[American Institute of Aeronautics and Astronautics 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition - Orlando, Florida ()] 48th AIAA

American Institute of Aeronautics and Astronautics

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Drag Prediction on NASA CRM Using Automatic Hexahedra Grid Generation Method

Atsushi Hashimoto1, Keiichi Murakami2, Takashi Aoyama3, Kazuomi Yamamoto4, Mitsuhiro Murayama5 Japan Aerospace Exploration Agency (JAXA), Chofu, Tokyo, 182-8522, JAPAN

and

Paulus R. Lahur6 Research Center of Computational Mechanics, Inc. (RCCM), Shinagawa, Tokyo, 142-004, JAPAN

To shorten the time to simulate flow, we develop an automatic and fast grid generator, HexaGrid, that produces Cartesian/prism hybrid grids. The objective of this study is to apply it for drag prediction on NASA Common Research Model (CRM). First, we compare the generated grid with the DPW gridding guideline and discuss about the capability and limitation of HexaGrid. Then, we validate computational results with HexaGrid, comparing with other solvers and grid generators. HexaGrid results agree well with the other results, where the difference of predicted drag is less than 5 counts except for the stall angle. Additionally, we compared the separated flows at attack angle of 4°. The separation lines and Cp distribution are largely affected by grid topologies.

I. Introduction rid generation process tends to be the most time consuming task in CFD procedures, since it usually requires manual operation. For complex geometry problems, the time required for the grid generation increases more.

To tackle this problem, one of the most effective ways is to make it automatic. To generate a grid automatically within a short time, a method based on Cartesian grid1 is chosen. It is well known for its efficiency and speed in filling a computational domain, and particularly good at generating cells with low aspect ratio (i.e. around one). Although this property is desirable for far-field region, it actually becomes a weakness when handling a boundary layer on solid surface, where high aspect ratio is preferable to resolve a thin boundary layer using appropriate number of grid. Therefore, we employ a hybrid grid2, 3 which is prismatic grid near solid surface and Cartesian grid for far-field region. Cartesian grid is generated around a solid object, and then prismatic grid is generated from the boundary of Cartesian grid towards the solid surface. We employed this approach and developed a software, “HexaGrid4, 5”. Our target is aerodynamic force prediction of aircrafts using HexaGrid. Usually, experienced technicians work on grid generation since know-how such as gridding guideline is necessary, and this work is time consuming even for trained people. If we build the automatic method into CFD procedures, we can accelerate the analysis work and reduce its cost. However, the automatic method is an immature technique and it is rarely applied for drag prediction that requires high-quality grid. Therefore, we evaluate the automatic method for drag prediction in this study. The benchmark problem of 4th Drag prediction workshop (DPW4)6 is chosen for validation. In the DPW4 held in June 2009, a new model, NASA common research model (NASA CRM)7, is employed to discuss trim drag. The model has been developed recently since new types of high-quality experimental data are necessary to better understand

1 Researcher, Numerical Analysis Group, Aerospace Research and Development Directorate, member AIAA. 2 Associate Senior Researcher, Numerical Analysis Group, Aerospace Research and Development Directorate, member AIAA. 3 Section Leader, Numerical Analysis Group, Aerospace Research and Development Directorate, Senior member AIAA. 4 Section Leader, Civil Transport Team, Aviation Program Group, Senior Member AIAA. 5 Researcher, Civil Transport Team, Aviation Program Group, Member AIAA. 6 Engineer, Technical Development Department, AIAA Member.

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

AIAA 2010-1417

Copyright © 2010 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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flow simulations as related to the accuracy of drag and moment predictions. Recently we have improved the HexaGrid capability for viscous flow simulations and carried out preliminary studies of drag prediction with it on simple geometries such as ONERA M6 (Ref. 5). In this paper, we apply HexaGrid for a more practical model, NASA CRM, and validate the results through the comparison with other solvers and grid generators.

II. Automatic Grid Generation Method (HexaGrid) We use the grid generation software, HexaGrid, in this study, which is a commercial software jointly developed

by Japan Aerospace Exploration Agency (JAXA) and Research Center of Computational Mechanics, Inc. (RCCM). It is capable of automatically generating hexahedra grid around solid surface discretized in triangles, in STL format. Virtually all CAD softwares can output surface geometry in STL format. The procedures of the grid generation are outlined below. The detail of method is described in Refs. 4 and 5. First, Cartesian grid is generated by means of successive local refinement. This step starts with one cell that covers the whole computational domain, whose size is set by users. In three-dimensional space, each refinement divides a cell isotropically into eight child cells of equal size and shape. At the beginning, the grid is locally refined until the size of cells intersecting solid surface is smaller than “a maximum grid size” set by users. The grid around the solid object is also refined according to the surface grid size (Fig. 1(1)). Then the grid is further refined until the size of cells intersecting the solid surface with large curvature reaches either a satisfactory level or “a minimum grid size” set by users (Fig. 1(2)). In addition, we can also control grid size anywhere using “Refinement Box” with HexaGrid GUI (Fig. 1(3)). In this study, we do not use the Refinement Box for simplicity, and, therefore, we set only the two parameters, maximum and minimum grid size, for Cartesian grid generation. Next, Cartesian grid cells intersecting the solid object and those around the solid surface are removed. The surface of Cartesian grid is then snapped onto the solid surface by moving each node of a quad surface to the closest location on the solid surface. A number of prismatic grid layers are constructed on the snapped surface. Users define the thickness of the first grid layer, and the expansion factor of thickness. Finally, the grid quality, which is measured mainly in terms of face flatness and convexity, is improved by means of grid smoothing and cell splitting.

Figure 1 Refinement process of Cartesian grid

III. Flow Computational Method (TAS-code) As an unstructured flow solver, the TAS (Tohoku university Aerodynamic Simulation) code8, is used in this

study. It is a well-validated code and used in a variety of aerospace applications9. In TAS, full Navier-Stokes equations are solved on the unstructured grid by a cell-vertex finite volume method. The HLLEW (Harten-Lax-van Leer-Einfeldt-Wada) method10 is used for the numerical flux computations. The LU-SGS (Lower/Upper Symmetric Gauss-Seidel) implicit method is used for time integration. The second-order spatial accuracy is realized by a linear reconstruction of the primitive variables with Venkatakrishnan’s limiter11 and Unstructured MUSCL-scheme12 (U-MUSCL). As for turbulence model, the Spalart-Allmaras model13, is used and the boundary layer is fully turbulent. The equations for the turbulence model are also solved using the second-order scheme.


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IV. Computational Grid A) Model

The model is NASA Common Research Model (NASA CRM)7 shown in Fig. 2. It is a simple wing/body/tail configuration with a wing-body fairing. The supercritical wing is employed for the main wing. The fuselage is representative of a wide/body commercial transport aircraft. Four configurations are provided by DPW4; the three of them have horizontal tail wings with different angle (iH=-2°, 0°, 2°) and the other is a model without the tail. In this paper, we focus on the model with horizontal wing of iH=0°. Reference values of this model are shown in Table1. The coordinates are the CAD coordinates of IGES file that is downloadable from DPW4 website6. B) Grid generated with HexaGrid

Three different grids (coarse, medium, and fine grids) are generated with HexaGrid. The medium grid is shown in Fig. 3. In addition, close-up views of wing root, wing tip, and horizontal tail are shown in Figs. 4-6, where coarse, medium, and fine grids are shown for each part. These grids are generated with input parameters shown in Table 2. As mentioned in Chapter II, we can control the Cartesian grid using the two parameters that are maximum and minimum grid sizes on solid surface. For the coarse grid, we set the same size of 5in for the max. and min. sizes, and a uniform-size grid is generated (Figs. 4(a), 5(a), 6(a)). For the medium grid, we set the max. size of 5in and the min. size of 1.25in(=5/22). The grid is further refined until the grid size with large curvature reaches the minimum grid size. In this case, the grid near the leading and trailing edges is refined two more times (Figs. 4(b), 5(b), 6(b)), where the grid size becomes half after grid is refined once and becomes quarter after twice. For the fine grid, the minimum grid size is set to be half of the medium one. The grid near the leading and trailing edges is refined one more time (Figs. 4(c), 5(c), 6(c)). For prismatic layers, we employed the same expansion ratio and thickness of first prism layer for the three grids (Table 2).

Figure 7 shows the grids of main wing at the cross-section of η=0.50, where η is a ratio of distance between the body axis and the cross-section to the span. The grid size around the airfoil is uniform in the case of coarse grid, whereas the grid is refined more around the leading and trailing edges in the cases of medium and fine grids. The total thickness of prism layer depends on the Cartesian grid size. Hence, the prism layer is relatively thin around the trailing edge for medium and fine grids (Fig. 8(b),(c)).

Table 3 shows the grid generation results. The numbers of generated grids are approximately 3.2 million, 11million, and 37 million for coarse, medium, and fine grids, respectively. The numbers of nodes and cells are almost same since hexahedral element is dominant. These grid are generated on Fujitsu SPARC Enterprise M9000 of JAXA Supercomputer System(JSS). For example, the generation time of medium grid is 120 minutes, which is quite faster than that of manual method as we see later.

Gridding guidelines are provided by DPW49. Although we follow as many guidelines as possible, some of them are neglected due to the restriction of HexaGrid. We describe the detail as follows,

• We use the thickness of first prism layer and the growth rate in the layer that are defined in the guideline.

HexaGrid can control them with input parameters for prismatic grid generation. • As for the outer boundary, HexaGrid can locate the boundary anywhere you want. However, the domain

shape has to be cubic, otherwise anisotropic grids will be generated. • We can not maintain the same stretching factors and the same topology among the three grids since

HexaGrid generates grids by means of successive local refinement. • The guideline defines that the chord-wise spacing at the leading and trailing edges for the main and tail

wings is less than 0.1% local chord for the medium grid. Hence, the grid sizes have to be changed according to the local chord length. However, HexaGrid can not control the grid size like this.

• The grid is rather coarse near the leading edge to keep the reasonable number of grid. For example, 0.1% of local chord is 0.11-0.47in for the main wing and 0.09-0.22in for the tail wing. However, the actual grid size is uniformly 1.25in for the medium grid. The grid size near the wing tip is approximately ten times coarser than that defined in the gridding guideline.

• Minimum 12 cells are required across the trailing edge base for the medium grid. We neglect this requirement due to the HexaGrid limitation. As shown in Fig. 8, 5-6 nodes are located across the base at the cross-section of η=0.50 for the three grids. However, it is difficult to allocate more grid across the base since the total thickness of prism layer becomes thin when fine Cartesian grid is generated near the trailing edge.


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Additionally, HexaGrid has a function of feature capturing4. It is employed for the three grids. As shown in Figs. 4 and 6, the feature line of wing-body juncture is clearly captured except for the coarse grid. The feature capturing becomes difficult when grid is coarse because the distance between the feature line and the grid node to be moved is relatively long. However, if we employ sufficiently fine grid, we can overcome this problem.

HexaGrid can automatically generate grids using a few parameters of Table 2. Therefore, even CFD beginners can generate grids easily and the time needed for training is short. In addition, the same grid is generated using the same parameters and less dependent on user’s experience since the generation process is fully automatic.

Figure 2 NASA Common Research Model (wing/body/tail (iH=0°))

Table 1 Reference geometry

Sref (reference area) 594,720 in2 Cref (reference chord length) 275.80 in

Xref, Yref, Zref (moment reference center) 1325.90 in, 0 in, 177.95 in

(a) Overall view (b) Upper surface of main wing

Figure 3 Medium grid generated with HexaGrid


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(a) Coarse grid (b) Medium grid

(c) Fine grid

Figure 4 Grid around wing root


(c) Fine grid

Figure 5 Grid around wing tip


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(c) Fine grid

Figure 6 Grid around horizontal tail

Table 2 Grid generation settings Setting parameter Coarse Medium Fine

Max cell size on solid surface 5 in 5 in 5 in Min cell size on solid surface 5 in 1.25 in (=5/22) 0.625 in (=5/23)

Expansion ratio of prism layers 1.25 1.25 1.25 Thickness of first prism layer 9.85E-4 in 9.85E-4 in 9.85E-4 in

Cubic domain size 100 Cref 100 Cref 100 Cref

Table 3 Grid generation results Coarse Medium Fine

Cartesian grid finest level 13 15 16 Number of prism layers 35 29 26

Node count 3,213,783 11,055,602 36,601,899 Cell count 3,644,942 12,654,764 41,630,191

Boundary node count 105,686 295,394 757,593 Boundary face count 106,272 297,697 762,131 Prismatic cell count 1,932,525 7,145,542 17,752,826

Generation Time (JSS) 20 min 120 min 380 min


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(c) Fine grid

Figure 7 Main wing grid at the cross-section of η=0.50


(c) Fine grid

Figure 8 Main wing grid near trailing edge at the cross-section of η=0.50


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C) Grid generated with MEGG3D In order to compare the results, we generate a tetrahedron-based grid as well. A hybrid unstructured grid is

generated using an unstructured surface/volume grid generator, MEGG3D14,15. First, ridges (feature lines) of the STL data are set manually with GUI of MEGG3D. The surface grids are triangulated with the direct advancing front method based on the ridges and STL data. After that, isotropic tetrahedral volume grids are generated using the method of Delaunay tetrahedral meshing. Then, prismatic layers are added on the non-slip walls. This grid generator was well-validated and used for many applications such as drag prediction9. A medium grid generated with MEGG3D is shown in Fig. 9. The grid is well clustered near the leading edge and trailing edge compared with that generated with HexaGrid. However, it is difficult to generate fine grid across the trailing edge base since the grid is basically isotropic. The MEGG3D grid in the middle of wing is relatively coarse, whereas HexaGrid is uniformly fine (Figs. 3(b), 9(b)). The grid information is shown in Table 4. We spent about 6 hours setting ridge lines and about 6 hours generating space grid. The surface grids are generated in a few minutes. The flow solver of TAS code is used for this grid. D) Grid generated with Gridgen®

A multi-block structured grid is also generated for comparison using a commercial software, Gridgen®. A medium grid generated with Gridgen is shown in Fig. 10. The grid is well clustered near the leading edge and trailing edge compared with those generated with HexaGrid (Fig. 3) and MEGG3D (Fig. 9). The grid is fine in the chord-wise direction, whereas it is coarse in the span-wise direction. The grid information is shown in Table 4. We spent about 5 days generating the grid.

As for a flow solver, UPACS16 is used for this grid. This flow solver is based on a cell-centered finite volume method. In this study, the second-order scheme of the Roe’s flux difference splitting for convection terms is used with the MUSCL extrapolation and the van Albada’s limiter. The viscous terms are discretized using a scheme based on 2nd-order central difference. Time integration is carried out using the MFGS implicit method. The Spalart-Allmaras model13 is used for a turbulence model.


Figure 9 Medium grid generated with MEGG3D


Figure 10 Medium grid generated with Gridgen


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Table 4 Grid information (Medium grid)

MEGG3D Gridgen Zones 1 283

Number of prism layers 34 40 Node count 13,490,678 9,903,509 Cell count 36,250,881 9,006,808

Boundary node count 369,541 294,408 Boundary face count 798,336 275,608

V. Results A) Grid convergence

Drag coefficients are computed at CL=0.500 for the coarse, medium, and fine grids. The free stream Mach number is 0.85 and the Reynolds number based on the reference length, Cref, is 5×106.

Contours of Cp computed with HexaGrid and TAS code are shown in Fig. 11, where the grid is medium and the angle of attack is 2.388°. The pressure at the wing-body juncture is smooth since the feature line is successfully captured.

Drag coefficients for the three grids are shown in Fig. 12, where HexaGrid+TAS is compared with MEGG3D+TAS and Gridgen+UPACS. Total drag is a summation of pressure drag and friction drag. The horizontal axis is a grid index, 1/(number of grid)(2/3). The result of HexaGrid shows good grid convergence and the predicted drag coefficients are between the results with MEGG3D+TAS and Gridgen+UPACS (Fig. 12(a)). In other words, the drag predicted with the automatic grid generation has almost the same accuracy as that with the manual grid generation. The pressure and friction drags of MEGG3D+TAS and Gridgen+UPACS have good convergence tendency, whereas those of HexaGrid show different trend with finer grid (Fig. 12(b)(c)). One of the reasons is the difference of the three grids used for this study. MEGG3D and UPACS grids have same stretching factors and a same topology among the three grids. However, HexaGrid can not generate such a family of grids. The medium and fine grids generated with HexaGrid are locally refined near the leading and trailing edges (Figs. 4-6).

B) Comparison of Cp and Cf distributions

Here, we compare the medium grid results at CL=0.500, which are the same results used in the grid convergence study. The Cp and Cf distributions at η=0.20, 0.50, and 0.95 of the main wing and at η=0.30 of the tail wing are shown in Figs. 13-16, and the cross-sections are illustrated in Fig. 17. These cross-sections are a part of the data required by DPW4, and the detailed definitions of the cross-sections are described in Ref. 9. Generally, HexaGrid+TAS results agree well with the other results except for the cross-section near wing tip, though the grid generated with HexaGrid is coarser near the leading edge than those generated with MEGG3D and Gridgen. The shock wave is well captured in the case of HexaGrid (Fig. 14(a)), where the grid generated with HexaGrid is isotropic and relatively fine compared with the others (Fig. 3(b), 9(b), 10(b)).The friction coefficient of HexaGrid becomes less than the others behind the shock wave location, since the boundary layer is affected by the larger pressure gradient (Fig. 14(b)). The difference of shock strength and the related friction cause the difference of friction drag. In fact, the friction drag obtained with HexaGrid is less than the others as shown in Fig. 12.

The three results are different near the wing tip as shown in Fig. 15. In order to see the effect of grid topology, Cp contours and Cp contours with grid are shown in Figs. 18 and 19, respectively. The grid of HexaGrid is isotropic and uniformly fine, and its Cp distribution seems to be the best among the three results (Figs. 18(a), 19(a)). However, the suction peak at the leading edge is not well resolved in the case of HexaGrid due to the insufficient grid resolution (Fig. 15(a)). The grid of MEGG3D is fine near the leading and trailing edges, but it is coarse in the middle of wing. As a result, the Cp distribution is smeared due to the coarse grid (Figs. 18(b), 19(b)). The grid of Gridgen is also fine near the leading and trailing edges, but it is anisotropic and relatively coarse in the span-wise direction. Therefore, the pressure distribution is smeared in the span-wise direction (Fig. 18(c), 19(c)). The merit of using uniformly fine grid is pointed out from the comparison.

The Cp distributions of the three grids are almost same at the tail wing (Fig. 16(a)). However, the friction of HexaGrid is less than the others probably due to the coarse grid at the leading edge (Fig. 16(b)). This deficiency may be covered by the local refinement using the refinement box (Fig. 1).


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C) CL-α, CL-CD polar curve Figures 20 and 21 show the CL-α graph and CL-CD polar curve, respectively, where the three results using the

medium grid are compared. The result of HexaGrid+TAS agrees well with those of MEGG3D+TAS and Gridgen+UPACS except for the attack angle of 4°. Figure 21 shows the CL-CD polar curve. The result of HexaGrid+TAS is close to that of Gridgen+UPACS except for the attack angle of 4°, where the difference is less than 5 counts.

The flow is separated at the attack angle of 4°. The separation line is largely different among the three grids as shown in Figs. 22 and 23. The result of HexaGrid+TAS shows the largest separation and its flow pattern is close to Gridgen+UPACS result. This difference will be discussed again after the planed wind-tunnel measurement is accomplished.

(a) Upper surface (b) Lower surface

Figure 11 Cp contours (HexaGrid+TAS, medium grid, CL=0.500)

0.0265

0.0270

0.0275

0.0280

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0.0295

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0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05

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(a) Total drag (b) Pressure drag

0.0105

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0.E+00 1.E-05 2.E-05 3.E-05 4.E-05 5.E-05

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(c) Friction drag

Figure 12 Comparison of grid convergence


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Figure 13 Main wing Cp and Cf distributions at the cross-section of η=0.20

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(a) Cp (b) Cf

Figure 16 Tail wing Cp and Cf distributions at the cross-section of η=0.30

η=0.95

η=0.50

η=0.20η=0.30

Figure 17 Cross-sections of Cp and Cf distributions

(a) HexaGrid+TAS (b) MEGG3D+TAS (c) Gridgen+UPACS

Figure 18 Wing tip Cp contours (CL=0.5, medium grid)


Figure 19 Wing tip Cp contours with grid (CL=0.5, medium grid)


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Figure 20 Comparison of CL-α Figure 21 Comparison of CL-CD


Figure 22 Cp contours (ΑοΑ=4 °, medium grid)


Figure 23 Oil flow (ΑοΑ=4 °, medium grid)

VI. Conclusions In this paper, we applied the automatic hexahedra grid generation code, HexaGrid, for a more practical model,

NASA CRM, and validated the results through the comparison with other solvers and grid generators. We compared the generated grid with the DPW gridding guideline and discussed about the capability and

limitation of HexaGrid. It can generate prism layers with desired first-layer thickness and growth rate. Furthermore, it can capture the feature line of wing-body junctures clearly. However, it is difficult to generate a family of grids used for grid convergence study, where they are required to have same stretching factors and a same topology. Additionally, the grid is rather coarse near the leading edge to keep the reasonable number of grid.

Despite such deficiencies, the lift and drag predicted using this automatic grid generator agree well with those using other manual methods, where the difference of predicted drag is less than 5 counts except for the stall angle. The pressure and friction distributions obtained with HexaGrid are a little less accurate near the leading edge due to


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the grid coarseness. However, the shock wave in the middle of wing is well captured with HexaGrid. The friction coefficient of HexaGrid becomes less than the others behind the shock wave location due to the difference of shock strength. This is one of the reasons of lower friction drag. Additionally, we compared the separated flows at attack angle of 4°. The separation lines and Cp distribution are largely affected by grid topologies. This difference have to be discussed more with experimental data.

HexaGrid was validated thorough the comparison with other results. The automatic method can quickly generate grids that are comparable to the grids with manual methods, and it can save time of making grids. HexaGrid becomes a powerful tool to accelerate CFD processes.

Acknowledgement The authors would like to thank Mr. Hideki Kitajima and Mr. Masafumi Yamamoto of Research Center of

Computational Mechanics, Inc., Mr. Taisuke Nanbu of Waseda University, Mr. Kentaro Tanaka and Mr. Tohru Hirai of Ryoyu Systems Co., Ltd. for their support of the sequence of CFD processing.

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AIAA paper 97-1980, 1997. 3Wang, Z.J. and Chen, R.F., “Anisotropic Solution-Adaptive Viscous Cartesian Grid Method for Turbulent Flow

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4999, 2005. 13Spalart, P. R. and Allmaras, S. R., “A One-Equation Turbulence Model for Aerodynamic Flows,” AIAA Paper 92-0439,

1992. 14Ito, Y., Shih, A. M. and Soni, B. K., "Unstructured Mesh Generation Using MEGG3D - Mixed-Element Grid Generator in

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15Ito, Y., Murayama, M., Yamamoto, K., Shih, A. and Soni, B., “Efficient CFD Evaluation of Small Device Locations with Automatic Local Remeshing,” AIAA paper 2008-7180, 2008.

16Takaki, R., Yamamoto, K., Yamane, T., Enomoto, S. and Mukai, J., “The Development of the UPACS CFD Environment,” High Performance Computing, Proc. of ISHPC 2003, Springer, pp. 307-319, 2003.

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[American Institute of Aeronautics and Astronautics 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition - Orlando, Florida ()] 48th AIAA