NASA Technical Reports Server (NTRS) 19930018275: Adaptive EAGLE dynamic solution adaptation and grid quality enhancement

14 Pages·0.9 MB·English

by  NASA Technical Reports Server (NTRS)

ADAPTIVEEAGLE N 9 3 " 2:7 4 G
DYNAMIC SOLUTION ADAPTATION AND GRID
QUALITY ENHANCEMENT S...3
Phu Vinh Luong
Naval Oceanographic Office
Stennis Space Center, MS 39529
J. F.Thompson, B. Gatlin, C. W.Mastin, H. J. Kim
NSF Engineering Research Center for Computational Field Simulation
Mississippi State University
Mississippi State, MS 39762
grid isdesired. The CFD code passe, itscurrent solution to this
EAGLE routine via ascratch Rle. This structure eliminates the
Intheeffort described here, theelliptic gridgeneration proce. needfor compatibility between CFD andgridarrays. One restric.
dure inthe EAGLE gridcode hasbeen separated from themain tions isthattheinitial gridmust begenerated bytheEAGLE sys.
code intoasubroutine, andanewsubroutine which evaluatel _v- tern,orbeprocessed throughthatsystem. Thisprovides theneces-
eral gridquality measm'_ ateach lgidpoint halbeen added. The saryparameters andstructural information tobe readfromfliesby
elliptic gridmuline cannowbecalled, either byaCFD code tosen- theadaptive EAGLE routine.
eraie anewadaptive gridbased onflowvariables andqualityinca-
rares through multiple adaptation, o¢bythe EAGLE main code Inthepresent work, thecontrol function approach istumlas
togenerate agridbasedonqualitymeasure variableathroughlitai,, the basicmechanism for theadaptive gridgeneration. The static
ic adaptation. Arrays of flowvatinblea can be read into the and multiple adaptive gridgeneration technique_ are investigated
EAGLE gridcode foruseinstatic adaptation u welL These major byformudafin8thecontrol functions intermJof either gridquality
changes in theEAGLE adaptive gridsystem make iteasier tocon- meamr_ the flow mlution, orboth.
vert anyCFD code thatoperates on ablock-slructured grid(or
single-block grid) into amultiple adaptive code. Previomwork (2,3)withtheadaptive EAGLE system allowed
thegrid toonly adapt to thegradient of avariable. The work de-
scribed here hasextended thk adaptive mechanism to also allow
IHI_,DD1/ClIDB
adaptation to thecurvature of avariable orto the variable itselL
Theaystem provides fordifferent weight funcdom ineach coordi-
The requirements of accuracy and efficiency for obtaining
nate direction. Inaddition, themechanism nowincludes the abil-
solutions to PDE's havealway1been aconflic_innumerical meth-
itytocalculate theweight functions mweighted averages ofweight
odsfor solving field problems. On theone hand, itkwell known
functions fi'om several flow variables and/or quality measures.
thatincreasing thenumber ofgridpointsimplies de_em_ng thelo-
Thisallowstheadaptationtotakeinto accounttheeffectofmany
cal truncation error. This, however, results in bng computation
of the flowvariable, instead ofjust one. The construction of the
timedue tolargenumbers ofgridpoints. Ontheother hand, short-
weighted average offlow variables andquality measures, and the
ercomputation timecanbeachieved bydecreau i thenumber of
grid points, butthe resultisale_ accuratesolution. choice of adaptation togradient, curvature, orvariable, areallcon-
trolled iu each coordinate directions through inputparameters.
Adaptive gridgeneration techniques areameans forresolving Thequalitymeasures nowavailable intheEAGLE gridsystemare
thisconflict. Formanypractical problenm, theinitial gridmay not skewness, aspect ratio, arc length, and smoothnem of the grid.
bethe bestmired for aparticular phydcal problem. Fc_ example, These grid qualitymeasures, and theresulting control and weight
thelocation of flow features, such mshocks, botmdmy andshear function values, canbeoutput for graphical contouring.
layenk andwake regions, arenot known before the gridisgener-
ate& Inmultiple adaptive gridgeneration, gridpoints aremoved
continually torespond tothese featurel intheflow fieldmthey de. ADAPTIVE MECHANISM
velop. Thinadaptation canreduce theoscillatiom umciated with
inadequate resolution oflarge gradients, allowing sharper shocks
Control Function Annroaeh
andbetter repreaentation of bouada_ iayenL Th,,- itiaposthole
to achieve both efficiency and high acx:uracyfor numerical mlu.
tiom of PDE'L Several basictechnique, involved inadaptive grid The control funcdon approach to adaptation isdeveloped by
generation aredik-'u_d inRef. (1). noting thecorreqxmdence between the 1Dform ofthe system,
Inthe earlier form ofthe adaptive EAGLE _em (2, 3),the
coupling oftheadaptive gridsyutemwithaCFD code required the
xw+ ex, - 0 (1)
encaplmlation of both theentire gridcode andtheCFD flow code
into separate subroutines, and theconstruction of adriverto call
each. _ wu inefficient thatitincluded mine mmeceua_ parta
of the grid code and required significant umdification, and per- and the differentia] form of the equidism'bution principle,
Imp,remmmutng, of theCI:D code.Inparticular, the flow code Wxe = ¢omtant,
azr_ and/o¢ thegridcode mTayshadtobemodified tobecompat.
l_le inmucture.
wxw + w,z, = 0 (2)
The convemion procedure il accomplished byadding the el-
liplic grid generation subroutine, andcertain other mbroutinel
from the EAGLE grid,ystem thatare invoh,ed intheelliptic grid
generation protest, totheflowcode. TheCFD code maythencall wherePk thefunction to control thecoordinate line sparing, and
the eRiptic gridgeneratiou routine at each lime stepwhen anew Wi, theweight function.
From(I)and Grid Oualitv Measures
(2)the control function canbedefined interms
of the weight function and itsderivative as
The objective of tiffspart of the investigation was to develop
a means of evaluating grids through the computation of certain
gridproperties that arerelated togridqualityand to develop tech.
r w, (3)
niques for estimating the truncation error. Following Kerlick and
Klopfer (Re£ 8), andGatiin, et. al. (Ref. 9), the gridquality mea-
sures are taken asskewangle, aspect ratio,gridLaplacian, and_'c
This equation can beextended inageneral 3D form as length. Techmques forestimating the truncation error due tothe
work of Mastin (ReL I0) are also included. At each grid point in
ageneral 3D grid,eachproperty can havethree values associated
w,, (4)
withthethree directions. The approach taken inthisinvestigation
istotreat each surface ofconstant _ separately forease ingraphi-
cal interpretation.
Thisapproach wassuggested byAnderson (Ref. 4,5),andhasbeen
applied with _ for 2D configurations by Johnson and
Thompson (ReL6)andfor3D configurations byKimandThomp-
sun (Re£ 2) andby'Ihand Thompson (Ref. 3). The minimum skew angle between intersecting grid lines is
one ofthe most important measurable gridproperties. This angle
The complete generalization of (4)was proposed byEiseman
canbe expressed interms of the covariant metric elements as
(_f.7) as =
E
where W_isthe weight function chosen for the _tdirection. This Sincegn - &=, 8n = 8nandsn = 8_, thethree skew angles
definition of thecontrol functions provides aconvenient means to associated with each gridpoint ina3D gridare 0t_ 03 and 031'
specify three separate control functions, with one ine_ coordi-
nate direction.
asimaxatlo
Inorder topreserve the geome_ characteristics of the ex-
isting grid,itispractical toconstruct the control functions inmuch Since aspect ratio isthe ratio of the length of thesidesof agrid
amanner thatthe control functiondsefined by(5)areadded tothe cell, itcanbe defined intwo different ways. Forexample, on_:
initial setof control functionosbtained from thegeometry, i.e.,
face ofconstant _, this ratiocan be expressed in terms ofme_c
elements 8uand 8_as
P_ -(P_; + C_P_. _ = 1,2"3 (6)
_ (lOa)
A_, l
where
F
(P,). : control function based on geometry or
(P_). : control function based Onweight function
Inthese equations theweight function Wcan be computed by
different formulas for different adaptive mechanisms:
Large changes inaspect ratio ofgridsfrom one partof the fieldto
Adaptation to
another may inhibit the convergence of viscous flow solutions to
Var_b/¢ : W = 1 + IV! asteady state.
Gradien: : W - ! + IVVI
Curvature : W = (I +plgl)Jl + alVVI 2 Ansefid measure ofthe smoothness ofagridisthe Laplacian
of the curvilinear system, _, / l 1,2,3, which is simply the
(7) rateofchange of gridpoint density inthe grid. Foraperfectly uni-
form grid,the gridLap]acian wouldvanisheverywhe_, but ex-
where Vcan beeither aflowsolution variable oragridqualitymea- ceedinglylarge values may arise in highly stretched grids. The
mathematical representation of the gi'idI..aplacian isdefined in
sure. Here aand flare on therange 0-1, and
terms of the contravariant metric elements 80,the contravariant
basevectors at and theposition vector •as
. T'V (s)
X" (1 + IVVP)_
isthecurvatureofthevariablVe,
Using these definitions of the control functions, the elliptic
generation system becomes anadaptivegridgeneration system.
This system is then solved iteretlvely inadaptive EAGLE bythe Another important inc.*mareofthegridqualityisthe localrate
pointSOR method togenerate the adaptive grid. atwhich gridspacing changes. Onacoordinate surface ofconstant
_3,and along acoordinate lineof constant _z thegridspacing can
bedefined as
Assume that there are twonumerical solutions oforder pac-
d, - [(x,.i - x__+ (y,+_- y_ + (-,÷_- .0q½ (12a) curacyfor (13b) thathavebeen computed onafine gridand on a
coarse grid,withgridspacing handnh, respectively, ineach coor-
dinate direction. Assuming thatthe samepth order method isused
Thenormalized rateatwhich gridspacing changes (ARCL) isthen inboth cases, then the relation between the two numerical solu-
tions and the actual solution uof the PDE can beestablished as
d, - dH
(ARCL), = ½(d, + d,-O (12b)
u = U. ÷ R(8)' (15a)
and
The objective of this section isto present heuristic error esti.
mates which give order of magnitude appro_dmations for the
truncation error and the solution error inthe numerical solution - = u,=+ R(nhy (15b)
of theElder equations forcompres_le flowandother systems of
conservation laws. Anyconservation lawcanbewritten inagener-
alformu From these equations, anextrapolated value ofu canbecomputed
as
u,+f.+&, + h. = 0 (13a)
n,u. - u_ (15c)
u = (w,- 1)
The transformation ofthis system toanarbitrary curvilinear coor-
dinate system is Thus the estimate ofthe error inthenumerical solution computed
on the h gridis:
U, + F,_ + G,z + HO = 0 (13b)
. - u, _ (15d)
= (n*- I)
where 4"gistheJacobian of the transformation and
RESULTS AND DISCUSSION
The adaptive grid generation system based on the control
G -/_ _ + _ + _:h) function approach as described inthe previo_ chapters hasbeen
used togenerate static and multiple adaptive grids forseveral ge-
ometries(gel 11). Some ofthese results arepresented inthissec-
Lethbe the spacing of the fine grid,and nh bethespacing of the tion. Thestatic adaptive gridswere obtained byadapting theinitial
coarse grid. Let L,be the difference approximation operator on grids to either gridquality m_ variables or to existing flow
thefinegrid,and L., bethe difference approximation operator on solution variables. Themultiple adaptive procedure wastested on
the coarse grid. Then the finite difference approximation of the several different configurations with the adaptive MISSE Euler
PDE canberepresented on thefine gridas flow code (Re£ 12) fortransoni-"and supersonic flow cases, and
with the adaptive INS3D flow code (Reg. 13) for incompressible
flow.
u, + f, + g, + h, = LdF, G,H) + T(h)" (13c) Ada_ntaflon to Oualit vMeasures
Some examples of the grid quality adaptation are shown in
andon thecoane gridas Figure 1for adaptation to various quality measures. (In ReL 11,
colorcontour plotsof the qualitymeasures andthe other adaptive
features aregiven.) Figures 2a-d shows the difference of theav-
u,+f= + &,+ h, = L.,(F,G,I'/)+ T(nhy (13d) erage skewangle between theinitial andadaptive grids. The same
number of total adaptive iterations were run in each case. The
control functJom were updated based onthe geomeUy of the pre-
where aisaninteger. vious grid,rather than the initial grid"ateach adaptation.
From (13¢:)and (13d), the estimate ofthemmcation error on Comparison ofFigure lb with Figure la shows that adapta-
the fine grid canbe computed as tion tothe skewness iseffective inreducing the skewness inone re-
gion, while increasing the skewness inother regions of thegrid. A
small improvement inaspectratiooccursb,utthe smoothness of
the grid isdecreased.
T(hF - _ (14)
Comparison ofFigure lcwith Figure laindicates that adapta-
tiontoaspect ratiodoes improve both aspect ratioand smoothness
Aaimilar procedure can beused to compute the error inthe of the grid; the skewnem isincreased, however. Comparison of
numerical solution. Such aprocedure haslong been used in the Figure ldwithFigure ia shows that adaptation tosmoothness ira.
numerical solution of ordinary different_l equations and is re. prove=the skewness andaspect ratioof thegrideffectively, butthe
ferred to as Richardson extrapolation. Even though numerical adaptive gridisnot as smooth asthe initial elliptic grid.
solutions must be computed onboth fine andcoarse grids, the er-
Figure le shows the beneficial effect of including adaptation
rorestimates which result donot havethe largepeaks at solution
to aspect ratio, arc length,aad smoothnea, with adaptation to
s/nguiarities which can be encountered with the truncation error
skewness: the skewness isreduced more bythecombination than
computed from difference approximation of higher derivatives. with skewness adaptation alone. Alittle improvement oco.n's in
Thus thesolutioenrror estimates may sometimes be more useful aspect ratio; the =moothness of the grid does, however, decrease.
in the construction of adaptive grids.
Resuflrtosmtheseexampslehsow Theinitialgrid,adaptedtothegradientofthecombination of
that the adaptation tothe
density and pressure inthe _ direction only isshown inFigure 7e.
combination of all gridquality measures, or to each individually,
TotalnumberofadaptationswasSwith CI = 0, C2 = 0.9,and
can improve some gridproperties while damaging others. For ex- updates were applied to the previous control functions. Pressure
ample the adaptation tothe Laplacian of this particular gridcan
contours obtained from this adaptive gridareshown inFigure 7£
:educe the skewness, but the resulting adaptive grid is not as
smooth as theinitial grid. The choice ofthe adaptive variable for From these figures, the representation of the shocks on the
the adaptation very much depends on what property of the grid adapted coarse grid is much sharper and closer to the fine grid
needs tobe improved andtheconfigurations of thegrids. solution than the nonadaptive coarse grid.The total C'PUtime for
obtalning 300 time steps solution forthe adaptive gridwas approx-
imately800 seconds foreach adaptive mechanism, nearly50% sav-
Results of multiple adaptation performed with the adaptive ing time compared to thatof the fine grid.
MISSE Euler flow code are shown in Figures 3-8. In allthese The adaptation tothe combination ofdensity and pressure in
plots, NIT is the total number of time steps, INT indicates the Idirection andtothegradient of thiscombination in_=direction
number of time steps atwhich the tint adaptation isperformed,
ofFigure 6gives asmoother behavior of the pressure coeff'tcient
NCL is the number of time steps between each adaptation, and
behind the shock than the adaptation to the gradient of pressure
MAXINT indicates the number of time steps at which the last
alone ofFigures 7cand7d. The adaptation tothecurvature of Fig-
adaptationis performed. Values of weight functions
ures 7aand "Togives abetter result, however with alittle overpre-
(AWT, AW'/'_ weight coefficients (C1, C_O,adaptive variables
diction of the pressure coefficient right behind the shock. The
density (RH01, RttO_ pressure (PRE$_, PRE$_) ate given for
adaptation to the gradient of the combination of the density and
the _ and the _= directions, respectively. For example, __=
pressure in]_ direction only inFigures 7e and 7fgives the closest
AWT = GRAD, CURV, C, = 0.5, C2 - 0.3, ,t_/O - t,0,
solution tothe fine gridsolution.
PRE5 = O,Iand a - I, ,8 = I,canbe interpreted asadapta-
tion to the density gradient in the _ direction with C_ = 0.5, From the= results,clearly multiple adaptive gridsproduce a m
and tothe curvature of the pressure in the _=direction with better representation ofthe shockregions, aswell astheexpansion
C= = 0.3 with coefficients of gradient and curvature regions, thanthat of the same nonadaptive grid. Among these
a = 1,/3 = l, respectively. adaptive mechanisms, theuse of the better results than theuse of
single variable. Another advantage that shouldbe mentioned here |
double w_lm_ (mnerxanic Euler) isthe controlling of the direction inwhich adaptation isapplied.
=
Results obtained fromasupersonic flow atMach- 2overfine As shownabove, the adaptation inonlyone direction (_2) gives the
closest solution tothe finegridsolution. Moreover, the gridinthis
(121 x41) andcoarse (81x31) double-wedge grids areshown in
Figures 3-7. Figure 3shows the pressure contours obtained from adaptive mechanism is not being disturbed as much as by the !
300time steps ontheinitialandadaptive grids(121x 41). Thegrid adaptation inboth directions. The minimum skew angle for this
was adapted tothe density gradient inthe flowdirection (R/-/O= case ishigher compared tothoseof adaptation inboth directions.
Ofcoune, thisistrueonly foracertain number ofadaptations and
1,0)with C1 - 0.7 and to the pressure gradient in the normal
direction with C2 = 0.5. Atotal of 4adaptations was used for aparticular vahm ofweight coefficients.
this case,with control functions updated from the previous grid. wind tunnel (m_rr_nic Euler_
Figure 4shows thepressure coefficients onthelowerwall, and Results fromthe supersonic flowatMach= 2inawind tunnel
convergence histories of the two solutions areshown inFigure 5. areshowninFigure 8.These resultswere alsoobtalned in300time
InFigure 5,thehigh peaks at each adaptation are due to the use steps. Figure 8aisthe initial grid, Figure 8b isthe adaptive grid
of the previous solution onthe newadapted gridwithout integra- adapted tothe error estimation inboth directions, and Figure 8c
tion onto the new grid. From these figurer, clearly the adaptive istheadaptive gridadapted togradient of the combination ofden- im
gridgivesamuch better representation oftheshock regions aswell sityand pressure in both directions. The number of adaptation
astheexpansion region=. Shocksaremuch sharper forthesolution wa= 5 for both cases, with Ct = 0.6, C: = 0.55 for the
obtained on the adaptive grid. A record of the CPU time on an adaptation to gradient of the combination. Shocks are much
IRIS4D/440VGX machine shows thatthe total CPU timefor the sharper for solutions obtained on the adaptive gridsthan on the
initialgrid(121x 41)without adaptation was1481.51 CPU seconds nonadaptive gridsforthisconfiguration insupersonic flow aswell.
and for the adaptive grid(121 x41)was 1599.02 CPU seconds, an Results fromthese examples show thatmultiple adaptive grids
8%increase. captured very well major features of the flow field in supersonic
Contour plots onthe pressure ofthe initial fine grid(121x4i), flow for these particular configurations. The adaptations to the
theinitial coarse grid(81x31),andthe adaptive grid(81x31)are combination ofthe gridqualitymeasures, such asskewness ofthe
shown in Figure 6. The coarse gridwas adapted tothe combina- gridand the flow solution, for these particular gridsnot only make
the _ more skewed butalsoresulted in poor resolution of the
tion ofdensity andpreumre in _tdirection, withweight coefficient
major features of thefiowfield. Ontheother handthe adaptation
Ct - 0.5, andtothe gradient of thiscombination in_=direction
to the error estimation and the use of the weighted average in
withweightcoeRicient C= - O.5,(AWT - VAK GP,,4D,RHO =
1,t, PRES= t,1). weight functions computed from several flow variables does, in
fact, improve the solutions.
Different adaptivemechanisn_appliedtothe coarsegridin
The computation oftheweight functions andthechoice ofthe
themultiple adaptationprocessareshowninFigure7. FigureTo
adaptive solution variable are independent from one direction to
showsthepressurecontoursobtainedontheadaptivegridofFig-
another thus enabling the usersto havemore freedom inchoosing
tare7a. The initial gridwas adapted to the curvature of the com-
suitable adaptive mechanism foreach kind of flow. Forexample,
bination ofdensity andpressure inbothdirections (AW/"= CURE', in the case of boundary layers and shocks occurring inthe same
CURV,RHO = 1,1,PRES = 1,1). The total numberof adaptations
flow field, the usenmay choose to adapt the gridto the velocity
was4withC| = 0.7. C= " 0.7. Thecoeffldantsofthegradi-
magnitude gradient inthe normal direction tocapture thebound-
ent and curvature were a = I and fl = 0.5, respectively, and
an/layer regions andtothepressure gradient inthe flow direction
the updates were from the original control functions. tocapture theshocks.
Figure 7¢1show=the pressure contours obtainedonthe adap- l_.t, mtrJL_tdatm_
tive gridofFigure 7c. The adaptative mechanism forthis casewas 0neomnrtsslble Navinr-Stoke_
presmm=gradient inboth directions with Ct " 0.7, C= - 0.7
end total number of adaptations wes 4,(AWT = GRA_ GRAD, Remtts of multiple adaptation performed with the adaptive
PJ_O=0,0,pm_s =t,l). INS3D incompressible flow code are du3wu in Figures 9-12.
These results areobtained for incompress_le laminar flow for a
2-block backward facing step, (grid size for the firstblock is(21 Figure 16shows thevelocity magnitude contours obtained on
x35) and (81x41) for the second block). The Reynolds number the initial andanother adaptive grid. The initial coarse gridwas
used in thisinvestigation for the backward facing stepwas 183.32 adapted tothe combination ofvorticity andqualitymeasure aspect
for comparison with experimental data. ratioofthegridinthedirection normal totheflowdirection, (AWT
= VAR, VAR, VORR ,, 0,1, ASPE = 1). Here
The gridconstructed forthe backward facing stepconsidered
Cl " 0.3, C2 = 0.5. Figures 17and 18show theskinfriction
inthiscase isthe same asthe geometry ofthe experiment. Howev-
and pressure coefficients of the inner end outer wails obtained
er, the step length downstream of the grid isonly 30 times the from'coarse,fine end adaptive coarse grids.
length ofthe stepheight, while the steplength forthe experiment
wasmuch larger. Figure 9¢shows thevelocity magnitude contours Figure 17shows that the behavior ofthe skin friction coe_-
obtained from 5000time steps ontheinitial gridofFigure 9a, Ve- cient of the outer wall isalmost identical to that of the fine grid.
]ocityvectors areshown inFigure 9b.. Figure 10shows thevelocity The representation oftheskinfrictionoftheinnerwall issmoother
magnitude, vorticity contours end velocity vectors obtained from than that of thenonadaptlve gridbutwith alarge change afterthe
5000 time steps ontheadaptive grid. The initial gridwasadapted separation region toward theoutlet of theduct. Figure 18,again
at500end I000 timesteps tothe vorticity magnitude inthe direc- indicates that the adaptation did not help in the improvement of
tinnnormaito the walls with C_ = 0, Cz - I,(AWT-VAR, the pressure coefficients for thiscase either.
FAR,FORR ="0,I). Total number of adaptations was 2for this Arecord of the CPU time on an IRIS 4D/440VGX machine
case, andupdates were applied tothe initial control functions. shows thatthe total CPU time for the initial grid (111 x51) was
Skinfriction coeffic/ents onthe lower andupper wails (begin- 23870.61 CPU seconds and for the adaptive grid (111 x31) was
ning at the step) obtained from initial end adaptive grids are approximately 13800 CPU seconds for each adaptive mechanism.
plotted inFigure 1I. Velocity profiles atthe stepandseveral Ioca- FromFtIDtres13andi6,itcan beseen thatinboth adaptations the
gridsget finer atthe turn. Correspondingly theskinfriction coeffi-
tions downstream (nearest to the experimental data) along with
digitized experimental data areshown in Figure12. cinnts obtained from adaptive grids have higher pick at the turn
andcapture separation region well, asshown inFigures 14and17.
Results from these figures show that the velocity profiles ob- Moreover, the reattachment point obtained from adaptive gridof
tained from theadaptive old are closer tothe experimental data Figure 14iscloser to thatof the fine grid than the adaptation of
than for thenonadapfive grid. However, there are some wiggles Figure 17end thenon-adaptive grid.
oftheskin_ction coefficient obtained from the adaptive gridoc-
curringattheseparation regionofthe lower wall. Thismaybedue
to the redistn_oution of gr/d spacings inthisregion. Digitized val-
ues of the reattachment length from Figure ll areappro_nately The widely-used EAGLE gridgeneration system (Ref. 14)
7.67 for both solution& while theexperimental value was 7.9 for hasbeen extended andenhenced sothatitcan bereadilycoupled
this particular Reynolds number. The difference ofthese values with existing PDE solverswhich operate onstructured gridstopro-
maybedue tothe difference of the steplength of theexperiment vide aflexible adaptive gridcapability. The adaptive EAGLE grid
and thegriddownstream. code canbe used forgenerating notonly algebraic gridsandellip-
ticgridsbutstatic adaptive gridsaswell. Inthes_aticadaptation,
Arecord oftheCPU time onaCray2machine shows thatthe thegridcanbeadapted toan e_stlng PDE solution ortogridqual-
totalCPU timeforthe initial gridwithout adaptation was.25956.26 itymeasuresor toacombination ofboth. The testcases show that
CPU seconds andfortheadaptive gridwas26363.74 CPU seconds. some gridproperties can beimproved bythestatic adaptation to
Since there isonly2adaptive iterations theincrease intimeforthis gridqualitymeasures.
case is 12%.
In this study, the weight functions can be formulated as
180 de_m-eeturn around duet weighted average of weight functions from several flow variables
(incompressible Navim'.-Stokes) orseveral qualitymeasures orthecombination ofboth. Different
weight functions and adaptive variables can be applied in each
Most flow solvers for incompreuffole flow require grid lines
direction. These operations are controlled through the input pa-
which arepacked closely to thewallsinorder toresolve thebotmd-
rameters instatic aswell as multiple adaptation mode.
arylayerregions. This results inalargenumber ofgridpoints and
hence longcomputer times. The multiple adaptation canbeused There are several mccesd_ incoxporationsof the adaptive
toreduce thecostofcomputer timebyallowing theuse ofacoarser EAGLE packed subroutines into flow codes, including INS3D
grid. Inthe present investigation, afinegrid(111x51)with spacing from NASA Ames andtheMISSE Euler solver developed atMls-
offthe wallsof 0.002 andacoarse grid(111x31)with spacing 0.004 siutippi State University. Several configurations are considered
off thewalls are considered for the turnaround duct. The result for each of these adaptive flowcodes for the investigation of the
of theadaptation onthecoarsegridiscompared withthenonadap- newweight fun_om formulations andgridquality measures in
tlve fine grid solution, while the Reynolds number for the turn the multiple adaptation.
around ductwas 500. Results obtained from 6000 time steps on Results obtained from the adaptive MISSE Euler flow code
fine, coarse and adaptive grids forturnaround duct areshown in show considerable succe_ asmeasured byimprovements inshock
Figures 9-18. resolution oncoante gridsinthe eompress_le flows. Some success
Figure 13shows thevelocity magnitude contours obtained on hasbeen made incapturing separation regions oncoarse gridsof
the initialand adaptive grids. The initial coarse gridwasadapted theadaptive INS3D flowrode inincompreuible flows. Forfurther
to the velocity magnitude gradient at 1000, 1500, 2000 and2500 study, the interpolation of the previous solutions to the new
adapted gridswouldbe recommended, especially forthe adaptive
time steps, inthedirection normal tothe flow direction, (AWT-
GRAD, GRAD, VOMA = 0,1). Total number of adaptations was INS3D flow code and the implementation of arbitrary block
adaptation in multi-block eonfigurations.
4with CI = O. 1, Cz = 0.5, and the updates were applied to
the initial control functions. Figures 14and15show the skinfrlc-
tion andpressure coeffi_ents oftheinner and outerwailaobtalned AC_O_E_E_NT
frommane, fine and adaptive coarse grids.
This work was supported in part by Grant
Figure 14shows that thebehavior ofthe skin friction coe_-
F08635-89-C-0209 fromtheAirForceArmament Directorate,
ctents fortheadaptive gridaremuch closer tothefinegridsoin_on
EgfinAFB, (Dr. Lawrence IAjewski,monitor) end inpartbyCon-
than the nunadapttve coarse grid.Figure 15showuthat theadapta-
tract NA_-36949 from NASA Marshall Space Flight Center
tionfor thiscase didaot help significentlyin theimprovement of
(Dr.Paul M_mnaughey, monitor).
the preumra coefficients, however.
[lllptl¢ _ld _| It4 Jrid
Its21 i11ttUbt t, ILLo1
6¢ _,,l', _. S,. S, o#I_m,
Figure la_ Elliptic Grid
Figure le. Adaptation to _ aspect ratio, andLaplacia.n.
_t tm grSd ...........
AVERAGE SKEW ANGLE
N
mm
ia
i
84
Im
i"
7,t
l|m| Itm_,_
Ta
_|. 3,|.$./_1
TO
Figure lb. Adaptation toskewness. its
#41
• _ (Ulm_i)
44
Is,i) _14
_.I •. 6. 4. .? .I.......O......I0 II lJl IS 14 16 i4 IT 18 |g i
Figure 2a. The difference of *ve_e _anitle between
initial and adaptive grids.
AVERAGE _ ANGLE
_yL_
m
limit • 21, _ • I u
M
m
ll_Bt |w Irid
m
, | .... | .............
II I 4 $ • T II • 10il Ul_ll141111tll'f|lllll*NILt
ITtt_ll • it, _ • a F_ttre2b. The _1[ of average skew a_le ia the
it_er._, |..So 0# Immm adapt,doeto,q)ea ratio.
F_,ttre ld. Adaptation to I.,apladsn.
O*i'i-l.".,"J"NA,_' PAQE !S :
OF POOR QUALITY
A_GE SK]_Y A_GLF
mu_ va. _
U
711
t"
74
3b
HITs3Qil
IB 121m41
N PI_SSUm
a
N
• $ • • • 7 • • I0 |1181214161117181g N II
Figure2c.Theincreasinogfaverageskewangleinthe
adaptationtoLaphc/anofthegrid.
AVERkGE _ _GLE 3C
NITs_QQ
N ................... 1 121N41
_f_HJt
N
n
71
"
_ • i I
IB
go
NI"l'n"lm='% Intstan, ©¥1s20o 0mMtnt :2m 3d
m,¢ul.lead, cualll. ?,ft. S, r_ea I. m#itqtmn_
41
Figure 3. Contour plots of pressure on initial
IlO ....... andadap_veSr_
• $ • • I ? • I li |l lJ111411|l IT1111101
BAntl_
pressure Coefficient (Lower wall)
Figure 2d. Average skew angle inthe adaptation to all grid I_m-adatptaU_.. Vs.Ji4apt,ll_
quality meuur_
U
-4LI
[lliptlc I,rld
-4.4
-4k4
•_uzta4n)
-- &4Sl_lll_4l )
-4.?
-4.4 i i , I n t
°lg -4 • l IO II i
3a
HITs'mn
I_lmr41 Figure 4. Adaptation with mw- _ _ PRE$ - 1,1,
cw - 0.7, 0.7.
ORl_,,i_,qLP;.OZ _S
OF POOR QUALITY
Convergence History llill#llul grid
_et.mmr ap._IL_lO_. I_HZ_O, I.ev_..|
|.0
O.6
0.0
"-4.6
-l.0
01x3i 6C
*lJ5
N|T=3OOoinlzll_l,cvl: "wl
-8.41 \\_, PRFHliRF
-ILil
-11.0
-SJI
-4.0 bdl_d (!111x41)
-4A
4J N IN 150 H4 m IIII :200. ill Ixp:rlOo prllss I, 1 _f
Fmeathm _b_" il #_ir. 8_o _0. S,O.S. _ I, I
Figure 5. Convergence history of the initiLl and Figure 6. Contour plots of the pressure on fine, coarse
adaptive grid mlutionL i
[|1 igt IC grid /_4g! IH grill
w
............... NITiqnn Oix31, i|_ |, I_l|lll. 5 78
121141 NITz_. Intsl_.cgl:_
F_(SSmK PA(SSO_
NI1T2x13N04o1 (_ _lmEucu_r_.IntsI_c.ur_.mf ICUupSuO_,Im7;.,0.7.pi-ag_8I1, I•I
(lllptic 0rid
N8I1xI3#1_ll/ _i_ NIT830U01.x31 I_l# Inn. ©vl:2g 7C
PR(|m Pfl(IIORf
Nlt=_ao 6d 7d
Nil lists liio, Ol ilqesnO, P¢'INIIII I•I
1111131 m,I =ormd0 grid, ©_ii. 7. O.7
RdIptl_Qrid _l[|p|le Irld
llllW_lt _C
NiTli_ma, inl |Inn, i:ilil _9
PI_FS_IW
NT_C,_Oe 9a
0tl.0S0iletiIS.le:l$3.)2
IRLOC IW
aE,
r.
nimlntsZnn, attlq_lnlI.O rlls=O.1 7f
i,¢sgclil.e¢_4hcl_O.O.O.g, rhosa. !
Figure 7. Contour plots o(pz_l;u_ onilc_t.iva grids.
?
(lllptl¢ irld NTWiamSII_ 9b
. O|t. O_, I_t_5, R_ 113.32
V(L_ I_' n_._ 11Folk
Ida 8a
Ill/pI ll..IJ Irlld
NTIIIIC= 5000
Die, _,ktt.mS. I_tm 113, 32
Figure 9. Contour plots of velocity magnitude, and velocity
vectorsoninitiaglrids.
i 8b
idlpt hm Irld Ildept li.I w'id
i sc
Figure 8. Contour plota of _ on initial and
w'nvmsam, rm. tON lOa
Dtm •IS 0Iml_S °o li_tlm •3_
U[I.OCI1_ Y*le_lty pruflle
vVrr r r J
lilJ!iiilll i i i.... i zJ
radium-, uer. c_ll..O.S lob u
_I_ Ntll41 _
-_--U =u= u u u L. u, u, u
.,.l _,-. w_-. m.ll..II.S |01_ u
Fiju:e 10. Contour plots ofvelacity mai_itude and velocity I
cut
Ill
U • ! a_
m.eu..c_m.4J)
-.4dl _ u u u IJI LII IJ I.II u
8k_ F-rleUon CoeHl_4mt in BF8
z
m A,Wen, 12b
UE
ldl
t_
t.4
t-
-4.1m
u
-4LIII
• • 4 41 Ill 10 *n W 14 tl N
Figure 11. Sign friction of the lower andupper walls obtained
from initial and adaptive pidL
12c