This 3-sentence summary provides the high-level information from the document: The document describes a zonal approach for simulating external vehicle aerodynamics using a panel method for the inviscid flow field, a boundary layer method for viscous effects, and a wake model to simulate separated flow. The methods are coupled iteratively, with the potential flow solved using a panel method, boundary layer separation determining the starting point for the wake model, and the wake influencing subsequent boundary layer and potential flow solutions. The approach aims to provide a fast simulation tool for vehicle aerodynamics in early development phases when analyzing multiple variants.

1. SAE TECHNICAL
PAPER SERIES
Simulating External Vehicle
Aerodynamics with Carflow
Norbert Griin
TESlS GmbH
Reprinted from: Vehicle Aerodynamics: Wind Tunnels, CFD,
Aeroacoustics, and Ground TransportationSystems
(SP-1145)
mA The Engineering Society
For Advancing Mobility
-land SeaAir and Space,
I N T E R N A T I O N A L
InternationalCongress & Exposition
Detroit, Michigan
February26-29,1996
400CommonwealthDrive,Warrendale, PA15096-0001U.S.A. Tel: (412)776-4841 Fax:(412)776-5760

2. Theappearanceofthe ISSNd eatthebottomofthispageindicatesSAE'sconsent
thatwpiesof the w e r maybemadefor personalor internaluseofspecificclients.
Thiswnsent isgi;e"onth&ndition however,thatthecopierpaya$7.00perarticle
mpy fee through the CopyriQhtClearance Center. inc. Operations Center, 222
RosewoodDrive.Danvers.MA01923forcopyingbeyondthatpermittedbySections
107or 108of U.S. copyri$htLaw. This consent does not exiendto other kindsof
copying such as copying for general distribution, for advertising or promotional
purposes, for creating new collectiveworks, or for resale.
SAE routinely stocks printed papers for a period of three years following date of
publication. Direct your orders to SAE Customer Sales and Satisfaction
Department.
Quantii reprint rates can be obtainedfrom the Customer Sales aod Satisfaction
Department.
To request permissionto reprintatechnicalpaperor permissiontousecopyrighted
SAE publicationsinother works, contactthe SAE PublicationsGroup.
AllSAEpaPer.5, standards,andselected
bwks an,abstracted and indsxedin lhe
GlobalMobilwDafabase.
Nopart of this publicationmay bereproducedinany form, inanelectronicretrieval
systemor otherwise, without the prkr writtenpermissionof the publisher.
ISSN.014&7191
Copyright 1996 Society of Automotive Engineers, Inc.
Positions and opinions advanced in this paper are those of the author(s) and not
necessarilv those of SAE. The author is soielv reswnsible for the content of the
paper. A irocess isavailablebywhichdiscus~onshillbe printedwiththe paper if
itispublishedinSAETransactions. Forpemlissionto publishthis paper infullor in
part, Wntact the SAE PublicationsGroup.
Personswishing to submit papersto beconsideredfor presentationor publication
through SAE should send the manuscriptor a 300 word abstract of a proposed
manuscriptto: Secretary, EngineeringMeetingsBoard, SAE.
Printed inUSA 9 ~ 4 x 9

3. Simulating External Vehicle
Aerodynamics with Carflow
Norbert Griin
TESlS GmbH
Copyright 1996 Societyof AWornotive Engineen,Inc.
ABSTRACT
A zonal approach to simulate the incompressible and
steadyflow aroundautomobilesis presented.
The method incorporates two componentsforthe invis-
cid and viscous domain of the flow field at high Reynolds
numbers. Although in many details it is especially well
suited for automobile aerodynamics, the code may also be
appliedto trains, submarines or aircraft.
For the inviscid part of the flow field, a first order
panel method is used which is also able to simulate the
separatedflow downstream of the vehicle's base, by means
of free shear layers. A constant source density is assigned
to solid body panels, while a bilinear doublet distribution,
equivalent to a panelwise constant vorticity, is used on the
wake panels, which represent the freeshear layers.
The main objective of this wake model is to simulate
the influence of separation on the vehicle's pressure dis-
tribution, rather than reproduce the wake structure in
detail.
Viscous effects are accounted for by a threedimen-
sional integral boundary layer code, working in general
surface coordinates. The generation of a structured surface
grid is part of the method. Besides friction drag and vis-
cous results on the surface,the boundary layer analysis
yields a three-dimensional separation line as the starting
locationfor the wake simulation.
As the present approach requires only a discretization
of the vehicle's surface, the effort of model generation and
data handling is reduced substantially compared to field
methods.
Depending on the level of model details, typical tnrna-
mund times of two or three days for the generation and
analysis of fiveto ten variantsmay be achieved.Parametric
modifications of the panel model are enabled by the code
itseIf.
Often, when conducting wind tunnel tests, it is not
quite understood where differences between variants come
from, and the success of shape modifications depends on
the skills and experience of the experimental aerodyna-
micists. The exampleapplicationsdemonstratesomeof the
optionsto gain insight into detailsof the flowfield.
Comparisons with experimental results show a good
agreement of pressure distributions in regions of attached
flow. A larger difference arises between measured and
computed absolute drag coefficients, due to the extreme
a m c y requirements,discussed in chapter 1.
However, satisijing results are obtained when the ef-
fect of shape modifications or the drag ranking of different
vehiclesorvariants is concerned.
INTRODUCTION
In the early phase of vehicle development a fast tool is
required to predict aerodynamic characteristicsin order to
reduce the number of variants finally realized in hardware
and tested in a wind tunnel.
Current Reynolds solvers are still too time consuming
in mesh generation and flow field analysis to keep pace
with thefrequent creationof shapevariantsat this stage.
Alter a thorough analysis of the requirements and the
stateof the art in CFD concerningexternalvehicleaerody-
namics [I], the BMW AG initiated the creation of the code
presented in this paper.
The method has been continuously improved parallel to
its production application, leading to a fast tool especially
tailored for the needs of externalautomobileaercdynamics.
1 ACCURACY CONSIDERATIONS FOR THE
PREDICTIONOF ABSOLUTE DRAG LEVELS
Before presenting the numerical method, it is useful to
discuss some basic topics of the mechanisms and para-
metersof draggeneration.
The following remarks should emphasize the extreme
a m c y requirements, any CFD code has to cope with, if
the absolutelevel of drag coefficient
Fo I
C D = ~with q,=Tp-v.vZ, (1)
is to be predicted.

4. In contrast to aerospace definitions, in automobile
aerodynamics the drag force F, is the component along
the longitudinalaxisx of the vehicle parallel to the ground,
regardless of yaw angle or incidence. As reference area
A,,, usually the frontal area of thevehicle is chosen.
From the physical point of view, total drag can be split
up intofrictiondragCwand pressure drag C,.
If shear stress7 and pressure distributionp are known,
the total drag can be evaluated by integrating over the ve-
hicle's surface
FD=j.rXd4+I-@-p-)n,d4 , (2)
or in nondimensionalform
where Cfi=7,/q, and Cp=@-p,)/q, are the coeffi-
cients of skin friction and static pressure and n, is the
x-component of the nondimensional surfacenormal vector.
A geometry-based viewpoint is to distinguish the con-
tributions upstream and downstream of the maximum
cross-section of a vehicle, i.e. forebodydrag C,, and after-
body dragc,.
In the ideal case of inviscid (potential)flow,the suction
forces on fore- and a f t e w y cancel exactly and the total
dragvanishes, independentof the body's shape.
Wmus effectsin real fluids disturbthis balance. Even
without separation the displacement effectof the boundary
layer causes a lower pressure level on the afterbody, lead-
ing to a resulting force pointing downstream (friction in-
, ducedpressure drag).
If separation can be avoided on the forebody, then the
theoretical lower limit of forebody drag is nearly reached,
because boundary layer displacement effects are very small
in this region. Independent of geometry details, the fore-
body pressure drag depends only on the lengthdiameter
ratio of the whole body [2,3]and is always CD~,F2 0.The
possiblemaximum of CD.E. =0 is reached for semi-infinite..
bodies.
On bluff bodies, Like road vehicles, massive separation
occurs on the afterbody and leads to a drastic deviation of
the pressure distribution compared to inviscid flow, also
upstream of separation. The forebody drag is also affected
by separation, because the wake increases the effective
length-diameterratio towardsa semi-infinitebody.
Inuarticular. the total ~ressurein the wake is below the
freest- val"e and thus decreases the static base
pressure.
Introducinga total pressurecoefficient defined as
PI-PI..
cpt=, , (4)
the energyequation
P, = Pr- + A p t
I
p+fpv2 = p_+fpvl + Cpl;ip~l (5)
local = beestream + loss(<O)
-,I ... ..
li i ,.*: .::. . .
~,.~,> ...;;.,,, ;,. ::.
,. . ,!*. . .?.'~.,:, ..
canbe used to expressthe static pressure d c i e n t by the7:''':
velocityand the pressure loss
F-P- Y 2
Cp =, +Cpt . (6)
Definition Energy Equation
In incompressible inviscidflowthere is no pressure loss
so that Cpl=0 and Cp= 1-(V/V,)~.If a pressure loss oc-
curs, as for instance on the base of an automobile, then
C,, <0 and the staticpressure is additionallydecreased.
m i d orders of magnitude of the terms in Eq.(6) in
the separated region on an automobileare
This simple estimation demonstratesthat the influence
of the total pressure loss on the static pressure in separated
regions is almost two orders of magnitude higher than the
influence of the velocity. In addition it is worthwile to
point out, that the static pressure does not depend on the
directionof thevelocityvector.
Thus the crucial parameter for numerical methods to
predict the absolute value of drag correctly is the pressure
loss in the wake.
Most CED codes use nondimensional notation. Some-
times total pressure lossesare expressedas Ap~lpt, in [%].
A simple considerahon reveals that, although nondimen-
s~onal,this quantity may be a very misleadingmeasure.
Eq (4) canbe written as
where the ratio in brackets dependson the velocity v, .
Free StreamVelocity V, Krn!h]
. .
Fig.1 TotalPressure CoefficientCpl vs. Velocifyv,
fordifferent Values ofAptlp* [%]
. . ., ,.,
., . ~.... .

5. For example, at v, =200 km/h with p, =101325N/m2
and p, =1.225 kglm3
Under these conditions, already 1% total pressure
loss changes the static pressure d c i e n t according to
Eq.(6)by AC, =-0.55 (s. Fig. 1).
The contniution of the separated region (base) to the
totalpressure drag is
i.e. the sensitivityof the drag d c i e n t to errors in base
pressure dependson the ratio of separ;ltedareato reference
area.
This value varies from at least 0.5 for fastback cars to
almost1.Oforvantypem
Topredict the absolutevalueof base dragwith anaccn-
racy of ACo, =0.005 (which is about what experimental
aerodynamicis& demand) requires an accuracy in static
pressure of at least AC, =0.01. which will hardly be met
byanyCFDwdeinthenearfuhue.
-
Fig.2 PressureDrag Integration along the Longitudinal
Axis of a Vehicle (NumericalResult)
Another uncertah@ concerningthe aa~racyof total
drag &ses when inte-g the surface pressure dishibu-
tion m.3).
While the skin friction generates only positive wn-
tributions, the static pressure d c i e n t and the &
normal component n, vary their signs over the &ce,
leading to an up and down of the drag d c i e n t , when
plotted overthelongitudinalaxisofthe vehicle fFig.2).
The resulting final value is the sum of positive and
negative contributions(or the difference of fore-and after-
body drag), sothat errors in thepressure field aswell asin
theintegrationalgorithmmay add orcanceleachother.
The numerical examplein Fig.2 alsodemonstrata that
pressure drag is almost completely generated on theafter-
body dueto the differencein staticpressure with andwith-
out separation. Despite the elliptical character of sobsonic
flow, theupstream iduence of thewake fadesvery fast
2. THEORETICALAPPROACH
At high Reynolds numbers, flow field domains can be
distinguished where certain phenomena are predominant
or may be neglected fFig.3). Each of these zones is gw-
emed by simpli6ed versions of the equations of motion,
which can be solved separately. Theirphysical interaction
hastobe accountedfor in aniterativeprocess.
Fig.3 Domainsof the Flow Field arounda Vehicle
at high ReynoldsNumbers
The method presented here is sucha zonal approach to
simuiate the complete flow field around vehicles using
singlecomponentsatthehighestlevel of simpli6cation
Panel Method
0 BoundaryLayerMethod
WakeModel
The following table gives an overview, which method
is usedto simulatewhich physical phenomenon in thedif-
ferentdomainsof Fig.3.
m'EluuL mcous v o m c .
........................
panelMeUld ip:$m@........... ....... ....,....~..
..*:................. ~.,::.:<.<*:: ~$i,:$:.......:..::<: *,$:$ $;:><$. ~:............ ........
BoundaryLayer :~~Bi:;;~,:,i,.. ...... ...... ii..::,q$j:+j:. ~ . 3;;.jxES:ES:ES:iljF:::...::.:.................. .. .. ..... . ...~........
w&eMdel $-iBgB,:;::!jii NO '<..::::~>>-"...........................................~....~.........

6. The panel method yields an inviscid and irrotational
flow field. V i u s effects on solid walls, simulatedby the
boundary layer methcd, introduce vorticity. At the separ-
ation line the wall boundary layer turns into a L?ee shear
layer. The convective hmsport of the vorticity contentsin
thislayeris simulatedbythewakemodel. Asthiswake
model is inviscid again, no dissipation of vorticity occurs.
Nevertheless its diffusive propagation may be accounted
for by an increasing thickness of the shear layer's vortical
core.
2.1 COUPLING OF THE ZONAL SOLUTIONS
An iterativestrategywiththreenested loops asdepicted
in Fig.4 is used to accountfor the interaction of the differ-
ent solutions.
Startingfrom the potential flow field, the wall bound-
ary layer is calculated and a iint separation line is gener-
atedasthe startinglocationforthewake model.
The initial wake is then created along the strramlines
oftheknownflowfield.
In the hermost loop the vorticity contents of the
wake, which depends on the velocity field and the wake
shapeitself, is iteratedto convergence.
Next the wake shape may be relaxed at fixed vorticity
contentsand separationlineusingthe velocityfield includ-
ingwakeinfluence.
If wake shape and strength have converged, the veloc-
ity field has changed. The following boundary layer cal-
culation now produces a separation line including wake
influence and the two inner loops, integrated in the panel
method, canbe iteratedagain.
(nun) (rm)
Fig.4 GeneralStrategyof CouplingZonalSolutions
2.2 POTENTIAL FLOW
Potential flowisirrotational, i.e.
andpreserves continuiv
Introducinga scalar velocitypotential@
which is irrotationalper definition, because
and insertingEq.(13) into Eq.(12) leadsto thebasic equa-
tion of linearized potential flow theory, the Laplace equa-
tionforthevelocitypotential
Thisequationcanbe solved by a linear combination of
the potential cp, of the free stream and a disturbance po-
tential cp
+ + + -+ -+@(x)=cp-(x)+cp(x)=v,* x +cp(x) , (16)
or expressedin terms ofvelocityusingEq.(13)
Fig.5 Contributionsto the Disturbance Velocity
According to Green's theorem the volume integral of
the Laplace equation can be replaced by surface integrals
over the boundary S of the flow field (Fig.5), so that the
I d velocityis composedby [4]
where

7. The generalfunctionscr(8 andp(8 in Eq.(19) haveto
be adjusted to the boundary conditiom of the paaiculat
problem.
Thevelocitycontribution$ inEq.(19) isthe induction
of sources (or sinks) with density r~(8[(d/s)I)lnZ]M b -
uted over surface S. It canbe shown, that thispart of the
flowfield isirrotationaleverywhere.
<and are inducedby.a doublet dishiiution p(8
[m5s]. In repiom where p is constant, $mishes, be-
cause the integrand contains the gradient of the doublet
d i s h i i o n
The last contnion apprsonly if surfaceS isnot
closed, because it's evaluationmpim a line integralalong
the edge aS of surface S. Although the velocity field, in-
+ . .duced by <and v a IS m general irrotational, it contains
vorlicity at least in singular surfaces or lines, where
doubletshavebeenplaced.
2.2.1 The Equivalenceof Doublets and Vorticity
To demonstrate the equivalence between doublets and
vorlicity, let's consider a shear layer of finite thickness 6
- + .withvorticity dishibution o(Fig..6).
which reducesthethree dimensionalvorlicitydistribution
to avorticitylayerof vanishingthiclmess, representinga
shearlayeratRe +m (Fig.7).
+
Fig.7 LayerElement with Vo~tiicityContents
Now it requires only the evaluation of a surface
integral
to calculatethe inductionofthe shearlayer.
A further integration of the vorticity contents within
the shear layer's plane (Fis.8) leads to a line vortex with
the circulation
Fig.6 Volume Element with ~orficify$
At anarbi- point 2the whole shearlayerinduces
whoseinductionisexpressedby a lineintegral
++ ++ -3 + + +v,(x)=&JjJw(xY)x$~.with r = x - X Y (20)
v Fig.8 tine Element with Circulation r
We cannow introducea quantity calledvorticitycoo- Ifwe now compare Eq.(19) with Eq.(22), the analogy
tents Q[S] by integratingperpendicularto the shear layer between doubletsand vorlicity isclear

8. and the componentsof the vorticity-wntentsvector in local
&ce cw~dinates(s,t,n) i i d l y read
-+
which shows that the vector lies in the plane of the
shearlayer(Q,=O).
Simulatingpotential flow numerically requires solving
the functions a(S)andMS)in Eq.(19) according to the
boundaryconditionsof the m c u l a r problem.
For this purpose, S is discretized by finte mrkce el-
ements, calledpanels. Sincethe panel model (s.Fig.9)does
not have to be a structured grid, complex configurations
may be modeled easily.
fig.S Panel Modelof a Vehide Surface
Generally one of the functions a Q andp(S) can be
p m a i i art,iWy and the other is treated as the un-
known [6].
Theboundaky S of the flow field shall be composed of
S=B+W, where B standsforthe surface of solidbodies and
Wrepresentsfkshearlayers.
The panelwise distniution of sourcesand doublets on
Band Winthisfirstorder method isselectedtobe
A linear doublet distribution, equivalent to a constant
vorticity distribution (Fig.10) is chosen to achieve a con-
sistentorder of accnracyon sourceand doublet panels [6].
Fig.10 PanelwiselinearDoublet Distribution
Using a constant doublet distriiutioq the vorticity
would vanish within each panel and only the jump in
doublet strength at adjacent panel edges would remain ac-
tiveaslinevorticeswith circulation T=AII (Fig. 1I).
Fig.11 Panelwiseconstant Doublet Distribution
2.2.2 Solid BodySimulation
Sincethe doublet strength p=0 ispreassigned to solid
bodypanels, only oneboundary condition
is necessary(and possible) to calculatethe sourcedensitya
This v.Neumann type condition p r e s c n i the normal
component of the resulting velocity on surface B, which
canbe
. .
c 0 : flowintothe snrface
v. = 0 : impermeablesurface . (28)
> 0 : flowoutofthesurface
.. . ., . .. ..~ . .
.:.

9. It is importantto emphasizethat thiscondition is only
p e d at the positive side of sutfacenormal 2.In the in-
terior of the pane1 model (n<O) a flow field is produced,
which hasnophysicalmeaning.
Inserting Eq.(18) in Eq.(27) leads to an integral
equation
for theunknown sourcedistributioncs (s. Eq.19).
If vortex layers are part of the model, their influence
+ +v, +va appearson the right hand side of Eq.(29). Doublet
mength CI can not be simultaneously solved, because the
Analytical solutionsforthekernelintegrals
in a local coordinate system (s,t,n, s.Fig.12) for planar
quadrilateralpanelsare given in [fl.
Theboundaxyconditions(Eq.29) for allN p e l s con-
stitutea systemof linear equations
+ + + +[ ~ d(Cj) =(v, -ni .(v., +v, +va)) , (34)
wake-&& is part of the solution its&. For the moment,
the wake influence is assumed to be Itnoown as an outer in-
where the matrixme0icientsare
fluence,likethefreestreamvelocityv, .
In dkrelked form the integral overB is replaced by a
sum over the inductionof all sourcepanels
where 2isthe vector fromthe smface elementd 4 ofthe
inducing panel 6)tothe control point 2of the influenced
panel (i).
As in this firstorder method aj is constant over Bj it
canbe written outsidethe integraland a purelygeometrical
exvression
remains asintegrand, sothat Eq.(30) becomes
The calculation of these influence me0icients is the
most time consuming part of the whole panel method De-
pending on the distance between in€luencing and in-
fluencedpanel, two approximationsfor thekernel integrals
[fl,up to the induction of a simple point source, are used
to speed up matrix generation
For realistic models the linear equations system is too
largeto be kept in core. Duetothe strongdominanceof di-
agonalelementsEii, which containthe influenceofa panel
on it's own controlpoint, theiterative Gauss-Seidel method
is used for solution. This method requires only one line of
the matrix in core at a time. However, depending on the
platform, a variablenumberof matrix lines(uptothe com-
plete mafix)may be kept in core to reduce the number of
vooperations.
generic automobilemodels usually between 10and
20 iterations are necessary to bring the maximum residual
++ N ---f
below v,,Ev., =10". A p k r convkgence rate oarurs only
v,(xi) = 2 oJCV. (32) at very complex configurations with strong influence be-
i=1
+ tween adjacentp e l s . Divergenceof the iterationhas al-
Thevector C,.represents the induced velocity at panel ways been an indication of possible errors in the panel
i for a unit sourcedensity(csj= 1)onpane1j. model, for instancethe wxrrence of identicalpanels.
Once Eq.(34) has been solved, the source densities csj
are known and the resulting velocity (Eq.18) canbe evalu-
ated at each smfacep e l , using the influenced c i e n t s
(Eq.31). An externalflow fieldsurvey is alsopossible now
by evaluating Eq.(30) for arbitmy points off the panel
model's surface. The static pressure d c i e n t follows
fromEq.(6) with Cfl=0 outsideof thewake.
2.2.3 Wake Simulation
The basic idea of wake simulation is the followingde-
finitionofseparationat high Reyn01ds numbers [8] :
At separation the boundmy layer leaves the wall and
turns into a ofe shear lnyer, convecting the vorticity
Fig.12 Local CoordinateSystem of a Quadrilateral that has been generated by viscousforces at the solid
Panelw%hconstant SourceDensity wall.

10. Inthis sense,anyflowfield arounda !kitebody,devel-
opinga boundarylayer, showsseparation 1
This includes for instancethe flowoff a rvlngs nailing
edgewithout the cccurrenceof recirculation, a case usually
called attached flow. Strictly attached flow would turn
around the trailing edge and leave the surface on the suc-
tion side, producing no lift (irrotational potential flow).
FTemiiinga Kutta condition at the trailing edgeis already
a form of wake simulation.
Once separation has m e d , viscous effects are ne-
glected in the wake, so that the vorticity contents of the
frez shear layer remains constant andisequal to that of the
boundarylayerat separationFig 13).
. , . ... .. . . ~..
I,, &,-&afor theboundarylayer(where ui=6jt$ey :,:+:'.
vorticitycontentsSLt perpendicular to the main flow direc- :
tion dependsonlyon thevelocity at the outer edge
regardless of the detailed velocity profile u(n). The local
component IL, parallel to the main stream vauishes, be-
cause in the cross flow profile v(n) of a boundary layer
v, =0 andv. =0 andno kinematicactivevorticityremains
[51.
Forthe numericalwake simulationthefree shearlayer,
originating at the separation line, is discretized by j,.
nodes in circumferential and i, nodes in -wise
W o n , forming j, -1 stripes of i, -1 doublet
panels(Fig.15).
Fig.15 Wake Discretization
Fig.13 Vorticity Transport at Separation
Applying the definition of vorticity contents (Eq.21) in
localstreamlinecoordinates(s,t.n) to a shearlayer (Fig.14)
yieldsforthe mainflowprofileu(n)
FreeSh~orLqwn=u.-u,
Fig.14 VorfiicjtyContents of a ShearLayer
Thefirstnodes (i=1)areplaced on the separationline
at a distance of n =612 to the surface, i.e. in the center of
the boundary layer. AU other nodes (i=2...i,) are lc-
mted on streamlinesof an intermediatevelocity field
For h=O the wake is simply aligned with the free
stream velocity, while for h =1 the wake follows exactly
the streamlines of the potential flow field (which run very
close to the surfacewhen started at half the boundary layer
thickness at separation).
Because on most realistic automobile configurations
thewake shapeiteration (seeFig.4) does not converge, it is
convementto usea fixedwake shape, createdwith a blend-
ing factor of h =0.5 in Eq.(38).
As the shearlayercarriesawaythevorticitycontentsof
the separating boundary layer @54.37), according to the
equivalence shown in Eqs.(25) and (26),the local doublet
gradient in meamwise direction s at the j,, starting
nodes alongthe separationline is simply

11. Startingwith =0 (thevalue on solidbody panels) the
doublet strength at the wake nodes downstream follows
from
bj ij( u ~ ~ ~ ) ~~ i jWithst;lj =0 , (40)
because in an inviscid model the vorticity contents, repre-
sented by the streamwise doublet gradient, remains
constant
In reality the dissipation of vorticity leads to a fading
kinematic activity of the shear layer dowmtmam. The
wake model simulatesthiseffect by an increasingradius of
the vortical core (Rankine vortex, see Fig.16). At separ-
ation the initial core radius is half the boundary layer
thickness on both sides of the wake panels, representing
exactlythe extensionof thevortical region in thismodel.
model is left open instead of closingit or addinga starting
vortex (whattheHelmholtzlawswould require).
Since the velocity u, varies along the separation line,
the evaluation of Eq.(40) finally leads to a bilinear dis-
tniution of the doublet strength on the wake model
(Fig.17).
Each variation of the streamwise doublet gradient
Jll/as=u. on adjacent lines leads to a doublet gradient
all/at in crossflow direction, equivalent to a meamwise
vorticity component B, (see Eq.26). This is analogue to
the generationofinduced dragon liftingwings.
To avoid longitudinal vortices in the wake the velocity
at separationshouldbe asconstantaspossible.
In the limiting case of potentialflow, separationoccurs
at the rear stagmtion (detachment) point with velocity
u, =0.The correspondingwake degenerates to a line with
vorticity contentsQ =0.
According to Eq.(6) the total pressure coefficient is
needed to calculate the static pressure coefficientat panels
in the separatedregion.
In steady state the static pressure on both sides of a
shearlayer can be assumedto be the same @.=pi), while
thevelocityp r d e revealsa discontinuity(Fig.18)
v, = ii + u,
vi = ii -ui (41)
Fig.18 Discontinuous Velocityacross a ShearLayer
In potential flow the limits of the inducedvelocities for
6+0 dependonthevorticity contents of the shearlayer
Fig.16 VorticalCore ofthe Wake
so that usingEq.(41) the differenceintotal pressureis
or in nondimeusionalform (s.Eq.4)
Fig.17 BilinearDoublet Distributionon the Wake
Because the kinematic ac!ivity of the wake va-
nishes downstream due to the increasing core radius, the
The total pressure loss vanisha if either the m a ve-
locityii orthevorticitycontents Q of the shear layeris0.
Immediately at separation, with ii =v,, and Q=v,,
thisleadsto the simplerelation

12. which means that the pressure loss is identical to the dy- which sometimes makes it diEcult to find reasonable
namicpressure at separation, analogueto the flowout of a initial conditions.
containerintothe atmosphere. These £irst order equations neglect cunature effects.
Again in the limiting case of potential flow with v= 0 They hold as long as the boundary layer thickness 6 is
at the detachmentpoint the effect of thewakevanishes. smallcomparedto the curvatureradiusr.
Unfortunatelythis analytical approach did not lead to
satisfyingresults. Thereforethe dishibutionof near-wrhce
total pressure inthewake oftwovehicleswas measured
Surprisinglyit turned out, that the averaged total pres-
sureweflicientwas almost the same, despitevery different
waketopologies. Withthis empiricallyfoundvalue thebest
results concerning drag ranking of different vehicles or
variantshavebeenobtained.
2.3 BOUNDARYLAYER FLOW
If theReynoldsnumber
is high enough, viswus effects are confinedto the immedi-
ate vicinity of solid flow field boundaries. From experi-
ments and special solutionsofthe Navier-Stokes equations
it is known, that in this case wall normal dimensions and
velocity components in theboundary layerarepropoxtional
to I/&. Introducingtheboundarylayer stretching
when normalizing wall normal dimensions and velocity
components in the Navier-Stokes equations, also brings
thesetermsin the order of 1.Now if allterms with the fac-
tor 1/Re are dropped, the equations of motion reduce to
Prandtl's firstorderboundary layer equationsin nondimen-
sionalform
a" a" a w - 0z + z + g j ; --. -, ..
a" a"uz + V- + ,"& - ap a2u
av & - - z + F
a" a" ap a+ . (48)u- + v- + w- - -- + -* av & - av a9
0 = -ap
az
The most significant simplification appears in the
z-momentum equation. It says that within the boundary
layer the wall normal pressure gradient ap/Jz vanishes or
p(z) =pa=const. This implies that the pressure is no
longer an unknown. Its wall tangential derivatives
aplaxamiaplay have to be prescribed as an external
boundarycondition.
In contrast to the elliptic Navier-Stokes equations, the
boundary layer equations are a parabolic system of partial
dBerentia1 equations. This means that a solution canbe
generated by a space marching technique. Starting from
prescribed initial velocity profiles the development of the
boundary layer is integrated downstream under the influ-
ence of theknown externalflowfield. h d y the velocity
components of the initial profiles have to fulfillEq.(48),
6lr << 1 (49)
This condition canbe used as an a posteriori check of
solutionvalidity.
Numerical solutions using finitedSerences have to
discretizethe boundary layer spatially. Another approach
are integral methods 191, like the one used here, which is
based onthetheorydescribed in [lo].
Before solvine: numericallv. the wuations are inte-
grated in wall no&al directiontermby ierm. Introducing
parametric laws for the velocity profiles reduces the three
dimensionalproblem to a quasi two dimensional case,be-
cause the wall normal direction vanishes as independent
variable. For instance the following integral parameters
(for simplicitygiven here for 2D) or comb'mtions thereof
areused asunknowns, insteadof thevelocity.
DisplacementThickn. 6 , = j(1 -e)dn
MomentumLoss Th. 62 = j(1 -$I$&
EnergyLoss Thickn. 63 = j ( 1 - $)($)Zdn
. (50)
ShapeParameter H12 = 61/62
In two dimensionalflow the velocityvectors within the
boundary layer all lie in the plane defined by the surface
normalandthe externalvelocity vectors.
VelociiyProfie of a 3D Boundary Layer

13. Lateral pressure gadients in three dimensions produce
an additional cross flow componen< which leads to a
skewedvelocity profileas depictedinFig.19.
It should be emphasized that the three dimensional
displacement thickness cannot be calculated from a local
velocity profile, not even in a plane of symmetry without
cross flow. Its determination requires the integration of
another partial differential equation (mass flow balance)
over the whole computationaldomain [ll]. In contrast to
plane boundary layers, a negative displacement thichess
may occur in the vicinity of longitudinalvortices within a
threedimensionalboundary layer.
Vanishing skin friction is not the only criterion for
separation, as in 2D. Squeeze-off separation can be in-
duced by the convergence of boundary layer fluid at finite
wall shear stress. A number of phenomena indicates the
ocamence of 3D separation [8,12] :
Rapidlydecreasingwall shearstress
Rapidlyincreasing shapeparameter
Rapidlyincreasingprofie skewing
Convergenceof skinKction lines
Local minimaof wall shearstress
Localmaximaof displacementthickness
These indicators have been comb'med to calculate a
nondimensional separationtendency,which is used to gen-
erate a 3D separation line as startinglocationfor the wake
simulation However, in many applications it turned out,
that by hr the most dominant criterion is still the wall
shearstress.
The integration of the boundary layer equations with a
space marching technique requiresa description of the do-
main under consideration in surface oriented cudinear
coordinates, which can be mapped onto a rectangle in the
computationalplane 0 <x15 1;0 <x2S 1 (s.Fig.20).
Fig.21 PanelModelwith SurfaceNormals
before (right) and affer(leff) Self Ohentation
In physical space this grid forms two hnilies of
one-parametric lines on the vehicle's surface. Lines with
x' =const. are the lines in circumferential (cross flow)
direction with xl=O at the upstream and x l = l at the
downstream end of the vehicle. x2=const.-lines run in
longitudinal (main flow) direction with x2=0 at the upper
and xZ= 1at the lower symmetry line.
Inviscid h c e meamlines andlor plane sections of
the panel model are used as a data base for the generation
of this grid 113,141. The codeoffers a number of optionsto
adapt the grid either to the external flow field directionor
to characteristiclinesofthevehicle.
3 EXAMPLES
3.1 POTENTIAL FLOW CALCULATIONS
Before using a panel model the first time some checks
to detect errors which would affect the solution, such as
identical panels, should be exercised. For this purpose the
code canbe run in an idle mode, where all actions not re-
quiringasolutionarestillpossible.
Not every grid generation system guarantees a uniform
orientation of the surhce elements. Therefore a self-orien-
tating algorithm is Med in any case if one of the panels
on the front, rear, top, side or bottom faceis recognizedto
be pointing inside. Fig.21 shows a panel model with sur-
face normals before (left half) and af&er(right half) this
procedure.
Fig.20 Surface Coordinates forBoundaryLayer
Calculationin Physicaland ComputationalSpace
Another typical option executed in idle rum is the pa-
rametricdeformationof thevehicle's surface.
A whole series of modifications may be performed se-
quentiallyin a single run. Each canbe confinedto a differ-
ent region of the model defined by a Cartesian coordinate
box. One of the coordinatescanbe shiftedor scaledwithin
this box. The maximum deformation is placed on an edge
or somewhere inside the limiting box, defined separately
for each coordinate direction Several intrinsic or an

14. arbitrary user defined function are used to control the
decayof deformation.
This way in a few minutes a whole family of variants
may be generated to investigate the effectof changing a
certaingeometryparameter.
In Fig.22 the original model is modified by boattailing
the rear half, tapering the front end, blowing out the
wheelhousesand changing the slopeof hwd, rear window
and trunklid
Fig.22 Modiricationof a Vehide Surface
7
this requires less additional panels and the flow field on
the ground is directly part of the solution. This may be
needed for h c e to calculatethe boundary layer on the
floor of awind tunnel.
The potential flow analysis of this model took 62 CPU
seconds (matrix generation 74%, solution 4%) on a
CRAY-YMP 41256 .A performance of 125MJXOPS was
achievedusing 1 CPU.
One symmetry half of the model including ground
plane has np =4413 panels. Core memory needed for the
panel methud, dimensioned for 15000 panels, is 14
MWords. To reduce the number of UO operations 10 ma-
tnx hues are always kept m core simultaneously. During
executionthe externalstorageof influence matnces m tem-
porary files on the SSD reqwes 7n; words, whlch are
around 120MWordsforthispanel model.
Onworkstations the code canbe comigured tovirtually
hold the complete matrices. However, it turned out that
keeping only parts (i.e. 10 lines) of the matrices m core
and handling 110explicitly by unformattedREAD/WRITE
is fasterthan systemswapping
The porcupine plot in Fig. 24 compares the potential
flow pressure distribution with experimental data in the
symmetryplane of a research model.
Fig.23 displays the pressure distribution of a potential
flow analysis together with surface streamlines. The sur-
face Sxamhes have been terminated on isobars C,=0.8
around the iiunt and rear stagnation point. They will be
used later as a data base for the generation of surface
coordinatesforboundary layer analysis.
Fig.23 Pressure Distribufionand Surface Streamlines
(PotentialRow Analysis)
To simulate the ground effect, a corresponding plane
has been added to the vehicle model. Compared to the use
of an image of the vehicle model below the ground plane,
Fig.24 Comparisonof measured and computed
Pressure Distribution (PotentialHow Analysis)
A good agreement can be observed from the front
stagnationpoint over the upper side almost up to the edge
of the base. Without wake simulation there is of muse a
suction peak at the rear edge and a pressure recovery on
thebase.
This points out again that the wake's influence in up
streamdirectionfadesrapidly and pressure drag generation
isfocused on thevehicle's base (comparewith Fig.2).
The larger difference on the underbody is due to the
higher inviscid mass flow in the duct between vehicle and
ground, compared to reality. Drag is only slightly affected
by this deviation and lift predictions do not make much
sensewith a completelyflatunderbody.
This comparison proves that already the potential flow
field can be usedto answer questions like where to put in-
takes and exhausts for cooling & ventilation or getting
forces and moments on parts of the surface and the dis-
tributionof aerodynamicloadsfor structural analysis.

15. Once the sourcedistnition on the vehicle's surfaceis
known,the flow field can also be evaluated at any offsur-
facepoint
Fig.25 Pressure Distributionin the ExternalFlow Field
(PotentialFlowAnalysis)
Fig.25 displaysan externalflowfield survey.The three
control planes at x, y and z have been generated by the
code itself. Such evaluationsare typically performed in a
postprocessing run.
Fig.26 CoolingAir Streamtube
If the mass flow of cooling air is known, its inhence
onthe outer flow field canbe simulated. After defining in-
takeand exhaust areasand theflow rate, the corresponding
h c e normalvelocity componentsareassigned asbound-
ary conditionto thesepanels.
In Fig. 26 the cooling air streamtube is visualized by
streamlines, integrated upstream from the vertices of in-
take panels. This cooling air streamtube has been used to
design a specialtest facilityfor coolingsystems.
At low speedshot coolingair reentersthe intakeand af-
fects the efficiency of the cooling system. This phenom-
enon of external recirculation can be investigated with
potential flowanalysis, if the flow rate is knownas a func-
tion of speed.
Fig.27 shows cooling air streamlinesat increasingvel-
ocities. This simulation enables the prediction of the limit-
ing speedfor external recirculation.
0,5 kmlh
Fig.27 CoolingAir Streamlinesat differentVelocities
At zero velocity there is no k e stream. The only
sourcefor air motion is the fan inside the engine compaxt-
ment In thiscasethe ilow must be simulatedin an inertial
systemwith prescribed vehiclevelocity (hereH).

16. 3.2 BOUNDARYLAYER CALCULATIONS
As discussed in chapter 2.3 the boundary layer method
requires a description of the vehicle in general &ce
coordinates. The generation of this grid is part of the
boundary layer module. Inviscid streamkes andlor plane
vehiclesectionsareused as a database.
Fig.28 showsapposed panel model and boundary layer
coordinates, generated using the e e s depicted in
Fig.23. The first (x' =0 ) and last (x' =1)circumferential
grid linesare placed on isobars. It is not possible to start
the grid immediately at the front stagnation point because
linex' =0 would degenerateto a point, leadingto a singu-
lar metric. At the rear end the grid must only cover those
edgeswheretheboundary layerwill surelyseparate.
Fig.28 BoundaryLayerCoordinatesand PanelModel
Laminar initial values £rom an analytical stagnation
point solutionareused on the first line to staa the bound-
ary layer analysis If the integration is started in acceler-
ated external flow the influence of initial values on the
solutionis known tobe small.
Fig.29 displays the wall shear stress distribution and
the boundary layer thickness perpendicular to the surface
(scaledby 10forbettervisibility) under the iduence of the
potential flowfield. Transition is indicated in the region of
the pressure minimumaroundthe front end.
A small local separation occursat the lower end of the
windshield around the centerlinedue to the strong adverse
pressure gradient. This leads to a sudden increase in
boundary layer thickaess. An automatic internal restart at
thisposition avoids a gap in the solution do-.
At the rear end separationoccurs on the upper side at
the end of the windshield and on the sideandunderbodyat
the edgeof the base.
Fig.29 Wall Shear StressDistribution
and BoundaryLayer Thickness (x10)
A clearviewof the boundary layer's reaction onthe ex-
ternal pressure distributionis given in Fig.30, where some
viscous results are plotted together with C, along the
upper centerline.
UpperSymmetryUrn,
Fig.30 StaticPressure and BoundaryLayer~arameiers
alongthe upper Symmetrytine
Transition at x' =0.2 is accompanied by a sudden in-
crease in wall shear stress and a decrease in shape para-
meter. The positive pressure gradient on the rearwindow
causes the wall shear stressto drop rapidly until separation
atx' =0.7

17. The illmtion of the limiting wall streamlines (skin
fiction lines), based on the field of wall shear stress
vectors is a vivid way to visualize the inviscid flow
(Fig.31). They compare to the experimental technique of
oil flowpictures in a wind tunnel.
Fig.32 Wake Model, Separation tine and
TotalPressure Distributionon the Vehicle
Fig.31 Limiting Wall streamlines (Slbn Friction tines)
and lnviscid SurfaceStreamlines(PotentialFlow)
Thisboundary layer analysiswith a resolution of 101x
101surfacenodes took 576 CPU secondson a worktion
of type SGI INDIG02 (R4400, 150 h4Hz). Execution on
CRAY is not faster becausethe integrationof the boundary
layer equations with a Runge-Kutta 4th order scheme is
more or less a scalar problem Vectorized grid generation
and evaluation of the raw results require only about 7% of
the completetime.
3.3 WAKE SlMULATlONS
A complete wake simulationruns throughthe loops de-
picted in Fig.4 to accountfor the interactionof the inviscid
and viscous solutions. Since on most realistic automobile
confirmrationsthe wake shape iteration runs into a chaotic
defor&tion, a frozen wake shapehas been used, generated
along streamlines of an intermediate flow field (s.Eq.38)
between thefree stream and the potential flowwithout sep-
aration
The wake model is shown in Fig.32 together with the
separation line and the total pressure distribution on the
vehicle.
Within a radius of two boundary layer thicknesses
around the separation line the total pressure coefficient is
smeared out by a transitional function. Otherwise the
sudden onset of pressure loss would cause an unsteady
staticpressuredistribution.
The boundary layer calculation is nm twice to account
for the influence of the wake on the separation line. One
wake simulationtook 390 CPU secondson a CRAY Y-MP
(matrix generation 12%, 8 inner loops for vorticily iter-
ation 54%).
Staticpressureand skinfiction with and withoutwake
simulation for the critical region on the rear window cen-
terlineareplotted in Fig.33.
x-coomate [mi
Fig.33 Effect of Wake Simulationon
StaticPressure and Slbn Friction
The open symbols denote pressure and corresponding
friction of the potential flow analysiswithout wake simula-
tion The adverse pressure gradient on the rear window
causes the wall shear stress to drop to 0, i.e. separation.
With wake simulation (filled symbols) the static pressure
levelsoff d o m e a m of this separationat C,=-0.I...-0.2.
Under the influence of this external flow field the
boundary layer would not separate at all. Therefore in the
outer loop of Fig.4 the separation line is never allowed to
move downstreamof apreviousseparation.

18. The staticpressure distriiutionswith and without wake
simulationare compared in Fig.34. Whilein pure potential
flow there is a strong suction peak at the base edge and a
recompressionto C, = 1,wake simulationleadsto a neatly
constant pressure of Cp=0 on the base. Upmeam of sep-
arationthe influenceof wake simulationvanishesrapidly.
Fig.34 Static Pressure Distribution withand without Wake
In section 2.2 it was mentioned that the usage of em-
pirical values for the total pressure loss leads to better re-
sultsthan an analytical approach. Insteadof using Eq.(45)
to compute the static pressure from the velocity field with
Eq.(6), an experimentallyfoundtotal pressure coeflicient is
usea
The average of measured Cpt values in the separated
regions of two vehicles turned out to be almost the same,
despitevev differentwake topologies. Therefore this aver-
agevalue hasbeen usedfor the simulationon all wnfgur-
ations, depicted in Fig.35, where measured and computed
total drag coefficientsfora research model arewmpared.
8aseSbMAnele Idssl
Fig.35 Comparisonof measured and computed
TotalDragCoeRcients(Research Model)
This model has exchangeablerear end tops so that the
intluence of base slantangleon drag canbe displayed.
Exprimental values show the typical curve
minimum at about 25" base slant angle. Numerical results
are shifted towa~dslower absolutelevels,but followclosely
the trend over base slant, giving confidence in the predic-
tion of drag differencesbetween variants or even different
vehicles. The analytical approach (Eq.45) failed to reprc-
ducethistrend over base slant angle.
Gmxl agreement with experiment has also been ob- , . ;
tained simulatinga rather small but veq effective aerody-
namic device. Fig.36 shows the rear end of a roof, on the ...:
lefIhalf without and onthe right with a typical spoiler.
Fig.36 Panel Model without and with Spoiler
at the Rear End of the Roof
The simulation predicted a drag reduction due to the
spoiler of ACn =-0.043, while in the wind tunnel the ef-
fect was measured tobe ACD=-0.041 .
The reason for such a large effect gets cleat fromplot-
ting C, =CD/A=-Cp *n, (the local drag distribution) on
thevehicle's surface (Fig.37).
Fig.37 lnfloence of a Spoiler on the Distributionof the
Local DragContribution C,=CdA=-C, n,

19. Without spoilerthe round rear end of the roof is gener-
ating a low pressure region, resulting in a downstream
pointing force (high positive C,). As separation later
downstream cannot be avoided on realistic automobile
shapes, the corresponding recompression is lacking. The
spoiler fixes separation earlier and destroys the suction
peak, thus reducing the drag contribution of this region
significantly, which can be recognized clearly from the
lower C,-values in Fig.37.
SUMMARYAND CONCLUSIONS
A zonal approach for the incompressible and steady
flow around closed bodies like automobiles, trains, aircraft
or submarines has been presented. Coupling simplified
methods for different domains of the flow field created a
fast and versatile tool, in many details especially tailored
for the needs of automobileaerodynamics.
Comparisonswith experimental results lend confidence
to the prediction of drag differences between shape vari-
ants. Turnaround times of two or three days for the cre-
ation and analysis of several v-ts enable a useful
application of this method in the early phase of aerody-
namic automobiledevelopment.
ACKNOWLEDGEMENTS
The author would l i e to thank Dr.Gerhard Krukow
fromBMW AG and Frank Wernerfrom ADAM OPEL AG
for providing some of the results and the permission to use
their panel models.
The exampleapplicationshavebeen mon the CRAY-
YMP and on workstationsof the BMW AG.
NOMENCLATURE
Cartesiancoordinates m
Local streamlimecoordinates m
General surE.cecoordinates -
Surfacenormal vector -
Velocityvector mls
Surface normalvelocity m/s
Shear stressvector Nlm2
Staticandtotal pressure Nlm2
Density kglm
Dynamicpressure N/m2
Staticpressure coefficient -
Totalpressure coefficient -
Drag coefficient -
Pressuredragcoefficient -
Friction drag coefficient -
Forebodydragcoefficient -
Afterbodydragcoefficient -
Reference area (fmntalarea) m2
Sepamted(base) area mz
Vorticityvector 11s
Vorticitycontentsvector mls
Circulation m2/s
Sourcedensity (per area) mls
Doublet strength m2/s
Unit sowceinduction -
Influence coefficient -
ReferenceLength m
Kinematicviscosity m21s
Reynolds number -
Boundary layerthickness m
Displacementthickness m
Momenhun lossthickness m
Shapeparameter
Profile skewing deg.
Nabla Operator

20. REFERENCES
[I] Stricker R,Krukow G.,
"Enhvicklungsstandund Einsatzmaglichkeiten
derAerodynamik-Simulationin der
Fahrzeugenhvicklung':
i n D i h m i d W.(Hrsg.),
"Rechenmethodenin derFahrzeugentwicklung",
Vieweg-Verlag,Braunschweig 1992,pp. 87-108
[2] Morel T.,
"Theoreticallower limitsof forebody drag",
Deville M. (Ed.), Notes on Numerical Fluid
Mechanics, AeronauticalJournal, January 1979,
pp. 23-27.
[3] Paul J.C., LaFond J.G.,
'Ynalysisand Design ofAutomobileForebodies
using PotentialFlow Theoryand a Boundary Layer
Separation Criterion",
SAEPaper No. 830999, 1983
[4] Hoeijmakers H.W.M.,
"ComputationalVortexDynamics",
VKI Lecture Series, 1986
[S] Hirschel E.E, Fornasier L.,
"Flowfieldand VorticilyDistribution near Wing
TrailingEdges",
AIAA Pa~erNo. 844421. AIAA 22nd Aerosoace
Sciences~ e e t i n ~ ,lanu& 9-12, 1984,Reno, Nevada
[q Hess J., Johnson F.T., Rubbert P.E.,
"Notebookof theProfessionalStudySeminar on
PanelMethods",
Seattle, 1978
[q Brettbauer N., Kraus W., Sacher P.,
"DasMBB-UFE UnterschallPanelverfafiren
(Teil l-3)",
MBB-ReportNo. UFE 632-, 633; 634-70, 1970
[8] Hirschel E.H.,
"DreidimensionaleGrenzschichtablcisung",
Leauresat the Technical Universityof Munich,
1987-88
191 CousteixJ..,
"Three-DimensionalBoundary Layers,
lntroductionto CalculationMethods",
AGARD Report No.741, 1987,pp.l.1-1.49
[lo] CousteixJ..,
'Ynalyse theoriqueet moyens de Prevision de la
CoucheLimite turbulentefridimensionelle",
ONERA Publication No. 157, 1974
Ill] Hirschel E.H., Kordulla W.,
"ShearFlow in Surface Oriented Coordinates",
Notes on Numerical Fluid Mechanics, Volume 4,
Vieweg-Verlag, Braunschweig 1981
[12] Hirscbel E.H., Kordulla W.,
"Localpropertiesof three-dimensionalseparation
lines",
ZFW Vo1.4, 1980,pp.295-307
. . . ~ . ..::. ~ . :.
. . . . '?,.:,,.
. . .: . , , . . < ~,..:.,: 2 ?
. . ,,.::
1131 Griin N.,
.:. ... , .~. .
"HybridCoordinatesfor the Calculationof
see-~imensional~oundary~ayers",
in Sengupta S. et a1(Eds.),
"NumericalGrid Generationin ComputationalFluid
Mechanics",
Pieridge Press, 1988,pp. 835-844.
[I41 Griin N.,
"S@&mungsfeldangepmsteOberfldichenkoordinaten
mr Berechnung dreidimensionalerGrenzschichten",
FortschrittberichteReihe 7, Nr. 187, VDI-Verlag,
Diisseldorf, 1991