA0012 wing. J. Fluid

Mech 496 , (2003), 63-72.

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1570-1576.

[12] M. -S. Liou. A sequel to AUSM, part II: AUSM+ -up for all speeds. J. Comp. Physics , 214 , (2006),

137170.

[13] J. Flaherty, R. Loy, P.C. Scully, M. S. Shephard, B. K. Szymanski, J. D. Teresco and L. H. Ziantz,

Load Balancing and Communication Optimization for Parallel Adaptive Finite Element Computa-

tions. Proc. XVII Int. Conf. Chilean Comp. Sci. Soc. , (1997), 246255.

[14] R. J. McGhee, B. S. Walker and B. F. Millard, Experimental results for the Eppler 387 airfoil at

low Reynolds numbers in the Langley Low-Turbulence Pressure Tunnel. NASA TM 4062, NASA,

(1988).

[15] Montelpare and Ricci, Wind tunel aerodynamic tests on six airfoils for use on samll wind turbines.

Transactions of the ASME 126 , (2004), 9861001.

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[16] L. Pauley, P. Moin and W. Reynolds, The structure of two-dimensional separation. J. Fluid Mech

220 , (1990), 397-411.

[17] Jean-Francois Remacle, Ottmar Klaas, Joseph E. Flaherty and Mark S. Shephard, Parallel Algorithm

Oriented Mesh Database. Engineering with Computers 18 , (2002), 274284.

[18] M. S. Selig and B. D. McGranahan, Wind tunnel aerodynamic tests on six airfoils for use on samll

wind turbines. Transactions of the ASME 126 , (2004), 9861001.

[19] L. Tang, RANS simulation of low-Reynolds-number airfoil aerodynamics. AIAA Paper 2006-249,

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Mesh Number of Vertices Boundary Layer dmin /c Boundary Layer NM1 168,942 0.00014 17M2 336,603 0.00010 25M3 664,824 0.00007 35

Adaptive 392,250 0.00010 25

Table 1: Description of computational meshes used in the present work. dmin is the minimum normalmesh spacing on the airfoil surface.

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Figure 1: The time-averaged model of laminar separation bubble sketched by Horton [9].

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Figure 2: The computational unstructured coarse mesh M1 for the ow past an Eppler 387 airfoil with168,942 vertices.

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x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Mesh M1Mesh M2Mesh M3Mesh Adaptive

Figure 3: Mesh convergence of the pressure coefficient distribution over the Eppler 387 airfoil at = 6

and Re=60,000.

t

C l

0.7 0.7 02 0.7 04 0.706 0.708 0.71 0.712 0.714 0.7160.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

t

C d

0 .7 0.7 02 0.704 0 .70 6 0.708 0.71 0 .7 12 0.714 0.7160

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Figure 4: Variation of lift and drag coefficients with time around an Eppler 387 airfoil at = 6 and

Re=60,000.

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t = 0

t = 1 / 4T

t = 2 / 4T

t = 3 / 4T

Figure 5: Several snapshots of vorticity magnitude contours around an Eppler 387 airfoil at = 6 andRe=60,000.

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x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0

x/c

C f

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0 .08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

t=0

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=1/4T

x/c

C f

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0 .08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

t=1/4T

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=2/4T

x/c

C f

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0 .08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

t=2/4T

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=3/4T

x/c

C f

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0 .08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

t=3/4T

Figure 6: Computed pressure (left) and skin friction (right) coefficients for an Eppler 387 airfoil at severaldifferent time levels t = 0, t = T/ 4, t = 2 T/ 4 and t = 3 T/ 4 at = 6 and Re=60,000.

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Figure 7: Computed mean vorticity contours (upper), pressure contours (middle) and streamtraces (bot-tom) around an Eppler 387 airfoil at = 6 and Re=60,000.

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x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

PresentMcGhee et al.

x/c

C f

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Figure 8: Computed mean pressure and skin friction coefficients around an Eppler 387 airfoil at = 6

and Re=60,000.

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t

C l

0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.820.72

0.76

0.8

0.84

0.88

0.92

t

C d

0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.820.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 9: Variation of three-dimensional lift and drag coefficients with time around an Eppler 387 airfoilat = 6 and Re=60,000.

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Figure 10: Computed vorticity magnitude iso-surfaces around an Eppler 387 airfoil at = 6 andRe=60,000.

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Figure 11: Computed vorticity magnitude contours at z/c = 0 .25 plane for the ow past over an Eppler387 airfoil at = 6 and Re=60,000.

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x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.764

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.772

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.780

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.788

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.796

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.804

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.812

x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

t=0.820

Figure 12: Computed pressure coefficients at z/c = 0 .25 at several different time levels for the ow pastover an Eppler 387 airfoil at = 6 and Re=60,000.

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x/c

C p

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Present 2DPresent 3DMcGhee et al.

x/c

C f

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Present 2DPresent 3D

Figure 13: Computed mean pressure and skin friction coefficients at z/c = 0 .25 around an Eppler 387airfoil at = 6 and Re=60,000.

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Figure 14: Computed mean vorticity contours around an Eppler 387 airfoil at = 6 and Re=60,000.

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P o w e r

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

/c

P o w e r

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

/c

Figure 15: Spectral analysis of the the kinetic energy along the spanwise line at point (0 .75c, 0.1c) (upper)and point (1 .0c, 0.1c) (lower) for an Eppler airfoil at = 6 and Re=60,000.

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P o w e r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

/c

P o w e r

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

/c

Figure 16: Spectral analysis of the the kinetic energy along the spanwise line at point (0 .75c, 0.1c) (upper)and point (1 .0c, 0.1c) (lower) for an Eppler airfoil at = 6 and Re=60,000 with 0 .0 z/c 1.0.

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Sahin Eppler

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