NASA Technical Reports Server (NTRS) 20120001451: Effects of Mesh Irregularities on Accuracy of Finite-Volume Discretization Schemes

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by  NASA Technical Reports Server (NTRS)

Effects of mesh irregulaties on accuracy of finite-volume
discretization schemes
Boris Diskin∗ JamesL. Thomas†
Theeffectsofmeshirregularitiesonaccuracyofunstructurednode-centeredfinite-volumediscretizationsare
considered.Thefocusisonanedge-basedapproachthatusesunweightedleast-squaresgradientreconstruction
withaquadraticfit. Forinviscidfluxes,thediscretizationisnominallythirdorderaccurateongeneraltrian-
gularmeshes.Forviscousfluxes,theschemeisanaverage-least-squaresformulationthatisnominallysecond-
orderaccurateandcontrastedwithacommonGreen-Gaussdiscretizationscheme.Gradienterrors,truncation
errors, anddiscretizationerrorsareseparatelystudiedaccordingtoapreviouslyintroducedcomprehensive
methodology. The methodology considers three classes of grids: isotropic grids in a rectangular geometry,
anisotropicgridstypicalofadaptedgrids,andanisotropicgridsoveracurvedsurfacetypicalofadvancing-
layergrids. Themesheswithintheclassesrangefromregulartoextremelyirregularincludingmesheswith
randomperturbationof nodes. Recommendations aremadeconcerningthediscretizationschemesthat are
expectedtobeleastsensitivetomeshirregularitiesinapplicationstoturbulentflowsincomplexgeometries.
I. Introduction
Traditionalmesh-quality metrics tend to assess meshes without taking into account the type of equations being
solved, solutions, or the desired computational output. The most widely-used mesh quality metrics are geometric
in nature, considering shape, size, angles, aspect ratio, skewness, Jacobian, etc., of the mesh elements. Additional
considerationsincludevariationsbetweenmeshelements,suchascell-to-cellandface-to-faceratios,linesmoothness,
etc. Some schoolsof thoughtarguethat an accurateand efficientsolutioncan be obtainedonly on“pretty” meshes
similartoeitherstructuredCartesianmeshesortomeshescomposedfromidenticalperfectelements(perfecttriangles,
tetrahedrals,etc.)ThisideaisindirectcontradictionwiththecommonComputationalFluidDynamics(CFD)practice,
in whichaccurate solutionsare computedon practicalmeshesthat wouldbe characterizedas unacceptableby most
geometric mesh quality metrics. Moreover,the most powerfulstate-of-artmethodfor improvingsolution accuracy,
output-based mesh adaptation,1 tends to produce “ugly” meshes, but provides vast improvements of the accuracy-
per-degree-of-freedom ratio.2 It is widely recognized today that mesh quality indicators should involve sufficient
information about the solution.3–5 In fact, it is envisioned that the connection should go even deeper and involve
modern error-estimation techniques that take into account the specifics of the discretization method in use and the
desiredcomputationaloutput.
Historically, mesh quality analyses were first performed for finite-difference and finite-element methods. It is
notstraightforwardtotranslatethoseapproachestofinite-volumediscretizations(FVD)thatrepresentthestateofart
in CFD computations. While there is no doubt that certain mesh characteristics critically affect accuracy of CFD
solutionsandgradients,theprecisenatureofthisinfluence(whataffectswhat)isfarfromclear.
Forfinite-differenceapproachesmostofthemeshqualitymethodstrytoestablishconnectionsbetweenmeshand
truncationerror.6,7 ThetruncationerroranalysisisoftenappliedtoFVDschemesaswell.8 However,ithasbeenlong
∗NationalInstituteofAerospace(NIA),100ExplorationWay,Hampton,VA23681,USA,[email protected]. Also
DepartmentofMechanicalandAerospaceEngineering,UniversityofVirginia,Charlottesville,VA22904,USA.SupportedbyNASAFundamental
AeronauticsProgram,SupersonicsProject,NRAContractNNL07AA23C(PI:Prof.N.K.Yamaleev).
†ComputationalAeroSciencesBranch,NASALangleyResearchCenter,MailStop128,FellowAIAA,[email protected].
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known,thattruncationerrorsofFVDschemesonunstructuredgridsarenotreliableestimatorsofdiscretizationerrors.
Thesupra-convergenceofdiscretizationerrorsobservedandstudiedforatleast50yearsindicatesthatdesign-order
accurate FVD solutions can be computed on unstructured grids even when truncation errors exhibit a lower-order
convergenceor,insomecases,donotconvergeatall.9–11
The theory and applications of mesh quality assessments are well developed and widely used within the finite-
element community. While initial groundbreaking work focused on pure geometrical mesh-quality metrics, such
as large angles,12,13 the later developmentstake solution into account.14 The standard finite-element estimates use
Sobolevnormsthat simultaneouslyestimate errorsin the solutionand its derivatives. These estimates mightbe too
conservative because recent finite-volume computationsindicate that accurate solutions can be obtained in spite of
formallypooraccuracyofgradients.15–17
Previously,theeffectsofmeshirregularitiesonaccuracyofunstructuredfinite-volumediscretizationswereeval-
uated for various common node-centered and cell-centered schemes.15,16,18–20 The focus here is on an edge-based
node-centeredapproachthatusesunweightedleast-squaresgradientreconstructionwitha quadraticfit. Forinviscid
fluxes, the discretizationis nominallythirdorderaccurateongeneraltriangularmeshes.21,22 Forviscousfluxes, the
schemeisanaverage-least-squaresformulationthatiscontrastedwithacommonGreen-Gaussdiscretizationscheme.
These schemescan be formulatedas edge-basedschemesongeneralsimplicial grids.23 The inviscidand averaged-
least-squaresviscousdiscretizationscan be formulatedas edge-basedon arbitrarygrids, includinggridsused in ag-
glomerationmultigrid.24,25
Gradienterrors,truncationerrors,anddiscretizationerrorsareseparatelystudiedaccordingtoapreviouslyintro-
ducedcomprehensivemethodology.15,16Alinearconvectionequation,
(a·∇)U =f, (1)
withavelocityvector,a,servesasamodelforinviscidfluxes. Poisson’sequation
∆U =f, (2)
subjecttoDirichletboundaryconditionsservesasamodelforviscousfluxes. Themethodofmanufacturedsolutions
is used, so the exact solutionsand forcing functions, f, are known. Solutions are chosen to be smooth on all grids
considered,i.e.,noaccuracydegradationoccursbecauseofalackofsolutionsmoothness.
The paper is organized as following. First, grids, FVD schemes, and accuracy measures are briefly described.
Then,numericalstudiesoftheFVDaccuracymeasuresarereportedforgridsofthreeclassesrepresentingisotropic,
adapted,andturbulent-flowgrids. Conclusionsandrecommendationsareofferedattheend.
II. Grids
Computationalstudiesareconductedontwo-dimensionalgridsrangingfromstructured(regular)gridstoirregular
gridscomposedofarbitrarymixturesoftrianglesandquadrilaterals. Highlyirregular(badquality)gridsaredeliber-
atelyconstructedthroughrandomperturbationsofstructuredgrids. Threeclassesofgridsareconsidered. Class(A)
involvesisotropic grids in a rectangular geometry. Class (B) involveshighly anisotropic grids, typical of those en-
counteredingridadaptation. Class(C)involvesadvancing-layergridsvaryingstronglyanisotropicallyoveracurved
body,typicalofthoseencounteredinhigh-Reynoldsnumberturbulentflowsimulations.
Four basic grid types are considered: (I) regular quadrilateral(i.e., mapped Cartesian) grids; (II) regular tri-
angulargridsderivedfromtheregularquadrilateralgridsbythesamediagonalsplittingofeachquadrilateral;(III)
randomtriangulargrids,inwhichregularquadrilateralsaresplitbyrandomlychosendiagonals,eachdiagonalorien-
tationoccurringwithaprobabilityofhalf;and(IV)randommixed-elementgrids,inwhichregularquadrilateralsare
randomlysplitornotsplitbydiagonals;thesplittingprobabilityishalf;incaseofsplitting,eachdiagonalorientation
ischosenwithprobabilityofhalf.Nodesofanybasic-typegridcanbeperturbedfromtheirinitialpositionsbyrandom
shifts,thusleadingtofouradditionalperturbedgridtypeswhicharedesignatedbythesubscriptpas(I )-(IV ). The
p p
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randomnodeperturbationineachdimensionistypicallydefinedas 1ρh,whereρ ∈ [−1,1]isarandomnumberand
4
histhelocalmeshsizealongthegivendimension.TherepresentativegridsofClass(A)areshowninFigure1.
(a)Type(I):regularquadri- (b)Type(II):regulartrian- (c)Type(III): randomtri- (d) Type (IV): random
lateralgrid. gulargrid. angulargrid. mixedgrid.
(e) Type (Ip): perturbed (f) Type (IIp): perturbed (g)Type(IIIp): perturbed (h) Type (IVp): perturbed
quadrilateralgrid. triangulargrid. randomtriangulargrid. randommixedgrid.
Figure1. Typicalregularandirregulargrids.
III. Finite-VolumeDiscretizationSchemes
TheFVDschemesarederivedfromtheintegralformofaconservationlaw
(F·nˆ) ds= fdΩ, (3)
I Z
∂Ω Ω
whereΩisacontrolvolumewithboundary∂Ω,nˆistheoutwardunitnormalvector,anddsistheareadifferential.The
generalFVDapproachrequirespartitioningthedomainintoasetofnon-overlappingcontrolvolumesandnumerically
implementingequation(3)overeachcontrolvolume.
Node-centereddiscretization schemes are considered, in which solutions are defined at the primal mesh nodes.
Thecontrolvolumesareconstructedaroundthemeshnodesbythemedian-dualpartition: thecentersofprimalcells
areconnectedwiththe midpointsofthesurroundingfaces. Thesenon-overlappingcontrolvolumescovertheentire
computationaldomainandcomposeameshthatisdualtotheprimalmesh.Thenode-centereddiscretizationschemes
employthesamedegreesoffreedomongridsofalltypes;thus,accuracyisnotinfluencedbyavariationindegreesof
freedom.
ForinviscidEq.1,thenumericalflux,
Fh·nˆ ≡Uh(a·nˆ), (4)
atacontrol-volumeboundaryiscomputedacco(cid:0)rdingto(cid:1)theflux-difference-splittingscheme,26
1 1
Uh(a·nˆ)= (U +U )(a·nˆ)− |(a·nˆ)|(U −U ), (5)
L R R L
2 2
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where first and second terms represent the flux average and the dissipation, respectively; U and U are the “left”
L R
and “right” solutions linearly reconstructed at the edge midpoint by using solutions and gradients defined at the
nodesconnectedbythe edge. Theedge-basedflux integrationscheme approximatesthe integratedflux throughthe
two faceslinkedatthe edgemidpointby Uh(a·n), wheren is the combined-directed-areavector. The integration
schemeiscomputationallyefficient. Forexactfluxes,theintegrationschemeprovidesthird-orderaccuracyonregu-
larsimplicialgridsoftype(II),second-orderaccuracyonregularquadrilateralandgeneralsimplicialgridsoftypes
(I),(III),(II ), and (III ), and first-order accuracy on mixed-elementand perturbed quadrilateralgrids of types
p p
(IV),(IV ), and (I ).18,19,27 Second-orderunweighted least-squares (ULSQ) gradient reconstructionemploying a
p p
quadratic fit is used. It was shown that with such gradients, third order discretization accuracyis achieved on sim-
plicialgrids.21,22 Theresultsinthereferenceswereobtainedwiththeweightedleast-squaresgradients,butthethird
order accuracy has been confirmed by the authors of this paper for ULSQ gradients as well. In (infrequent) cases
whentheleast-squaresstencilofthenearestedge-connectedneighborsisnotsufficientforaquadraticfit,thestencil
isexpandedtoincludeneighborsofneighbors. Notethatfiveneighborsaretypicallysufficientforaquadraticfit. On
triangulargridsconsideredin this study, the averagenumberof edge-connectedneighborsis six; and the minimum
numberofedge-connectedneighborsforaninteriornodeonanygridisfour.
ForviscousEq.2,thenumericalfluxisdefinedas
Fh·nˆ ≡(∇rU ·nˆ), (6)
(cid:0) (cid:1)
where ∇rU is the gradient reconstructed at the face of the control volume. Two gradient reconstruction schemes
areconsidered: (1)TheGreen-Gauss(GG)scheme15,28 computesgradientsattheprimalelementsandusesthemin
face-gradientcomputationsatcontrol-volumeboundaries.TheGGschemeiswidelyusedinnode-centeredcodesand
equivalenttoaGalerkinfinite-element(linear-element)discretizationfortriangular/tetrahedralgrids.(2)Theaveraged
least-squares(Avg-LSQ)schemeaveragesthe ULSQ gradientsatthe nodesto computethe face gradient.24,25 Both
schemesusetheedgegradienttoaugmentthefacegradientandincreasetheh-ellipticity29ofthediffusionoperator15,23
and,thus,avoidcheckerboardinstabilities. Thegradientaugmentationisintroducedintheface-tangentform.25 Note
that when the edge is normalto the face, the edge gradientis the only contributorto the flux. For the GG scheme,
the augmentationdoes not affect the face gradientwithin a triangular element. It has been shown18,19,24,25 that the
schemespossesssecond-orderaccuracyforviscousfluxesongeneralmixed-elementgrids.
IV. Accuracy Measures
TheaccuracyofFVDschemesisanalyzedforknownexactormanufacturedsolutions. Theforcingfunctionand
boundaryvaluesarefoundbysubstitutingthissolutionintothegoverningequations,includingboundaryconditions.
Thediscreteforcingfunctionisdefinedatthenodes.Boundaryconditionsareover-specified,i.e.,discretesolutionsat
boundarycontrolvolumesand,possibly,attheirneighborsarespecifiedfromthemanufacturedsolution.
IV.A. Discretizationerror
Themainaccuracymeasureisthediscretizationerror,E ,whichisdefinedasthedifferencebetweentheexactdiscrete
d
solution,Uh, ofthediscretizedequations(3)andtheexactcontinuoussolution,U, tothecorrespondingdifferential
equations,
E =U −Uh, (7)
d
whereU issampledatmeshnodes.
IV.B. Truncationerror
Anotheraccuracymeasure commonlyused in computationsis truncationerror. Truncationerror, E , characterizes
t
theaccuracyofapproximatingthedifferentialequation(2). Forfinitedifferences,itisdefinedastheresidualobtained
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aftersubstitutingtheexactsolutionU intothediscretizeddifferentialequations.30 ForFVDschemes,thetraditional
truncationerrorisusuallydefinedfromthetime-dependentstandpoint.31,32 Inthesteady-statelimit,itisdefined(e.g.,
in[33])astheresidualcomputedaftersubstitutingU intothenormalizeddiscreteequations(3),
1
E = − fhdΩ+ Fh·nˆ ds, (8)
t V Z I
(cid:0) (cid:1)
 Ω ∂Ω 
whereV isthemeasureofthecontrolvolume,
V = dΩ, (9)
Z
Ω
fh isanapproximationoftheforcingfunctionf onΩ,andtheintegralsarecomputedaccordingtosomequadrature
formulas.Truncationanddiscretizationerrorsarerelatedthroughtheerrortransportequation34
L (E )=−E , (10)
h d t
whereL isthelinearizationofthediscretegoverningequations.
h
Ithasbeenlongknownthatconvergenceoftruncationerrorsseverelydegradesonirregular(low-quality)meshes.
Suchdegradation,however,doesnotnecessarilyimplyalessthandesignorderconvergenceofdiscretizationerrors.
Plentifulcomputationalevidenceandasolidbodyoftheoryconcerningtheeffectofsupra-convergenceindicatethat,
on irregular grids, the design-order discretization-error convergence can be achieved even when truncation errors
exhibitalower-orderconvergenceor,insomecases,donotconvergeatall.9–11
IV.C. Accuracyofgradientreconstruction
The accuracy of the gradientapproximationis also important. The accuracy for an ULSQ gradientis evaluated by
comparingthereconstructedgradient,∇rU, with theexactgradient,∇U, computedatthenode. Theaccuracyofa
GG gradient is evaluated at an element. The reconstructed and exact gradients are compared at the element center
computedastheaverageoftheelementvertexes. Inaccuracyevaluation,thegradientreconstructionusesasampling
oftheexactsolutionU atthenodesonagivengrid. Theerrorinthegradientreconstructionismeasuredas
E =kǫ k, (11)
g g
wherefunctionǫ istheamplitudeofthegradienterrorevaluatedatanode,
g
r
ǫ =|∇ U −∇U|, (12)
g
andk·kisanormofinterest.
V. ClassA:IsotropicGridsinRectangular Geometry
V.A. Gridandsolutionspecifications
Sequencesof consistently refined19 grids with 52,92,172,332,652,1292, and 2572 nodesare generatedon the unit
square[0,1]×[0,1].Irregularitiesareintroducedateachgridindependently,sothegridmetricsremaindiscontinuous
onall irregulargrids. With the randomperturbationrangelimited bya quarterofthe localmesh size, the anglesof
triangularelementscanapproach180◦andtheratiooftheneighboringcellvolumescanbearbitrarilyhigh.
The exact solutions is U = sin(πx−2πy), so for the inviscid Eq. 1 with a = (2,1), the force is f = 0, and
fortheviscousEq.2,f = −5π2sin(πx−2πy). Thenumericaltestsevaluatingtruncationanddiscretizationerrors
are performed with boundary conditions over-specified from the manufactured solution for all nodes linked to the
boundary.
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V.B. Gradientreconstructionaccuracy
Figure2showsthevariationoftheL normofthegradienterror.Asexpected,theULSQgradientreconstructionwith
1
aquadraticfitissecond-orderaccurateonallgrids. TheGGgradientreconstructionissecondorderonlyonperfect
gridsoftype(I); onallothergrids,theGGgradientsarefirst-orderaccurate. Allequivalent-ordermethodsprovide
verysimilarerrors.Thus,nomeshqualityeffectsareobservedfortheL normofthegradienterroronisotropicgrids.
1
Althoughnotshown,theobservedL normsofthegradienterrorsconvergewiththesameordersasthecorresponding
∞
L norms, but the L norms of GG gradient error on grids of types (III ) and (IV ) are an order of magnitude
1 ∞ p p
greater than the L normsof other first-order errors. The latter effectis caused by gradientaccuracydeterioration
∞
ontriangularelementswithobtuseanglesapproaching180◦. Theoretically,withaninfinitesimalprobability,theGG
gradienterrormaybecomeinfinitelylargeatanelementwithavanishingvolume.
101 101
(I) (I)
(II) (II)
(III) (III)
100 (IV) 100 (IV)
(I) (I)
p p
m) (IIp) m) (IIp)
Gradient error (L − nor11100−−21 ((123IIsnrIVdItdp p oo)o)rrrddd... Gradient error (L − nor11100−−21 ((123IIsnrIVdItdp p oo)o)rrrddd...
10−3 10−3
10−4 10−4
10−2 10−1 100 10−2 10−1 100
effective meshsize effective meshsize
(a)ULSQ-quadraticfitatnode (b)GGatelement
Figure2. Accuracyofgradientreconstructiononisotropicgrids.ManufacturedsolutionisU=sin(πx−2πy).
V.C. Truncationerrors
Thetruncationerrorsareextremelysensitivetomeshquality.ConvergenceratesoftheL normoftruncationerrorsfor
1
inviscidandviscousfluxesareshowninFigures3and4,respectively.Theinvisciderrorsconvergewiththethirdorder
onlyonregulartriangularmeshesoftype(II). Onirregulartriangulargridsoftypes(III),(II ),and(III )andon
p p
perfectquadrilateralgridoftype(I), theinviscidtruncationerrorsconvergewiththesecondorder. Irregularitieson
gridswithquadrilateralelements(types(IV),(I ),and(IV ))leadtothezeroth-orderconvergence.
p p
SimilarsensitivityisobservedforthetruncationerrorsofviscousfluxesdiscretizedbytheAvg-LSQschemewith
the second-orderaccurateULSQ gradients(Figures4(a)and 4(b)). Thesecond-orderconvergenceis observedonly
onperfectlyregulargridsoftypes(I)and(II). Theconvergencedeterioratestothefirstorderonirregulartriangular
gridsand to thezerothorderonmixed-elementandperturbedquadrilateralgrids. For theviscousfluxesdiscretized
with theGG scheme(Figures4(c)and4(d)),truncationerrorsdonotconvergeonanybutperfectlyregulargridsof
types(I)and(II). NotethatGGschemeproducesidenticaldiscretizationsongridsoftypes(I),(II), and(III).15
Thus,correspondingGGsolutionsandtruncationerrorsongridsoftypes(I)and(II)arealwaysidentical. Different
resultsongridsoftype(III)areexplainedbythedifferencesinthedualvolumes.
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100 100
(II) (I)
(III) (IV)
10−1 ((IIIIpIp)) 10−1 ((IIpV)p)
1st ord. 1st ord.
Truncation error (L − norm)1111000−−−432 23nrdd oorrdd.. Truncation error (L − norm)1111000−−−432 23nrdd oorrdd..
10−5 10−5
10−6 10−6
10−2 10−1 100 10−2 10−1 100
effective meshsize effective meshsize
(a)Triangulargrids (b)Mixedandquadrilateralgrids
Figure3. Inviscidtruncationerrorsonisotropicgrids.ManufacturedsolutionisU=sin(πx−2πy).
V.D. Discretizationerrors
ConvergenceratesoftheL normofdiscretizationerrorsforinviscidandviscousfluxesareshowninFigures5and6,
1
respectively. Discretization errors for viscous fluxes show no sensitivity to mesh quality. The errors for both Avg-
LSQandGGschemesarepracticallyidenticaltotheplottingaccuracyforallgrids.Theaccuracyofinviscidsolutions
deterioratesonmesheswithquadrilateralelements.Thisisnotasurprisebecausetheinviscidschemeusedinthisstudy
isdesignedtobethirdorderonlyonsimplicialgrids.21,22 Theedge-basedintegrationschemeusedinthisschemeis
knowntodeterioratetofirstorderongridsoftypes(I ),(IV),and(IV ).18,19,27 Ontriangulargrids,thediscretization
p p
accuracyofinviscidsolutionsisnotsensitivetomeshquality. Ifanything,discretizationerrorsaresomewhatsmaller
ontopologicallystructuredgridsoftypes(II)and(II ).
p
VI. ClassB: AnisotropicGridsinRectangular Geometry
This section considers FVD schemes on irregular stretched grids generated on rectangular domains. Figure 7
showsanexamplegridwiththemaximalaspectratioA=1000.Asequenceofconsistentlyrefinedstretchedgridsis
generatedontherectangle(x,y)∈[0,1]×[0,0.5]inthefollowing3steps.
1. AbackgroundregularrectangulargridwithN =(N +1)×(N +1)nodesandthehorizontalmeshspacing
x y
h = 1 isstretchedtowardthehorizontalliney = 0.25. They-coordinatesofthehorizontalgridlinesinthe
x Nx
tophalfofthedomainaredefinedas
y =0.25; y =y +hˆβj−(cid:16)N2y+1(cid:17), j = Ny +2,...,N ,N +1. (13)
N2y+1 j j−1 y 2 y y
Here hˆy = hAx is the minimalmesh spacing between the vertical lines, A = 1000 is a fixed maximal aspect
ratio,andβ isastretchingfactorwhichisfoundfromtheconditiony =0.5. Thestretchinginthebottom
Ny+1
halfofthedomainisdefinedanalogously.
2. Irregularitiesare introduced by random shifts of interior nodes in the vertical and horizontal directions. The
verticalshiftisdefinedas∆y = 3 ρmin(hj−1,hj),whereρisarandomnumberbetween−1and1,andhj−1
j 16 y y y
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101 101
(II) (I)
(III) (IV)
(IIp) (Ip)
100 (IIIp) 100 (IVp)
1st ord. 1st ord.
Truncation error (L − norm)11100−−21 23nrdd oorrdd.. Truncation error (L − norm)11100−−21 23nrdd oorrdd..
10−3 10−3
10−4 10−4
10−2 10−1 100 10−2 10−1 100
effective meshsize effective meshsize
(a)Avg-LSQ;triangulargrids (b)Avg-LSQ;mixedandquadrilateralgrids
101 101
(II) (I)
(III) (IV)
(IIp) (Ip)
100 (IIIp) 100 (IVp)
1st ord. 1st ord.
Truncation error (L − norm)11100−−21 23nrdd oorrdd.. Truncation error (L − norm)11100−−21 23nrdd oorrdd..
10−3 10−3
10−4 10−4
10−2 10−1 100 10−2 10−1 100
effective meshsize effective meshsize
(c)GG;triangulargrids (d)GG;mixedandquadrilateralgrids
Figure4. Viscoustruncationerrorsonisotropicgrids.ManufacturedsolutionisU =sin(πx−2πy).
andhj areverticalmeshspacingsonthebackgroundstretchedmesharoundthegridnode. Thehorizontalshift
y
is introducedanalogously, ∆x = 3 ρh . With these random node perturbations, all perturbedquadrilateral
i 16 x
cellsareconvex.
3. Eachperturbedquadrilateralisrandomlytriangulatedwithoneofthetwodiagonalchoices;eachchoiceoccurs
withaprobabilityofonehalf.
Sequences of consistently refined stretched grids with maximum aspect ratio A = 1000 including 9 × 65,17 ×
129,33× 257, 65 × 513, and 129 × 1025 nodes have been considered. The corresponding stretching ratios are
β ≈1.207,1.098,1.048,1.025,and1.012.Theaspectrationeartheexternalhorizontalboundariesisabout2.7.
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10−1 10−1
(II) (I)
(III) (IV)
10−2 ((IIIIpIp)) 10−2 ((IIpV)p)
m) 1st ord. m) 1st ord.
Discretization error (L − nor1111000−−−543 23nrdd oorrdd.. Discretization error (L − nor1111000−−−543 23nrdd oorrdd..
10−6 10−6
10−7 10−7
10−2 10−1 100 10−2 10−1 100
effective meshsize effective meshsize
(a)Triangularmeshes (b)Mixedandquadrilateralmeshes
Figure5. Invisciddiscretizationerrorsonisotropicgrids.ManufacturedsolutionisU=sin(πx−2πy).

(I) (I)
(II) (II)
(III) (III)
10−2 (IV) 10−2 (IV)
(I) (I)
m) (IpI) m) (IpI)
Discretization error (L − nor11100−−43 ((123IIsnrIVdpItdp p oo)o)rrrddd... Discretization error (L − nor11100−−43 ((123IIsnrIVdpItdp p oo)o)rrrddd...

10−2 10−1 100 10−2 10−1 100
effective meshsize effective meshsize
(a)Avg-LSQ (b)GG
Figure6. Viscousdiscretizationerrorsonisotropicgrids.ManufacturedsolutionisU=sin(πx−2πy).
InthetestsongridsofClass(B)performedwitheitherthemanufacturedsolutionsin(πx−2πy)orextendedover-
specificationusedintestsongridofClass(A),theasymptoticbehaviorofthediscretizationserrorsforviscousfluxes
wasnotobservedon coarsegrids. Theexhibiteddiscretizationerrorswere uncharacteristicallylow oncoarsegrids,
butdidnotconvergewiththeasymptoticorder.ThemanufacturedsolutionhasbeenchangedtoU =cos(πx−2πy).
Also only solutions at boundary nodes are over-specified. With these changes, the asymptotic behavior of the dis-
cretizations errors for the GG viscous fluxes is established on relatively coarse grids. Note that the force term for
inviscidequationsisstillf =0fora=(2,1).
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0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure7. Stretchedgridoftype(IIIp)with9×65nodes.
Theconvergenceplotsoftruncationerrorsarenotshownfortestsof Class(B). Thequalitativebehavior(orders
ofconvergence)oftruncationerrorsisthesameasonisotropicgrids,showninFigures3and4Themagnitudeofthe
errorsisincreasedbythreeordersofmagnitude,proportionaltotheaspectratioA=1000.
VI.A. Gradientreconstructionaccuracy
103 103
(I) (I)
(II) (II)
102 (III) 102 (III)
(IV) (IV)
(I) (I)
p p
m) 101 (IIp) m) 101 (IIp)
Gradient error (L − nor∞ 111000−−021 ((123IIsnrIVdItdp p oo)o)rrrddd... Gradient error (L − nor∞ 111000−−021 ((123IIsnrIVdItdp p oo)o)rrrddd...
10−3 10−3
10−4 10−4
10−3 10−2 10−1 10−3 10−2 10−1
effective meshsize effective meshsize
(a)ULSQ-quadraticfitatnode (b)GGatelement
Figure8. AccuracyofgradientreconstructiononstretchedgridswithmaximumaspectratioA=1000.ManufacturedsolutionisU=cos(πx−2πy).
A recent study20 assessed the accuracy of gradient approximations on various irregular grids with high aspect
ratio A = hx ≫ 1. The study indicates that for rectangular geometries and functions predominantly varying in
hy
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