NASA Technical Reports Server (NTRS) 20040110939: Geometric Model for a Parametric Study of the Blended-Wing-Body Airplane

9 Pages·0.33 MB·English

by  NASA Technical Reports Server (NTRS)

Geometric Model for a Parametric Study
(cid:3)
of the Blended-Wing-Body Airplane
y
C. Wayne Mastin
z
Robert E. Smith
x
Ideen Sadrehaghighi
{
Michael R. Wiese
ABSTRACT NOMENCLATURE
A parametric model is presented for the blended- D0;D1 Dirichlet boundary conditions
wing-body airplane, one concept being proposed N0;N1 Neumann boundary conditions
for the next generation of large subsonic trans- P Point on NURBS surface
ports. The model is de(cid:12)ned in terms of a small P(cid:22) NURBS control points
set of parameters which facilitates analysis and X Surface coordinate
optimization during the conceptual design process. C Airfoil chord length
The model is generated from a preliminary CAD F1;F2 Airfoil thickness parameter
geometry. From this geometry, airfoil cross sections M1;M2 Airfoil camber parameters
are cut at selected locations and (cid:12)tted with analytic N NURBS basis function
curves. The airfoils are then used as boundaries for P1;P2 Airfoil camber parameters
surfaces de(cid:12)ned as the solution of partial di(cid:11)erential T Airfoil thickness parameter
equations. Both the airfoil curves and the surfaces h Wing section span
are generated with free parameters selected to give u;v Surface parametric variables
a good representation of the original geometry. The w NURBS weight
original surface is compared with the parametric x;y;z Surface coordinates
model, and solutions of the Euler equations for x(cid:22);y(cid:22) Airfoil coordinates
compressible (cid:13)ow are computed for both geometries. xe;ye Airfoil trailing edge
The parametric model is a good approximation of y^ CAD airfoil coordinate
the CAD model and the computed solutions are y(cid:22)c Airfoil camber
qualitatively similar. An optimal NURBS approxi- y(cid:22)t Airfoil thickness
mation is constructed and can be used by a CAD (cid:18) Twist angle
model for further re(cid:12)nement or modi(cid:12)cation of the (cid:24);(cid:17) Wing section parametric variables
original geometry. (cid:27) PDE weighting factor
1. INTRODUCTION
ManyofthebusiestairportsinAsiawillrunoutof
spaceinthenearfuture. Onesolutiontotheproblem
proposedby the aircraftindustry is the development
of huge airliners carrying up to 800 passengers.
Along with the tradition aircraft con(cid:12)gurations,
1
some manufacturers, including McDonald Douglas ,
(cid:3)ThispaperisdeclaredaworkoftheU.S.Governmentand
have advanced a (cid:13)ying wing con(cid:12)guration similar to
isnotsubjectedtocopyrightprotection intheUnitedStates.
yNRCSeniorResearchAssociate,NASALangleyResearch the B2 bomber. This unconventional blended-wing-
Center,Hampton,Virginia23681-0001. body (BWB) concept for future subsonic transports
zSenior Research Engineer, NASA Langley Research Cen-
is an aerodynamically e(cid:14)cient airplane with the
ter,HamptonVirginia23681-0001,AssociateFellow,AIAA.
xResearch Associate, Department ofMechanical Engineer- interior passenger volume distributed over a large
ing,OldDominionUniversity,Norfolk,Virginia23529-0247. portion of the centerbody.
{MemberoftheTechnicalSta(cid:11),ComputerSciencesCorpo-
ration,3217N.ArmisteadAvenue,Hampton,VA23666.
1
New concepts in aircraft designs often begin with ure 2. Four of these cuts were on the main horizon-
only a rough sketch on the designers note pad. tal section of the wing and two were on the vertical
Theseconceptualdesignsalwayshaveto bemodi(cid:12)ed winglet. These cross sections were output as a dense
due to structural and aerodynamic constraints. set of points. The airfoils in the parametric model
For this reason, it is convenient to have a model were then generated with the parameters selected to
which is de(cid:12)ned by a set of parameter values so the give the best (cid:12)t to the CAD airfoils. The paramet-
the model can be easily modi(cid:12)ed as more design ricairfoilequationsde(cid:12)ningthecoordinatesx(cid:22)((cid:24))and
variables are considered. This report will describe y(cid:22)((cid:24)) of the points on an airfoil are
the application of a conceptual design tool that will
facilitate aerodynamic analysis in the early stages
x(cid:22)((cid:24))=Csin(cid:25)(cid:24); y(cid:22)((cid:24))=y(cid:22)t((cid:24))+y(cid:22)c((cid:24));
in the development of an aircraft. The parametric
model which is generated in this manner now can T
y(cid:22)t((cid:24))=(cid:0) (sin2(cid:25)(cid:24)+F1sin4(cid:25)(cid:24)+F2sin6(cid:25)(cid:24));
be optimized to improve the aerodynamic charac- 2
teristics of the aircraft by varying a small set of 8 M1 2
pexaarammineetderst,oodrettheermcoinmepalelltepposasriabmleetfeearssipbalecedceasingnbse. >>>>>< (PP122M(cid:0)(2P1P1)13r((cid:0)r(cid:0)rP)2)2if(2rr(cid:20)+PP12(cid:0)3P1)
lBsetysasgpeleiskrefoolyfrmttoihneghadaveemvetoloroepmmcaoekmnetp,mlettaehjoearndadeleyvsseigilsonpaecthrtahwnegileelasrbilnye y(cid:22)c((cid:24))=>>>>>: (cid:0)(1(cid:0)(iMPfP22P2M(cid:0))12P2(1<1)3(cid:0)r(r<2(cid:0)PP2P2+1)22(P22rr+(cid:0)Pr12)(cid:0)3iPf2r)(cid:21)P2;
the (cid:12)nal Computer-Aided Design (CAD) models to
r=sin(cid:25)(cid:24)
be used for Computational Fluid Dynamics (CFD)
analysis and wind tunnel testing. 0(cid:20)P1 (cid:20)P2 (cid:20)1; 0(cid:20)(cid:24) (cid:20)1: (1)
Thedesignparametersintheseequationsallowsuf-
This report will describe an application of the
(cid:12)cient freedom to give a good approximation to all
conceptual design tool Rapid Airplane Parametric
2;3 the airfoil crosssections used for the model. The pa-
Input Design (RAPID) in creating a parametric
rameterC is the section chord length. The thickness
model for the the BWB airplane. Since RAPID
was developed for modeling conventional airplane curve y(cid:22)t((cid:24)) is de(cid:12)ned with the thickness parameter
geometries, modi(cid:12)cations were necessary to apply T and Fourier coe(cid:14)cients F1 and F2. A general-
ization of the piecewise quadratic camber curve in
the techniques to the current application. The
the RAPID system has been developed to model the
input parameters for a geometry may come from
wing sections of the BWB con(cid:12)guration. The cam-
a variety of sources and range from crude sketches
to detailed drawings or surfaces generated from a ber curve y(cid:22)c((cid:24)) is a piecewise cubic curve which can
have both a local maximum and a local minimum.
CAD system. For the BWB geometry a preliminary
CAD model was available and was used to gener- The camber curve has relative extrema of M1 and
atemostoftheparametersusedintheRAPIDmodel. M2 at locations P1 and P2, respectively. The cam-
ber locations are the fractions of total chord length
2. AIRFOIL APPROXIMATION measured from the airfoil trailing edge. If P1 = P2
and M1 = M2 this expression reverts to the original
The construction of the parametric model for the
piecewisequadraticcambercurveinRAPID.Thead-
BWB begins with the de(cid:12)nition of airfoil cross sec-
ditional parameters are needed for the BWB model
tions. The actual CAD data was used to de(cid:12)ne the
to approximate the forward camber and aft re(cid:13)ex of
airfoil parameters in RAPID. The number of airfoils
the centerbody airfoil.
de(cid:12)ning the model determines the total number of
The selection of a set of design parameters to (cid:12)t a
parameters in the model and also e(cid:11)ects the (cid:12)delity
given airfoil can be accomplished by solving a non-
of the model. If a large number of cross sections
linear optimization problem. Assume that a set of
are made, the CAD model can be more accurately
approximated by the RAPID model. However, the values (cid:24)i; i = 1;(cid:1)(cid:1)(cid:1);m have been selected based on
some desired criteria for distributing points around
number of parameters increases with the number of
the airfoil. From these values, a correspondingset of
airfoils, and there is the ever present danger of os-
cillations in the model that is inherent in this as values x(cid:22)i; i = 1;(cid:1)(cid:1)(cid:1);m can be calculated from the
(cid:12)rst equation in (1). Now use the airfoils from the
well as many other interpolation and approximation
CADdataandcalculatecorrespondingy coordinates
schemes. From the CAD surface given in Figure 1,
six cuts were made at the positions indicated in Fig- y^i; i = 1;(cid:1)(cid:1)(cid:1);m, by interpolating the data if neces-
sary. Observe that on the upper and lower surfaces
2
of the airfoil the value of y(cid:22) can be given as a single- example, if the airfoil generated in Section 2 corre-
valued function of x(cid:22). The chord length C can be sponded to (cid:24) = 0, then the Dirichlet boundary con-
measured and will not be included in the set of vari- dition would be
ables for the optimization problem. Thus, the best 8
(cid:12)t to the given CAD data is the airfoil de(cid:12)ned by < xe+x(cid:22)((cid:24))cos((cid:18))+y(cid:22)((cid:24))sin((cid:18))
equation (1) with the remaining design parameters D0((cid:24))=: xe(cid:0)x(cid:22)((cid:24))sin((cid:18))+y(cid:22)((cid:24))cos((cid:18))
h
T;F1;F2;M1;M2;P1;P2 chosen to minimize
Xm The Neumann boundary conditions in(cid:13)uence the
2
[y(cid:22)i(T;F1;F2;M1;M2;P1;P2)(cid:0)y^i] shape of the wing surface and must be determined
i=1 in part from the CAD data. From a geometric point
ofview,theNeumannboundaryconditionscanbein-
The solution of this nonlinear least-squares problem
terpretedasatangentribbonwhichde(cid:12)nes theslope
de(cid:12)nes the airfoil shapes for the parametric model.
of the wing surface at the airfoil section. This tan-
This optimization problem can be easily solved by
gent ribbon, as depicted in Figure 3, is constructed
a number of methods. The method which has been
by o(cid:11)setting the cross sections in Figure 2. For each
usedinthisworkisthetruncatedNewton method in
4 airfoil,ano(cid:11)setdistanceintheparametricvariable(cid:17)
the TNBC package developed by S. G. Nash . The
is selected. A new airfoil is constructed at the o(cid:11)set
software is designed for minimization with bound
locationusingtheRAPIDparameterswhichwillgive
constraints and can be obtained from netlib. Other
a good approximation of the actual CAD cross sec-
dataisnecessarytopositiontheairfoilsintheproper
tionat thesamelocation. Althoughtherearealarge
locationinthreedimensionalspace. Fromtheairfoils
numberofparameterswhichdeterminetheshapeand
in Figure 2, the spanwise locations, h the locations
location of an airfoil, for calculating the o(cid:11)set airfoil
of the trailing edges, xe and ye, and the twists, (cid:18),
it is su(cid:14)cient to consider only changes in the trail-
can be determined. It is assumed that each airfoil
ing edge, (xe;ye), chord length, C, and thickness, T.
alongthefuselageandwingsectionsarecontainedin
Changes in other airfoil parameters are neglected in
a z =constant plane and that both airfoils de(cid:12)ning
de(cid:12)ning the tangent ribbon. The o(cid:11)set airfoils now
the winglet are in a y =constant plane. Once the
determine the changes in x and y with respect to
airfoils are de(cid:12)ned in space, the wing surfaces can
z, that is, @x=@z and @y=@z at each point. Now if
be constructed to span the airfoils and create the
@z=@(cid:17) is given, then @x=@(cid:17) and @y=@(cid:17) can be cal-
airplane surface.
culatedfrom @xe=@z, @ye=@z, @C=@z,and @T=@z by
constructing the o(cid:11)set airfoil and applying the chain
3. WING SURFACES
rule. Thus, the procedure for generating the Neu-
The RAPID methodology creates wing surfaces in mann boundary conditions can be described in the
xyz-space in the form following steps: (1) calculate airfoil parameters for
the o(cid:11)set airfoils with chord, thickness, and trailing
X=(x((cid:24);(cid:17));y((cid:24);(cid:17));z((cid:24);(cid:17))); 0(cid:20)(cid:24) (cid:20)1; 0(cid:20)(cid:17) (cid:20)1
edge coordinates given by
The coordinates of surface points are obtained by @C @z @T @z
C+ ; T + ;
solving the fourth order partial di(cid:11)erential equation
@z @(cid:17) @z @(cid:17)
(cid:20)(cid:27)2 @2 + @2 (cid:21)2X=0; (2) xe+ @xe @z; ye+ @ye @z;
@(cid:24)2 @(cid:17)2 @z @(cid:17) @z @(cid:17)
where(cid:27) isaconstantweightingparameter. Dirichlet resectively;(2)constructtheo(cid:11)setairfoilwithcoordi-
and Neumann boundary conditions are imposed at nates(xo((cid:24));yo((cid:24)));and(3)settheNeumannbound-
(cid:17) =0 and (cid:17) =1. The boundary conditions are: ary conditions
8
X((cid:24);0)=D0((cid:24)); X((cid:24);1)=D1((cid:24)); < xo((cid:24))(cid:0)x((cid:24);0)
N0((cid:24))= yo((cid:24))(cid:0)y((cid:24);0)
: @z
X(cid:17)((cid:24);0)=N0((cid:24)); X(cid:17)((cid:24);1)=N1((cid:24)): @(cid:17)
The solution is periodic in (cid:24).
The spanwise changes in trailing edge coordinates,
The Dirichlet boundary conditions, de(cid:12)ned by the chord length and thickness can be easily estimated
functions D0((cid:24)) and D1((cid:24)), are determined from the fromthe CAD data. However,an estimate of @z=@(cid:17),
airfoils which form the edges of the surface. For which gives the spanwise in(cid:13)uence of the Neumann
3
boundary condition, is not apparent from the data. 4. CFD ANALYSIS
Thus some trial and error was necessary to obtain
The RAPID system can output surface grids with
the best (cid:12)t to the CAD surface. In most cases,
speci(cid:12)ed grid point distributions in each coordinate
setting @z=@(cid:17) equal to the span of the wing section
direction. Thesesurfacegridscanbeuseddirectlyin
is a good starting value.
a potential (cid:13)ow analysis code or they can be used as
boundary surfaces to generate a volume grid for an
Once the boundary conditions are determined, Euler or Navier-Stokes calculation. Similar volume
equation (2) is solved by generating a Fourier series gridswereconstructedfor theregionabout theCAD
expansion. An approximation of the CAD surface and RAPID models. In order to guarantee com-
using the RAPID technology appears in Figure 4. parable grids, a grid was (cid:12)rst generated about the
RAPID surface and then this grid was distorted to
generate the grid about the CAD surface. The grid
The total number of parameters will now be sum-
distortion process was carried out as a application
marized. There are a total of six airfoils de(cid:12)ning the
of the Coordinate and Sensitivity Multidisciplinary
BWB geometry. Each airfoil is de(cid:12)ned by eight pa-
5
Design and Optimization (CSCMDO) system .
rameters
With this system, the di(cid:11)erences in the volume
C; T; F1; F2; M1; M2; P1; and P2: grids is no greaterthan the di(cid:11)erences in the surface
grids. The CSCMDO system could also be used
The location and orientation of each airfoil in space to generate new volume grids after making small
mustbede(cid:12)ned. Thisisdonebyspecifyingthespan- changesin designparameters. Solutionsto the Euler
wise location of the airfoil, h, the coordinates of the equations for compressible (cid:13)ow were computed on
trailingedge,xe andye,andthetwistangle,(cid:18). There grids for the RAPID and CAD geometries. The
are additional parametersfor each airfoil used to de- following comparisons are for a free stream Mach
(cid:12)ne the Neumann boundary conditions. These are number of 0.85 and an angle of attack of 2 degrees.
Figure 8 contains comparisons of contour plots of
@z @xe @ye @C @T the local Mach number on the upper surfaces of the
; ; ; ; and :
@(cid:17) @z @z @z @z CAD and RAPID geometries. The similarity in the
solutions is evident, especially the shock structure
Thus, there are a total of 17 parameters for each
along the wing segment. Further comparisons reveal
of six airfoils. Now deleting the spanwise location
the e(cid:11)ects of the di(cid:11)erences in geometry. Figures 9
of the centerbody airfoil which we assume to be at
and 10 are pressure coe(cid:14)cient plots near the center
z = 0, the BWB airplane, as de(cid:12)ned by RAPID, is
of the fuselage cross section and on the wing. The
generated with a total of 101 parameters.
pressure coe(cid:14)cient plots are superimposed over
plot of the airfoil cross sections, so that di(cid:11)erence
Since considerable e(cid:11)ort has gone into attempting in solutions can be compared with the di(cid:11)erences
to approximate a given CAD surface with a para- in geometries. It was interesting to note that the
metricmodel, somecomparisonsoftheoriginalCAD RAPID surface geometry gave smoother plots and
surface and the RAPID surface will be presented. a larger lift coe(cid:14)cient than the solution computed
It is evident from comparing Figures 1 and 4 that with the CAD geometry.
the gross geometry of the BWB airplane has been
captured in the RAPID model. Figures 5 and 6 5. NURBS APPROXIMATION
compare the models at two cross sections used in
NonUniform Rational B-Splines (NURBS) have
de(cid:12)ning the RAPID model. The approximation
become a standard surface representation in the
of the supercritical airfoil section in Figure 6 is 6
CAD industry . For this reason, a method for
excellent, but there is noticeably more error in the
accuratelyrepresentingtheRAPIDsurfacegeometry
approximation of the centerbody airfoil in Figure 5.
by a NURBS surface has been investigated. As with
A comparison of the planforms appears in Figure 7.
the RAPID methodology, one of the advantages in
Whether this degree of (cid:12)delity in the approximation
a NURBS representation of a surface is the fact
is su(cid:14)cient would of course be a judgement call of
that the surface is de(cid:12)ned by a set of parameters
the designer. Further accuracy could be achieved
which can be manipulated to change the shape of
with the RAPID technology, but at the cost of
the surface. However, there is a di(cid:11)erence in the
increasing the complexity of the model and the
two representations. The RAPID parameters are
number of parameters.
actual geometric lengths and distances, whereas the
4
NURBS parameters are the control points, weights, and knots is solved using the TNBC optimization
and knots. routine.
6. SUMMARY AND CONCLUSIONS
A NURBS surface of degree p in the u-direction
anddegreeq inthev-directionisapiecewiserational The RAPID methodology has been used to
function of the form generate a parametric model for the BWB subsonic
PIi=0PJj=0Ni;p(u)Nj;q(v)wi;jP(cid:22)i;j tisraanlsipmoirttt.oWthitehaocnculyraacsyminaltlhseetaopfpproaxraimmaettieorns,otfhtehree
P(u;v)= PIi=0PJj=0Ni;p(u)Nj;q(v)wi;j original CAD model. In examining the di(cid:11)erences
in geometry and CFD solutions with the CAD and
where P(cid:22)i;j are the points of the control net, wi;j Rapid models, it should always be noted that this
are the weights, and Ni;p(u) and Nj;q(v) are the B- methodology is intended for aerodynamic analysis
spline basis functions de(cid:12)ned on the knot vectors at the conceptual stage of the design process. At
u(cid:22)0;(cid:1)(cid:1)(cid:1);u(cid:22)I+p+1 and v(cid:22)0;(cid:1)(cid:1)(cid:1);v(cid:22)J+q+1 with this stage, the CAD geometry may only be a crude
approximation of the (cid:12)nal con(cid:12)guration. The ob-
u(cid:22)0 =(cid:1)(cid:1)(cid:1)=u(cid:22)p = 0 =v(cid:22)0 =(cid:1)(cid:1)(cid:1)=v(cid:22)q+1 jective in constructing the RAPID model is to have
u(cid:22)I+1 =(cid:1)(cid:1)(cid:1)=u(cid:22)I+p+1 = 1 =v(cid:22)J+1 =(cid:1)(cid:1)(cid:1)=v(cid:22)J+q+1: a parametrization of the BWB con(cid:12)guration. The
RAPIDmodelcanthenbeusedinaparameterstudy
to predict the e(cid:11)ects of changesin the geometryand
Let Xm;n; m = 1;(cid:1)(cid:1)(cid:1);M; n = 1;(cid:1)(cid:1)(cid:1);N be a sur-
improve upon the original design.
face grid generated by RAPID. Surfaces as NURBS
are de(cid:12)ned on the unit square and can be evaluated
7. ACKNOWLEDGEMENTS
at an array of points (um;vn); m = 1;(cid:1)(cid:1)(cid:1);M; n =
1;(cid:1)(cid:1)(cid:1);N in the uv-plane. The optimal NURBS ap- This work was performed while the (cid:12)rst author
proximation is obtained by selecting the control net, held a National Research Council-NASA Langley
weights, and knots which yields the minimum value Research Center Research Associateship.
of
XM XN 8. References
[Xm;n(cid:0)P(um;vn)]2: 1 Liebeck, R. H., Page, M. A., and Rawdon, B. K.,
m=1n=1 \Evolution of the Revolutionary Blended-Wing-Body,"
Transportation Beyond 2000: Technologies Needed for
The optimal NURBS will depend on the selection of
Engineering Design, NASACP 10184, pp431-459, 1996.
the arrayof parametric variables u and v. Both uni- 2 Smith, R. E., Bloor, M. G. I., Wilson, M. J., and
form and arclength parameters have worked well in Thomas, A. M., \Rapid Airplane Parametric Input
practice. Design (RAPID),"AIAAPaper 95-1687, June,1995.
3
Smith, R. E., Cordero, Y., and Jones, W. T., \Au-
An example of a NURBS approximation of the tomated Airplane Surface Generation," Numerical
BWB airplane and the control net is plotted in Grid Generation in Computational Field Simulations
Figure 11. This surface is de(cid:12)ned using piecewise pp. 293-301, NSF Engineering Research Center for
Computational Field Simulation, Mississippi State, MS,
cubic basisfunctions in both directionsand an 8(cid:2)8
1996.
controlnet. The NURBS surface is therefore de(cid:12)ned 4 Nash, S. G., User’s guide for TN/TNBC, Technical
by 64 values for each coordinate of the control net, Report 397, Department of Mathematical Sciences, The
64 weights, and four knot values for each direction. Johns HopkinsUniversity,Baltimore, MD, 1984.
5
Jones, W., and Samareh-Abolhassani, J., \A Grid
Thus the NURBS surface is de(cid:12)ned with a total of
GenerationSystemforMulti-disciplinaryDesign," AIAA
264 parameters.
Paper 95-1689, June, 1995.
6
Tiller, W., \Rational B-Splines for Curve and Surface
Representation," IEEE Computer Graphics and Applica-
The NURBS approximation requires the solution
tions, 3, pp. 61-69, 1983.
of a nonlinear optimization problem. The solution
is computed using a two-stage process. First, the
optimal control net is constructed using unit weights
and uniform knots. This is a linear least-squares
problem that is solved directly using a QR matrix
factorization. Theresultingsetofcontrolpointswith
unit weights and uniform knots are initial values for
calculation of the optimal NURBS. Next, the full
nonlinearproblemwithvariablecontrolnet,weights,
5
Fig. 1 Blended-wing-bodyCADgeometry Fig. 3 Tangent ribbonsfor RAPIDmodel
Fig. 2 Airfoil cross sections for RAPIDmodel Fig. 4 Blended-wing-bodyRAPIDgeometry
6
+RAPID
(cid:0)CAD
: RAPID
(cid:0) CAD
Fig. 5 Comparison of CAD andRAPIDmodel
for centerbody airfoil
+RAPID
(cid:0)CAD
Fig. 6 Comparison of CAD andRAPIDmodel
for wing airfoil Fig. 7 Comparison of CAD andRAPIDplanforms
7
Pressure Coefficient Comparison at Z = -1
1.0
400
300 Surface RAPID
Cp RAPID
200 Surface CAD
Cp CAD 0.5
100
0
RAPID Y Cp
-100 0.0
-200
-300
-400 -0.5
-500
-1500 -1000 -500 0
X
Fig. 9 Pressure coe(cid:14)cient at fuselage cross section
Pressure Coefficient Comparison at Z = -1016
0.5
100
Surface RAPID
Cp RAPID
Surface CAD
Cp CAD
50 0.0
Y Cp
CAD
0
-0.5
Fig. 8 Mach contours of CADand RAPIDmodels
-50
-1.0
-500 -400 -300 -200
X
Fig. 10 Pressure coe(cid:14)cient at wing cross section
8
Fig. 11 NURBS approximationof RAPIDgeometry
9