American Institute of Aeronautics and Astronautics 092407

1

Assessment of a Kinetic-Eddy Simulation Turbulence Model

for 3D Unsteady Transonic Flows

Jeremy J. Bain1 and Sumit S. Mishra

2

INTECH Software Solutions, Inc.

Lakshmi N. Sankar3 and Suresh Menon

4

School of Aerospace Engineering, Georgia Institute of Technology

The Kinetic-Eddy Simulation (KES) turbulence model has been evaluated for a variety of

unsteady and oscillating wing transonic flows. KES is a two equation VLES/LES model that

solves for the unresolved kinetic energy and the local turbulent length scales without any

dependency on grid spacing. KES was used in OVERFLOW and compared to standard

models. KES showed improved ability to capture the negative lift curve slope and separated

flow of the NACA0036. In dynamic stall, KES improved prediction of peak lift, drag, and

pitching moment. In transonic flows, KES was able to improve modeling of separated flow at

the wing-body connection of the DLR-F6 wing-body configuration but over predicted drag.

Unsteady oscillating transonic flow over a scaled F-5 wing also showed improved prediction

with KES.

I. Introduction

NE of the most difficult challenges in Computational Fluid Dynamics (CFD) is accurately modeling separated,

unsteady, compressible, high Reynolds number flow. Most fixed wing aircraft are designed to avoid large

regions of separated flow but this is often not possible for rotorcraft. Rotorcraft hubs and pylons are often blunt

bodies that are sometimes covered with thick airfoils to reduce separation to a limited degree. They are located in

the unsteady wake of the main rotor further complicating accurate flow prediction. The retreating blade is often

operated at or beyond its stall limit. The motion of the rotor blade is coupled between the aerodynamic forces and

the rotor flap, torsion, and lead-lag motion. Accurate prediction of these flows is extremely challenging. They are

highly unsteady and require time accurate prediction of forces and moments.

Most CFD codes in industry and academia are based on the compressible Reynolds Averaged Navier-Stokes

(RANS) equations and use one or two equation turbulence models. This typically yields acceptable results for

steady, attached flow. RANS turbulence models fail to accurately predict separated flow structures because they

attempt to model all scales and frequencies of turbulence. Direct Numerical Simulation (DNS) can model all scales

of turbulence but requires extremely dense grids and comes at a computational cost that is prohibitive for most flows

of interest. Large-Eddy Simulation (LES) computes the largest turbulence structures and then models the sub-grid

unresolved structures. LES has shown improved predictions but still require too dense of grids for routine

engineering calculations.

Over the past decade, researchers have been working on creating hybrid turbulence models that can capture some

of the benefits of LES models but use standard RANS CFD codes. Spalart extended the Spalart-Allmaras (SA)

model to compute eddy viscosity near the wall using traditional SA model1 and switched to a pseudo Smagorinsky

LES model away from the wall2. This approach, called Detached Eddy Simulation (DES), has been implemented

with other turbulence models like k-ω and Menter’s SST3,4,5,6

.

1Research Engineer, 5655 Peachtree Pkwy., Suite 200 Norcross, GA 30092, Student Member, AIAA

2Proposal Manager, 5655 Peachtree Pkwy., Suite 200 Norcross, GA 30092, Student Member, AIAA

3Regents Professor and Associate Chair (Academic), School of Aerospace Engineering, Georgia Institute of

Technology, Atlanta, GA, 30332-0150, Associate Fellow, AIAA. 4Professor, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0150, Associate

Fellow, AIAA.

O

26th AIAA Applied Aerodynamics Conference18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-7176

Copyright © 2008 by Jeremy Bain, Sumit Mishra, Lakshmi Sankar, and Suresh Menon. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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The k-ω-SST model has been demonstrated to work well for a number of steady and unsteady applications.

However, in some cases where massively separated flows occur, this model has a tendency to under-predict the flow

separation even on extremely fine grids7. In an attempt to overcome these deficiencies, the present researchers and

their coworkers have been exploring a variety of turbulence models that fall under the classification of “hybrid

RANS-LES (HRLES) models.” One particular version of the HRLES model, called the subgrid Kinetic-Eddy

Simulation (KES) has been implemented in a number of in-house flow solvers at Georgia Tech and in the NASA-

Army flow solver OVERFLOW 2.08,9,10

. The KES model was found to work well in 2-D static stall, 2-D dynamic

stall, and 3-D dynamic stall occurring on helicopter rotors in forward flight.

II. Research Objective

The motivation behind this work to apply the KES model to a variety of steady and unsteady flows is to identify

its strengths and limitations, and to identify areas of future research. The NACA0036 thick airfoil was chosen due to

available computational results with other turbulence models in OVERFLOW. NACA0012 dynamic stall tests were

chosen to be representative of helicopter airfoils. The DLR-F6 wing-body configuration from the 2nd

AIAA CFD

Drag Prediction Workshop was chosen due to the difficulty in prediction of separated flow at the wing-body

connection and to asses the ability of KES to predict a transonic flow. The F-5 wing was chosen to asses the ability

of KES to predict unsteady oscillating transonic flow similar to the advancing rotor blade in high speed forward

flight. In all these simulations, KES results are compared with the Spalart-Allmaras1 (SA) turbulence model. SA is

considered the baseline due to its wide spread use in OVERFLOW and robustness11

.

III. Numerical Formulation

The KES model is briefly described below. The reader is referred to Ref. 8 and 9 for a more complete description

of this model. Kinetic-Eddy Simulation (KES) is a significant departure from the strategies in either HRLES12,13

or DES-k-ω−SST. The KES method directly estimates the local unresolved kinetic energy and the local

characteristic length scale corresponding to this unresolved kinetic energy via the solution of the two transport

equations for k and kl. This model solves for the subgrid ksgs

and klsgs

for the subgrid velocity scale, sgs

k and

the subgrid length scale, lsgs

. The transport equations for ksgs

and klsgs

are

( )3 2

,Pr

sgssgssgs sgs

sgsi i t

ij k sgs

i i i k i

ku k uk kC

t x x l x xε

ρ νρ ντ ρ ρ

σ

∂ ∂∂ ∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂

ɶ ɶ (1)

( ) ( ) ( ) ( )3 2

,Pr

sgs sgs sgs

sgs sgs sgsi i t

l ij kl

i j i kl i

kl u kl kluC l C k

t x x x xε

ρ ρ νντ ρ ρ

σ

∂ ∂ ∂∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂

ɶ ɶ (2)

While KES is an LES methodology, in situations where the local grid resolution is inadequate, the KES model

smoothly transitions from LES to Very Large-Eddy Simulation (VLES) with only the very largest scales resolved. In

a very fine grid where the subgrid length scale and subgrid kinetic energy goes to zero, KES asymptotically becomes

a DNS simulation. The ability to smoothly transition from LES to VLES is a unique feature of the KES model,

which can be considered a VLES-LES model. Furthermore, there is no a priori assignment of where VLES or LES

dominates since the transition between the two methods occurs naturally as a part of the solution.

The KES model does not depend on non-physical parameters like grid spacing to determine the length scales of

the turbulent eddies. It has relatively few tunable constants, allowing the application of this model to broad variety

of configurations and flows, ranging from airfoils to helicopter rotors to 3-D mixing layers. It has the following

additional features:

• Subgrid length scale is solved locally

• Dissipation of kinetic energy depends on the SGS length as well as length scale gradient

• No wall-distance is employed as in conventional Detached Eddy Simulation schemes

A “law of the wall” type approach is used to specify the ksgs

and lsgs

values at the first point off the wall using

the following equations,

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( )0.5

110.25

sgs

i ik v u u= ∆ɶ ɶ (3)

( )2

10.53

sgs sgsl k v= ∆ (4)

where ∆1 is the grid length scale of the first cells and ν is the kinematic viscosity. These values are based off of

energy dissipation rates at the wall8. At the wall, they are set to zero. At the far field boundary condition, k

sgs is set to

( )2

0.016u∞

and lsgs

is set to the local grid length scale.

The numerical calculations used the compressible Reynolds Averaged Navier-Stokes computational fluid

dynamics code OVERFLOW14,15

. OVERFLOW uses overset structured grids and has been used for a wide variety

of fixed wing and rotorcraft problems16,17

. OVERFLOW has numerous spatial and temporal options. Version 2.0y

with modifications made under the Defense Advanced Research Projects Agency (DARPA) Helicopter Quieting

Program was used18

. These modifications include the addition of HRLES and KES turbulence models as described

in Ref. 10.

OVERFLOW was run on an eight node, dual 2.4 GHZ AMD Athlon Processor cluster. This solver has been

parallelized using the Message Passing Interface (MPI) library, and was run on multiple processors for all

calculations except for the relatively simple NACA0012 calculations. All the simulations reported here were done

with 4th

order central differencing in space, and dual timesteps. It was necessary to use 4-10 Newton sub-iterations.

IV. Numerical Simulations and Discussions

A. NACA0036

The NACA0036 has been the subject of recent computational and experimental studies. Experimental studies

have been performed at the NASA Ames Fluid Mechanic Lab (FLM) 32”x48” low speed open-draft wind-tunnel19

and the California Polytechnic State University – San Luis Obispo (Cal Poly) 3x4 ft low speed, open-return wind

tunnel20,21

. Computational studies of this airfoil are particularly difficult due to the highly separated flow that occurs

at angle of attack. At AFDD, Potsdam studied this airfoil in OVERFLOW with numerous turbulence models in both

2D and 3D studies22

. The 2D grid consists of a 573x81 body fitted O-grid and a 140x80 Cartesian grid to model the

wind tunnel walls with inviscid boundary conditions. In two dimensions, none of the standard RANS one and two

equation turbulence models tested in the OVERFLOW were able to capture the negative lift curve slope. A 3D grid

was created and run with the hybrid RANS/LES (DES) models available in the newest versions of OVERFLOW4,11

.

These models showed improvement in capturing the effects of the separated flow and the wind tunnel wall effects

but at greatly increased computational cost. The 2D grids were run with KES in time accurate mode using the same

nondimentional timestep of 0.001 that was used on the 3D runs by AFDD. KES was able to capture the negative lift

curve slope and had similar results to the SSTDES fine 3D grid results as shown in Figure 1. Instantaneous

streamlines in Figure 2 show the flow highly separated on the upper surface. While KES has higher cost in 2D due

to the requirement for physical time steps and longer convergence time, it is significantly cheaper than the full 3D

runs.

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Figure 2. Instantaneous Streamlines with KES of NACA0036 at 10 degrees Angle of Attack

B. NACA0012

The NACA0012 is a common airfoil in helicopters for its moderately high maximum lift and low pitching

moment. Dynamic stall is a dominating factor in forward flight and can limit the operational envelope of a

helicopter. In forward flight, the advancing blade encounters additional velocity due to the free stream and the

retreating blade encounters a reduced velocity. The lift generated on the advancing and retreating side needs to be

balanced to avoid a large rolling moment necessary to trim the vehicle. This leads to the retreating blade operating

near its maximum lift capability while the advancing blade is at low or negative incidence. This causes the blade to

undergo dynamic stall cycles as the blade rapidly goes from low to high incidence causing large negative pitching

moments, high vibration, and buffeting. The high negative pitching moment that occurs causes an elastic

deformation in the rotor blade that causes further pitch changes. This presents significant challenges in modeling due

to the effect of leading and trailing edge shedding. In the experiment of McCroskey et al23

, several airfoils are

studied in light, moderate and deep stall. A C-grid was generated to have y+=1, 547 chordwise points, and 105

Figure 1. KES 2D Results Compared to 2D (Left) and 3D (Right) Results using Various Turbulence

Models (Ref 22)

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points in the normal. The static CL was calculated using SA and KES compared for a slowly increasing angle of

attack (k=0.001) with 150,000 timesteps in Figure 3. KES captures the maximum lift of 1.29 at 13.4° that is within

the experimental scatter. SA predicts a maximum lift of 1.43 at 14.8°. After the stall, SA smoothly decreases

smoothly while KES shows the unsteady fluctuation of vortex shedding. To capture the unsteady effects of dynamic

stall, 35000 timesteps per cycle were run for 2.5 cycles. Figure 4 compares SA and KES results for deep stall. At

low angles of attack, the two models have similar results as expected. The SA model overestimates the peak lift,

drag, and pitching moment while the KES model does a significantly better job predicting these critical quantities.

Figure 4. NACA0012 Dynamic Stall M=0.3, k=0.1, αααα=15° + 10° sin(ωωωωt)

Figure 3. Static NACA0012 Lift with SA and KES

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Table 1. Results for CL=0.5, Mach 0.75

CL CD CM

Experiment 0.500 0.0275 -0.121

SA 0.501 0.0284 -0.0907

KES 0.4999 0.0347 -0.0923

Figure 5. 2

nd AIAA CFD Drag Prediction Workshop Wing Body Medium Grid

C. Drag Prediction Workshop Wing Body Configuration

Since 2000, the AIAA has organized four CFD Drag Prediction Workshops to test the ability of a variety of CFD

solvers to calculate the drag on generic transport aircraft configurations24

. OVERFLOW was used extensively on the

various configurations on a series of progressively finer grids25,26

. The DLR-F6 wing-body configuration from the

second workshop was selected for further study with KES. This configuration had separated flow at the wing-body

juncture that was over predicted using the SA1 and Baldwin-Barth turbulence models

27.

The flight condition selected represents the cruise design point of Mach number = 0.75, CL = 0.5, and Reynolds

number of 3 million. An initial run was completed using the SA turbulence model and the medium grid provided to

the authors. The medium grid consists of twelve overset grids and approximately 6.8 Million points, shown in

Figure 5, created following accepted best practices28,29

. The grids were created using Chimera Grid Tools and

PEGASUS30,31,32

. More details of the grid generation can be found in Ref. 28. KES was run with the same

computational options as the SA baseline but with a physical time step of 0.01 nondimensionalized by freestream

velocity and no multigrid acceleration. The physical time step is required by KES. The KES run took approximately

twice as long to run due to the greater number of physical time steps required for convergence.

As shown in Table 1, KES overestimates the drag. Figure 6 has the surface pressure contours and Figure 7 has

the locations of supersonic flow. The shock location using KES occurs upstream of the SA prediction. The suction

peak was also under predicted compared to SA. The separation bubble at the wind body connection was better

predicted with KES. The width of the bubble is shown in Table 2 and instantaneous streamlines in Figure 8.

Table 2. Separation Bubble Width

Separation

Size (mm)

∆mm

Experiment (Ref 25) 90.9 -

SA 96.8 5.9

KES 89.4 -0.6

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Figure 6. Upper Surface CP Distribution with SA (left) and KES (right) for CL=0.5

Figure 7. Predicted Locations of Supersonic Flow (Grey) near the wingtip with SA (left) and KES (right)

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Figure 8. Instantaneous Streamlines near Wing Body Connection with SA (left) and KES (right)

Figure 9. F-5 Grid (Every 2

nd Point in each direction removed)

D. F-5 Wing

A scaled F-5 wing has been tested for a variety of transonic steady and unsteady conditions. The experiment of

Tijdeman et al. had eight spanwise stations instrumented to capture steady and unsteady pressures33,34

. The wing has

a semi-span of 0.6476m, a taper ratio of 0.281, leading edge sweep of 31° and a modified NACA 65-A-004.8 airfoil.

A variety of types of solvers have studied this configuration including potential flow35

, Euler36

, Navier-Stokes37,38

,

and hybrid Navier-Stokes/Full-Potential39,40

solvers.

The F-5 grid was created utilizing best practices in OVERFLOW grid generation including those established

during the Drag Prediction Workshop28,29

. The grid corresponds to a medium grid and consists of approximately 3

million points and is shown in Figure 9. For hole cutting, xrays were used30

.

To simulate the combined bending and torsion aeroelastic response of the F-5 wing, grid deformation routines

were added to OVERFLOW. The method used followed the method of Morton41

that was used by previous

researches and the authors in OVERFLOW42,43

. The deformed surface is calculated using the measured mode

shapes44

. The points in the normal direction are then calculated to maintain grid orthogonality. As the distance from

the surface increases, a cubic spline is used to blend the deformed grid and the original grid. This maintains the grid

spacing and outer boundary of the original grid.

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Figure 10. Surface Pressure Contours over the Upper Surface of the F-5 Wing using SA (left) and KES

(right) at Zero degrees Angle of Attack and Mach 0.9

Figure 11. Surface Pressure Distribution over F-5 Wing at 35.2% Span (left) and 72.1% Span (right) at Zero

Angle of Attack, Mach 0.9

Steady runs were done at zero degrees angle of attack at Mach 0.9. The upper surface pressures are shown in

Figure 10 and compared with experimental data at two span stations in Figure 111. The real and imaginary

components of surface pressure during a 20Hz oscillation are shown in Figure 12. The SA results show a stronger

shock than the experimental and KES data. The unsteady results using KES show good correlation with

experimental data. The SA results showed the effect of overestimating the shock around x/c=0.6.

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V. Conclusions

The Kinetic Eddy Simulation turbulence model has been evaluated for a variety of unsteady low speed and

transonic flows. In areas of separated flows, KES showed improved predictions over standard RANS turbulence

models while smoothly transitioning between VLES and LES. In transonic flows, KES overestimated drag from

improper shock placement. This deficiency may stem for specifying the ksgs

and lsgs

at the first point off the wall.

The HRKES45

model overcomes this problem by using k-ω-SST near the wall and then smoothly transitioning to

KES away from the wall. Work is also ongoing to reduce and eventually eliminate the six empirical constants in

KES.

Acknowledgements

This work was funded under a NASA SBIR project entitled “Advanced Turbulence Models for Unsteady and

Stalled flows.” Ethan Romander is the technical monitor. The authors would like to thank Mark Potsdam of AFDD

for providing the NACA 0036 grids and Joe Morrison of NASA Langley for access to the AIAA CFD Drag

Prediction Workshop grids. The authors also thank Mina Zaki for discussions on KES and the HRKES model.

References

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Exhibit, Reno, NV, January 2007, AIAA-2007-260 25 Sclafani, A., J., DeHaan, M.A., Vassberg, J.C., “OVERFLOW Drag Prediction for the DLR-F6 Transport Configuration; A

DPW-II Case Study” 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2004, AIAA 2004-393 26 Sciafani, A., Vassberg, J., Harrison, N., DeHaan, M., Rumsey, C., Rivers, C., Morrison, J., “Drag Prediction for the DLF-

F6 Wing/Body and DPW Wing Using CFL3D and OVERFLOW on an Overset Mesh” 45th AIAA Aerospace Sciences Meeting

and Exhibit, Reno, NV January 2007, AIAA 2007-257 27 Baldwin, B.S., Barth, T.J., “A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded

Flows,” AIAA 1991-0610, January 1991. 28 Vassberg, J.C., DeHaan, M.A., ”Grid Generation Requirements for Accurate Drag Predictions based on OVERFLOW

Calculations” AIAA 2003-4124 16th AIAA Computational Fluid Dynamics Conference, Orlando, FL, June 2003. 29 Chan, W.M., Gomez-III, R. J., Rogers, S.E., Buning, P.G., “Best Practices in Overset Grid Generation” 32nd AIAA Fluids

Dynamics Conference, St. Louis, MO, June 2002. AIAA 2002-3191 30 Chan, W.M., Rogers, S.E., Nash, S.M., Buning, P. G., Meakin, R. L., Boger, D.A., Pandya, S., “Chimera Grid Tools

User’s Manual, Version 2.0” http://people.nas.nasa.gov/~wchan/cgt/doc/man.html, NASA July, 2007. 31 Chan, W.M., Rogers, S.E., Nash, S.M., Buning, P.G., Meakin, R.L., “User’s Manual for HYPGEN Hyperbolic Grid

Generator and the HGUI Graphical User Interface” NASA-TM 108791, NASA October 1993. 32 Rogers, S.E, Shus, N.E., Dietz, W.E., Nash, S.M., Onufer, J.T., “PEGASUS User’s Manual Version 5.1k” October 2003,

http://people.nas.nasa.gov/~rogers/pegasus/guide.html 33 Tijdeman, M., van Nunen, J.W.G., Persoon, A.J., Poetkoke, R., Roos, R., Schippers, P., Siebert, C.M., “Transonic Wind

tunnel Tests on an Oscillating Wing with External Stores. Part I. General Description.” Air Force Flight Dynamics Laboratory,

AFFDL-TR-78-194, Part I, Wright-Patterson AFB, Ohio, December 1978 (Also NLR-TR-78106-U,Part I) 34 Tijdeman, M., van Nunen, J.W.G., Persoon, A.J., Poetkoke, R., Roos, R., Schippers, P., Siebert, C.M., “Transonic Wind

tunnel Tests on an Oscillating Wing with External Stores. Part II. The Clean Wing.” Air Force Flight Dynamics Laboratory,

AFFDL-TR-78-194, Part II, Wright-Patterson AFB, Ohio, March 1979 (Also NLR-TR-78106-U,Part II) 35 Malone, J.B, Sankar L.N., Sotomayer, W.A., “Unsteady Aerodynamic Modeling of a Fighter Wing in Transonic Flow”

AIAA 17th Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Snowmass, CO June 1984, AIAA-1984-1566

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1987 Vol. 24. No. 1 37 Gargoloff, J.I., Cizmas, P.G.A., Strganac, T.W., Beran, P.S., “A Parallel Multigrid Algorithm for Aeroelasticity

Simulations” 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2007 AIAA 2007-330 38 Obayashi, S., Guruswamy, G.P., Goorjian, P.M., “Streamwise Upwind Algorithm for Computing Unsteady Transonic

Flows Past Oscillating Wings” AIAA Jornal Vol 29, No 10, 1990 39 Mello, O.A.F., Sankar, L.N., ”An Improved Hybrid Navier-Stokes/Potential Method for Computation of Unsteady

Compressible Flows,” 26th AIAA Fluid Dynamics Conference, San Diego, CA 1995 AIAA 1995-2160 40 Mello, O.A.F., “An Improved Hybrid Navier-Stokes/Full-Potential Method for Computation of Unsteady Compressible

Viscous Flows” Ph.D. Dissertation, Aerospace Engineering Georgia Institute of Technology, Atlanta, GA 1994 41 Morton, S. A., Melville, R. B., Visbal, M. R., ”Accuracy and Coupling Issues of Aeroelastic Navier-Stokes Solutions on

Deforming Meshes” Journal of Aircraft, Vol. 35, No. 5, September – October 1998. 42 Kerho, M., “Adaptive Airfoil Dynamic Stall Control” 43rd Aerospace Sciences Meeting and Exhibit, Reno, NV, January

2005 AIAA-2005-1365 43 Bain, J., Sankar, L. N., Prasad, JVN, Peters, D., He, C., ”Computational Modeling of Variable Droop Leading Edge in

Forward Flight”, 4th Flow Control Conference, Seattle, WA, June 2008 AIAA 2008-3872 44 Tijdeman, H., et al, “Results of Transonic Wind Tunnel Measurements on an Oscillating Wing with External Stores” NLR

TR 78030 U (1978) 45 Zaki, M., Menon, S., Sankar, L., “A Hybrid RANS/KES Model for External and Internal Flow Applications” 44th

AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 2008, Hartford, CT AIAA 2008-5216

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