a des procedure applied to the flow over a naca0012 airfoil

A DES Procedure Applied to the Flow Over a

NACA0012 Airfoil

Radoslav Bozinoski∗ and Roger L. Davis†

University of California at Davis, Davis, California, 95616

This work describes a detached-eddy simulation (DES) for massively separated flowover a NACA 0012 airfoil at off design conditions. Simulations were performed using theMBFLO1,2 simulation procedure. The Navier-Stokes equations are solved using an explicit,multi-grid procedure for the steady-flow Reynolds-averaged (RANS) computations. Adual time-step procedure is used for unsteady Reynolds-averaged (URANS) or DES. Atwo-equation k − ω turbulence model is used for the entire turbulence field in the RANSand URANS simulations as well as in the near-wall regions of the DES. The DES resultsare obtained using the model presented by Bush and Mani3 and are compared with theunsteady Reynolds-averaged Navier-Stokes solutions and experimental data for the NACA0012 airfoil at an angle of attack of 60◦. The results of the two- and three-dimensionalcomputations are compared to each other as well as to experimental results. Differencesbetween two- and three-dimensional computations are observed in terms of the coeffiientsof lift and drag as well as vorticity and entropy contours. The two-dimensional DES showedan improvement in lift and drag predictions when compared to two- and three-dimensionalURANS, however, the three-dimensional DES lift and drag are closer to the experimentalresults in the literature as expected. A method accounting for the three-dimensional DESeffects in two-dimensional DES is also explored.

Nomenclature

E total energye internal energyH total enthalpyh static enthalpyk turbulent kinetic energyp pressurePr Prandtl numberPrt turbulent Prandtl numberSij mean strain-rate tensorui velocity componentV velocity magnitudeμ coefficient of viscosityμT turbulent coefficient of viscosityω turbulent dissipation frequencyτ̂ij total shear stress tensorτij laminar shear stress tensorτRij Reynolds shear stress tensorρ densityldis dissipation length scaleCdes DES coefficients

∗Graduate Student, Mechanical and Aeronautical Engineering Department, Student Member AIAA.†Professor, Mechanical and Aeronautical Engineering Department, Associate Fellow AIAA.

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American Institute of Aeronautics and Astronautics

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-921

Copyright © 2010 by Radoslav Bozinoski & Roger L. Davis. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction

Simulation of “steady” and “unsteady” flow of aerodynamic bodies has matured a great deal over the pastdecade. Aerodynamic performance and flow structures can be predicted with acceptable accuracy except

in the complex flow regions of mixing and at off-design conditions near stall. In these regions, complexflow structures with multiple eddies that mingle and mix are often not predicted well due to inadequatecomputational grid density and a breakdown of turbulence models.

II. Governing Equations

The unsteady, Favre-averaged governing flow-field equations for an ideal, compressible gas in the right-handed, Cartesian coordinate system using primary variables are used in the MBFLO code. The three-dimensional continuity, momentum, and energy equations can be written in a conservative form as follows,

∂ρ

∂t+

∂ (ρuj)

∂xj= 0 (1)

∂ρui

∂t+

∂ (ρujui)

∂xj= − ∂P

∂xi+

∂τ̂ij∂j

(2)

∂E

∂t+

∂ (ρujH)

∂xj=

∂

∂xj

[uiτ̂ij +

(μ

Pr+

μT

Prt

)∂h

∂xj

](3)

τ̂ij = τij + τRij = (μ+ μT )Sij (4)

Sij =1

2

(∂ui

∂xj+

∂uj

∂xi

)− 2

3

∂uk

∂xkδij (5)

Since we are dealing with compressible flows, we require an equation of state to relate the energy withpressure and enthalpy.

E = ρ

[e+

1

2V 2

](6)

H = h+1

2V 2 =

γ

γ − 1

P

ρ+

1

2V 2 =

E + P

ρ(7)

Additional governing equations, as developed by Wilcox4–6 , are solved for the transport of turbulentkinetic energy and turbulence dissipation rate in regions of the flow where the computational grid or globaltime-step size cannot resolve the turbulent eddies. In regions of the flow where the larger-scale eddies can beresolved with the computational grid, techniques borrowed from large-eddy simulation are used to representthe viscous shear and turbulent viscosity. The large-eddy sub-grid model described by Smagorinsky7 ismodified according to the detached-eddy considerations described by Strelets8 and Bush and Mani.3

III. Numerical Techniques

The conservation of mass, momentum, and energy equations are solved using a Lax-Wendroff control-volume, time-marching scheme as developed by Ni,9 Dannenhoffer,10 and Davis.11,12 Numerical solutions ofunsteady flows can be performed with either the explicit9 or a dual time-step procedure.13 These techniquesare second-order accurate in time and space. A multiple-grid convergence acceleration scheme9 is used forsteady, Reynolds-averaged solutions and the inner convergence loop of the unsteady simulations using thedual time-step scheme. The approach is called MBFLO and has two-dimensional,2 axi-symmetric14 (withand without swirl), and three-dimensional1 versions. The two- and three-dimensional procedures, MBFLO2Pand MBFLO3P (two/three-dimensional, parallel), and results for a DES for flow over a NACA 0012 airfoilat an angle-of-attack of 60◦ are described here.

It is recognized that two-dimensional DES lack the three-dimensional effects resulting from the interactionof eddies in the third dimension15 and therefore, tend to under-predict diffusion. However, two-dimensional

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design is an important part of the overall design process of wings, turbomachinery blades, and other similarlifting bodies so that a robust two-dimensional simulation capability is highly desirable. The hope is tocompare the two-dimensional and three-dimensional URANS and DES and to possibly account for the three-dimensional eddy-interaction effects through additional modeling in the future. One approach for modelingthese differences is introduced and discussed below.

� �

� � ��

�

��

��

��

��

��

��

��

��

��

��

� ��

��

Figure 1: Typical Speed-Up Factoras a function of Number of Processes

The combined second- and fourth-difference dissipation model ofJameson16 is used in the current procedure for both the mean flowand turbulence equations. The fourth-difference dissipation is scaledby the inverse of the absolute value of the mean strain rate squared.This function decays the numerical dissipation in all viscous flowregions, including boundary layers, wakes, large eddies, secondaryflows, etc.

In the MBFLO suite of codes, parallelization is performed usingthe Message Passing Interface (MPI) library.17 Figure 1 and Table 1show the typical speed-up and associated efficiencies as functions ofthe number of processors for the MBFLO3P code. The configurationused to generate this data was similar to that shown below in theresults section where the computational grid consisted of 1.80 milliongrid points. The data was generated on a Linux cluster consistingof 3.6GHz Intel Xeon processors. Figure 1 shows that a speed-upfactor of 18.06 is realized with 20 processes yielding a 90% parallel efficiency. In Table 1 we can see thatefficiencies of 90% and higher can be obtained if no less than 100, 000 grid points per process are used.

Table 1: Typical Speed-Up and Efficiencies

Processes Speed Up Efficiency Pts/Process

1 1.00 100% 1,780,440

2 2.00 100% 892,440

4 4.00 100% 448,440

8 7.79 97% 226,440

16 14.60 91% 115,440

20 18.06 90% 93,240

40 31.39 78% 48,840

IV. Results and Discussion

To determine the solution capability of the URANS and DES procedures for massively separated flows,the flow over a NACA 0012 airfoil was performed. These simulations were run at an angle-of-attack of 60◦,a freestream Mach number of 0.2, and a Reynolds number of 100, 000 with a freestream total temperatureset to 463.68◦R. Both two- and three-dimensional simulations were performed to assess the importance ofthree-dimensional flow structures over two-dimensional geometries. The dual time-step procedure for theunsteady URANS and DES was utilized with a global time-step of 1.0×10−4 seconds. For the DES procedure,a CDES

1 coefficient of 0.65 was used. Figure 2 shows the computational grids used in these simulations.The two-dimensional grid consisted of 74, 185 points with the far field approximately 50 chords away. Forthe three-dimensional simulations the two-dimensional grid was used and extruded one half chord into thespan-wise direction resulting in a 1, 854, 625 point grid. Using Table 1, it was determined that the three-dimensional grid could be decomposed into 20 blocks to operate MBFLO3P at 90% parallel computationalefficiency.

The temporal periodicity and unsteady behavior was initially studied using the two-dimensional URANSprocedure. The information obtained was then used to determine the number of time steps necessary toresolve a minimum of 10 periodic cycles for the subsequent two- and three-dimensional URANS simulationand DES at the given global time-step. Figures 3(a) and 3(c) show the signal history for the instantaneouslift, CL, and drag, CD, coefficients as well as the Power Spectral Density (PSD) as a function of the number

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(a) Stream-wise view (b) Span-wise view

Figure 2: NACA0012 computational mesh (every fourth point shown).

of time-steps. The PSD of lift and drag was plotted here as a function of the number of time-steps to moreeasily see time-periodicity in the flow. In Fig. 3(c), we see that the dominant signal is repeated approximatelyevery 260 time-steps, which corresponds to a frequency of 39 Hz. This frequency was used to determine theperiod over which the solution was time-averaged. Once the period was determined, this case was run for aminimum of 20 periodic cycles and time-averaged. Using the information obtained from the two-dimensionalURANS simulation, a two-dimensional DES was performed. Figure 3(b) shows the signal histories for thetwo-dimensional DES. Here, we notice that the CL and CD signals contain signal power at more frequenciesthan their URANS counterparts, which is also reflected in Figs. 3(d) and 3(f). However, the PSD showsa dominant signal repeating every 300 time-steps. Using this information the two-dimensional DES wastime-averaged for 16, 000 time-steps, which is well over 20 periodic cycles. A dominant PSD frequency of 38Hz was noticed for the two-dimensional DES.

Figure 4 below shows the corresponding three-dimensional CL and CD signal histories as well as theirPSD’s for URANS and DES. In Fig. 4(a), we can see the periodic lift and drag signals for the URANScase. Due to the high dissipation of the turbulence model, there was no significant three dimensionalityobserved and the solution had identical streamwise contours in the spanwise direction. It was reported3

that no three-dimensionality would be noticed for the NACA0012 airfoil case unless the spanwise domainis extended past two chord lengths. Figure 4(c) shows that the dominant power signal for the URANS wasrepeated every 300 time-steps corresponding to a frequency of 35 Hz. The lift and drag signals for the two-and three-dimensional URANS simulations produced very similar PSD plots with dominant frequencies atapproximately 30 and 60 Hz shown in Figs. 3(e) and 4(e).

The DES lift and drag signals, shown in Fig. 4(b), for the three-dimensional simulation showed signalpower at more frequencies and reflected the three dimensionality of the flowfield. In Fig. 4(d), we cansee that there are three dominant signals repeating every 100, 230, and 420 time-steps, and these can beattributed to the three-dimensional flowfield in the separation region on the suction side of the airfoil. Itshould be noted that a Hanning window function was used to ensure no artifical frequencies are present dueto signal clipping. When comparing the PSDs in Figs. 3(f) and 4(f), we can see that both the two- andthree-dimensional DES have a larger range of frequencies present in the lift and drag signals compared toURANS. An interesting observation when comparing the PSD for the URANS and DES is that the dominantpower signals for the lift and drag are at approximately the same frequency. The two-dimensional DES hasa dominant lift and drag power signal at 38 Hz and the three-dimensional DES at 42 Hz.

Based on the results found by analyzing the lift and drag signals for the two- and three-dimensionalURANS and DES, the total number of unsteady time-steps for each case was chosen to allow for the time-averaging of a minimum of 10 periodic cycles. The global time-step was also set so that the dominantfrequencies being resolved would be within the 35 to 45 Hz range. Table 2 shows a comparison of the timeaveraged CL and CD coefficients against experimentally measures values. Here we can see that the two- andthree-dimensional URANS as well as the two-dimensional DES predicted similar results and over predictedboth CL and CD. The two-dimensional URANS predicted CL and CD values of 1.33 and 2.25, respectively.The two-dimensional DES predicted CL and CD values of 1.25 and 2.12, respectively, and did show a slightimprovement over the URANS procedure at the given time-step. The time-averaged three-dimensional CL

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0.8

1.0

1.2

1.4

1.6

1.8

2.0

CL

Avg. CL = 1.33, CD = 2.25

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Seconds

1.61.82.02.22.42.62.83.0

CD

(a) URANS Signals

0.0

0.5

1.0

1.5

2.0

2.5

CL

Avg. CL = 1.25, CD = 2.12

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Seconds

0.00.51.01.52.02.53.03.54.0

CD

(b) DES Signals

0.0

0.5

1.0

1.5

2.0

PSD

CL

×106

0 100 200 300 400 500Period [iterations]

0.0

0.5

1.0

1.5

2.0

2.5

PSD

CD

×106

(c) URANS PSD

0.0

0.2

0.4

0.6

0.8

1.0

1.2

PSD

CL

×106


012345678

PSD

CD

×105

(d) DES PSD

0.0

0.5

1.0

1.5

2.0

PSD

CL

×106

0 20 40 60 80 100Frequency [Hz]

0.0

0.5

1.0

1.5

2.0

2.5

PSD

CD

×106

(e) URANS PSD (Hz)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

PSD

CL

×106

0 20 40 60 80 100Frequency [Hz]

012345678

PSD

CD

×105

(f) DES PSD (Hz)

Figure 3: NACA0012 lift and drag history for two-dimensional simulations.

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1.0

1.5

2.0

CL

Avg. CL = 1.40, CD = 2.37

0.30 0.35 0.40 0.45 0.50 0.55 0.60Seconds

1.0

1.5

2.0

2.5

3.0

3.5

4.0

CD

(a) URANS Signals

0.40.60.81.01.21.41.61.8

CL

Avg. CL = 1.09, CD = 1.89

0.10 0.15 0.20 0.25 0.30 0.35 0.40Seconds

1.0

1.5

2.0

2.5

CD

(b) DES Signals

0.0

0.5

1.0

1.5

2.0

2.5

3.0

PSD

CL

×105


0.00.51.01.52.02.53.03.54.0

PSD

CD

×105

(c) URANS PSD

0.0

0.5

1.0

1.5

2.0

2.5

PSD

CL

×104


0.0

0.5

1.0

1.5

2.0

2.5

3.0

PSD

CD

×104

(d) DES PSD

0.0

0.5

1.0

1.5

2.0

2.5

3.0

PSD

CL

×105

0 20 40 60 80 100Frequency [Hz]

0.00.51.01.52.02.53.03.54.0

PSD

CD

×105

(e) URANS PSD (Hz)

0.0

0.5

1.0

1.5

2.0

2.5

PSD

CL

×104

0 20 40 60 80 100Frequency [Hz]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

PSD

CD

×104

(f) DES PSD (Hz)

Figure 4: NACA0012 lift and drag history for three-dimensional simulations.

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and CD predictions for the URANS procedure were higher than the two-dimensional URANS with values at1.40 and 2.37, respectively. Similar observation were reported by Bush et al.3 and this may be due to therelatively coarse spanwise spatial resolution. The three-dimensional time averaged DES predicted a CL of1.09 and CD of 1.89 and matched the experimental CL and CD values much more closely.

Table 2: NACA0012 CL and CD comparisons

CL CD

2D URANS 1.33 2.25

2D DES 1.25 2.12

3D URANS 1.40 2.37

3D DES 1.09 1.89

Experiment 0.90 1.60

A comparison of the instantaneous two- and three-dimensional vorticity contours for the URANS andDES can be seen in Figs. 5 and 6. The three-dimensional plots for both URANS and DES have beenspatially averaged over the span. In Figs. 5(a) and 6(a), we can see the larger vortical structures with agreater level of dissipation when compared to Figs. 5(b) and 6(b). We can also see a larger range of scalesin the DES vorticity contours when compared to the URANS simulations. Similar results can be seen in theentropy contours in Figs. 7 and 8.

(a) 2D URANS (b) 2D DES

Figure 5: NACA0012 instantaneous vorticity contours.

Instantaneous vorticity contour comparison between three-dimensional URANS and DES is shown in Fig.9. Here, we see span-wise vorticity contour slices of the flow domain at y/c = 0.15, 0.35, and 0.55 normal tothe surface of the NACA0012 airfoil. As expected, the 3D URANS procedure produced a two-dimensionalflow structure with no noticeable variation in the spanwise direction. The 3D DES procedure, however, startsto show three-dimensional vortical structures close to the surface at y/c = 0.15 that continue throughoutthe wake. To further highlight the three-dimensionality in the flow domain of the DES, Fig. 10 shows aiso-surface contour plot of the instantaneous vorticity color by pressure.

As previously mentioned, two-dimensional simulations are considered a valuable tool during design. How-ever, these 2D simulations lack the three-dimensional flow effects as previously discussed. A technique toinclude local 3D effects in 2D simulations would be very useful for improving two-dimensional simulationaccuracy. There are various ways to model three-dimensional effects in two-dimensional calculations. Crudetechniques include modeling aerodynamic geometry with quasi-two-dimensional extensions using streamtubeheight or area variation. Another approach could be the use of source terms similar to deterministic stressmodels18,19 used previously to model unsteady effects in steady simulations. For instance, let T (u) be thetime-averaged solution from a URANS or DES simulation and S(u) be the steady solution obtained from a

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Figure 6: NACA0012 instantaneous vorticity contours.


Figure 7: NACA0012 instantaneous entropy contours.


Figure 8: NACA0012 instantaneous entropy contours.

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(a) y/c = 0.15 (URANS) (b) y/c = 0.35 (URANS) (c) y/c = 0.55 (URANS)

(d) y/c = 0.15 (DES) (e) y/c = 0.35 (DES) (f) y/c = 0.55 (DES)

Figure 9: NACA0012 spanewise vorticity contours

Figure 10: NACA0012 airfoil non-dimensional vorticity isosurface (colored by pressure) for instantaneousDES.

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RANS simulation that satisfies the conservation equations Ds(u) = 0. A discrete deterministic stress modelfor unsteady effects in a steady simulation can be generated by initializing the flow field with T (u), runningone iteration in the steady RANS simulation, Ds(T (u)), and capturing the residual, Rs(u). The residualcan be thought of as a deterministic source term that is required to drive the steady procedure to producethe time-averaged solution.

Now consider using a similar approach to drive a two-dimensional, unsteady procedure to produce athree-dimensional, unsteady solution. Complete temporal similitude would require a discrete source termfor every time-step in the simulation. This is a considerable task and outside the scope of what is consideredan engineering model. However, it would be feasible to model the three-dimensional, unsteady effects in atwo-dimensional procedure with time-invariant source terms determined from the time-averaged solutions.A discrete model for the three-dimensional effects could be computed using the following steps:

1. Initialize the flow field of the unsteady, two-dimensional simulation with the three-dimensional locally,spanwise-averaged, time-averaged flow field, T3D(u). Note that if the three-dimensional flow field isperiodic in the spanwise direction, then the spanwise-average, time-averaged flow field should not haveany spanwise velocity component. This is the case for the simulations shown here. However, in general,it would not be the case and the spanwise velocities would need to be neglected.

2. Run one iteration of the unsteady, two-dimensional simulation, Dus(T3D(u)), and capture the residual,R̄us−3D(u). This residual represents the discrete deterministic stresses necessary to drive the two-dimensional unsteady solution to the three-dimensional time-averaged solution. The residuals areessentially the time-average of the instantaneous deterministic stresses:

R̄us−3D(u) =∑

R′us−3D(u)/T (8)

3. We are interested in determining the time-invariant deterministic stresses that when applied to theunsteady, two-dimensional simulation and then time-averaged, produces the three-dimensional time-averaged solution instead of the two-dimensional time-averaged solution. In addition, the instantaneoustwo-dimensional simulation with these additional modeled 3D effects would also produce a solutionsimilar to the locally spanwise averaged three-dimensional instantaneous solution. An approximationto these discrete unsteady source terms can be found by assuming that the three-dimensional effects pertime-step are time-invariant and equal to the time-averaged deterministic stress, R̄us−3D(u). Numericalresults, however, have shown that the instantaneous deterministic stress, R′

us−3D(u), applied to eachtime-step should be a fraction of the time-averaged value, R̄us−3D(u). As shown below, a value ofR′

us−3D(u) = Δt/T R̄us−3D(u) works reasonably well.

Figure 11 shows the discrete time-averaged three-dimensional deterministic stresses that arise using thistechnique with the previously described detached-eddy simulations. A Δt/T fraction of these discrete 3Deffects are subtracted as source terms from the left-hand side of the unsteady, two-dimensional Navier-Stokesequations. Ultimately, the development of an analytical model for these effects, either in the form of sourceterms for each equation or as additional stresses in the momentum and energy equations, is the goal of thisresearch. The results shown below demonstrate that the use of this type of modeling approach for 3D effectsin two-dimensional simulations is viable.

(a) Continuity (b) Energy (c) X-Momentum (d) Y-Momentum

Figure 11: Discrete three-dimensional effects.

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Figures 12 and 13 show a comparison between the two-dimensional RANS, two-dimensional time-averagedDES, and three-dimensional time- and space-averaged DES for entropy and vorticity. The 3D time-averagedDES entropy and vorticity contours are more similar to the 2D steady RANS contours than the 2D time-averaged DES contours. The next step is to determine if the differences between the 3D time/spanwise-averaged DES and 2D time-averaged DES can be modeled using the deterministic stress approach describedabove.

(a) 2D RANS (b) 2D TA-DES (c) 3D TA-DES

Figure 12: Time-averaged and RANS entropy contours.

(a) 2D RANS (b) 2D TA-DES (c) 3D TA-DES

Figure 13: Time-averaged and RANS vorticity contours.

Figure 14 shows the results of applying the discrete 3D effect source terms, shown in Fig. 11, on theinstantaneous and time-average 2D DES. We can see that by adding the source terms, both the time-averaged vorticity and entropy contours more closely resemble the time-averaged 3D DES. Further researchand numerical experimentation is required, however, to determine an appropriate analytical model for local3D effects based on the discrete source terms resulting from many simulations.

(a) Instantaneous entropy (b) Time-averaged entropy (c) Instantaneous vorticity (d) Time-averaged vorticity

Figure 14: NACA0012 airfoil 2D DES contours with modeled 3D effects.

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V. Conclusion

A general Reynolds-averaged/detached-eddy simulation procedure was used to predict the flow structureover a NACA0012 airfoil at 60◦. As expected, the three-dimensional DES procedure produced the bestresults when compared to the experimental values for CL and CD. With the use of the model describedby Bush et al.,3 the detached-eddy simulation showed significant improvement over the Reynold’s-averagedNavier-Stokes simulation. This can be attributed to the DES model’s ability to better capture the flowsinherent three-dimensionality and to resolve the smaller vortical structures. The two- and three-dimensionalURANS, as well as the two-dimensional DES, over-predicted both the lift and drag with the two-dimensionalDES showing no improvement when compared to the URANS procedures. The two-dimensional DES did,however, capture the smaller vortical structures and showed less dissipation in the wake of the NACA0012airfoil. Finally, a new approach for modeling the effects of local three-dimensional flow in two-dimensionalsimulations has also been introduced and discussed.

Acknowledgments

The authors would like to thank Dr. John Clark and the managers of the turbine branch at the Wright-Patterson Air Force Research Laboratory in Dayton, Ohio for their support of this effort under contract09-S590-0009-20-C1. We would also like to thank the managers of Pratt & Whitney Rocketdyne for donatingthe computer resources used to conduct these simulations.

References

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2Davis, R. L. and Dannenhoffer, J. F., “A Detached-Eddy Simulation Procedure Targeted for Design,” AIAA Paper No.2008-534 , January 2008, 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV.

3Bush, R. H. and Mani, M., “A Two-Equation Large Eddy Stress Model for High Sub-Grid Shear,” AIAA Paper No.2001-2561 , June 2001, 15th AIAA Computational Fluid Dynamics Conference, Anaheim, CA.

4Wilcox, D. C., Turbulence Modeling for CFD , DCW Industries, Inc., 3rd ed., Nov. 2006.5Wilcox, D. C., “Formulation of the k-w Turbulence Model Revisited,” AIAA Journal , Vol. 46, No. 11, November 2008,

pp. 2823–2838.6Wilcox, D. C., “Reassessment of the Scale-Determining Equation for Advanced Turbulence Models,” AIAA Journal ,

Vol. 26, No. 11, Nov. 1988, pp. 1299–1310.7Smagorinsky, J., “General circulation experiments with the primitive equations,” Mon. Weather Review , Vol. 91, No. 3,

March 1963, pp. 99–164.8Strelets, M., “Detached Eddy Simulation of Massively Separated Flows,” AIAA Paper No. 2001-879 , Jan. 2001, 39th

Aerospace Sciences Meeting and Exhibit, Reno, NV.9Ni, R., “A Multiple-Grid Scheme for Solving the Euler Equations,” AIAA, Vol. 20, No. 11, Nov. 1982, pp. 1565–1571.

10Dannenhoffer, J. F., Grid Adaptation for Complex Two-Dimensional Transonic Flows, Ph.D. thesis, MassachusettsInstitute of Technology, Aug. 1987.

11Davis, R. L., “Cascade Viscous Flow Analysis Using The Navier-Stokes Equations,” AIAA Journal of Propulsion andPower , Vol. 3, No. 5, Sep.-Oct. 1987, pp. 406–414.

12Davis, R. L., Hobbs, D. E., and Weingold, H. D., “Prediction of Compressor Cascade Performace Using a Navier-StokesTechnique,” ASME Journal of Turbomachinery, Vol. 110, No. 4, June 1988, pp. 520–531.

13Jameson, A., “Time Dependent Calculations Using Multigrid with Applications to Unsteady Flows Past Airfoils withWings,” AIAA Paper No. 1991-1596 , June 1991, pp. 14, 10th Computational Fluid Dynamics Conference, Honolulu, HI.

14Andrade, A. J., Davis, R. L., and Havstad, M. A., “A RANS/DES Numerical Procedure for Axisymmetric Flows withand without Strong Rotation,” AIAA Paper No. 2008-702 , January 2008, 46th AIAA Aerospace Sciences Meeting and Exhibit,Reno, NV.

15Spalart, P., “Young Person’s Guide to Detached-Eddy Simulation Grids,” Tech. Rep. CR-2001,211032, NASA, 2001.16Jameson, A., Schmidt, W., and Turkel, E., “Numerical solution of the Euler equations by finite volume methods using

Runge Kutta time stepping schemes,” AIAA Paper No. 1981-1259 , June 1981, 14th Fluid and Plasma Dynamics Conference,Palo Alto, CA,.

17Gropp, W., Lusk, E., and Skjellum, A., MPI: A Message-Passing Interface Standard , Scientific and Engineering Com-putation Series, The MIT Press, 1994.

18Busby, J., Sondak, D. L., Staubach, B., and Davis, R. L., “Deterministic Stress Modeling of Hot Gas Segregation in aTurbine,” ASME , Vol. 122, No. 1, January 2000, pp. 62–67.

19Sondak, D. L., Dorney, D. J., and Davis, R. L., “Modeling Turbomachinery Unsteadiness with Lumped DeterministicStresses,” AIAA Paper No. 1996-2570 , 1996, 32nd ASME, SAE, and ASEE, Joint Propulsion Conference and Exhibit, LakeBuena Vista, FL.

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a des procedure applied to the flow over a naca0012 airfoil

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