Validation of Arbitrary Unstructured CFD Code for ... - J-Stage

Trans. Japan Soc. Aero. Space Sci. Vol. 53, No. 182, pp. 311–319, 2011

Validation of Arbitrary Unstructured CFD Code for Aerodynamic Analyses By Keiichi K ITAMURA,1Þ Keiichiro FUJIMOTO,1Þ
Eiji S HIMA,1Þ Kazuto K UZUU1Þ and Z. J. W ANG2Þ 1Þ

JAXA’s Engineering Digital Innovation Center, Sagamihara, Japan 2Þ Iowa State University, Ames, IA, USA (Received January 14th, 2010)

An unstructured grid CFD code capable of handling arbitrary polyhedra, named ‘‘LS-FLOW,’’ is developed for aerodynamic analyses of complex geometries. Through a series of numerical test cases, it is demonstrated that LS-FLOW can handle both structured and body-fitted/Cartesian hybrid unstructured grids successfully. Then, the code is validated by comparison with experimental data and theoretical solutions. In addition, it is shown that when a Baldwin-Lomax algebraic turbulence model is employed on the body-fitted/Cartesian grid, the portion of the body-fitted grid should be large enough to contain the whole boundary-layer. Finally, LS-FLOW is applied to a rocket configuration, and its future prospects are addressed. Key Words:

CFD, Unstructured Grid, Cartesian Grid, Compressible Flow, Turbulence Model, LS-FLOW

Nomenclature cf : cp : Cp : E: F; Fv:
: H: : M: ; t : p: Pr: Prt : Q: Re: : T: u; v; w:

skin-friction coefficient specific heat at constant pressure pressure coefficient total energy inviscid (Euler) and viscous flux vectors specific heat ratio, 1.4 total enthalpy thermal conductivity,  ¼ cp =Pr Mach number molecular and turbulent viscosities pressure Prandtl number, 0.72 turbulent Prantdtl number, 0.89 (conservative) state vector Reynolds number density temperature velocity components in x; y; z-directions, respectively x; y; z: Cartesian coordinates Subscripts n: face-normal component 1: freestream condition 1.

Introduction

With recent advancements in computational resources and methods, aerodynamic analyses are expected to support aerodynamic designs of launch vehicles or airplanes with complex geometries (e.g., Advanced Solid Rocket,1) Reusable Ó 2011 The Japan Society for Aeronautical and Space Sciences


Currently JAXA’s Engineering Digital Innovation Center, Tsukuba, Japan

Sounding Rocket2)). However, in their design, simulation parameters involving body configurations and flow conditions vary over a broad range, resulting in enormous computational burdens. In such aerodynamic analyses in which experiments are too expensive, Euler simulations are efficient but not accurate enough.3) Thus, high-Reynolds number viscous computations are essential to obtain accurate aerodynamic data efficiently. There are generally three major mesh categories4) for high-Reynolds number viscous flow computations: Structured grid methods; unstructured grid methods; and Cartesian grid methods. A structured overset grid5) or multi-block approach,6) e.g., UPACS,7) is widely used for such highReynolds number aerodynamic analyses. However, generation of a structured grid requires much more time compared to (usually automatically generated) unstructured grids, because this procedure requires a huge amount of labor and skill. Therefore, within the framework of structured grids, it is very costly to perform a parametric study including variation of configurations and flow structures. Unstructured grid algorithms are attractive and have gained popularity over the past decade8–11) because, in general, these grids have more flexibility to represent complex geometries than their structured counterparts. Specifically, the unstructured hybrid grid approach (e.g., prismatic/ tetrahedral,8) Cartesian,9,10,12,13) Building Cube Method,11) and Immersed Boundary Method14)) has matured for these high-Reynolds number flow simulations. This is partly because unstructured grid CFD can be equipped with the AMR (Adaptive Mesh Refinement)15,16) technique in which flow details of interest are well resolved by a locally refined mesh. However, it is also tricky and difficult to apply AMR to such hybrid grids with resolving important flow structures such as shock waves and vortical flows.

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A body-fitted/Cartesian grid approach has the capability for adaptive grid refinement15,16) while keeping accuracy for high-Reynolds number viscous flow computation. Thus, we developed an aerodynamic analysis tool consisting of a body-fitted/Cartesian grid generator, LS-GRID, and a cellcentered finite volume compressible flow solver on arbitrary unstructured grids, LS-FLOW. With the arbitrary polyhedra capability, LS-FLOW can be used on arbitrary grids including body-fitted/Cartesian grids, not to mention structured grids. This feature relaxes the complexity of the data structure and facilitates parallel computations especially for complex geometries, so wideranging applications are expected such as the aforementioned launch vehicles.1,2) This paper describes a validation study of LS-FLOW to demonstrate its reliability on both structured and unstructured grids. In Section 2, the governing equations and numerical methods are described. Then, Section 3 presents numerical test cases. First, the numerical accuracy of the code is assessed by vortex preservation benchmark tests. Second, laminar or turbulent boundary-layer over a flat plate is solved and the code is validated by comparison with theories and experiments. Third, laminar flow over an airfoil is solved and the result is compared with others. These three cases use structured grids, but we then use body-fitted/ Cartesian hybrid meshes in the two cases, treating a simple configuration of a sphere, and a rocket configuration as an application to complex geometries. Through series of test cases, some key features of body-fitted/Cartesian simulations are also addressed. 2.

2.1. Governing equations The governing equations are three-dimensional, compressible Euler/Navier-Stokes equations.

E 2 6 6 Fvk ¼ 6 4

0

3

uk H

7 lk 7 7 @T 5 um mk þ  @xk
 @ul @uk 2 @un lk ¼  þ lk   3 @xn @xk @xl

ð1aÞ 9 > > > > > > > > > > > > = > > > > > > > > > > > > ;


¼ 1:4. The Prandtl number is Pr ¼ 0:72. The molecular viscosity  and thermal conductivity  are related as  ¼ cp =Pr where cp is specific heat at constant pressure. In the Euler simulations,  is merely turned off ( ¼ 0). In the turbulence calculations, molecular viscosity  is replaced by ( þ t ), where t is turbulent viscosity. Likewise,  is replaced by ( þ cp t =Prt ), and Prt is turbulent Prandtl number, 0.89. 2.2. Numerical methods In the numerical code LS-FLOW, the system of equations (Eq. 1) is discretized and solved by the cell-centered finite volume method (FVM) for unstructured grids with arbitrary polyhedra. In this code, to avoid special treatments, a cell with hanging nodes is treated as having more than six faces: For example, a cell with one hanging node is treated as a nonahedron (nine faces), not as a hexahedron (six faces). Cell geometries such as volumes, centroids, and facecenters are calculated for arbitrary polyhedra according to Wang.17) For numerical methods (e.g., inviscid and viscous fluxes, and time evolution), LS-FLOW has many options and they are summarized in Table 1 (for details, see the literature). As shown in the next section, second-order spatial accuracy is achieved. Due to space limitations, only frequently used options are briefly described here. 2.2.1. SLAU numerical flux The general expression for AUSM-family flux functions is F 1=2 ¼

m_ þ jm_ j þ m_  jm_ j 
þ
þ p~ N 2 2


¼ ð1; u; v; w; HÞT ;

Governing Equations and Numerical Methods

@Q @F k @Fvk þ ¼ @t @xk @xk 2 3 2 3  uk 6 7 6 7 Q ¼ 4 ul 5; F k ¼ 4 ul uk þ plk 5;

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N ¼ ð0; nx ; ny ; nz ; 0ÞT

ð2aÞ ð2bÞ

Then, the mass flux of SLAU23) is written as    þ        1 m_ ¼ L VnL þ V n  þ R VnR  V n   p ð2cÞ 2 c
  L jVnL j þ R jVnR j V n  ¼ ð2dÞ L þ R

Table 1. Numerical methods in LS-FLOW. Governing equations

ð1bÞ Spatial Discretization

Three-dimensional, compressible Euler/Navier-Stokes (N-S) Cell-centered finite volume (CC-FV) method

Reconstruction Gradients

Least-Square (LSQ),

Slope Limiters

Venkatakrishnan20Þ

Green-Gauss (G-G)18;19Þ

ð1cÞ

Where,  is density, ui are velocity components in Cartesian coordinates (i ¼ 1, 2, and 3 corresponds to u, v, and w, respectively), E is total energy, p is pressure, H is total enthalpy (H ¼ E þ ðp=Þ), and T is temperature. Substituting 1, 2, or 3 into k, l, m, or n covers the full description of the 3D equations. The working gas is air approximated by the calorically perfect gas model with the specific heat ratio

(Wang’s modification21Þ ) Inviscid Term (Euler Fluxes)

SHUS,22Þ SLAU,23Þ Roe,24Þ AUSMþ -up,25Þ Ha¨nel26Þ

Viscous Term

Wang27Þ

Turbulence Models

Baldwin-Lomax (B-L),28Þ Spalart-Allmaras (S-A)29Þ

Temporal Evolution

TVD Runge-Kutta,30Þ LU-SGS31Þ

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K. KITAMURA et al.: Validation of Arbitrary Unstructured CFD Code for Aerodynamic Analyses

 þ   V n  ¼ ð1  gÞV n  þ gjVnL j;     V n  ¼ ð1  gÞV n  þ gjVnR j

ð2eÞ

g ¼  max½minðML ; 0Þ; 1  min½maxðMR ; 0Þ; 1 2 ½0; 1 ð2fÞ and the pressure flux is p~ ¼

þ  fpL j ¼0  fpR j ¼0 pL þ pR þ ðpL  pR Þ 2 2  pL þ pR  þ  j ¼0 þ fpR j ¼0  1 þ ð1  Þ fpL 2

ð2gÞ

_

 ¼ ð1  M Þ2

ð2hÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! _ 1 u L 2 þ v L 2 þ w L 2 þ u R 2 þ vR 2 þ w R 2 M ¼ min 1:0; c
2 ð2iÞ Vn unx þ vny þ wnz ¼ ð2jÞ c
c
8  1 > > if jMj  1 < 1  sign ðMÞ ; 2  ð2kÞ fp ¼ > > : 1 ðM  1Þ2 ð2  MÞ; otherwise 4 It is one of the features of SLAU that no cutoff Mach number25) is required for low-speed flow computations. This will broaden its application to a wide range of flows, e.g., turbopump internal flows. 2.2.2. Turbulence models LS-FLOW has two popular turbulence models: the Baldwin-Lomax (B-L) algebraic model,28) and the SpalartAllmaras (S-A) one-equation model.29) (A) B-L model: This algebraic (zero-equation), two-layer model is expressed as: M¼

t ¼ minð ti ; to Þ; t ¼  t

ð3aÞ

Inner layer:  

ð3bÞ

ti ¼ lmix 2 j!j; lmix ¼ y 1  exp  yþ =Aþ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   


@v @u 2 @w @v 2 @u @w 2 ð3cÞ    þ þ !¼ @x @y @y @z @z @x Outer layer:

to ¼ Ccp Fwake FKleb

Fwake ¼ min ymax Fmax ; Cwk ymax Udif 2 =Fmax  1 Fmax ¼ maxðlmix j!jÞ  y

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  Udif ¼ u 2 þ v2 þ w 2  u 2 þ v2 þ w 2 " FKleb ¼ 1 þ 5:5

max




y CKleb ymax

6 #1

ð3dÞ 9 > > > > > =

min

> > > > > ;

ð3eÞ

ð3fÞ

where, ymax denotes the value of y where Fmax achieves its maximum value. In the above expression, y is assumed to be exactly in the direction normal to the wall; in the actual three-dimensional

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applications, however, y is replaced by d which stands for distance from the wall. Closure coefficients are  ¼ 0:40, Aþ 0 ¼ 26, ¼ 0:0168, Ccp ¼ 1:6, Cwk ¼ 0:25, CKleb ¼ 0:3. We point out here that, because the model requires ‘‘searching’’ along a mesh line for maximum and minimum values of lmix j!j etc., these equations are considered valid only in structured grids or ones composed of a similar data structure. Thus, use of this model in the present code is confined to the ‘‘layer mesh’’ portion of the mixed grids (such as those generated by LS-GRID). In addition, since it is known that the transition term suggested in the original model (i.e., t ¼ 0 if ðt Þmax < 14, along a mesh line) sometimes causes difficulty of convergence, the term is not used in this work. (B) S-A model: This is a one-equation model used widely both on structured and unstructured grids, partly because searching along a mesh line is not required. D ~ ¼ cb1 ½1  ft2 S~ ~ Dt

1 þ r  ðð þ ~Þr ~Þ þ cb2 ðr ~Þ2    2 cb1

~  cw1 fw  2 ft2 þ ft1 U 2 ð4aÞ  d

t ¼ ~ fv1 ;

fv1 ¼

3 ; 3 þ cv1 3



~



~ S~ ¼ S þ 2 2 fv2 ; fv2 ¼ 1  1 þ  fv1 d 9  6 1=6 > 1 þ cw3 > > ; fw ¼ g 6 = 6 g þ cw3

~ > > ; g ¼ r þ cw2 ðr 6  rÞ; r  2 2 > ~ S d


!t 2 2 2 2 ft1 ¼ ct1 g t exp ct2 þ g d d t t U 2   ft2 ¼ ct3 exp  ct4 2

ð4bÞ ð4cÞ

ð4dÞ

ð4eÞ ð4fÞ

where, S is the vorticity magnitude (equivalent to ! in Eq. (3c)) and d is the wall distance. LS-GRID’s default output data set includes d. In the present code, ft1 and ct3 (and hence, ft2 ) are turned off ( ft1 ¼ ft2 ¼ 0). By omitting tripping terms, the code relies upon a ‘‘natural’’ transition from laminar to turbulent flows. Then Eq. (4a) is rewritten as  2 D ~

~ ~ ¼ cb1 S ~  cw1 fw Dt d

1 þ r  ðð þ ~Þr ~Þ þ cb2 ðr ~Þ2 ð4gÞ  Other constants are given as  ¼ 2=3, cb1 ¼ 0:1355, cb2 ¼ 0:622, cv1 ¼ 7:1,  ¼ 0:41, cw1 ¼ cb1 =2 þ ð1 þ cb2 Þ=, cw2 ¼ 0:3, cw3 ¼ 2. 2.2.3. Local time stepping The local time stepping technique used here where the time step takes its maximum allowable quantity in each local cell is written as:

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,"


ti ¼ CFL Vi

X

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# i; j Si; j

ð5Þ

j

where, Vi represents cell volume, i; j is spectral radius, and Si; j is interface area dividing cells i and j. When this technique is not used (global time stepping), the time step is expressed as follows for all the cells: t ¼ CFL= maxði; j =hi; j Þ i; j

ð6Þ

where hi; j is the distance between centers of cells i and j. 3.

Fig. 1. Solutions for stationary (left) and propagating (right) vortex, with 80 80 grid (t ¼ 2).

Table 2. Grid Refinement study for stationary vortex.

Numerical Examples

Grid

The following numerical methods are used as default options (Table 1), unless stated otherwise: Three-dimensional compressible N-S equations are solved with spatial reconstruction using Green-Gauss (G-G) with no limiter (second order in space), SLAU inviscid flux, Wang’s viscous flux, and temporal evolution method is LU-SGS with three inner iterations with three-point backward difference (second order in time). 3.1. Vortex preservation: accuracy study (structured grid) The static and propagating vortex problems, which are widely known benchmark tests for Euler equations,30,32–34) are solved by LS-FLOW to assess the spatial accuracy of the code. The selected options are: Euler equations and Shu’s TVD Runge-Kutta30) (second order in time); the rest of the methods are the same as the defaults. The computational setup is exactly as in Ref. 32). — Mean flow: ð; u; v; w; pÞ ¼ ð1; U1 ; 0; U1 ; 1Þ where U1 ¼ 0 for static vortex, whereas for vortex advection problem U1 ¼ 1; p ¼ 1 (not 1/
). — Perturbations representing isotropic vortex: 0:5ð1r2 Þ e fðz  z0 Þ; 0; x  x0 g; 2 ð
 1Þ 2 1r2 T ¼  e ; S ¼ 0; 8
2

ðu; v; wÞ ¼

ð7Þ

r 2 ¼ ðx  x0 Þ2 þ ðz  z0 Þ2 where the vortex center is located initially at ðx0 ; y0 ; z0 Þ ¼ ð5; 0; 5Þ in the computational domain of ½0; 10 ½0:5; 0:5 ½0; 10. In the vortex advection case, the vortex moves with the speed U1 ¼ 1 both in the x and z directions (thus, propagating in the diagonal direction) in the x–z plane. For the boundary conditions of what are called characteristic conditions,34) the exact solution is imposed at all ghost cells at each timestep. Uniform Cartesian grids are used with different time step size t to keep CFL number almost unchanged. The numbers of cells used in each grid are presented in Tables 2 and 3. Computations are continued until t ¼ 2:0 when the exact solution of the vortex center in the advection case is located at ð7; 0; 7Þ.

ðtÞ ðtime stepsÞ

L1 error

L1 order

10 10

0:2 10

9.57E-3

—

20 20

0:1 20

2.37E-3

2.01

40 40

0:05 40

4.91E-4

2.20

80 80

0:025 80

9.70E-5

2.25

160 160

0:0125 160

2.12E-5

2.14

Table 3. Grid Refinement study for vortex advection. Grid 10 10

ðtÞ ðtime stepsÞ 0:2 10

L1 error

L1 order

1.67E-2

—

20 20

0:1 20

5.31E-3

1.77

40 40

0:05 40

1.20E-3

2.10

80 80

0:025 80

2.74E-4

2.09

160 160

0:0125 160

6.62E-5

2.03

Figure 1 shows the computed flows at t ¼ 2 of stationary and propagating vortex cases. In Tables 2 and 3, the L1 norms in density difference between computed and exact values are presented for static and propagating vortex problems, respectively, and summarized in Fig. 2. We see that the formal (second) order of accuracy is achieved in L1 norm in both cases. 3.2. Turbulent boundary-layer over flat plate (structured grid) This benchmark test is available at NASA’s NPARC Archives.35) The flow conditions are M ¼ 0:2, Re ¼ 6:18 105 , with no angle of attack; computations were conducted with CFL ¼ 200 (except for S-A model: CFL ¼ 10) for 150,000 time steps using local time stepping (Eq. (5)). Four cases are considered by using different turbulence models: S-A model, B-L model, B-L model only in the limited region (B-L model (lmax ¼ 40), explained later), and no model (laminar). The grid system used here is also provided at the NPARC website,35) consisting of 110 (streamwise: wall-boundary starts from 11th cell) 80 (wall-normal) structured cells, covering 1:0 x 16:7 and 0 z 3:0. This mesh was converted to the LS-FLOW unstructured format (Fig. 3). The minimum spacing (width of cell closest to wall) satisfies zþ  1 over the turbulent region. The prescribed boundary conditions are: adiabatic, no-slip condition on wall (bottom of Fig. 3, x  0); inflow

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315

0.010

S-A model B-L model B-L model, lmax=40 Laminar (CFD) Turbulent (Exp. by 36) ) Turbulent (1/7th. Law) Turbulent (Prandtl-Schlichting) Laminar (Blasius)

0.008

cf

0.006

0.004

0.002

0.000 0.E+00

2.E+06

4.E+06

Re x

6.E+06

8.E+06

1.E+07

Fig. 2. Solution errors for stationary and propagating vortex (t ¼ 2). Fig. 4. Skin friction coefficient vs. Reynolds number over flat plate.

x 0

condition (left); outflow condition with fixed pressure (right and upper); symmetry (slip wall) condition for rest (spanwise, and bottom except for wall, x < 0). Figure 4 shows the skin friction profiles over the flat plate. Symbols indicate computed values except for experimental data by Wieghardt,36) whereas lines are analytical solutions. All the computations reproduce the corresponding analytical or experimental data both for laminar and turbulent cases, with only one exception for B-L model (lmax ¼ 40) case at high Reynolds numbers (Re > 6 106 ) where the computed cf deviates 20% at most from the other data. In the B-L model (lmax ¼ 40) case where the BaldwinLomax model is used only in the 40 cell-layers normal to the wall (only 0 < z < 4.3e-2), Fig. 5 shows that the development of the turbulent boundary-layer is confined to near the wall, leading to hanging of the evolution of (t =Þmax at a some value (about half the B-L model case, see Fig. 6). Thus, it is clear that unless the boundary of its searching region is sufficiently far from the wall, the Baldwin-Lomax model underestimates turbulent viscosity (and hence, skin friction). This fact restricts use of the model on body-fitted/Cartesian hybrid meshes (presented later), because its potential searching region where the model is

Fig. 5. Turbulent viscosity distributions, 1 < ðt =Þ < 300, (top) S-A model, (middle) B-L model, (bottom) B-L model (lmax ¼ 40).

500

S-A model, CFL= 10 B-L model, CFL=200 B-L model, lmax=40, CFL=200

400 (µ t/ µ)max

Fig. 3. Computational grid for flat plate simulations (110 80 cells), (top) overview, (bottom) blow-up view near leading-edge.

300 200 100 0 1,000

10,000 time steps

100,000

Fig. 6. Maximum turbulent viscosity histories.

valid is too limited on the body-fitted portion of the mesh, so the model underestimates turbulent viscosity, skin friction, total drag, and heat flux. 3.3. NACA0012 airfoil (structured grid) The flow conditions are M ¼ 0:5, Re ¼ 5;000, no angle of attack; computations were conducted with CFL ¼ 20 (10,000 time steps), and SLAU flux, Green-Gauss reconstruction (without limiter), and LU-SGS time integration (three inner-iterations) were used. The grid is a two-dimen-

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Table 4. CD values and separation points for NACA0012 computations.

Fig. 7. Computational grid for NACA0012 airfoil (201 51 points) simulations: (left) overview, (right) blow-up view.

Method

Elements

CD

Separation point

LS-FLOW (2nd-order CC-FV)

200 50 (10,000)

0.0569

84.9%

2nd-order SD33Þ

144 48 (6,912)

0.0545

88.6%

2nd-order DG39Þ

64 16 (1,024)

0.0501

(N/A)

4th-order SD33Þ

288 96 (27,648)

0.0548

81.4%

4th-order DG39Þ

64 16 (1,024)

0.0501

(N/A)

CV-FV40Þ

256 64 (16,384)

0.0554

81%

CC-FV40Þ

256 64 (16,384)

0.0556

80.9%

CC-FV40Þ

512 128 (65,536)

0.0553

81.4%

320 64 2 (40,960)

0.0569

83.4%

Triangle scheme 4 ¼ 1=6441Þ

(a)

(b) Fig. 8. Computed flow field, three sub-iterations (M1 ¼ 0:5, Re ¼ 5;000): (a) Iso-Mach-contours (0 < M < 0:59), (b) u-velocity contours; blow-up view of separation region near trailing-edge (0:01 < u < 0).

sional, O-type, structured grid, and the topology is shown in Fig. 7; 201 points are used in the circumferential direction, whereas there are 51 points in the radial (wall-normal) direction; the minimum spacing near the wall is 1.0e-3, based on the chord length of 1. (This spacing achieves sufficient resolution for the boundary-layer considered here.) The far field boundary is 50 times the chord length away from the wall. The computed flowfield is shown in Fig. 8 where the boundary-layer and wake recirculation region are visible. From Table 4 showing drag coefficients CD and separation points computed from a variety of methods, a laminar boundary-layer separation reportedly occurs at some location scattered within 80.9%–88.6% of the chord length in the literature, and CD is 0.0501–0.0569. The present computation yielded a separation point at 84.9% of the chord length and CD ¼ 0:0569, both in the range of the reference data. An extension of LS-FLOW to low Mach number flows and a detailed survey of preconditioning issues were conducted in Ref. 37), using other combinations of time integration methods and flux functions on the present grid. 3.4. Flow past sphere (body-fitted/cartesian hybrid grid) To demonstrate the performance of LS-FLOW in a bodyfitted/Cartesian mixed grid, a steady viscous flow around a sphere was computed here as in Ref. 33). The grid has

(a)

(b)

Fig. 9. Computational grid for sphere (a) overview (b) blow-up view.

158,342 cells (15 cell-layers around the body) for the half domain (y 0); the minimum spacing near the wall is 1.0e-3, based on the diameter of 1; the far field boundary is 20 times the diameter away from the wall (Fig. 9). The flow conditions are M ¼ 0:1, Re ¼ 118; computations were conducted with CFL ¼ 1;000 (10,000 time steps), and SLAU, Green-Gauss (without limiter), LU-SGS (three inner-iterations) were used. From the residual history (Fig. 10) and from the comparison to streamlines obtained in the experiment38) (Fig. 11), the computation successfully converged to a physically correct solution. 3.5. Application to rocket configuration (body-fitted/ cartesian hybrid grid) Finally, we present an example of a practical configuration. Figure 12 shows the grid generated around a rocket configuration, and it also demonstrates that LS-GRID can handle complex geometry. The grid consists of 4,673,642 cells; its minimum spacing near the wall is 3.0e-6 (based on the mainbody length L of 1); the far field boundary is 10 times the mainbody length and about 100 times the cross-section diameter ( 0:1) away from the wall.

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Fig. 10. Residual history for sphere computation.

L

0 H

x

(a)

(a)

H

0

x

(b) Fig. 12. Computational grid for rocket configuration simulation (a) overview and (b) blow-up view of nose-fairing.

(b) Fig. 11. Comparison of (a) computed and (b) experimental38) wake flows.

The flow conditions are M ¼ 0:7, Re ¼ 1:01 107 ; computations were conducted with CFL ¼ 1;000 for 10,000 time steps; SLAU flux, Least Square (for inviscid terms)/Green-Gauss (for viscous terms) methods without limiters, LU-SGS time evolution (no inner-iterations), and B-L model (only in the body-fitted mesh for x < L) were used. Although we demonstrated in 3.2 that the S-A model is preferred to the B-L model for this type of mesh, we adopted the B-L model here because it can take a much larger CFL than the S-A model, and also because the objective of this computation is to compare the calculated pressure profiles with measured data on the upstream portion of the body. In this region, the boundary layer is thin enough to be included in the current body-fitted mesh, and turbulence is considered to play a minor role.

The computed result (Fig. 13) shows that flow features such as compression and expansion are well captured, and that the pressure profile is in good agreement with the experimental data on the blunt-cone shaped nose-fairing and its vicinity. In particular, the stagnation pressure at x=H ¼ 0 (H is the nose-fairing length in the axial, x-direction) is in excellent agreement with the measured data, and expansion at the conjunction of the cone and the cylinder parts (x=H ¼ 1) is captured fairly. Another set of examples for supersonic cases with more realistic configurations are presented in Ref. 1), where the computed normal force, using the S-A model, on similar configuration shows good agreement with experimental data. Baseflow-related issues such as base pressure, recirculation, and axial force (drag at zero incidence) are future challenges, and we are preparing by incorporating more advanced and suitable methods of DES and DDES into LS-FLOW.42) Furthermore, it is costly to use two reconstruction methods of LSQ and G-G each for inviscid and viscous terms. The reason for this choice is explained in Ref. 18),

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(a)

Vol. 53, No. 182

—The case of the turbulent boundary-layer over flat plate indicated that use of the B-L model is not recommended in body-fitted/Cartesian grids, because turbulent viscosity (and hence, skin friction, total drag, and heat flux) over the wall is significantly underestimated (20% at most in present case). Recent work involves further improvements to LSFLOW by incorporation of DES, DDES,42) and a new reconstruction formula GLSQ.18) Extensions to multi-species gases, reacting flows, and cavitating multi-phase flows are being considered. A broad range of applications are expected, such as aerodynamic analysis of the Advanced Solid Rocket (currently Epsilon Launch Vehicle)1) and Reusable Sounding Rocket.2) Acknowledgments This work has been partly conducted as joint research between JAXA/JEDI and Iowa State Univ. We are grateful to Kozo Fujii (JAXA/ISAS, former Director of JEDI) and staff members at both organizations for relevant support. We thank Taku Nonomura (Univ. of Tokyo/ISAS, currently JAXA), Kengo Asada (Univ. of Tokyo/ISAS), Yoshinori Namera, Yuki Yamazaki (alumni of Tokyo Univ. of Science/ISAS), and Nobuyuki Tsuboi (JAXA/ ISAS, currently at Kyushu Institute of Technology) for constructive discussions on computational issues. We also thank Satoshi Nonaka, Tomoko Irikado (JAXA/ISAS), and Moriyasu Fukuzoe (JAXA) for providing us with reference data on the rocket configuration.

References (b) Fig. 13. Computed results for rocket configuration (M1 ¼ 0:7): (a) flowfield and (b) pressure profile on nose-fairing and its vicinity (H stands for axial length of fairing).

and the new method named GLSQ,18) combining these two methods in a unified manner, is currently incorporated in the code. These updates are not covered in these test cases, but will be included in the future publications. 4.

Summary

To verify and demonstrate the current status and recent development of an unstructured grid CFD code named ‘‘LSFLOW’’ for arbitrary polyhedra, a series of numerical tests have been conducted. The results are summarized as follows: —Accuracy study of stationary and propagating vortex cases demonstrated that the formal (second) order of spatial accuracy was achieved. —Computations on structured grids for flows over a flat plate (both laminar and turbulent) and an airfoil (laminar) validated the code by comparison to theories and/or experiments. —Computations on body-fitted/Cartesian mixed grids for flows over a sphere and a rocket configuration, along with the validation, demonstrated that the code successfully handle such an unstructured mesh.

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Feb. 2011

K. KITAMURA et al.: Validation of Arbitrary Unstructured CFD Code for Aerodynamic Analyses

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