A93 *47397 AIAA-93-349541

NAVIER-STOKES PREDICTION OF A DELTA WING IN ROLL WITH VORTEX BREAKDOWN

Neal hI Chaderjian* and Lewis B. ScliiEt

N4S.A .4mcs Research Center. Moffett Field. C.4 994035

Abstract

The three-dimensional, Reynolds-averaged, Navier-

Stolces (RAKS) equations are used to numerically simu1at)e

vortical flow about a 65 degree sweep delta wing. Sub-

sonic turbulent flow computations are presented for this

dclta wing at 30 degrees angle of attack and static roll an-

glcs up to 4% degrees. This work is part of an on going

effort to validate the R.4NS approach for predicting high-

incidence vortical flows. with the eventual application to

wing rock. The flow is unsteady and includes spiral-type

vortex breakdown. The breakdown positions, mean surface

pressures, rolling moments, normal forces, and streamwise

ccnter-of-pressure locations compare reasonably well with

experiment. In some cases. the primary vortex suction

peaks are significantly underpredicted due to grid coarse-

ncss. Nevert,heless. the computations are able to predict

the sanit. nonlinear variation of rolling moment with roll

angle that appcared in the experiment,. This nonlinear-

ity includes regions of local static roll imtability, which is

attributed to vortex breakdown.

Introduction

The High-speed Civil Transport (HSCT), National

Aerospace Plane (NASP), and modern tactical fighters

all operate at high-angle-of-attack flight conditions. The

HSCT and NASP have slender bodies and highly swept

wings which are designed for high speed flight. While land-

ing. their low airspeed will require a large angle of attack

to maintain a proper glide slope. Fighters also operate at

high anglcs of attack to achieve greater maneuveraljility

and agility performance. If the angle of attack is suffi-

cient,l~- high. large regions of vortical flow can form and

* Research Scientist, Computational Aerosciences

Branch. Senior Member AIAA.

tSpecial Assistant for High Alpha Technology, Fluid

Dynamics Division. Associate Fellow AIAA.

Copyright @ 1993 by the American Institute of Aeronau-

tics and Astronautics, Inc. N o copyright is asserted in the

United States under Title 17, U.S. Code. The U.S. Govern-

ment has a royalty-free license to exercise all rights under

the copyright claimed herein for Governmental purposes.

All other rights are reserved by the copyright owner.

interact with the aircraft's nmt,inn. rc>sulting ill wing r o ~ l i . ~

a sustained periodic motion in roll and yaw.

The high-arigle-of-attack flight regilric is 7-er)- com-

plex and may inclurle streamwise and crcwflon- parati ti oil.

vortex brealidon-11. autl vortex asymmetry. These pl1e1ion1-

ena are inherently nonlinear. often time dependent. and

can be very sensitire to small disturbancw in tlir, flon-.

Experimental investigations aimed at understanding wing

rock have been carried out on delta-wing geometries ~vitli

a single-degree-of-freedom in roll. These simplified geome-

tries exhibit the pertinent flow physics associated with

wing rock. Experimental data typically consist of flow 3.i-

sualization and time-dependent forces and ~nonients.~-';

More recentl>-. Hanff and his colleag~ies7-'I 11aw carried

out higll-qualit5 experimental investigations of ii large+ scale delta-wing/hody model undergoing large-aniplitutlc

roll motions, and include measllrements of time-dependent

surface pressures. ,-2rena ant1 Xelsonl' haw also m e a s ~ ~ r r d

time-dependent surface pressures on a highly swept delta

wing. These experiments provide additional dctail to help

understand wing rock; and provide a timely opportlmit~-

to validate computational fluid dynamic (CFD) ~llethuds.

Computational fluid dynamics is just beginning to

be utilized to investigate wing rock anti can provide more

flow-field information than is otherwise possible with ex-

periment alone. However, these computations require large

amounts of computer time. In order to reduce the com-

putational cost, some investigators hare simplified the

problem by using the conical and three-dimensional Eu- ler equations.13-'I' The conical form reth~ces the t,lirex-

dimensional physical flow int,o a txvo-dililel-isioiial coiiipll-

tation using similarit!- principles. The conical approach

can not produce vortex breakdown which usually accompa-

nies wing rock. The inviscid approximation further limits

the flow sin~ulation to a single primary \wries that fornii

at sharp leading edges. Thc inviscid approxim;rtioli call-

not account for the vortices that are linown to forni on

smooth surfaces. e.g.. forebody. secondary. ant1 tcrtiar!.

vortices. Moreover, the primary vortex has a question-

able core strength which can significantly affect the vortex

breakdown mechanism and location. Murman" has re-

cently shown that there are significant viscous effects on

the F-18 leading edge extension JLEX), even though it ha5

a sharp leading edge. An improvement in the prediction of

the LEX suction peak was obtained by refining the grid in the boundary layer. The approach adopted here is t,o solve

the t1lrc.c.-tlirnensi(j11i-tl. Reynolds-averaged. Navier-Stokcs

IRASS) cqliations. Even though this requires more com- p~itcr time. tlic RXSS equations contain all of the relevant

flow phj-sics that are applicable to a complete aircraft ge- onictry.

Thcre i5 an effort at S.4S.S -4nles Research Center to validate the R l l N S approach to simulate high-incidence vortical flows about delta wings. with the eventual applica- tion to wing rock. Chaderjian18 recently used the Kavier-

Stokes Simulation code (SSS) to simulate time-dependent. \-ortical flow about R 65' sweep delta wing undergoing large-amplitudt= roll oscillations. This case was investi-

gated esperimentally by Hanff and his colleague^.^-'^ Fig- ure 1 summarizes the computational and experimental rolling niornent coefficients obtained at 15 degrees angle of attack, and static roll angles up through 42 degrees. .?llso shown in Fig. 1 is a large amplitude (40 degrees), high rate of roll (7 Hz), forced dynamic motion. .4t this

anglr of attack, no vortex breakdown was observed in the experiment or the computations. The static and dynamic

computatio~ls are in excellent agreement with the exper-

imental results. In addition, computed surface pressures,

~iornlal forces, and streamwise center-of-pressure locations were also in excellent agreement with the experiment.

,,L,

.04 + Static Exp, Ref. 9 o Static CFD, NSS

.03 - Dyn Exp. Ref. 9

.02

- .o 1 U 0

-.O 1 -.02 -.03 -.04 -.05

-50 -40 -30 -20 -10 0 10 20 30 40 50

Fig. 1 Static and dynamic rolling moment coefficients

from Ref. 18. = 0.27, a = 15"> Re = 3.67 million,

and k = 0.14 for the dynamic case.

This paper builds upon the previous study and in- yestigates flows at 30 degrees angle of attack. xhere vortex 1~rralidow-11 does occur. Experimental1O static and dynamic rolling moment coefficients obtained at 30 degrees angle of attacli are shown in Fig. 2. Kotice the nonlinear behavior

of the static rolling moment coefficient for roll angles of -10" < Q < 10". This is very different from the near linear behavior seen at 15 degrees angle of attack. -4lso notice

that the dynamic case does not follow the static rolling

moment curve. This discrepancy has been attributed''

to time lags occurring in the vortex breakdown locations. Thus. the goal of this study is to verify that the RAXS approach can predict the nonlinearity of the static rolling

moment curve shown in Fig. 2. The effects of large time

lags observed in the dj-na~nic case mill be inrwtigated conl- pntat,ionall?- at a later timc.

Static

Fig. 2 Static and dynamic experimental rolling moment

coefficients from Ref. 10. l f m = 0.27, a = 30°, Rt= = 3.67

million, and k = 0.2 for the dynamic case.

A desc~iption of the governing equations. turbulence

model. numerical algorithm, colnputational grid. and nu-

merical boundary conditions is given below. This is fol-

lowed by a discussion of the compl~tational reiults and

concluding remarks.

Numerical Approach

Governing Equations

For the high Reynolds number flows considered in this paper, the thin-layer approximation'g to the RA4NS equations is employed together with body-fitted curvilin- ear coordinates. Assuming a general coordinate transfor-

mation from physical space (x,y,z) to computational space ([,v,C), and that viscous terms are only important in the body-normal (<-coordinate) direction, the governing equa- tions can be expressed in the following strong-conservation-

law form:''

where Q is the vector of conserved dependent variahlcs, E. A A

F, and G' are the inviscid flux rectors in the (, 7 ; itnd ( A

directions, respectively. and S is the thin-layer viscous flus vector. These equations have been ~lo~idilrlensioilalizetl by

C , the wing root chord, arid (L,; the free5tream speed of

sound. The coordinate transformation metrics allow for

a general moving and deforming grid system. The perfect gas law, Sutherland's viscosity law, and a turbulence lnodel

completes the RANS system of equations. The turbulence

~rlodcl will 1jv clrscril~ctl in the nest section. Further clct,ails

aljol~t the o;nverning equations can be fo1111c1 in Ref. 19.

Turbulence Model

The higli-incidence. higli-Reynolds-nunlhrr flows con-

sidered in this paper have significant crossflow sc3para-

tion on t l ~ r leeward side of the delta wing and there-

fore rcquirei a suitable t~trbulence model. The Bi~ldwili-

Lonlas algebraic nod el^^ together with the Degani-Schiff

~ n o d i f i c a t i o i l ~ ~ - ~ ~ is a n efficient isotropic eddy viscosity

rrlodel that properly accounts for the crossflow separation

in the prwrncc of strong vorticcs above the wing.

T h c Baldwin-Lornax model often chooses a length

scale associated with the vortical flow structure outside

the viscous boundary layer. This results in a n eddy vis-

cosity which can he one or two orders of magnitude too

large. and eff'ectively suppresses all but the most dominant

vortex structure. The modified model of Degani and Schiff

restricts the choice of length scale to the boundary layer

regioii in a ratiorlal ma~lner . and therefore gives correct

values for the eddy viscosity.

A11 viscous c o m p ~ ~ t a t i o n s presented in this paper as-

silnlr: f d l y tiirhident flow l~cginr~ing at the apex of the delta

wil~g. i.e.. transitiorial effects are ignored.

Numerical Algorithm

There is a choice of two implicit, approxinlately-

fx torcd . central clifferencr algorithms in the KSS code

to integrate Ey. (1). T h e first algorithnl is t l ~ e Beanl-

Warming a l g ~ r i t h m , ' ~ and the second is a diagonal ver-

sion of thc Beam-LVarrrling algorithni due to Pullianl and

C I i a u ~ s e c . ~ ~ Both algorithms use second-order-accurate

central differencing throughout, with added artificial dissi-

pation t o darnp out high frequency errors. Explicit artifi-

cial dis~ipat ion terms consist of blended second-order and

fourth-order differences. The terms have been modified to

reduce thc dissipation within the bolmdary laJ-er.Z"he

Beam-JYarnling algorithm uses second-order implicit arti-

ficial dissipation and thereforcx requires the solution of a

5 x 5 b l ~ c l i tridiagonal system of equations. The diagonal

algorithm uses an iniplicit dissipation that is idelltical to

the explicit seco~id/fortl~-order dissipation. and therefore

requires the solution of scalar pent adiagonal equations.

The diagonal algorithm requires fewer computational op-

erations per time s t ~ p than the Beam-Warming algorithm.

and is therefore more computationally efficient. However.

Levyz6 has shown that the diagonal algorithm has a n in-

herent directional hins. If the angle of attack is too large.

this ma?- lead to a no~iphysical vortex asymmetry in the

flow. The choice of algorithm is case dependent. Both

algoritlims use Eulcr implicit timc tliff~sc~~icillg. n-hicli pro-

vides first-order tinie ;LCCII~B(-J- . Both algcritl~nis in the

XSS code treat the viscous tcrnis in ail csplicit maliner.

The KSS codc can also treat conlplrs gconictries.

such as a complete F-16-4 fighter aircraft." Thi.; is accom-

plished b y using a zonal grid approacl~. The zonal gritl ca-

pal~ility will facilitate the cveutual application of this cmdr

to a maneuvering aircraft e q x x i e ~ ~ c i n g \x-ing rock. -4 zollal

grid approach can also h e used to reduct, ill-core inelnosy

requirements on computers with ext,endetl s~conclary rntm1-

ory. A brief sulnnlary of the zonal interface procecl~~rr is

given in the numerical boundary condition sc:ct,icm.

Computational Grid

Extensive wind-tunnel investigations were carried

out by Hanff and his colleague^^-^^ for the 65' sweep delta

wing shoxvn in Fig. 3. This wing is symmetric frolli tol)

t o bottom and side to side. with double bevels running

along all its edges. A fuselage/sting houses instrumen-

tation and pro\-ides for a olie-tlcsrc-1.-of-frcecloni in roll.

Time-dependent surfact. presslxrei arc, o h t a i ~ ~ e t l acrois the

span of the wing at thr. 73% rnot chord positiol~.

b = 22.83 in.

Fig. 3 Thrw-vien dra~virig of the n-intl-tunnel model.

A three-tlil~iensiorinl hyperl~olic grit1 gcmeratos'%\-as

used to gcnerate a spherical gritl for this clrlta n-inq ge-

ometry. A perspecti\-e vicv of the viing surface gricl ant1 a

portion of tlic s t i~ ig is i11on.n in Fig. 4. A4 porrioli of the vis-

cous gricl clustering liormal t o thc n-ing at tllcl trailing edge

is also showli ill this figure. The ~l111c.rical axis t.steuds up-

wind from the wing apes. The far-field l~m~~lt lar ic . ; arc not

shown in the figure, but extend two root chord lengths lip-

wind and downwind of the wing l~ody , and five root chord

lengths in the body-normal direction. The grid consists of

67 points in the streamwise direction, 208 points in the cir-

cumferential direction, and 49 points in the body-normal

direction. totaling about 700,000 grid points. In order to

resolve the leeward-side vortices, there are more grid points

on the leeward side of the wing (in the circumferential di-

rection) than on the windward side.

Fig. 3 Perspective view of the computational grid.

There are two Cray C-90 supercomputers and one

Cray Y-YIP superco~nputer available at NASA Anles Re-

search Center. Each of these machines is configured with

different internal memory and extended secondary mem-

ory (SSD) capacities. In order to take full advantage of

each machine, the single delta wing grid was split into four

zones in the streamwise direction. This reduced the inter-

nal memory requirements so that each machine could be

fully utilized.

Numerical Boundary Conditions

The no-slip condition (zero velocity relative to a solid

surface) is imposed on the wing and fuselage/sting surfaces

while density and pressure are found by extrapolation.

The total energy per unit volume is then computed from

the perfect gas law. Uniform flow is imposed at the far

field. while a zero-gradient condition is used at the outflow

boundary. Flow variables.are averaged across the wake

cut, which extends from the wing trailing edge downwind

to the outflow boundary. Boundary conditions are imposed

on the spherical axis by averaging flow variables that are

one point off the axis, and located circumferentially around

the axis.

Zonal boundary conditions are updated sequentially

with the most recent data available. .Adjacent zones over-

lap by three grid cells so that centrally differenced fourth-

order dissipation can be used across zonal bourldaries.

Since the zonal grids were originally constructed from a

single grid, zonal interfaces have coincident surfaces with

identical grid points. Data transfer from one grid to an-

other is accomplished by direct injection. A more complete

description of the zonal interface bou~iclary conditions is

given in Ref. 27.

Results

The time-dependent, RAYS equations are ilu~rit~ri-

cally integrated to simulate viscous flow al~out the 63 dc-

gree sweep delta wing shown in Fig. 3. Each computa-

tion had a fre~streain Mach nuinher JI, = 0.27. angle

of attack a = 30°, and Reynolds numbrr l~a.;ed on thr

wing root chord Re=3.67 million. These flow conditions

correspond to experimental data provided by Hanff and

his colleagues.7-" Six static (fixed) roll angles are inves-

tigated, i.e., q3 = 0. -5, -7, -14, -28, and -42 degrees. The

direction of positive roll and the coordinate axis orienta-

tion are shown in Fig. 4. -4 positive roll angle corrcsponcli;

to the right wing down and the left wing up, from a pi-

lot's perspective. The flow evolution is solved ill a time-

accurate manner with a constant nondimensional time strp

of AT = 0.005. This value provided good time-accurntc rc-

sults in Ref. 15.

Initially the diagonal algorithm was used to compute

the flow about the delta wing at zero roll angle. This

resulted in a vortex asymmetry that was not present in

the experiment. Levyz6 has shown analytically that there

is a directional bias in the diagonal algoritlm~. Furthcr.

he has also demonstrated computationally that this bias

can lead to spurious vortex asymmetry at sufficirntly high

angles of attack. The Beam-Warming algorithm prod~lcecl

a symmetric vortical flow, consistent with the experiinent.

and has therefore been used for all of the cases presented

below.

Instantaneous particle traces are used to visualizr thc

leeward side vortices. A planform view of these vortices are

shown in Fig. 5 for GJ = 0 and -5 clegrecs of roll. The vortes

cores are indicated in the figure. E ~ p e r i m e n t a l ~ - ~ ~ surface

presure data is available at the 75% root chord location.

which is also indicated in Fig. 5. .4t zero roll, a spiral-

type vortex breakdo~vil is indicated by a rapid espasssiosl

of the vortex diameter and the ..corliscreiv" path of thc

vortex core. At d = -5 degrees, an asyrrimetric vortex

breakdown occurs. In this case the left wing is down, re-

sulting in a reduction of the effective leading-edge sweep.

while the effective leading-edge sweep of the right ning is

increased. Consequently thc left x-ilig I-ortcs is stroi1gc.r.

than the right one and breaks down sooner. It is intewst-

ing to note that even though there is vortes asynm~etr!-.

the computed rolling rnoment for this roll angle is nearly

zero.

It can be difficult to precisely quantify the location of

vortex breakdown, because there is no univrrsally accepted

(b) 4 - -5O Fig. 5 1,-isualization of leeward-side vortices with install-

taneous particle traces. -11, = 0.27. a = 30'. Re = 3.67

n.~illion.

definition for breakdown. Breakdown is usually visualized

in an rxperirrlent by natural condensation occurring within

the vortices. or by injecting smoke into the vortex cores.

The shear layer lcaving the leading edge of thc delta \\-i~lg

rolls up and entrains the smoke. rendering the vortex visi-

blr. -4 b u l ~ l j l c - t j - ~ ~ 11rc.akdon-11 i i irrinir~cli;\ti-l~- c)i)~-ioui I ) , , -

c;lilse of the tlisprriior~ of i lnu l r tlon-wtic~;rlt~ of'Ij~c~akc!i~:\-t~.

The core v c l o ~ i t ~ will also go to zero ir t l j r (~ ; i l ;~ l i~n .~~ . C ' O ~ I I ~

putationally. n l~~~l j l> le - ty~)c . Lri~alirlr~n-11 , . ; l l r i~lw) 1i1, c,;t-il>- . .

idrntificd by siniilar tw1111i(~uc< ~15111; l l l ~ t i l l l r ; t l l ( ~ o i 1 . I ) ~ I t i -

cle trac.c2s a n d e s a n i i ~ ~ a t i o n of t11~ core ~-r ' lo( . i t~- .

Figure G shows tllc axial location of the esljc~rillici~t;rl

and computational vortex brealidow11 p o s i t i o ~ ~ s oii tlit. lcft

wing at variolw roll angles. T l lv : i i l a l o ~ o ~ ~ ~ in fo~i l~ i \ t i t )~ l

for the right wi11g can be f11jtaitte:tl 11)- l i , f l ( ~ , t i ~ ~ ~ thi> (lata

a110iit d = 0'. The c o ~ n p a r i w i ~ l>rtn-cen t11r r ~ ~ ~ t i i ~ ) i ~ r a t i i ) t ~ ; i l

and experinlcutal reslllts i i good n-hcli l ~ l t ~ a l i i l o i ~ x occnus

~lpstrearli of thc n1i~11~1ti)rcl pi15itio11. ;III(I i - on1~- f1~i;~lit:i t i~-c

x l i m l ~ r ~ ~ a l i ~ l o ~ w oc.c,i~ri 011 I aft 1 i , 111 of ~ l t c \xvi~:c. . .

It should be noted t h a t the l~rc~;rlitlon~i~ pc~-lt~c>:i i- \-('rj-

sensitive to roll angle over tlir range - 10" ( o 5 3'. n l ~ e r c

the breakdown position moves rnpitll!- froin a ryqicm \-cry

ncar the v ing apcx (nose) t o thi' wing ti.iiiling etl;c. T11~t.r

1.2 1 Left Wing

Fig. 6 Cornpalison of cor~~~)l l tecl and r q , c ~ i n i c ~ ~ ~ t a l ~ o r r i ~

hrealitlown location for differmt btatic rill1 ailglc~s. .\I, =

0.2'7. a = 30". Rc. = 3.67 million.

.. ' N C = 0.75 2,5 ( a ) @ = O 9

I

.~.. Exp. Ref. 10 . . , . .-. ,

2.0 , , . Max. NSS , .. 8 ,

.:' , ',., Mcm. NSS ,:' . o '.., I ',

.... A I , n ', ... Min.NSS ,.:" o o ' ~

iq alio :: riiljitl clin~lgc. of s loIx~ in the. experimental ljreak-

(1i1m1 c i ~ r ~ - ( > :I~;II c) = -.To. v:lli(:l~ is &I e\-irlcnt in tll t

c(>1il1iilt~~iO11;d cilrr-c.

At zcro roll angle. the c o m p u t ~ t l mrali pressiu.ci

agree only cli~:ilitativc.ly wit11 the eslii~ri:neut:rl 1-a1ur.q. Tlita

conlputat,io~ls do not picli 1-11) tliv large mca11 ilictic~~i peal;

of thc grirnnrj- vortcs. The grid qhon-11 ill Fig. 4. ant1 iisecl

liert. was constriicted to resolvc ~ o r t c x streiigt,lls that rc-

sult in si~ction pea l s ncnr 1.5. Tbi.; g;r\-r3 1.cl.y acci1r;rtc

results a t 15 tlegrecs angle cif att;~cli." Howe~-r'r for t h ~

present case. thc rr~asilrnull trnllmral C',, ii alxn1t 2.2. :I fi11c.r grid is nreded to sustain tlie very lligli temporal SIIC-

tion pealis recjiiiretl t o captin-? the 11l(~ali presllrei. Tlli'

same t r m d can 1)e noted on the wing's left sit!e for 0 = -5"

ant! the wing's right sidc for o = - 7". Each of tlirse c~;c,c~s

correspond to a side in wliicli t , l~erc are largc meail C',,. On the uther lland. x-hm thc suction pralis arca smallrr.

i.c.. the right side fur o = -3'. tlia left side for o = -7". and l~o t l i sirlei f i r L? = -14. -2s. iilltl - 42 t1egrcc.s. the

aejreerr~ent is good. For roll ariglcs of 7 . -14. '7s. ant1 -42 degrees. tlie flat portion of the llleilll CI, c1lrvt.s arc2 cap-

tured. The flat surface pressure regious are due to the:

.i comparison of mean computation;d arid rsperi-

111ental rolling nlonitmt rocfflcients ( C ' ( ) are slion-n in Fig. 8.

The coniputcd rcsults for positive roll angles are ohtair~etl

by rc'flc~ctiol~ of tlir rcsillts f r u ~ n negative roll a~lgles. In

genc~al . co~nputed C'! are in clualit,ativi~ agreement with

the esperirnent and exhibit the same nonlinear behavior

for - 10' 5 o < 10'. The corrlputations indicate that there

are tlnec stable static trim points presmt, a t o = -13. 0, anti 1.5 d e g r ~ e s . There are alio rr~gions of posirivc slol)c;

whcrc tlics wing is statically unstable. Even thnilgll thc

q i r l is crx+l+l. tlir R.AXS ecltlations arc nl,le t o moclcl the

l)h~.hical ~ i i e c l ~ i ~ ~ ~ i s ~ i ~ s r ( , ~ p o n ~ i l ~ l r for t11<> ncmli~i~~al i ty .

Fig. S Cornparison of rlieaii co~nputecl and eslwrinle~ltal

rolling n~onierlt coc,fficients for diffcsent static roll angles.

Jrl, = 0.27, (i = 30". Re = 3.67 r~rillion.

Recirll that at 15 tlegrwi angle of attack. Fig. 1. tlie

rollirig moment varies alniost lil~early wit,li roll aiiglc arid is

.itatically stable over the entire roll angle range of --42" < o < 42". 111 this case there is no vortex br.c~alitlown. iil~tl t h ~

static stability can be explaincJ in the following manner.

For positive roll angles. thc low right wing has a rcduced

rffectivt: bvwt'p xvllile tlrt high lcft wing has a n increasrd

eff'ertix-e .sn-pep. This causcs the right wing vortcs to 11e

itrongrr tliari the left wing vortex. rcsdt ing ill a nrgative

l s t a l ~ i l i ~ i r l ~ ) rolling ~noincnt . Similar arguriients can h r

give11 for r~cgativc, roll ;triglcs. A more cletailcd description

call h~ found in Ref. 18.

The situation is very different a t 30 degrees angle of

attack. due to vortex breakdown. For small positive roll

angles (less than 10 degrees) the right wing should have

a stronger x-nrtex than the left wing due t o sweep effects.

This would result in a negative (stabilizing) rolling mo-

nlent. However. tile right wing vortex is observed com-

putationally to break down closer to the delta-wing apex

than the lcft wing. see Fig. 6. This causes a loss of lift

on the right wing, resulting in a positive (unstable) rolling

r n o ~ n e ~ i t . This accounts for the positive slope of the rolling

The mean coinputational rlormal force cot$ficir~its.

C,y, are coniparccl -.\-it11 cxperirilcmtal v;ill~t~; : I n o l i ~ i l o i ~ ~

the roll angle range, -42" 5 o 5 1'7". ill Fig. 9 , Tlie

;malogous cmmpariioli for thc mrari s t r e i i ~ ~ l ~ v i w r.1 n t ~ r - o f -

p r e s a ~ c location. -y,Ij. is 511on.11 in Fig. 10. Oi-(~rall . tlr<>

comparisoris are good. Tile computc~d C'.\ c.orlll);irc

c ~ p t i o ~ ~ a l l y well with esperirr~ent for large roll ; in~les . For

low roll angles; CV is undcrprcdictetl tliie to ~irrrc~wlr-ctl

vortex suction peaks. The streamwise ccnter-of-l~rcssiirc

locations are rrlativelj- constant and agrcr wirli thc, c.xl,<'l-

imcntal values within 3%

Fig. 9 Co~riparison of mean cornputetl ant1 cspr,rimeli-

ta l normal force coefficiel~ts for different static roll angles.

r\f, = 0.27. or = 30". Re = 3.67 million.

T h e Beam-Warming algorithrrl. with a block periodic

solver in the circumferential direction, requires 26 p s l g r i d

point/timc step on a Cray C O O superconlputcr. Each

static case was solved in a time-accurate nlannt3r. Aipprox-

imately 8000 time steps were required to insure the initial

start-up transients were 110 longer p r r smt . This rcqnirctl

46 hours of C-90 compilt?r time.

Fi5. 10 C o m ~ > ; \ r i s o ~ ~ of mean computrd and ~ s p c r i ~ n e n t a l

streamwise cc.11ter-of-pressi~rf: locations for different static

roll ariglcs. -11, = 0.27. cr = 30'. Re = 3.67 million.

Conclusions

C'omln~tations fi)r a 6.5' sn-eep clc'lta wing at 30 tle-

grer-s anglr of a t t ~ c l i and 3.67 million Reynolds n u m h r ner r ~ x e ~ ~ i t ( d i ~ ~ i n g the time-dependent . three-dirrlellsional

RrlTS ecji~iitiol~s. Static (fixed) roll angles were invcs-

tigated up tlirollgli 42 drgrees of roll. The flon- field

was fount1 to 11e ~ ~ n s t e a d y at a11 roll angles. and included

;I sI>iral-tYIw vortex hrcakdown. The vortex breakdown

position. nlc~ari surface pressures. rolling moment coefi-

cicmts. normal force cocfficirnts. ant1 strearnwise center-of-

1)reisurc locations coniparecl reasonal~1~- well with rsprr i -

~ i lcn t . In some casc>s. the corilputational and experimental

I-orri1j;irisorii wcle only qldi tat ive1~- good. Finer grids are ~ieet l (d t o hr t tcr rc-solve. tlir c~ff-surface vortices and their c o r r I o i ~ l i f a r s r ti a l s Ncvcrtlle-

1 ~ s i . the ~lolllinc~nr variutioli of rolling n lo~r l r~ l t with roll

a~lgl(. n-;ls nr l l pretlictc~tl. Details xvrre prt'sc:ritcd on lion-

tllc vor tc ,~ l j r t : t~ l i< l<) \~~~ r a ~ i ~ d the: 11o11linear region5 of local

~ t a t i c roll k t ;hilit!-.

T l ~ r author.: would like to t h ~ t ~ l l i Drs. Hanff and

Himnq of the Canadian Ir~st i tutr of Aerospace Research for thcir 11i.lpfid disrussio~ls and for graciously proriding

their cspc~rirr~ental data .

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