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High-Accuracy Viscous Simulations of Gust Interaction with

Stationary and Pitching Wing Sections

Vladimir V. Golubev1 and Brian D. Dreyer

2

Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA

Nikolay V. Golubev3

Central Aerohydrodynamic Institute (TsAGI), Zhukovsky, Russia

Miguel R. Visbal4

Air Force Research Laboratory, WPAFB, OH 80906, USA

The paper reports on the progress in the development of a high-accuracy prediction tool for the analysis of

unsteady viscous fluid-structure interaction phenomena. The numerical procedure employs a high-order low-

pass filter operator which selectively damps the poorly resolved high-frequency content to retain numerical

accuracy and stability over a wide range of flow regimes. An efficient analytical model is developed to

introduce an unsteady incompressible 2D vortical flow perturbation inside the computational domain

through a source term in the momentum equations. The unsteady aerodynamic and aeroacoustic responses of

stationary and pitching airfoil sections subject to an unsteady viscous flow with imposed harmonic gust are

examined in test studies. The results are compared with available experimental and previously validated

numerical predictions.

I. Introduction

The current work advances the previous inviscid analyses of the nonlinear unsteady response of oscillating

airfoils subject to incoming flow disturbances, initiated in Ref. [1] and further extended in Ref. [2]. The problem is

of significant importance for various external and internal aerodynamic applications, such as the unsteady wing

loading in maneuvering aircraft and vortex interference of lifting surfaces, or unsteady rotor-stator interactions. In

particular, the gust-airfoil interaction problem serves as one of the CAA benchmarks (e.g., Ref. [3]) modeling the

unsteady aerodynamic and aeroacoustic responses of a lifting surface to an incident flow turbulence or a wake-

induced flow unsteadiness. Moreover, for low Reynolds number unsteady flows typical of MAV applications, the

gust encounter can induce a particularly significant aerodynamic and aeroelastic responses. While the inviscid

nonlinear airfoil response phenomena may result, e.g., from high-amplitude flow and structural excitations as well as

shocks, the viscous nonlinear unsteady response effects are generally dominated by the dynamic boundary-layer

separation and generation of vortices, which results, e.g., in complex hysteresis phenomena observed in unsteady

aerodynamic characteristics of oscillating airfoils.

In an effort to develop a highly accurate prediction tool for a fully nonlinear, viscous, unsteady analysis of fluid

interaction with a flexible structure, this work examines a set of benchmark cases that involve an airfoil response to

the impinging gust and a pitching airfoil motion. An efficient model to introduce the vortical, incompressible

unsteady flow perturbation inside the computational domain through a source term in the momentum equations is

developed, which conveniently eliminates the need to impose the perturbation at the upstream boundary. In

particular, this allows to employ grid stretching throughout the farfield regions to remove any spurious boundary

reflections.

The gust-airfoil interaction studies focus on the comparative analysis of the nonlinear viscous effects on both

unsteady aerodynamic and aeroacoustic airfoil responses. Two benchmark test cases are considered corresponding

to the low-speed unsteady flow around a pitching SD7003 airfoil, and a high-speed, high-Reynolds number unsteady

flow around a stationary Joukowski airfoil. Numerical predictions are compared with an experimental database and

previously validated computational results.

1 Associate Professor, Associate Fellow AIAA

2 Graduate Research Assistant

3 Lead Engineer, Helicopter Division

4 Technical Area Leader, Associate Fellow AIAA

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-11

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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II. Governing Equations and Numerical Implementation

The governing equations for the numerical simulations are the compressible Navier-Stokes equations represented in

strong, conservative, time-dependent form in the generalized curvilinear computational coordinates (ξ,η,ζ,τ)

transformed from the physical coordinates (x,y,z,t):

SHGFHGF

J

Q vvviiir

rrrrrrr

=∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂+

∂

∂][

Re

1)(

ζηξζηξτ (1)

The solution vector ),,,,( ewvuQ ρρρρρ=r

is defined in terms of the flow density ρ, Cartesian flow velocity

components (u, v, w), and flow specific energy,

)(2

1

)1(

222

2wvu

M

Te +++

−=

∞γγ,

with assumed perfect gas relationship 2/ ∞= MTp γρ connecting the flow pressure p, temperature T, and the

freestream Mach number M∞ (γ is the specific heat ratio). The other variables in (3) include the inviscid flux vectors

defined by

−+

+

+

+

=

pupe

puw

puv

puu

u

F

t

z

y

x

i

ξρ

ξρ

ξρ

ξρ

ρ

ˆˆ)(

ˆˆ

ˆˆ

ˆˆ

ˆ

r,

−+

+

+

+

=

pvpe

pvw

pvv

pvu

v

G

t

z

y

x

i

ηρ

ηρ

ηρ

ηρ

ρ

ˆˆ)(

ˆˆ

ˆˆ

ˆˆ

ˆ

r,

−+

+

+

+

=

pwpe

pww

pwv

pwu

w

H

t

z

y

x

i

ζρ

ζρ

ζρ

ζρ

ρ

ˆˆ)(

ˆˆ

ˆˆ

ˆˆ

ˆ

r, (2)

the transformation Jacobian, ),,,(/),,,( tzyxJ ∂∂= τζηξ , the metric quantities defined, e.g., as

xJx ∂∂= − /)(ˆ 1 ξξ , etc., and the transformed flow velocity components,

wvuw

wvuv

wvuu

zyxt

zyxt

zyxt

ζζζζ

ηηηη

ξξξξ

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

+++=

+++=

+++=

(3)

The viscous flux vectors, vFr

, vGr

and vHr

, are defined, e.g., in Ref. [4]. Sr

represents the source term (further

described below) developed in the current work to generate the incompressible vortical perturbation upstream of the

wing section. All flow variables are normalized by their respective reference freestream values except for pressure

which is nondimensionalized by2

∞∞uρ .

Note that the governing equations are represented in the original unfiltered form used unchanged in laminar,

transitional or fully turbulent regions of the flow. Refs. [5-7] provide details on the Implicit LES (ILES) procedure

employed in the numerical code in which a high-order low-pass filter operator is applied to the dependent variables

during the solution process, in contrast to the standard LES addition of sub-grid stress (SGS) and heat flux terms.

The resulting filter selectively damps the evolving poorly resolved high-frequency content of the solution.

The numerical simulations are conducted using a high-order Navier-Stokes solver FDL3DI which has been

extensively validated for a variety of complex unsteady flows, e.g., in Ref. [8], and for computational acoustics

problems, e.g., in Ref. [9]. The code employs a finite-difference approach to discretize the governing equations, with

all the spatial derivatives obtained using the high-order compact-differencing schemes from Ref. [10]. For the wing

section computations of the current paper, a sixth-order scheme is used. At boundary points, higher-order one-sided

formulas are utilized which retain the tridiagonal form of the scheme. In order to ensure that the Geometric

Conservation Law (GCL) is satisfied, the time metric terms are evaluated employing the procedures described in

detail in Ref. [8]. Finally, the time marching is accomplished by incorporating a second-order iterative, implicit

approximately-factored procedure as described in Refs. [5, 6].

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III. Gust Interaction Model

For numerical simulations of unsteady flow interaction with an airfoil, we select a two-dimensional vortical

perturbation velocity harmonic (gust), described in the form,

)cos(

)cos(

tyxv

tyxu

gvg

gug

ωβαε

ωβαε

−+=

−+=, (4)

where the gust component amplitudes are,

,22

+−=

∞

βα

βεε

ug

u ,

22

+=

∞

βα

αεε

ug

v (5)

εg is the gust intensity relative to the mean flow, α and β are the gust wave numbers in the x and y directions, ωg is

the imposed gust frequency, and ∞u is the convective freestream velocity. Note that ∞= ug /ωα and χαβ tan= ,

where χ is the angle between the normal vector of the gust phase front and the x-axis.

Figure 1: Gust-airfoil interaction problem.

IV. Gust Source Implementation

The approach proposed in this work to generate a 2D gust inside the computational domain extends the analysis of

Ref. [11] where the analytical 1D gust source was developed. In contrast to the standard procedure used, e.g., in

Refs. [1, 2, 12], the proposed method avoids the difficulty of imposing the proper vortical flow disturbance at the

upstream boundary, and allows using stretched computational grids throughout the farfield to minimize spurious

reflections.

Following Ref. [11], we assume that the gust source is located in a region of uniform flow aligned with the x-

direction, and that the interactions between the gust and other waves are negligible. Furthermore, since the gust to be

generated downstream of the source region is incompressible and vortical of the form (4), the source terms should

only be added to the momentum equations in (1). To this end, to obtain the required solution for )( , vu ssS =r

, we

examine the gust momentum equations,

v

gg

u

gg

sx

vu

t

v

sx

uu

t

u

=∂

∂+

∂

∂

=∂

∂+

∂

∂

∞

∞

(6)

Note that, in order to satisfy the divergence-free condition for the incompressible gust, the source components are

constrained by 0// =∂∂+∂∂ ysxs vu . We then seek the solution in the form,

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)sin()('),,(

)cos()(),,(

0

0

ψβω

ψβωβ

+−=

+−=

ytxKgtyxs

ytxKgtyxs

gv

gu , (7)

where g’(x)=dg/dx and ψ0 and K are constants. The solution is obtained in the complex plane where (6) reduces to

)('~~

)(~~

xKgu

ivi

dx

vd

xKgu

uidx

ud

g

g

g

g

∞

∞

−=+

=+

α

βα

, (8)

and

)}](exp{)(~[),,(

)}](exp{)(~[),,(

0

0

ψβω

ψβω

+−ℜ=

+−ℜ=

ytixvtyxv

ytixutyxu

ggg

ggg

Focusing for the moment on the first equation in (8), the solution can be expressed as follows,

∫∞−∞

−=x

g dgxixiu

Ku σσαα

β)()exp()exp(~ , (9)

Now, introducing the function g(x),

>−

≤−−+

=

bxx

bxxxxb

xg

s

ss

π

π

||,0

||)}),(cos{1(2

1

)( , (10)

the integration of (8) is performed to obtain,

)sin()}(exp{)(

~22

2

buxxi

b

bKu

g

s

g

g

∞

−−−

−=πω

ααω

β

The second equation in (8) can be integrated in a similar fashion. One may then deduce that in order to generate the

gust solution of the form (4) downstream of the source region bxx s /|| π≤− , we should impose the following

source components in the momentum equations from (1),

)sin()()('),,(

)cos()()(),,(

sgv

sgu

xytyxKgtyxs

xytyxKgtyxs

αβωλ

αβωλβ

−−=

−−=, (11)

where the constant

)sin(

)(

2

22

22

2

bub

buK

gg

∞

∞ −

+=

πωα

βα

αε

and the function λ(y) is selected to provide a smooth transition in the y-direction to provide a compact region of the

uniform gust distribution, e.g.,

)]}(3tanh[)](3{tanh[2

1)( ss yyyyy −−+=λ (12)

Note that a shear layer is generated in the region where λ(y) varies, but the resulting pressure waves have generally

very moderate amplitudes. The generated gust solutions are well-behaved as long as the parameters are selected to

avoid very high values of the constant K.

V. Results and Discussion

The purpose of the current investigation is to examine the code performance for two cases corresponding to low-

and high-speed unsteady flows. In the present test study, the numerical code parameters and the baseline 649 × 395

× 3 grid are similar to those employed in Ref. [13] in an ILES study of the low-Re steady-state flow around a

stationary SD7003 wing section slice. To this end, the code ability to perform numerically stable simulations over a

wide range of flow regimes using the code’s advanced low-pass high-order spatial filtering procedure is tested. The

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efficiency of parallel MPI implementation is illustrated in Figure 2 that includes comparison of the code scalability

performance using several compilers tested on different computer clusters, including ERAU’s 262-processor

Beowulf Zeus cluster (64-bit, 3.2 GHz Intel Xeon, 4GB RAM systems).

Figure 2: Code Parallel Performance.

A. Gust Source Validation Test computations to validate the gust source model are first performed for stationary and pitching SD7003

airfoil section in the laminar flow regime with M∞ =0.1 and 000,10/Re == ∞ µρ cu , where c is the airfoil chord.

Recall that in the numerical implementation, the variables are non-dimensionalized by the airfoil chord, flow

density, and flow velocity.

A fixed time step of 5105 −×=∆t is chosen for numerical simulations, which corresponds to CFL~10. For the

current analysis, the computations are parallelized using an efficient grid partitioning with 20 blocks, thus taking

about 1 CPU sec per a time step.

Figure 3. Contour plot for flow vorticity at t=5.6 sec.

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(a) y=-1

(b) y=0

(c) y=1

Figure 4. Flow velocity and vorticity after t=6 sec obtained along the lines y=-1, 0, 1.

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The pitching motion is prescribed in terms of the harmonic variation of the airfoil angle of attack with frequency

ptω relative to the pitching center xp,

(13)

which translates to the following time-dependent grid motion:

(14)

where (X0,Y0) corresponds to the reference airfoil position at α0. Test simulations are performed to compare two

approaches to move the grid. In the first approach, the whole grid is moved in a rigid fashion, while in the second,

the grid motion according to (14) is imposed in a circular region surrounding the airfoil, with the motion gradually

eliminated in the far field. Both approaches produce practically identical results in the regions of interest.

In this part of the study, test computations are performed for the following parameters of the pitching

motion: cxpptm 25.0,8,5.21,0 00

0 ==== ωαα .

For all the results shown in this and subsequent sections involving upstream flow unsteadiness, a 2D gust

convecting with 045=χ is selected. In the current test study, the gust oscillates with reduced

frequency 22/ == ∞uck gg ω . Note that with the code non-dimensionalization, the gust wavenumbers and

frequency are thus .42 ==== gg kωβα

Figure 2 shows the time snapshot of the contour plot for the flow vorticity yuxvz ∂∂−∂∂= //ϖ obtained at

t=5.6 sec. The 2D gust is generated using the momentum source (11) in a region specified by xs=-3.5, ys=2, and b=5

in equations (10) and (12). Notice the vortical patterns generated by the gust and the pitching airfoil.

Since the grid is significantly stretched in the selected gust source region, the generated gust amplitude appears

to be distributed non-uniformly in the y-direction. Another set of results was produced with xs=-1.5. Figure 3 shows

the comparison of the analytical and numerical predictions for the flow velocity components u, v and vorticity zϖ

after t=6 sec, obtained along the convection lines y=-1, y=0 and y=1. Notice that the computed velocities (solid

lines) deviate from the analytical predictions due to the impact of the potential pressure waves propagating from the

airfoil; however, the vorticity plots match very well.

B. Interaction of Unsteady Low-Speed Flow with Pitching SD7003 Airfoil In an effort to examine the code performance for low-speed unsteady viscous flows, we consider a flow regime

with the upstream mean flow velocity is V=7.5m/s and the flow Reynolds number Re=100,000. Two cases of

pitching motion are considered for the SD7003 airfoil, specified with the following parameters:

Case 1: cxpptm 25.0,77.3,10,3 00

0 ==== ωαα

Case 2: cxpptm 25.0,77.3,10,8 00

0 ==== ωαα

The pitching frequency thus corresponds to the Strouhal number Sr=0.7. Note that these flow conditions and the

airfoil pitching parameters correspond to results from the experimental database developed for the NACA-23012

airfoil in Ref. [14] and further examined in Ref. [2]. The latter are included in the comparisons below.

v (a) (b) (c)

Figure 5. Comparison of stationary aerodynamic coefficients Cl (a), Cd (b), Cm (c): Current predictions (red line, ‘+’) and Ref.[13]

(blue line, ‘x’) for SD7003 airfoil, Ref. [14] (black dashed line) for NACA-23012 airfoil.

0sin)( αωαα += tt ptm

)(cos)(sin)()(

)(sin)(cos)()(

00

00

tYtxXtY

tYtxXxtX

p

pp

αα

αα

+−−=

+−+=

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Figure 5 shows comparison of current predictions for the stationary aerodynamic coefficients of lift, drag and

pitching moment about the quarter-chord, with ILES predictions of Ref. [13] obtained for a 3D wing section with

the same airfoil at Re=60,000, and with results from Ref. [14] for the NACA-23012 airfoil. The results are obtained

after averaging data for a sufficient period of time. Figure 6 illustrates the time history of large fluctuations of Cl

coefficient at near-stall flow conditions, while Figure 7 shows the vorticity contour plots for selected angles of

attack. Note a massive flow separation pattern at α= 170. In addition, to further investigate the effect of Reynolds

number in current comparisons, Figure 8 shows the vorticity contours obtained for α=40

at Re=100,000 (a) and

Re=10,000 (b). Note significant differences in the boundary layer and vorticity shedding patterns that have an effect

on the airfoil aerodynamic performance.

Figure 6. Stationary lift fluctuations at different angles of attack.

(a) (b) (c)

Figure 7. Vorticity contour plots for α=17

0 (a), α=3

0 (b), α=-7

0 (c).

Figure 8. Vorticity contour plots for α=4

0: Re=100,000 (a), Re=10,000 (b).

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The aerodynamic hysteresis curves obtained for Cases 1 and 2 of the pitching airfoil motion are compared in

Figure 9. The results also illustrate effects of the unsteady upstream flow conditions and the Reynolds number. Note

that the effect of the pitching airfoil gust response is rather moderate for the 2D gust imposed with kg =1 and εg =0.1.

Surprisingly, the Reynolds number also shows a modest impact on the unsteady aerodynamic response. On the other

hand, the mismatch with experimental results is evident and requires further analysis, in particular, of the impact of

different airfoil geometry. The differences may be partially attributed to the flow separation conditions observed in

the experiment for the NACA-23012 airfoil, and especially to the near-leading-edge laminar separation bubble with

dynamic vortex that dominates the airfoil aerodynamic performance and finally separates at higher pitching

amplitudes. In the current simulations for the SD7003 airfoil, this effect is practically absent for the selected range

of the angles of attack as illustrated for the maxima of the pitching amplitudes in Case 1 in Figure 10, but it does

appear in the previously attempted RANS simulations [2] for the NACA-23012 airfoil.

(a) (b) (c)

Figure 9. Comparison of aerodynamic coefficients Cl (a), Cd (b), Cm (c). Case 1: baseline conditions (blue line), gust effect (red

dash-dotted line), effect of Re=10,000 (squares), Ref. [14] (black line). Case 2: baseline conditions (light blue line). Ref [14]

(black dashed line).

(a) (b)

Figure 10. Vorticity contour plots for Case 1 with α=13

0 (a) and α=-7

0 (b). Top: without gust; bottom: with 2D gust.

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C. Gust Interaction with Stationary Joukowski Airfoil

We now employ the same grid used in the previous sections but generated around the symmetric Joukowski

airfoil. The purpose of this part of the study is to compare viscous predictions with validated inviscid results of Ref.

[15] for a CAA benchmark gust-airfoil interaction case proposed in Ref. [3] for the unloaded Joukowski airfoil (at

zero angle of attack). Thus, we consider the same small-amplitude 2D gust generated upstream of the stationary

airfoil with reduced frequency kg = 1 but now convected by a mean flow with M=0.5. To further examine the code

ability to produce efficient filtering of unresolved turbulent fluctuation scales, we impose a more realistic flow

condition with Re=2x106.

Comparison of viscous and inviscid results for the airfoil surface pressure in Figure 11 shows the effect of

turbulent transition with large near-wall pressure fluctuations. The current essentially 2D grid geometry cannot fully

describe the 3D turbulent flow breakdown, but the transition process appears similar to the ILES results obtained in

Ref. [6] and references therein for the high-Reynolds number flow around Aerospatiale A-airfoil section.

(a) (b)

Figure 11. Airfoil surface pressure: mean (a), RMS (b). Current predictions (red line), Ref. [15] (green dashed line).

(a) (b) (c)

Figure 12. Directivity of RMS acoustic intensity at R=1 (a), R=2 (b), R=4 (c). Current predictions (red line), Ref. [15] (green

dashed line).

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(a) (b)

Figure 13. Pressure (a) and vorticity (b) contour plots with successive zoom (top to bottom) on the airfoil trailing edge.

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Figure 12 further compares directivity plots for the RMS acoustic intensity at three circles at distances from one

to four chords away from the airfoil midchord position. In the near field, the comparison appears satisfactory but

shows wake-induced fluctuations, more noticeable in the downstream part of the lobes. The mismatch with acoustic

directivity from Ref. [15] appears more significant in the farfield with characteristic lobes developing in the

upstream region. Although additional analysis will be required to more clearly identify the causes of such a pattern,

the resulting farfield lobes appear similar to the directivities observed in the LES study of the trailing-edge noise in

Ref. [16]. This hypothesis is supported by the pressure and vorticity contour plots from the current simulations

shown in Figure 13 with successive zoom on the airfoil trailing edge. The noise radiation pattern from the trailing

edge also closely matches results from Ref. [16], and suggests possible superposition of the leading and trailing edge

noise radiation observed in Figures 12.

IV. Conclusions

The current study reported on the progress in the development of an ILES prediction tool for the high-accuracy

viscous analysis of unsteady fluid-structure interaction phenomena. In contrast to the standard LES addition of sub-

grid stress models, the current numerical procedure employed a high-order low-pass filter operator applied to the

dependent variables during the solution process which selectively damped the poorly resolved high-frequency

content to retain numerical accuracy and stability for a wide range of flow operating conditions.

For the gust-airfoil interaction study, an efficient model was introduced to generate a vortical, incompressible

unsteady flow perturbation inside the computational domain through a source term in the momentum equations,

which conveniently eliminated the need to impose the perturbation at the upstream boundary and thus allowed to

employ an efficient grid stretching throughout the farfield boundary to remove any spurious reflections.

To examine the performance of the employed numerical procedures in a wide range of flow regimes, two

benchmark test cases were considered corresponding to the low-speed unsteady flow around a pitching SD7003

airfoil, and a high-speed, high-Reynolds number unsteady flow around a stationary Joukowski airfoil, with the same

grid applied in both cases. The results were compared with available experimental and previously validated

numerical predictions and demonstrated success of the current numerical approach to maintain accurate, numerically

stable solutions.

Acknowledgements

The first two authors would like to acknowledge the partial support for this work through the AFRL-RB summer

research program.

References:

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Aerodynamic and Aeroacoustic Responses of Oscillating Airfoils in Non-Uniform Flow,” AIAA Paper

2008-0033.

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CAA Workshop on Benchmark

Problems, NASA CP-2004-212954, pp.45-48.

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and Deforming Meshes,” Journal of Computational Physics, Vol. 181, pp.155–185, 2002.

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9. Visbal, M.R. and Gaitonde, D.V., “Very High-Order Spatially Implicit Schemes for Computational

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Nonlinear Computational Aeroacoustics Code to the Gust-Airfoil Problem,” AIAA Journal, Vol. 44, No.2,

pp. 323-328, 2006.

13. Galbraith, M.C. and Visbal, M.R., “Implicit Large Eddy Simulation of Low Reynolds Flow Past the

SD7003 Airfoil,” AIAA Paper 2008-225.

14. Golubev, N.V., “A Database of NACA-23012 Airfoil Unsteady Cy, Cx, Cmz Characteristics,” Proceedings

of the 7th

Forum of Russian Helicopter Society, 2005.

15. Golubev, V.V., Mankbadi, R.R. and Hixon, R. (2004), “Simulation of Airfoil Response to Impinging Gust

Using High-Order Prefactored Compact Code,” NASA/CP – 2004-212954, Proceedings of the 4th

CAA

Workshop on Benchmark Problems, Cleveland, OH, October 2003, pp. 141-148.

16. Ewert, R. and Manoha, E., “Trailing-Edge Noise,” from “Large-Eddy Simulation for Acoustics,” by

Wagner, C., Huttle, T. and Sagaut, P., Cambridge University Press, pp. 293-333, 2007.