--- -----11 lm_

AIAA 2002-0838

TRANSONIC DRAG PREDICTION

USING AN UNSTRUCTURED

MULTIGRID SOLVER

D. J. Mavriplis

ICASE

MS 132C, NASA Langley Research Center

Hampton, VA

and

David W. Levy

Cessna Aircraft Co.

Wichita, KS

40th AIAA Aerospace Sciences Meeting

January 14-17 2002, Reno NV

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https://ntrs.nasa.gov/search.jsp?R=20020019227 2020-04-23T15:17:27+00:00Z

TRANSONIC DRAG PREDICTION USING

AN UNSTRUCTURED MULTIGRID

SOLVER

D. J. Mavriplis

[('ASE

IkIS 132('. NASA Langley Research Center

Hampt(m. \'A

_lll(I

David VV. Levy

C'essna Aircraft C_.

\\ichita, KS

Abstract

This paper sunmlarizes the results obtained with the

NSt'3D unstructured multigrid solver for the :\IA.\

Drag Prediction Workshop held in Anaheim. ('A, .hme

2001. The test case for the workshop consists of a

wing-body configuration at transonic flow conditions.

Flow analyses fi)r a conlplete test matrix of lift coeffi-

cient values and Math numbers at a constant Reynolds

number are performed, thus producing a set of drag

polars and drag rise curves whMl are compared with

exp(,rimental data. l-lesults wer(- obtain('d ind('t)('n-

dently by both authors using an identical baseline grid.

and different refned grids. Most cases w(,ro run in

parallel on commodity cluster-type machines while the

largest cases were run on an SGI Origin machin(, using

128 processors. The ot)jectiv(' of this t)aper is t() study

the accuracy of the subject unstructured grid solver

for predicting drag in the transonic cruise regime, to

assess the efficiency of the method in terms of conver-

gell('e, cpu tillle alld illelllory, and to (teternfino the

effects of grid resolution on this predictive ability and

its computational efficiency. A good predictive abil-

ity is demonstrated over a wide range of conditions.

although accuracy was found to degrade for cases at

higher Math numbers and lift vahtos where hwreas-

ing amounts of flow separation occur. The ability to

rapidly compute large numbers of cases at varying flow

conditions using an unstructured solver on inexpensive

clusters of commodity computers is also demons( rated.

Introduction

Computati(mal fluid dynanfics has progr(,ssed to the

point where Reynolds-averaged Navier-Stokes solvers

have become standard simtflation tools for predict-

ing aircraft aerodynamics. These solvers at'(' r()utinoly

('opyright ,_') 2002 by the American [n,,,titute of \erorlm, tics

a.rld \.,,tr'onautilb. Inc. -\11 riRht,b re_.erved.

] OF

us('d t,) pr(,dict aircraft flw('e coefficients su('h as lift.

drag and moments, as well as the changes in these

values with design changes. In order to be usofifl to

an aircraft designer: it is generally acknowledged that

the computati(mal niethod should I)(' cat)abh" of pre-

dicting drag to within several counts. Whih' Reynolds-

averaged Navier-Stokes solvers have made great strides

in accuracy and affordal)ility over the last decade, the

stringent accuracy requirements of the drag prediction

task have proved difficult to achi('ve. This difficulty is

('onll)ounde(t by the multitude of Navier-Stokes solver

fornmlations availabh.', as well as by the effects on

accuracy of turbulence modeling and grid resolution.

Therefore. a particular Navier-Stokes solver must un-

dei'go extensive validation including th(, deternfination

of ado(tuate grid resolution distribution, prior to being

trusted as a usefi.fl predictive tool. With these issues in

mind, the AIAA Applied Aerodynamics technical com-

mittee organized a Drag Prediction Workshop, held

ill Anahoiln CA..hm(' 2001.1 in order to assess th,'

predictive capabilities of a number of state-of-the-art

computational fluid dynamics methods. The chosen

configuration, denoted as [)LR-F4'-' and depicted in

Figure 1. consists of a wing-body geonwtr.v, which is

represontativ(' of a modern sup('rcritica[ swept wing

transport aircraft. Participants included Reynolds-

averaged Navier-Stokes formulations based on block-

structured grids, overset grids, and unstructured grids.

thus affording an opportunity to compare those meth-

ods Or!. all equal basis in terms of accuracy and (,ffi-

ciency. A standard mesh was supplied for each type

of methodology, with participants encouraged to pro-

duce rosuhs on additionally refined meshes, in order to

assess the effects of grid resoluti(m. A Math number

v('rsus lift (:oeffM(,nt (('I.) matrix of test cases was de-

fined: which included mandatory and optional cases.

The calculations were initially run by the participants

without knowledge of the experimental data. and a

l0

.\MERIUAN ]NS'FITI'TE OV _\ EnONAITI('q AND \_;II(ONAVTICS PAPER _2002 08fin

c(,npilation of all workshop results including a statis-

tical analysis of these results was perf()rmed t)3 the

conlnlit t(_(L :_

Fig. I Definition of Geometry for Wing-Body

Test Case (taken from Ref. _')

This paper describes the results obtained for this

workshop with the unstructured mesh Navier-Stokos

soh'er NSU3D. _-6 This solver has been well validated

attct is curr('ntly in use in both a research setting and

an industrial production onvironnlcnt. Results were

obtained independ(,1My by both authors on file base-

line w()rksh(q) grid. aim on two refined grids gencrate(t

independently by both authors. All required and op-

tional cases were run using the baseline grid and one

refined grid, whih" the most highly refined grid was

Oll[y full 1)I1 the ltlalldatory ('_ts('s, Th(' fUllS w('ro per-

formed on thr_x' different types of parallel machines at

two different h.)cations.

Flow Solver Description

Tlw NSU3D cod(' s()lv('s the Reynolds averaged

Navier-Stokes equations on unstructured meshes of

mixed element types which may include tetrahedra,

pyramids, prisms, and hexahedra. All elenu, nts of the

grid are han.dled bv a singh' unifying cdge-base(l data-

strttctur_ _ in the flow solver. I

Tetrahedral elements are employed in regions where

the grid is nearly isotropic, which generally correspond

to regions of inviscid flow. and prismatic (:ells aw em-

ph)y('d in regions close to the wall. such as in boundary

layer regions where the grid is highly stretched. Tran-

siti_m b_-tw('en prismatic and tetrahedral cell regions

occurs naturally when only triangular prismatic faces

art, exposed to the tetrahedral region, but requires a

small numt)er i)f pyramidal o'lls (calls fornwd by 5 ver-

tices) in cases where quadrilateral prismatic faces are

exposed.

Flow variables are stored at the vertices of the mesh,

and the gov(,rl|ing equations are dis(:r('tized using a

central (lifferenc,' finir,'-vo[mue technique with ad(h'd

artificial dissipation. The matrix formulation ()f the

artificial dissipation is employed, which corresponds

to an upwind scheme based on a Rue-Riemann solver.

The thin-layer fiJrm (ff the Navier-Stokes equations is

employed in all cases, and the viscous terms are dis-

cretized to second-order accuracy by finit_'-diff('r(mce

approximati(m. I F(_r muhigri([ calculati(ms, a first-

order discretization is employed for the convo('tive

[(H'nts on the coarse grid levels.

The basic time-stepping scheme is a three-stage ox-

plMt multistage s('hem_' with stag(' c(,'ffi('i(q_ts Ol)ti-

mized for high fr_,qUClWy dalnl)ing l)roperties. 7 an(t

a CFL number of 1.8. Convergence is accelerated

by a local block-.lacobi preconditioner in regions of

isotropic grid ('ells. which involvf,s inverting a 5 x 5

matrix for each verwx at each stage. _ 10 In bound-

arv layer r_'gions, whor_, the grill is highly stret('lw(t.

at line smoother is entphued, which involves inverting

a block tridiagonal along lines constructed in the un-

structured mesh by grouping together edges normal to

the grid str_,tching ([ire('tion. Th(' lint' sm(_othing te('h-

hi(lug' has been shown to relic're the numerical stiffness

associated with high grid anisotroI)y.[I

An agglomeration multigrid algorithm |` 1_ is used

to further enhance convergence to steady-state. In

this approach, coarse levels are c(mstruct(,d by flls-

ing together neighboring fine grid control volumes to

form a smaller number of larger and more complex

control volumes on the coarse grid. '['his process is

i)e|'fbrm(,d automaticalh" in a pre-processing stage by

a graph-based algorithm. A multigrid cych, consists

of performing a time-stop on the fine grid of the se-

quence, transferring the flow solution and residuals to

the coarser hwel. perfl)rming a time-step on the coarser

level, and then interpolating the corr('cti_)ns back from

th(' coarse lev('l to update the fine grid soluti(m. The

process is applied recursively to tit(., coarser grids of

the seqtl(_nce.

The single equation turbul(,nce model of Spalart and

kllmaras 1:_ is utilized to a('('OUllt for turbulence ef-

fects. This ('(luation is discrotized and solved in a

manner completely analogot|s to the flow equations.

with the exception that the convective terms are only

discretized t,) first-order accuracy.

The unstructur(,d multigrid solver is parallelized

by partitioning the domain using a standard graph

partitioner _ and communicating between the various

grid partitions running on individual processors us-

ing either the MPI message-passing library.l; or the

OpenMP compiler directives, t6 Since OpenMP goner-

ally has been advocated fi)r shared memory architec-

tures, and MPI for distributed memory archite('tures,

this dual strategy not only enables the sol\'(,r to run

('ffi('iently on both types of memory architectures, but

can also be used in a hybrid two-level mode, suitable

for networked clusters of individual shared memory

multi-processors. _ l%r tit(, results presented in the pa-

p_,r, the s,)lver was run ()n distributc(t /u('mury PC

2OF IO

,\MERIt'AN [N>;TITI'TE OF AERONAIFICS ANI) ;\_,rI'IIONAI'TI('% [)APEII 2002 (),N3N

clusters and an SG[ Origin 20(}0. using the MPI pro-

grarnming model exclusively.

Grid Generation

Th(' bas('lin(' grid supplied for th(" workshop was g('n-

crated using the VGRIDns package. *r This approach

produces fully tetrahedral meshes, although it is ca-

pable of generating highly stretched semi-structuredtetrahedral elements near the wall in the boundary-

lay('r r('_i()ll, and (,nlph)ys ll!.od(}rat(' spanwis(' str(,tt'h-

ing in order to reduce the total number of points. A

semi-span g(,ometry was modeled, with the far-field

boundary located 50 chords away from the origin, re-

suiting in +tt(+tal of 1.65 nfillion grid points. 9.7 millitm

t(-trah<'(lra, and 36.1)1)0 wing-body surfac(' points. The

chordwise grid spacing at the leading edge was pre-

scribed as O.25O mm and 0.500 nun at the trailing

edge. using a dimensional mean chord of 142.1 nml.

Th(- trailing ('dge is t)hmt, with a base thi('kn('ss of0.5 '/ chord, and the t)aselin(' mesh contain('d 5 grid

points across the trailing edge. The normal spacing atthe wall is 0.001 mm, which is designed to produce a

grid spacing corresponding to f- = 1 for a Reynolds

lllunb('r ()f 3 million. A stretching rat(' of 1.2 was pre-

s('rib('d for th(' gr()wth of cells in th(' llortnal directi(mnear the wall. in order to obtain a tllinimunl of 20

points in the boundary lay(,r.Becaus(" the NStTaD solver is optimized to run on

ufix(,d (,h'm('nt m(,sh('s, the fltlly totrahedral bas(,linc

mesh is subsequently converted to a mixed element

nlesh by merging the send-structured tetrahedral lay-ers in th(" boundary layer region into prismatic eh.'-

nlents. This is done in a pre-processing phase where

triph'ts ,>f tetrah('ttral lay('rs are identifi('d and m(,rg('d

into a single prisnlatic eh,ment, using information iden-

tif_ving these elements as behmging to the stretched

viscous layer region as opposed to the isotropic inviscid

tetrah(,dral region. The merging operation results in atotal of 2 million creat(,d prismatic eh_nt('nts, whih' thenuml)er of tetrahedral cells is reduced to 3.6 million,

and a total (.)f 10090 pyramidal elements are created to

nl(,rge prismatic elements to tetrahedral eh.'m('nts in

regions where quadrilateral faces from prismatic ele-

ments are adjacent to tetrahedral elements.

A higher resolution mesh was generated t)3: the sec-

ond author using VGRIDns with snlaller spacings in

ttl(' vicinity of the wing root. tip. and trailing ('dg(',

resulting in a total of 3 million grid points, and 7a.000

wing-body surface points. One of the features of this

refined grid is the use of a total of 17 points across the

wing trailing ('dge v('rsus 5 for th(' bas(,line grid. After

th,' m('rging operathm, this grid ('()ntained a t,Jtal ()f

3.7 million prisms and 6.6 million tetrahedra.An additional fine mesh was obtained 1)y the first

attthor through global retinement of the baseline work-

shop m(,sh. This strategy op(,rates (tir(,('tly on th('

mixed prismatic-tetrahedral mesh. and consists of sub-

dividing each element into 8 smaller self-similar eh>

ments, thus producing an 8:1 refinement of th(" originalmesh. Is The final mesh obtained in this manner con-

tain('d a total of 13.1 million points with 16 million

prismatic elem('nts and 288 million t(,trah('dral eh'-

ments, and 9 points across the bhmt trailing edge of

the wing. This approach can rapidly' generate very

large meshes which would otherwise be very time con-

suming t() construct using th(' original m,,sh generationsoftware. One drawback of' the current approach is

that newly generated surface points do not lie exactly

on the original surface description of the model geom-

etry, but rather along a linear interpolation between

pr,,vi()usly existing surface coarse grid points. D)r a

single level of refinement, this drawback is not ex-

pected to have a noticeable effect on the results. An

interface for re-projecting new surface points onto the

original surface geometry is currently un(h'r ('onsid(,r-ation.

The baseline grid was found to be sufficient to re-

solve all major flow features. Tile computed surface

pressure coefficient on the baseline grid for a Math

numb('r of 0.75, Reynolds numb(,r (>f 3 million, and

('t 0.6 is shown ill Figure 2. illustrating good reso-

lution of tile upper surface shock. A small region of

separation is also resolved in the wing root area. as

shown by the surface streamlin('s for the same flow

conditions, in Figure 3.

Table 1 Grids and Corresponding Run Times

Grid No. Points

Grid 1 1.65× 10°

Grid 1 1.65 × t0 G

Grid 1 1.65 x 10 (i

Grid 2 3.0 x 1(I(;

Grid 3 13 x 106

No. T('ts

2 x 1()(s

2 x 10°

2 x 1(1(;

3.7 x 10a

16 x 10o

No. Prisms3.6 x 10_

3.6 x I ()(_

3.6 x 10"

6.6 x 10"

28.8 x lt/°

Memory

2.8 Gbytes

2.1 Gbytes3.0 Gbytes

4.2 Gbytes

27 Gbytes

_tln Time

2.6 hours

8 hours

45 min.

8 hours

I hours

Hardware

16 Pentium IV 1.TGHz

I DEC Alpha 2126.1 (667MHZ)64 SGI Origin 21)I)t) (400MHz)

8 DEC Alpha 21264 (667MHZ)

128 SGI 02000 (100MHz)

3 oF It)

AMERIt'AN INSTITUTE OF :\ERONAI:T[(_S AN[) ASTRONAII'IL_S PAPER 2002 083s

Figuro 1 dopicts the computed y + vahLos at tho break

sovt i,m fl_r tho sanw flow conditions, indicating vahu,s

woll bvl(,w unity ovor the ,,nt iro lower surfaco arid a nla-

jority of thv upper surfaco. Tho convergenec history

fl_r this cast is showu in Fig;urv 5. Thv fl<_w is initializ_,d

as a _mif'.rm flow at freesu'vam conditions, and tan sin-

gh' grid cych's (no muliigrid) aro onlptoyed to smooth

the initialization prior to thv initiation of the nlulti-

grid [tvration proeeduro. A total t'vsidual reduction of

aplWOXintatvly .3 ordors of magnitudv is achivvvd ovor

500 multigrid cycles. COl'lvergen(:e in the lift coetti-

ciont is obtained in as little as 200 multigrid cycles

for Ihis cas_,, although all cases aro run a minimum of

.3[)[) nmLtigri([ cvclos as a consvrvativv cotlvot'g(qlt'v cri-

rotion. This convvre;on('e behavior is reprosentative of

thv majority ,_t' cases run. with somo of the high Math

number and high ('r. cases involving larger rogions of

soparation roquiring tip to 800 to 1000 multigrid cycles.

A flow solution oil the baselin(' grid roquiros 2.8 Gbvtcs

of tn,'m_,rv and a total of 2.6 hours of wall clock time

(fl)r 500 muttigrid cycles) on a cluster of comm<_dity

components using 16 Pentium IV 1.7 GHz processors

comnnmicating through 100 Mbit Ethernet. This case

was also run on 4 DE(' Alpha processors, roquiring 2.4

Gbytos of memor.v and 8 hours of wall clock timo. This

case was also benchmarked otl 64 processors (,I00MHz)

of an SGI Origin 2000. requiring 3 Gbytes of memory

and 45 minutes of wall clock time. The momory re-

quit'olnVnts aro iiMel)endont _)f the spvcific hardwar,'

and arv _mlv a functiotl of tho number of partitions

used in the, calculations.

Tho cases using the 3 million point grid woro run

on a chtstor _,f g DEC Alpha prot>ssors comnmnicat-

ing through It){) Mbit Ethernvt and roquired approx-

imatoly S hours of wall clock time and t.2 Gbytos of

mvmory.

Th,' 13 million p_Jinr xt'id casos w,,rc run on an S(;I

()rigin 2000. using 128 processors and roquired I hours

of wall clock time and 27 Gbyws of memory. A descrip-

tion of the three grids emph)yed and the associated

/'()lllptltatioll;ll rv([uirOlllOlltg oil various hardwar(' plat-

forms is givvn ill Tabh' 1.

Fig. 2 Baseline Grid and Computed Pressure

Contours at Mach=0.75, Cr = 0.6, Re ---- 3 mil-

lion

i o_" I 0

AME[tIUAN INSI'ITI" rE OF ,kF.RONAUTIUS ANI') ASTRONAUTICS [_APER 2002 I)83S

Fig. 3 Computed Surface Oil Flow Pat-

tern in Wing Root Area on Baseline Grid fi)r

Math=0.75, Cr = 0.6, Re = 3 million

_b

2

• i i i i i L

Nlllllber _q ('vde_,

Fig. 5 Density Residual arid Lift Coefficient

Convergence History as a Function of Multigrid

Cycles on Baseline Grid for Mach----0.75, ('L =

0.6, Re = 3 million

11

1

0.9

0.8

07

06

05

03

04

02

-- Lower Surface

Upper Surface

\\'\_ /

I I i i I I I i I I i i

025 05

XlC

iiIilll

0 75

Fig. 4 Computed y+ on wing surface at spanbreak section on baseline grid for Mach----0.75,

C:. : 0.6, Re : 3 million

Results

The workshop test cases compris('(t two rt,quired

cases and two optional cases. These cases are described

in "Fable 2. For all cases the Fleynol(ts numl)er is 3 mil-

lion. The first test ease is a singh, p_int at Math

0.75 and C:, = 0.5. The second test case involves the

computation of the drag polar at Math=0.75 using in-

ci(lencos from -3.0 to +2.0 degrees in increments of 1

degree. Th(' optional Casos 3 and 4 involve a matrix

of Math and ('t. valu(,s in order to ('onip_lto drag rise

curves. Since an automated appr()ach for computing

fixed Cr cases has not been implemented, a complet('

drag polar for each Math number was c'oml)ut('d fl)r

(!ases 3 and 4. For the baselin(' grid. the incidence for

the prescribed lift value was then int('rl)olated from

the drag t)olar using a cubic spline fit. and the flow

was recomputed at this prescribed incidencv. The fi-

nal force coefficients wero then interpolated from the

values computed in this ('ase to the pr(,scribed lift vai-

llCS. which ar(" very close to the last ('()In.l}uted case.

D_r the 3 million point grid. the force c()(4ficient val-

ues at the prescribed lift conditions wore interpolated

directly fi'om th(, 6 integer degree cases within each

drag polar.

5 ()F t0

.\MEIH('AN INSI'II'ITE OF ,\EI(ONAI'FI('S \ND ,\STItON:\ViI_'S [L\PER 2[)()]2-087{S

Table 2 Definition of Required and Optional

Cases for Drag Prediction Workshop

Case Descript ion

Case I ( tl eq u i t'('d )

Single l)oint

Case 2 (Ih,quirod)

Drag Polar

Case 3 (()pti<mal)

Constant ('t

Math Sw('ep

Case I (()ptional)

Drag Rise Curw, s

Math = 0.73. C-'t = 0.500

_lach = 0.73

+_ 3". 2". I",O °. 1".2 _'

._lach = .50..6().. 70.. 75_ 76_ 77..78..80

Cr = 0.300

lach = .50,. 60_ 70.. 75.. 76.. 77.. 78,. 80

Cr = 0. 100.0.500.0.600,

All cases wer(' computed using the baseline grid

(I.6 milli<m points), and the nu'dium grid (3 million

points). Only the required cases were computed using

the finest grid (13 million points) due to time con-

straints. Table 3 depicts the results obtained for Case

1 with the three different grids. Th(" drag is seen to be

('(mq)uted accurately by all thr('(' grids, although tlwr('

is a 10.6 count variation between the 3 grids. How-

ever. the incidence at which the prescribed C,+ = 0.3 is

achieved is up to 0.6 degrees lower than that observed

experimentally. This effect ix more evident in the ("L

v('rstts [n('i(h,nc(, plot of Figure 6, where the computed

lift values are consistently higher than the experimen-

tal values. Sittce this discrepancy increases with the

higher resohttion grids, it cannot be attributed to a

lack of grit[ res<)httion. Tit<' slope of th(' computed

lift. curve is about 5c/( higher than the experimentally

d<,termitled slope, attd is [argety unalfected by grid res-

olution.

Figure 7 provides a comparison of computed surface

pressure coefficients with experimental values at the

experimentally prescribed C',, of 0.G (where _t1<' effects

are more dramatic than at CL 0.5) as well as at the

experimentally prescribed incidence of 0.93 degrees, at

the 10.9 _ span location. When the experimental inci-

dence value is matched, the computed shock location

is aft ()f tit<' (-xp('rim('ntal values, and th(, computed

lift ix higher than the experimental value, whih, at the

prescribed lift condition, the shock is further forward

and the suction peak is lower than the experimental

val tit'S.

This bias in lift versus incidence was obsm'vod for a

majority of the nunwrical solutions submitted to the

workshop. :_ and thus might b,' attributed to a model

g_,,onletry eP[ect or a wind tunnel correction eIfect, al-

though all exact catlse has nol he('n determin+,d. \Vhen

plotted as a drag polar. Cr versus CD as shown in

Figure 8, the results compare favorably with expori-

n|ental data. Although the (trag polar was computed

independently by both authors using tile baseline grid.

the results of both sets of computations were identical

(as expected) and thus only one set of computations

is shown for the baseline grid. The ('<mH)utatiomd r('-

sults on this grid compar,, v(,ry well with exlwriment

in the mid-range (near Cr. = 0.3), while a slight over-

prediction of drag is observed for low lift vahtes, which

decreases as the grid is refined.

This behavior suggests an und('r-prediction ()f in-

duced drag, possibly due to inadequate grid resolution

in the tip r(,gion or ds(,wlwr< The absolute drag h'v-

('Is have heen found to b(' sensitive to the d('gr('(' ()f

grid refinement at the blunt trailing edge of the wing.

The drag h'vel is reduced by I count.s wh('n going from

the 1.6 million point grid. which has 5 points on the

trailing edge, to the 3 million point grid. which has

17 p()ints oil the trailing edge. Internal studies bv th('

second author using structured grids hay<, shown that

up to 33 points on the bhmt trailing edge are required

before the drag does not decrease any further. In the

current grid gelleratioll ('llvir()lUll('llt. 0.lid without the

aid of adaptive meshing techniques, the gen(.,ration of

highly relined trailing edge unstructured meshes has

be('r! f(.)und to be problenmtic, thus limiting our study

in this ai'(,a.

Figure 9 provides an estimate of the induced drag

factor, det('rmined experimentally and computation-

ally on tit(., three meshes.

Table 3 Results for Case 1; Experimental

Values I:ONERA, 2:NLR, 3:DRA; Gridl': Per-

formed by first author, Grid1+: Performed by

second author. Experimental data and 3 M

point grid results are interpolated to specified

Ct condition along drag polar.

Case

Experiment I

Experiment e

Experinmnt :_

Gridl ( 1.63lets)*

Gridl ( 1.6+llpts) +

Grid2(3.0Mpts)

Grid3( la3Ipts)

CL

0.5000

0.5000

0.5000

0.500,1

0 A995

0.5000

(}.5(}03

(_ CD

+.192 ° 0.02896

_-.15W 0.02889

+.179" 0.02793

-.241 '_ 0.02921

-.248'-' 0.02899

-.117 '_ 0.02857

.367" ().02815

C;t/-

-.13(}1

-.1260

-.1371

-. 13,19

-. 1348

-.1613

-.1657

6 OF 10

AMERICAN ]NNTITITE t')F AERONAI'TI(% AND ,\STRONAUTICS PAPER 2002 [),v,3S

0.8 0.8

0.7

0.6

0.5

(S04

0.3

0.2

0.1

0.0,

/_-(]D O M=.75, NLR-HST

__/_1 [] M=.75, ONERA-S2MA

o u_-.7_ORA-Sx8_ 1.6M Grid

13.1M Grid

1.6M Grid

----C-_- 3.0M Grid

.... I .... l .... l .... i .... ' ' ' '-2 -1 0 1 2

Alpha

Fig. 6 Comparison of Computed Lift as a

flmction of Incidence for Three Different Grids

versus Experimental Results

07

0.6

0.5

¢'J_0.4

03

0.2

0.1

0.0

_..t

_#- [No,.:,,io,,,,,oo.,d.,-,,,,,.w1AI_ I pr_-scribed b'Ip pattern; I

Lc_c__od,==,._,,_,turbulentL--o2,_.;2o

,_/ a M=.75,ONERA-S2MA_J,j/ O M=.75, DRA*8x8

^_r_ ---C,----- 1.6M Gridv• _ 13.1M Grid

_ 1.6M Grid

--- _,"-_ 3,0M Grid,01 i , , , I , , , , I , _ , , I

0.020 0.030 0.040 0.050

Co

Fig. 8 Comparison of Computed versus Exper-

imental Drag Polar for N'Iach----0.75 using Three

Different Grids

15

-05

n

0

05

NSU3D: Alpha = 0.93NSU3D: CL= 0.6

r_ll_._ ' • Experiment: Alpha=O,93

i ; I I i I I I I i I i i I .... I0 25 05 075

X/C

Fig. 7 Comparison of Computed Surface Pres-

sure Coefficients at Prescribed Lift and Pre-

scribed Incidence versus Experimental Values

for Baseline Grid at 40.9 % span location

0.6

[] M=.75, ONERA-S2MAj_7 O M=,75, DRA-Sx8

._z 7 _ 1.6M Grid

-_ 13.1M Grid

._')_ .... + 3.0_M Gddl,_I i _ I J L i I i

0-0-20 0.030 0.040 0.050

Co

0.5

0,4

0.3

0.2

0.1

0.0

Fig. 9 Comparison of Computed versus Exper-imental Induced Drag Factor for Math----0.75

using Three Different Grids

7 OF 10

._[E['[I('AN INb;I'I'TI'FE OF .\ERONAI-TI('t'-; AND ,_TR()NA[ ['[('N PAPER '200:2 (),S;{_

0.8

0.7

0,6

0.5

o0.4

0.3

0.2

0.1

00%

-4F'Coun"

_{ _l_ O "=.75, NLR-HST

_iI [] M=.75,ONERA-$2MA;; _a_- ,.;,;,G&..-,-

O o _ 13.1M Grid

_---- 3.0M Gdd

,,l:g,I, ,I,,,I , li ii|_ , , [, ; _ I i _ _

0,020 0.024 2 0.028 0.032Cop= C o - C,/(x AR)

Fig. 10 Comparison of Computed ver-

sus Experimental Idealized Profile Drag at

?viach=0.75 using Three Different Grids

-0"08 f O M=.75, NLR-HST !i

O M=.75, ONERA-S2MA

-0.10 O M=.75, DRA-ax81.6M Grid

13.1M Grid

--<2,---- 3.0M Grid on I

OOOOOOOOO

= .==oO.,ooooo_ I0

_0.14ooooo:S?:o>OOOOj / I-0.t6

.,I .... I, ,,1 .... I , ,1 ilJl I i I I I

0 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8. . , i , i

CL

Fig. 11 Comparison of Computed versus Ex-

perimental Pitching Moment for Mach=0.75

using Three Different Grids

For C'I-' up to ahout 0.36. when the flow is mostly

attached, induced drag is underpredicted by approxi-

mately 10 _X. as determined by comparing the slopes

of the computational and experimental curves (using

a linear ctlrve fit) in this wgion. Grid refinement ap-

pears to have litth' effect on the induced drag in this

region. At the higher lift values, the 3 million point

grid yields higher Cr and lower Cr) vahtes, which is

attributed to a slight delay in the amount of predicted

8oF I0

AMERI{'AN [NSTIrFI'TE (H" ;\I<I{ONAI I'I('S AND ;_;TRONAIT1CS PAPER 2002-0s:},s

flow separation. Results for the 13 million point grid

are not shown, since a fully converged solution t'ould

not be obtained at the highest incidonce. On the of her

hand. il should be noted tha.t the wind tllnnel ('xper-

iments used a boundary layer trip at 15'/_ and 25(X

chord on the tipper and lower surfaces, while all cal

cttlations were performed in a fully turl)ulent mode.

Examination of the generated eddy viscosity h.wels in

the calculations reveals appreciabh, levels b,'gi,ming

])etw,'en 5'X to 7(J chord. The exact influence of tran

sition h>cation on overall computed force coefficients

has not t)_,en quantified and requires further study.

Figure l(/ shows the idealized profile drag I_ which

is defined by' the tbrmula:

CDt, = CD - Cr-'/(rc.4R) (1)

where AR is the aspect ratio. Plotting Cr>r generally

results in a more compact representation of the data,

allowing more expanded scales. It also highlights the

characteristics at higher Cr, where the drag polar be-

comes non-parabolic due to wave drag and separation.

In the non-parabolic region, the error in drag is rela-

tively large at a constant Cr.

The pitching moment is plotted as a function of Cr

in Figure 11 for all three grids versus experimental

vahtes. The pit('hing niotnent is substan.tially mM('t'-

predicted with larger discrepancies observed for the re-

fined grids. This is likely a result of the over-prediction

of lift as a [hnction of incid_,nce, its mentioned earlier

and illustrated in Figure 6. Because the computed

shock location and suction peaks do not line up with

experimental values, the predicted pitching moments

can not be expected to be in good agreement with ex-

perimental values.

Figure 12 depicts the drag rise curves obtained for

Cases 3 and 4 on the baseline grid and the first refined

grid (a million points). Drag values are obtained at

four different constant Ct, values for a rang,, of Math

ntunbers. Drag values are predicted reas,mal)ly well

except at the highest lift and Math number conditions.

There appears to be no improvement in this area with

increased grid resolution, which suggests issues such as

transition and turbulence modeling may account for

these discrepancies. However. since the two grids have

comparable rosolution in various areas of the dolnain.

grid resolution issues still cannot be ruled out at this

stage.

The results obtained for Cases 3 and I can also be

plotted at constant Math number, as shown in the

drag polar plots of Figure 13. The plots show simi-

lar trends, with the drag being slightly, ow,rpredicted

at low lift values on the coarser grid and with the re-

fined grid achieving better agreement in these regions.

For the higher Math numbers, the drag is substantially

tmderpredicted at the higher lift values. These discrep-

ancies at the higher Math numbers and lift conditions

pointto anunder-predictionoftheextentof the S_'l)- 0.056

arated regions of flow in the numerical simulatkns

The comparison of idealized profile drag in Figure l() 0.052

also sugg_'sts that the drag due to flow sq)aration is

not predicted accurately at the higher lift conditions. 0.048

However. the character of the turves also suggest that0.044

the error may be due as well to the CL offset (shown

in Figure 6). Additional information c(mcerning the 0.040ta

regions of flow separation found in tlw wind tunnel O

would be needed to tll()I'(' accurately quantify the na 0.036

ture of the error.

The above rt,sults indicate that the current Llnstl'llC- 0.032',

tured mesh Navier-Stokes solver achieves a reasonably 0.028:

good predictive ability for the force coefficients on the'

basdine grid m'('r the majority of tlw flow conditions 0.024 I

considered. The overall agreement, particularly at t he

how lift values, is improved with added grid resolution, 0.0; 50while the more extreme flow conditions which incur

[argor amounts of s('t)aration arc more difficult t(> pr('-

dict accurately. On th(' other hand. th(' obscrv(,d bias

between computation and experiment in the lift versus

incidence values has an adverse affect on the prediction

of pitching moment. While the source of this bias is

not fully understood, it was observed fur a ma.jority of

independent numerical simulations tmdertaken as part

of the subject workshop a and can likely be attributed 0.8

to geometrical differences or wind tunnel corrt'ctions.

The resJlts presented in this paper involve a larg(.' 0.7

number of individual steady-state cases. For example

on the baseline grid. a total of 72 individual cases were 0.6

('<mH)ttt<,d. as shown in Figure 14. t,) ('1lab[(' th(, con-

struction of Figures 8_ 12, and 13. The majority of 9. 5these cases were run from freestream initial conditions

for 500 multigrid cycles, while s(.,veral cases particu- ¢_ 0.4

larly it). the lfigh Math nu.mb(,r anti high lift r(,gions

were run 800 to I000 cych,s to obtaill fldlv c(mv('rged

results. The baseline cases (.3(}(} multigrid c.vch's) re- 0.3

quired approximately 2.6 hours of wall clock tim(' on a

cluster of 16 commodity PC processors. Thin enabled 0.2

the entire set of 72 cast's t,> b<' comph't('d within a

period of (m(' week. This <,x(wcise illustrat(,s th(' possi- 0.1

bility of performing a large number of paramt.'t('r runs,

as is typically required in a design exercise, with a

state-of-the-art unstructured solver on relatively inex-

p(,nsive paralM hardwar('.

C L 3M Grid 1,6M Grid Exp<>

0 30 + - - -e - - o

040 _ 41

0 50 • •

0 60 _ .... O - - O

@

Notes: /

1) Wind tunnel data usa prescribed BL trip pattern. ,'

2) CFO data are fully turbulent //

3) On fine grid, even Ct data interpolated from I/

rz-sweep data using cubic splina, //

CL= .50 _1i_ A', ¢.

' C L = .40 _lll-l-l[....... Z _.7__Z2 ---- 27 :L

_- c_=.ao, , , I , i i_', , , , I , , i i I i L k (1), , , I

0.55 0.60 0.65 0.70 0.75 0.80Math

Fig. 12 Comparison of Computed versus Ex-

perimental Drag Rise Curves Three Different

CL values on Two Different Grids

c>:_ :U

q F 10 counts Q9 _

-_ .... M=.60, 3.0M Grid

5,_ + M_o, a.OMG.dr_J,,_ -e- - M=.60, 1.6M Grid

- -_- - M=.80,1,6M G_d___,1_' .... I ,, ,, I _ _, , I J ,, , I .... I

).0_0 0.030 0'040-oC 0.050 0.060 0.070

Fig. 13 Comparison of Computed versus

Experimental Drag Polars for Nlach=0.6 and

Mach:0.8 on Two Different Grids

90F I0

,-\MEP, I('AN ]NSTITI'TF OF :\ER{)NAI'TI(% AND ,\STIIONAI'TI('S ])APF21( 2()02-08:]S

0.8

0.7

0.6

05

0,4

0.3

0.2

-_F 10 c°unts _j/..,_O

/

----O--- M=0.50, 1.6M Gr¢l

------o----- M=0.60, 1.6M Gr_

_-- M--0.70, 1.6M Gr_

M=0.75, 1.6M Grid----c,--- M=0.76, 1.6M Grid

_-- M=0,77, 1.6M Grid

M=0.78, 1.6M Grid

_- M=0.80, 1,6M Grid

0.10.(_20 ' ' ' | .... , .... , , , ,0.030 0.040 0.050 0.060

Co

Fig. 14 Depiction of All 72 Individual Cases

rt|n on Baseline Grid Plotted in Drag Polar

Fo r m a t

Conclusions

A stat('-(ffth('-art unstructured multigrid Navier-

Stokes solver has d_'monstrated good drag predictiw,

ability for a wing-body configuration in the transonic

regime. Acceptable accuracy has been achieved on rel-

alivoly coarse meshes of the order of several million

grid points, whih, intpr()v(,d accuracy has been demon-

strated with increased grid resolution. Grid resolution

remains an important issue, and considerable exper-

tise is r(,quirod in specifying the distribution of grid

resolution in order to achieve a good predictive ability

without r(,sorting to (,xtr(,nwly larg(' mesh sizes. These

issu('s can be resoh, ed to somv degree by the use of au-

tomat it' grid adaptati(m procedures, which are planned

['or future work. The predictive ability of the numcri-

('a[ scheme was found to degrade for flow conditions

involving larger amounts ,)f flow si,paration. Slight

convergence degradation was observed on two of the

grids for the ca_s('s involving increased flow separation.

whih.' a fully conv(,rg_,d resuh, could not be obtained

,)n thv finest grid (13 million points) for the high('st

lift ('as_, at a Math number of 0.75. The curr('nt re-

suits utiliz('d the Spa[art-Alhnaras turbulence model

ox('lusively, and the effect of other turbulence mod-

els in this regime deserves additional consideration.

Tlw rapid converg('n('e of th,' multigrid scheme cou-

ph'(t with the paralM imph'mmltation on commodity

n_'tworked computer ('lusters has been shown to pro-

duce a useful design toot with quick turnaround time.

"G. I(edeker. I)I,R 1;I wing body couliguraLion, qeehuicat

report, August 199,1. :\(;.\R I) lleport ,\R 303. Vol II.

-_Xl. Hemseh. Statistical analysis of CI:I)solut.i(,us ['rein _.he

I)rag lh't_dietion \Vorkshop. .\1\ \ Paper 2002 0812..Jalmary

2002.

IlL J. 3,[avriplis and V. Venkat_krishnan. A uuilied nlulti

grid solver ['or the Na_ier-Stokes equatious ,:,n mixed elemeut

meshes. Intcrnltlzot_al aourncd ]o," (7omputut_onul l"h.d l)tlmlm-

ic_, R:217 263, 19,97.

_1)..I. Ma_riplis aud S. Pirzadeh. t,araescah, parallel tm

structured mesh computations fi)r 31) high-lil't analysis. \1\.\

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10 OF 10

,\MERIt'AN ]NSTITI'TE [)F AERONAUTICS AND ,\SI'RONAUFIt'S }>APEI{ '.2)002 (),X3 '4